Research ArticleAMaintenanceTask Similarity-BasedPrior ElicitationMethod forBayesian Maintainability Demonstration

Zhenya Wu and Jianping Hao

Shijiazhuang Campus Army Engineering University Shijiazhuang China

Correspondence should be addressed to Zhenya Wu wzy9012163com

Received 9 April 2020 Revised 5 June 2020 Accepted 23 June 2020 Published 11 August 2020

Academic Editor Alessandro Lo Schiavo

Copyright copy 2020 Zhenya Wu and Jianping Hao is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

Prior distribution elicitation is a challenging problem for a Bayesian inference-based mean time to repair (MTTR) demonstrationbecause if inaccurate prior information is introduced into the prior distribution the results become unreliable is paperproposes a novel maintenance task representation model based on the similarity of attributed maintenance items A novelsimilarity computation algorithm for maintenance tasks is then formulated on the basis of this model Optimistic and pessimisticvalues are ascertained from the time data for similar maintenance tasks to obtain a prior distribution e main idea is to separatemaintenance tasks into distinct items and use attribute sets to extract key features Each pair of items is then compared to quantifythe differences between reference and candidate tasks Candidate tasks with an acceptable difference from the reference task aretaken as prior information sources for constructing the prior distribution A case study involving a high-frequency (HF)transceiver MTTR Bayesian demonstration shows that the proposed method can effectively obtain maintenance tasks similar tothose of information sources for prior distribution elicitation

1 Introduction

Maintainability refers to the designed characteristics ofsystems or products that facilitate ldquothe relative ease andeconomy of time and resources with which an item can beretained in or restored to a specified condition whenmaintenance is performed by personnel having specified skilllevels using prescribed procedures and resources at eachprescribed level of maintenance and repairrdquo [1] For manylarge-scale systems the cost of system maintenance andsupport ranges from 60 to 75 of their total overall life-cycle cost [2] us ensuring a product has good main-tainability is a key concern for product developers and users

Maintainability demonstration is a formal processconducted by a product developer and an end customer todetermine whether specified maintainability requirementshave been achieved e mean time to repair (MTTR) is oneof the key metrics for describing system maintainability andthe main index that is drawn upon during a maintainabilitydemonstration According to MIL-HDBK-470A the

number of samples for an MTTR demonstration should notbe less than 30 However it is almost impossible to obtainenough samples for a maintainability demonstration duringoperational tests and evaluations because the tests are ex-pensive Generally the problem of insufficient samples canbe dealt with by using probabilistic and statistical ap-proaches such as Bayesian techniques [3ndash6] bootstrapmethods [7 8] and DempsterndashShafer (D-S) evidence theory[9ndash11] Of these Bayesian methods have increasingly be-come the de facto option in reliability and maintainabilityengineering [12]

In a BayesianMTTR demonstration how to construct anappropriate prior distribution is a key challenge Wang andZhou [13] divide this problem into two parts e first partfocuses on whether the prior information is accurate enoughto describe the actual behaviore second part is concernedwith translating this prior information into an appropriatemathematical form Existing research on MTTR Bayesiandemonstrations has mainly been focused on the latterproblem e accuracy analysis of prior information has not

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 2730691 19 pageshttpsdoiorg10115520202730691

received the same attention Zhang [14] Zhang [15] Zhu[16] Huang [17] Liu [18] and Wang [19] have all analyzedthe accuracy of prior information from the perspective ofdata consistency In their methods when the prior main-tenance time data come from the same type of the productthe accuracy of the prior data depends on whether the priordata and field data are from the same distribution by usingnonparametric and parametric test methodsWhen the priordata come from similar products the multilayer Bayesmethod and the Kullback information method are used tocalculate the similarity degree between the prior data andfield data However the consistency of the prior data andfield data is a necessary but insufficient condition to de-termine whether the prior data are accurate ere is apossibility that the maintenance actions of the two productsare not similar but their maintenance time happens to havethe same distribution In addition these methods all dependupon a certain amount of data to ensure validity However itis not always possible to obtain enough maintenance timedata in practice Hence judging whether the prior data areaccurate only through maintenance time data analysis is notreliable and feasible in some cases Chen et al [20] used theweighted sum of distances between product attributes tomeasure the similarity degrees between airplanes whichwere then converted to the fusion weights of the priordistribution However the chosen attributes include pas-senger numbers wingspan airplane length and load ca-pacity which are performance parameters and only have anindirect relationship to maintenance action It is obviousthat the similarity degree based on these attributes cannotappropriately reflect the accuracy of prior informationere in conclusion the limitations of existing methodsmean that analyzing prior information accuracy remains animportant topic of research for MTTR Bayesiandemonstrations

In this paper we present a novel prior distributionelicitation method for an MTTR Bayesian demonstrationRather than undertaking maintenance time data analysisour method analyzes the accuracy of prior informationbased on maintenance task similarity As the maintenancetasks directly reflect the maintainability characteristics of theproduct a similarity analysis gives a comprehensive over-view of the accuracy of the prior information especially forcases with limited maintenance time data Developing apractical prior elicitation method involves substantialchallenges First there is a question of how to abstractlyrepresent the original maintenance task while retaining thenecessary features for similarity analysis Second to the bestof our knowledge there have been no reported methods thatcan calculate a distance measurement to quantify the sim-ilarity between maintenance tasks Last but not least afterobtaining a similar maintenance tasks it is not clear how toconstruct the prior distribution based on the time data ofthese tasks

To tackle the above challenges we first develop a novelrepresentation model for maintenance tasks is model isbased on an attributed item sequence that uses the itementity attribute tuples and maintainability attribute valuevectors to extract operations andmaintainability features To

measure the similarity between maintenance tasks a novelsimilarity computation algorithm is developed based on therepresentation model is algorithm is able to quantify thedifference between maintenance tasks Next an optimisticand pessimistic value method is used to construct the priordistribution

e main contributions of this paper include thefollowing

(1) A novel representation model that can extract thekey features of a maintenance task

(2) A novel similarity computation algorithm that canmeasure the similarity between maintenance tasks

(3) A novel method for constructing the prior distri-bution based on themaintenance time data of similartasks

e remainder of the paper is organized as follows inSection 2 the proposed methodology is discussed en inSection 3 the application of this methodology is shownthrough a case study Section 4 provides the conclusions

2 The Structure of the Proposed Methodology

is section presents the methodology for constructing theprior distribution based on maintenance task similarityanalysis e basic feature of the methodology is illustratedin Figure 1 e methodology is based on three main tasks

(1) Construction of the maintenance task representationmodels

(2) Maintenance task similarity analysis(3) Elicitation of the prior distribution

21 Construction of the Maintenance Task RepresentationModel

211 Literature Review Previous research regardingmaintenance representation models mainly exists in thefield of automated disassembly which uses computers tosimulate the disassembly process and calculate the mostefficient sequence All automated disassembly techniquesrely on first developing a suitable product maintenancemodel [21] Srinivasan and Gadh [22] used a ldquowavepropagationrdquo approach to establish the relationship be-tween each component in an assembly is approach usestau and beta waves which are created during the process todetermine the sequence of operations necessary to removea specific component Homem de Mello and Sanderson[23] used an ANDOR hypergraph to give a compactrepresentation model of all possible assembly plans ismodel forms the basis for efficient planning algorithmswhich enables the selection of the best assembly plan andopportunistic scheduling Agu [21] proposed a graph-basedmethod for representing product assembliesis approachuses nodes to represent the individual components whilearcs represent the different types of connections betweencomponents e node variables provide information re-garding specific components and the arc variables provide

2 Mathematical Problems in Engineering

information on the physical connections between them Tofind the best assemblydisassembly sequence the mainemphasis of the above methods is on the location ofcomponents within the overall assembly which makessense for repairinterchange tasks However these methodsare not suitable for the analysis of fault confirmation andfault isolation tasks as these do not necessarily consist ofassemblydisassembly operations In addition thesemethods do not take into account other concerns that caninfluence maintenance such as environmental factors andhuman factors which can have a significant influence onthe maintenance process

In our method the maintenance task is seen as a series ofoperations concerning maintenance itemsis process thenforms the basis of a representation model referred to as anattributed item sequence that can be used to represent amaintenance task is representation model consists of anitem sequence item entity attribute tuples and itemmaintainability attribute value vectors

212 Problem Definition To ensure the clarity of this idearsquosexpression some definitions are given below

Definition 1 (maintenance item sequence) A maintenanceitem denoted by I is the specified level of an item that is thedirect object of a maintenance operation For exampleremoving screw and disconnecting plug en a mainte-nance item sequence S I1 I2 IN1113864 1113865 is a series of time-ordered maintenance items that represent the items in amaintenance task

Definition 2 (item entity attribute tuple set) An item entityattribute tuple E langp1 p2rang is a two-tuple where bothelements in the tuple are attribute pairs e parameter p1represents the type of item and p2 represents the corre-sponding maintenance operation For example a tupledescribing ldquoopen caprdquo is E lang(type cap) (operation

open)rang and ldquoscrew nutrdquo is E lang(type nut) (operation

screw)rang en an item entity attribute tuple setG E1 E2 EN1113864 1113865 is the set of item entity attribute tuplesfor items in a maintenance task

Definition 3 (item maintainability attribute value vectorset) An item maintainability attribute set U

U1 U2 UM1113864 1113865 is a set of maintainability attributes de-scribing the maintainability characteristics of an item Itscorresponding value vector is denoted by V (u1

u2 uM) en a maintainability attribute value vectorset A V1 V2 VN1113864 1113865 is the set of maintainability at-tribute value vectors for items in a maintenance task

Definition 4 (attributed item sequence) An attributed itemsequence M langS G Arang is a three-tuple representing amaintenance task

For example the maintenance task ldquoRepairinter-change--replace the transceiverrdquo for the troubleshooting ofthe airplane HF transceiver failure includes the followingprocedures [24]

(1) Unscrew the nut(2) Lower the nut(3) Pull the HF transceiver from the shelf and dis-

connect the electrical plug(4) Dismantle the transceiver(5) Place cap on the electrical plug(6) Clean the interface and adjacent area(7) Check the interface and adjacent area(8) Remove the cap from the electrical plug(9) Check the cleanliness and condition of the electrical

plug(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug(12) Screw the nut

en according to the above definitions the repre-sentation model for the task is constructed as shown inFigure 2 (assuming the maintainability attribute set includessix attributes)

213 Identification and Formulation of MaintainabilityAttributes As numerous complex factors can affect themaintenance process researchers have used a variety ofattributes or indicators to reflect maintainability charac-teristics Several researchers have made use of a compre-hensive evaluation method [2 25ndash30] to incorporate a rangeof attributes e maintainability attribute sets in the aboveresearch are shown in Figure 3

Determine candidate maintenance tasks

Identify and formulate maintainability attributes

1

Construction of the maintenance task representation models

Calculate item weights

Identify maintenance itemsIdentify maintenance operations

Specify sequence mapping cost threshold value

Perform sequence matching

Maintenance task similarity analysis

2

Construct prior distribution

3

Elicitation of the prior distribution

Calculate sequence mapping cost

Clustering of similar candidate maintenance tasks

Figure 1 Process of prior distribution elicitation based on amaintenance task similarity analysis

Mathematical Problems in Engineering 3

Although there are differences between the above lists ofmaintainability attributes they all provide a comprehensiveoverview of the attributes required to understand main-tainability In practice one of the above maintainabilityattribute sets can be chosen as required or according toexpert opinion

22 Maintenance Task Similarity Analysis After the con-struction of maintenance task representation models asimilarity analysis can be performed on these models In thissection we propose a similarity computation algorithm formaintenance tasks After that the clusters of similarmaintenance tasks can then be obtained

221 Literature Review Similarity search methods have beenused in a wide variety of applications areas such as data mining[31] face recognition [32ndash34] image classification [35] medicalengineering [36] and human behavior analysis [37] Mainte-nance tasks are usually performed by maintenance staff sosimilarity searches for maintenance tasks fall under the remit ofhuman behavior analysis Human behavior can be representedfrom many perspectives from a low level eg individualmotions to an abstract level eg business processes Zhanget al [37] proposed an extended semantic distance calculationmethod called linked data semantic distance (LDSD) forsimilarity searches in relation to human behavior is methodis based on a multilayered process model (MLPM) whichdecomposes behaviors into three layers a processtask layer anactivity layer and an action layer However it is difficult toemploy this model for maintenance tasks because of the dif-ficulty of obtaining enough detail regarding human behaviorNeumuth et al [38] proposed using surgical process models

(SPMs) to represent surgical interventions and introduced fivesimilarity metrics for comparing SPMs ese metrics relate tothe granularity content temporal aspects transitional featuresand frequency of transitions However no clear instructions aregiven as to how to combine these five metrics into a singlesimilarity value Obweger et al [39] proposed a generic simi-larity model for time-stamped sequences in complex businessevents is model calculates similarity on the basis of devia-tions between a query pattern and its representation in acandidate event Additionally thismodel assesses dissimilaritiesat the level of single events their order their timing and theabsence of events However the single-event similarity is de-rived from the semantic distance between mapped eventswhich is not suitable for maintenance task similarity analysis

In this section the similarity measurement betweenmaintenance tasks is converted to a sequence matchingproblem e difference between two item sequences isquantified from the perspective of maintenance time andmaintainability characteristics To allow some tolerance thatcan recognize acceptable differences mapping cost functionsbetween the attributed item sequences are introduced into thematching process After that similar maintenance tasks canbe obtained by specifying the mapping cost threshold value

222 Problem Definition To ensure the clarity of this idearsquosexpression some definitions are given below

Definition 5 (similar items) For a given two items they aredefined to be similar if and only if their entity attribute setsare exactly the same that is they have the same type andoperation

Two attributed item sequences M1 langS1 G1 A1rang andM2 langS2 G2 A2rang are given assuming they both have five

V12 = (u121 u122 u123 u124 u125 u126)

V11 = (u111 u112 u113 u114 u115 u116)

V10 = (u101 u102 u103 u104 u105 u106)

V9 = (u91 u92 u93 u94 u95 u96)

V8 = (u81 u82 u83 u84 u85 u86)

V7 = (u71 u72 u73 u74 u75 u76)

V6 = (u61 u62 u63 u64 u65 u66)

V5 = (u51 u52 u53 u54 u55 u56)

V4 = (u41 u42 u43 u44 u45 u46)

V3 = (u31 u32 u33 u34 u35 u36)

V2 = (u21 u22 u23 u24 u25 u26)

V1 = (u11 u12 u13 u14 u15 u16)

E12 = lang(type nut) (operation screw)rang

E11 = lang(type transceiver) (operation press)rang

E10 = lang(type transceiver) (operation install)rang

E9 = lang(type electrical plug) (operation check)rang

E8 = lang(type cap) (operation dismantle)rang

E7 = lang(type interface) (operation check)rang

E6 = lang(type interface) (operation clean)rang

E5 = lang(type cap) (operation place)rang

E4 = lang(type transceiver) (operation dismantle)rang

E3 = lang(type transceiver) (operation pull)rang

E2 = lang(type nut) (operation lower)rang

E1 = lang(type nut) (operation unscrew)rang

M = langS G Arang whereS = I1 I2 hellip I12G = E1 E2 hellip E12A = V1 V2 hellip V12

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 t

Figure 2 Representation model for the task ldquoRepairinterchange--replace the transceiverrdquo

4 Mathematical Problems in Engineering

items and every item in one sequence has a similar item inanother sequence en to quantify the difference betweentwo sequences a one-to-one correspondence is assigned tosimilar items between S1 I11 I12 I13 I14 I151113864 1113865 andS2 I21 I22 I23 I24 I251113864 1113865 as shown in Figure 4 (the items withthe same color are similar)

Definition 6 (virtual item) A virtual item is a nonexistentitem that is used for full-sequence matching when theexisting items of the two sequences cannot all establish aone-to-one correspondence For example there are differentitems (different types or operations) or redundant itemsbetween two sequences

e given two attributed item sequencesM1 langS1 G1 A1rang and M2 langS2 G2 A2rang which have threesimilar items a different item I21 and a redundant item I15are shown in Figure 5

en in order to quantify the impact of I15 and I21 on thesimilarity analysis two virtual items I11 and I25 are added intothe sequences to enable full-sequencematching (see Figure 6)

1 General(i) Standardization

(ii) Components functionally grouped(iii) Console layout(iv) Complexity(v) Self-test

(vi) Maximum-time-to-repair(vii) Auxiliary tools and test equipment

(viii) Labeling(ix) Weight(x) Calibration requirements

(xi) Repairreplace philosophy(xii) Maintenance procedures

(xiii) Personnel requirements(xiv) Trade-off studies

2 Handling3 Equipment racks-general4 Packaging5 Accessibility6 Fasteners7 Cables8 Connectors9 Servicing and lubrication

10 Panel displays and controls11 Test points12 Adjustments13 Parts and components14 Environment15 Safety16 Reliability17 Software

Blanchard et al [2]

1 Inherent attributesConnection modeVisibilityStandardizationEntity reachabilityModularizationSecurityMaterial selectionProcessing technologyProcessing convenience

2 External factorsWorking environmentQuality of consumablesThe technical level of operatorsMaintenance cycleMaintenance positionMaintenance actionMaintenance spaceWorkerrsquos wagesRaw material costStorage securityThe technical level of maintainers

Jian et al [26]

1 Design configurationAccessibilityErgonomic factorsAutomation and mechanizationNormalization and interchangeability

2 Maintenance supportOrganization locality personnel and trainingProvision of spares facilities test equipmentEnvironmental conditions

Tarelko [28]

1 DesignAccessibilityDisassemblyassemblyStandardizationSimplicityIdentificationDiagnosabilityModularizationTribo-concepts

2 PersonnelPersonnel including ergonomicsSystem environment

3 Logistic supportTools and test equipmentDocumentation

Wani and Gandhi [30]

1 DesignStandardizationModularizationSimplicityDiagnosabilityIdentificationAccessibilityAssemblibilityServiceabilityTestabilityPartscomponentsReliability

2 PersonnelAnthropologyHuman sensoryPhysiologicalPsychological

3 Logistic supportSpares procurementTools amp test equipmentDocumentationSoftware

4 Operation contextSafetySystem environmentOperationmission type

Tjiparuro and Thompson [29]

1 General attributesSimplicityIdentificationModularityTribologyErgonomicsStandardizationFailure watchRelation with the manufacturer

2 Specific attributesAccessibilityAssemblydisassemblyTrainingPersonnel organizationEnvironmentSpare partsMaintenance tools and equipmentsInterdepartmental co-ordinationDocumentation

Leon et al [27]

1 Physical designSimplicityAccessibilityAssemblydisassemblyStandardizationModularizationTest points layout

2 Logistics supportTest equipmentAssemblydisassembly tool or maintenance toolDocumentation

3 ErgonomicsFault and operation indicatorsSkills of maintenance personnelMaintenance environmentOther ergonomics factors

Chen and Cai [25]

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)

(i)(ii)

(iii)

(iv)(v)

(vi)(vii)

(viii)(ix)

(i)(ii)

(iii)(iv)(v)

(vi)

(i)(ii)

(iii)

(i)(ii)

(iii)

(i)(ii)

(iii)(iv)

(iv)(v)

(vi)

(vii)(viii)

(ix)(x)

(xi)

(i)(ii)

(iii)(iv)(v)

(vi)(vii)

(viii)

(i)(ii)

(i)(ii)

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)

(i)(ii)

(iii)

(iv)

(v)(vi)

(vii)(viii) (i)

(ii)(iii)(iv)(v)

(vi)(vii)

(viii)

(i)(ii)

(iii)(iv)(v)

(vi)(vii)

(viii)(ix)

(ix)(x)

(xi)

Figure 3 Summary of the existing sets of maintainability attributes in the literature

t

tI11 I21 I31 I41 I51

I12 I22 I32 I42 I52

M1

M2

Figure 4 e one-to-one correspondence between sequences M1and M2

t

t

Different (redundant) items

I21 I31 I41 I51

I12 I22 I32 I42

M1

M2

Figure 5 Two sequences with different (redundant) items

Mathematical Problems in Engineering 5

e sequence of virtual items added is called the extendedattributed item sequence

Definition 7 (cosine similarity) Cosine similarity is ameasure of the similarity between two high-dimensionalvectors at is given two vectors X and Y

cos(X Y) langX Yrang

XY (1)

where ldquolangrangrdquo indicates the inner product of two vectors andldquordquo indicates the L2 norm of the vector

Definition 8 (item mapping cost IMC) For a given twoextended attributed item sequences M1 and M2 under theone-to-one correspondence the item mapping cost fi be-tween two similar items is

fi wi 1 minus cos V1i V

2i1113872 11138731113872 1113873 (2)

and the item mapping cost between one item and its cor-responding virtual item is

fi wi (3)

where wi is the item weight which represents the relativelength of the mean maintenance time spent on that type ofitem V1

i and V2i are the maintainability attribute value

vectors of the two items 1 minus cos(V1i V2

i ) represents thedifference between the maintainability characteristics of twosimilar items

Equations (2) and (3) show that the greater the differencebetween the maintainability characteristics of two similaritems or the more maintenance time the item costs thegreater the impact of the difference on the similarity analysis

Definition 9 (sequence mapping cost SMC) e sequencemapping cost H between the sequence M1 and M2 is

H12 1113944N

i1fi (4)

where N is the number of items in each sequencee SMC reflects the difference between two mainte-

nance tasks based on the representation models In generalthe larger the value of H is the larger the difference betweenthe two maintenance tasks is

Definition 10 (reference maintenance task) When theequipment to be MTTR demonstrated is specified themaintenance tasks for this equipment are defined as the

reference maintenance tasks denoted by Pri (i 1 2 K)

where K represents the number of task types

Definition 11 (candidate maintenance task set) A candidatemaintenance task P is a task that is compared to the ref-erence task e candidate maintenance task set denoted byOi Pi1 Pi2 PiNi

1113966 1113967(i 1 2 K) is the task set for thesimilarity search of the reference task Pr

i where Ni repre-sents the number of tasks Possible sources of candidate tasksinclude maintenance tasks relating to equipment or com-ponents in the same system that have similar functions orthat take place in a similar location

Definition 12 (similarity calculation) For a given referencemaintenance task Pr with a corresponding candidatemaintenance task set O and a user-specified SMC thresholdof ε a similarity search will retrieve all maintenance tasksPj isin O such that

Hj le ε j 1 2 Ni (5)

where Hj is the SMC between maintenance tasks Pj and PrIf equation (5) holds it can be stated that Pr and Pj aresimilar to the ε boundary We can then obtain the cluster ofsimilar candidate tasks for the reference task which isdenoted as C e sequence mapping cost (SMC) threshold εis a user-specified value and it is obvious that the larger the εthe more candidate maintenance tasks will be determined tobe similar to the reference maintenance task and then moredata will be available for constructing prior distributionHowever a larger ε will make some candidate maintenancetasks that are less similar to the reference task similar enoughfor a prior distribution elicitation which in turn makes theobtained prior distribution unreliable Hence it is importantto achieve a balance between the quantity and quality of datawhen specifying the SMC threshold value e SMCthreshold ε value can be determined through discussion withexperts based on the SMC calculation result to obtain datafrom the equipment or components as similar as possibleunder the precondition of having enough data for con-structing a prior distribution

223 Calculation of the Item Weights In this study theexpert experience is used to estimate the weight coefficientw As human judgments can be vague or ill-defined a fuzzyanalytic hierarchy process (FAHP) is used to calculate theweight coefficient of each item is method is mature andeasy to use in engineering practice and can make the weightsmore scientific when combined with the fuzzy judgment of

Extended

t

t

Different (redundant) items

I21 I31 I41 I51

I12 I22 I32 I42

M1

M2

Virtual items

t

tI11 I21 I31 I41 I51

I12 I22 I32 I42 I52

M1

M2

Figure 6 Extended sequences for full-sequence matching

6 Mathematical Problems in Engineering

experts based on their experience e implementation ofthis procedure is described below [40]

First a priority matrix Q (qij)ntimesn needs to be con-structed where the value of qij can be acquired through thepriority matrix scale method shown in Table 1

According to the results of the comparison betweendifferent items a priority matrix for each item can beconstructed as shown in Table 2

en the overall priority matrix Q is given by

Q qij1113872 1113873ntimesn

q11 middot middot middot q1n

middot middot middot

middot middot middot

middot middot middot

qn1 middot middot middot qnn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

Now a fuzzy consistent matrix can be constructedwhere R (rij)ntimesn

First the fuzzy complementary matrix is summed line byline where Q (qij)ntimesn

qi 1113944n

j1qij i 1 2 n (7)

en the following transformation is implemented toconstruct the fuzzy consistent matrix R (rij)ntimesn

rij qi minus qj

2n+ 05 (8)

e next set of calculations begins with the weight vectorof R is is given by the following

gi 1113945n

j1rij

⎛⎝ ⎞⎠

1n

(9)

e weight vector gi is now normalized

gi gi

1113936ni1 gi

i 1 2 n (10)

Finally the weight vector w can be constructed asfollows

w g1 g2 middot middot middot gn( 1113857T i 1 2 n (11)

e similarity computation algorithm based on theabove definitions is shown in Algorithm 1

To illustrate the method the maintenance task ldquoFaultisolationrdquo for the troubleshooting of the HF transceiverfailure is taken as an example After referring to the trou-bleshooting manual the chosen candidate tasks and theirprocedures are shown in Table 3

Assume that the maintainability attribute set includesentity reachability visibility maintenance space toolstechnical level of the maintainers maintenance position andsecurity e indicators are scored with a number from 0 to10 e higher the score is the better the maintainability isen the representation models for the reference andcandidate maintenance tasks are constructed as shown inTable 4

ere are two types of items in the sequences circuitbreaker and pin Using fuzzy AHP the priority matrices forthe two items are

Q 05 0

1 051113890 1113891 (12)

en according to equations (7)sim(11) the item weightsare obtained as

w1 0366

w2 0634(13)

e sequence matching between the reference and twocandidate maintenance tasks is shown in Figure 7

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks is ascertained (the resulthas been multiplied by 1000 for better comparison)

H1 asymp 655

H2 asymp 1272(14)

After discussions with the experts the SMC threshold isspecified as ε 800 en because H1 lt 800 H2 gt 800 themaintenance task for the wiring between the transceiver pinand the ground terminal is determined to be similar to thereference task

23 Elicitation of the Prior Distribution Commonly usedmethods for constructing a prior distribution include eli-cited priors conjugate priors and noninformative priors[41] As the similar candidate tasks in each cluster onlycontain the maintenance time data for the correspondingreference task not the whole maintenance action we use anoptimistic and pessimistic value method to estimate theparameters of the prior distribution A normal distributionis the commonly used form of the prior distribution inMTTR Bayesian demonstrations [14 16 20 42] so in thisstudy we also assumed a normal prior distribution for theparameter of interest

Let X sim LogN(μ σ2) denote the maintenance actiontime distribution of a specified product e variance σ2will either be known from prior information or a reasonablyprecise estimate can be obtained e prior distribution of μ

Table 1 Priority matrix scale method

Scale Definition Illustration1 More time Ii consumes more time than Ij

05 Equal time Ii and Ij consume equal time0 Less time Ii consumes less time than Ij

Table 2 Priority matrix of each item

Q Q1 Qn

Q1 q11 qn1 Qn q1n qnn

Mathematical Problems in Engineering 7

is denoted as N(μπ σ2π) According to the properties of thelognormal distribution

θ eμ+σ22

(15)

where θ is the mean of the maintenance time distributionen μ can be calculated as follows

μ ln θ minusσ2

2 (16)

If Xi(i 1 2 K) denotes the time spent on eachmaintenance task and xi(i 1 2 K) denotes the cor-responding maintenance task time data set then

X 1113944K

i1Xi (17)

Two predictions of the mean of the maintenance actiontimemdashthe lower or optimistic value θL and the upper orpessimistic value θUmdashcan be obtained as follows

1113954θL 1113944K

i1xi(min) (18)

1113954θU 1113944

K

i1xi(max) (19)

where xi(min) and xi(max) are the minimum and maximumvalues respectively for the time data set corresponding tocluster Ci

According to equation (16) the two possible predictionsof μ are

Input Pri Oi εi

Output Ci

(1) for each Pri do

(2) Ci⟵empty(3) Construct representation models for Pr

i and maintenance tasks in Oi(4) for each Pij isin Oi do(5) Perform sequence matching between Pr

i and Pij(6) Calculate item weights(7) Calculate SMC Hij(8) if Hij lt εi then(9) Ci⟵Pij(10) end if(11) end for(12) end for(13) Return Ci

ALGORITHM 1 Similarity computation algorithm based on maintenance task representation models

Table 3 Reference and candidate maintenance task procedures

Reference task Candidate task

HF transceiverPr

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC4 5 and 6 of the transceiver

e wiring between the transceiverpin and ground terminal P1

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC45 and 6 of the HF transceiver

(3) Do a check for a ground signal at pinAC8 of the HF transceiver

VHF transceiver P2

(1) Do a check of the circuit breakerstatus

(2) Do a check for 28DC at pin AC2 ofthe VHF transceiver

t

tI1r I2r I3r I4r I5r

I1 I2 I3 I4 I5

Mr

M1

Mr

M2 t

tI1r I2r I3r I4r

I1 I2 I3 I4

Figure 7 Sequence matching between the reference and candidate maintenance tasks

8 Mathematical Problems in Engineering

1113954μL ln 1113954θL minusσ2

2 (20)

1113954μU ln 1113954θU minusσ2

2 (21)

It can then be assumed that the range (1113954μU minus 1113954μL) en-compasses 100 times (1 minus p) percent of the total possible valuesof μ and that the best estimate is at the midpoint of the rangeerefore the following prior distribution estimates can beused

μπ 1113954μU + 1113954μL

2 (22)

σ2π 1113954μU minus 1113954μL( 1113857

2

4 times Z2p2

(23)

3 Case Study

In this section the implementation of an MTTR demon-stration for an HF transceiver is once again used to illustrateour method

31 Selection of Candidate Maintenance Tasks An HFtransceiver is part of the HF system and is installed at thefront of the electronics rack in a plane After referring to thetroubleshooting manual and the aircraft maintenancemanual [24 43] we established candidate tasks for eachreference task ese relate to other components in the HFsystem or other equipment at the front of the electronicsrack A breakdown of the tasks is shown in Table 5

32 Identification and Formulation of the MaintainabilityAttribute Set and Evaluation Rules e maintainability at-tribute set developed by Jian et al [26] was used for thesimilarity analysis of the maintenance tasks e main-tainability attributes were tailored to the characteristics ofthe different tasks as shown in Table 6 e correspondingevaluation rules are shown in Table 7

33 Similarity Analysis between the Maintenance Tasks

331 Construction of the Maintenance Task RepresentationModels After referring to the maintenance manuals and theexpertsrsquo experience representation models for the referenceand candidate maintenance tasks were established as shownin Table 8

332 Calculation of the Item Weights On the basis of therepresentation models an item list for each type of main-tenance task was established as shown in Table 9

Using fuzzy AHP the priority matrices for the variousitems in each type of maintenance task were thenobtained

Q1

05 05 05

05 05 05

05 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2

05 0 0

1 05 0

1 1 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q3

05 1 1 1 1 0 05 1 1 05 1 05 1 1

0 05 0 0 05 0 0 05 0 0 05 0 0 0

0 1 05 0 1 0 0 05 0 0 05 0 05 0

0 1 1 05 1 05 1 1 1 0 1 05 1 1

0 05 0 0 05 0 0 05 0 0 0 0 0 0

1 1 1 05 1 05 1 1 1 05 1 05 1 1

05 1 1 0 1 0 05 1 05 0 05 0 1 1

0 05 05 0 05 0 0 05 0 0 05 0 0 0

0 1 1 0 1 0 05 1 05 0 05 0 05 05

05 1 1 1 1 05 1 1 1 05 1 05 1 1

0 05 05 0 1 0 05 05 05 0 05 0 05 05

05 1 1 05 1 05 1 1 1 05 1 05 1 1

0 1 05 0 1 0 0 1 05 0 05 0 05 05

0 1 1 0 1 0 0 1 05 0 05 0 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

After this equations (7)sim(11) were used to establish theweight vectors for the items in each type of maintenancetask as follows

w1 (0333 0333 0333)

w2 (0211 0335 0454)

w3 (0094 0044 0055 0091 0040 0099 0077

0046 0069 0099 0061 0096 0063 0066)

(25)

333 Clustering of the Candidate Maintenance Taskse sequence matchings between the reference and candi-date maintenance tasks are shown in Figure 8

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks was ascertained (the resultwas multiplied by 1000 for better comparison)

H11 269

H12 198

H13 599

H21 34640

H22 80325

H23 67239

H31 41494

H32 038

H33 142

(26)

Based on the SMC calculation result the thresholds werespecified as

Mathematical Problems in Engineering 9

Tabl

e4

Representatio

nmod

elsforthereferenceandcand

idatemaintenance

tasks

Referencetask

Candidate

task

Mr

I 1rI 2r

I 3rI 4r

t

Mr

lang

SrG

rA

rrangwhere

Sr

Ir 1

Ir 2Ir 3

Ir 41113864

1113865

Gr

E

r 1E

r 2E

r 3E

r 41113864

1113865

Ar

V

r 1V

r 2V

r 3V

r 41113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Er 1

lang(type

circuitbreaker)

(op

erationcheck)

rangV

r 1

(77865910

)

Er 2

lang(type

pin

)(op

erationcheck)

rangV

r 2

(8873899)

Er 3

lang(type

pin

)(op

erationcheck)

rangV

r 3

(78738910

)

Er 4

lang(type

pin

)(op

erationcheck)

rangV

r 4

(8973898)

M1

I 1I 2

I 3I 4

I 5t

M1

lang

S1

G1

A1rang

where

S1

I 1

I2

I 3I

4I 5

11138641113865

G1

E1

E2

E3

E4

E5

11138641113865

A1

V

1V

2V

3V

4V

51113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

eration

check)

rangV

1

(98747710

)

E2

lang(type

pin

)(op

erationcheck)

rangV

2

(88728810

)

E3

lang(type

pin

)(op

erationcheck)

rangV

3

(89728710

)

E4

lang(type

pin

)(op

erationcheck)

rangV

4

(8962879)

E5

lang(type

pin

)(op

erationcheck)

rangV

5

(9872879)

M2

I 1I 2

t

M2

lang

S2

G2

A2rang

where

S2

I 1

I2

11138641113865

G2

E1

E2

11138641113865

A2

V

1V

21113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

erationcheck)

rang

E2

lang(type

pin

)(op

erationcheck)

rang

V1

(87868910

)

V2

(7873899)

lowastTo

quantifytheim

pact

ofdifferent

numbers

ofchecks

atpins

ontheSM

Ccalculation

echecks

atpins

AC4A

C5A

C6and

AC8

aretreatedseparately

10 Mathematical Problems in Engineering

Table 5 Candidate maintenance tasks for each reference task

Reference task Componentequipment Candidate task

Fault confirmationcheckout Pr

1

HF antenna coupler P11 (1) Press the row key near the HF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menuHF antenna P12

Very high-frequency (VHF) transceiver P13

(1) Press the row key near the VHF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menu

Fault isolation Pr2

e wiring between the transceiver pin AC8and the ground terminal P21

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver

P22

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver(4) Do a check of the wiring from the circuit breaker to the

HF transceiver pins AC4 5 and 6

VHF transceiver P23(1) Do a check of the circuit breaker status

(2) Do a check for 28 DC at pin AC2 of the VHF transceiver

Repairinterchange Pr3

HF antenna coupler P31

(1) Disconnect the electrical plug(2) Place cap on the electrical plug

(3) Unscrew the nut(4) Lower the nut

(5) Dismantle the antenna coupler(6) Place cap on the electrical plug

(7) Clean the interface and adjacent area(8) Check the interface and adjacent area

(9) Dismantle the cap from the electrical plug(10) Check the cleanliness and condition of the electrical

plug(11) Install the antenna coupler on the shelf

(12) Screw the nut(13) Dismantle the cap from electrical plug

(14) Check the cleanliness and condition of the electricalplug

(15) Connect the electrical plug to the antenna coupler

Audio management unit (AMU) P32

(1) Unscrew the nut(2) Lower the nut

(3) Pull the AMU from the shelf and disconnect the electricalplug

(4) Dismantle the AMU(5) Place cap on the electrical cap

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the AMU on the shelf(11) Press the AMU to connect the electrical plug

(12) Screw the nut

VHF transceiver P33

(1) Unscrew the nut(2) Lower the nut

(3) Pull the transceiver from the shelf and disconnect theelectrical plug

(4) Dismantle the transceiver(5) Place cap on the electrical plug

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug

(12) Screw the nut

Mathematical Problems in Engineering 11

ε1 6

ε2 420

ε3 3

(27)

en according to equation (5) the clusters for similarcandidate tasks for each reference task were established

C1 P11 P12 P131113864 1113865

C2 P211113864 1113865

C3 P32 P331113864 1113865

(28)

e results showed that the candidate tasks for faultconfirmationcheckout were all similar to the reference taskbecause they had the same items as the reference task de-spite having slightly different maintainability characteristicsIn the case of fault isolation three candidate tasks had

different items to the reference task Task P22 had two ad-ditional items (I5 and I6) and P23 had two missing items (Ir

3and Ir

4) P21 however had only one additional item (I5)making it more similar to the reference task than the othertwo tasks In the case of repairinterchange task P31 hadseven different items (I1 I2 Ir

3 Ir11 I13 I14 and I15) while

the other two tasks all had the same items as the referencetask us task P31 had the most differences and was theleast similar to the reference task

34 HF Transceiver MTTR Demonstration Based onBayesian eory

341 MTTR Demonstration Methods Based on Bayesianeory e Bayesian maintainability demonstrationmethod proposed by Balaban [44] is used in our paper for

Table 6 Maintainability attributes tailored to maintenance tasks

Maintainability attributes Fault confirmation Fault isolation Repairinterchange CheckoutEntity reachability Visibility Maintenance space Tools Technical level of the maintainers Maintenance position Security

Table 7 Evaluation rules

Maintainability attribute Evaluation rules

Entity reachability (U1)

Good can observe the maintenance component comfortably and there is a wide observation angle(7ndash10)

General can generally see the outline of the maintenance component though it can easily cause eye andbody fatigue (4ndash6)

Poor prone to fatigue in this pose (0ndash3)

Visibility (U2)Good clear line of sight and there is enough light (7ndash10)

General the line of sight is blocked or the light is dark (4ndash6)Poor the line of sight is seriously blocked or the light is insufficient (0ndash3)

Maintenance space (U3)

Good no restrictions on the maintenance space for operating posture (7ndash10)General maintenance space for the body is basically enough but the operatorrsquos posture is abnormal

(4ndash6)Poor there is not enough maintenance space for a body (0ndash3)

Tools (U4)Good do not need the aid of auxiliary tools (7ndash10)

General need auxiliary tools sometimes (4ndash6)Poor dependent on auxiliary tools (0ndash3)

Technical level of themaintainers (U5)

Good familiar with the relevant technical knowledge and can quickly determine the operation processand solve the problem (7ndash10)

General is familiar with the relevant knowledge and can solve problems by referring to the operationmanual (4ndash6)

Poor only a general understanding of the situation and how to carry out the relevant work (0ndash3)

Maintenance position (U6)Good ground (7ndash10)

General have to climb to the machine (4ndash6)Poor need to stand outside the machine or in a similar position to the machine (0ndash3)

Security (U7)

Good no danger of being injured by heavy objects and there are no sharp edges that may scratch ordanger of electrical shock (7ndash10)

General there is a certain security threat sometimes there are sharp edges that may cause bumps orscratches (4ndash6)

Poor there is a security threat (0ndash3)

12 Mathematical Problems in Engineering

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

information on the physical connections between them Tofind the best assemblydisassembly sequence the mainemphasis of the above methods is on the location ofcomponents within the overall assembly which makessense for repairinterchange tasks However these methodsare not suitable for the analysis of fault confirmation andfault isolation tasks as these do not necessarily consist ofassemblydisassembly operations In addition thesemethods do not take into account other concerns that caninfluence maintenance such as environmental factors andhuman factors which can have a significant influence onthe maintenance process

In our method the maintenance task is seen as a series ofoperations concerning maintenance itemsis process thenforms the basis of a representation model referred to as anattributed item sequence that can be used to represent amaintenance task is representation model consists of anitem sequence item entity attribute tuples and itemmaintainability attribute value vectors

212 Problem Definition To ensure the clarity of this idearsquosexpression some definitions are given below

Definition 1 (maintenance item sequence) A maintenanceitem denoted by I is the specified level of an item that is thedirect object of a maintenance operation For exampleremoving screw and disconnecting plug en a mainte-nance item sequence S I1 I2 IN1113864 1113865 is a series of time-ordered maintenance items that represent the items in amaintenance task

Definition 2 (item entity attribute tuple set) An item entityattribute tuple E langp1 p2rang is a two-tuple where bothelements in the tuple are attribute pairs e parameter p1represents the type of item and p2 represents the corre-sponding maintenance operation For example a tupledescribing ldquoopen caprdquo is E lang(type cap) (operation

open)rang and ldquoscrew nutrdquo is E lang(type nut) (operation

screw)rang en an item entity attribute tuple setG E1 E2 EN1113864 1113865 is the set of item entity attribute tuplesfor items in a maintenance task

Definition 3 (item maintainability attribute value vectorset) An item maintainability attribute set U

U1 U2 UM1113864 1113865 is a set of maintainability attributes de-scribing the maintainability characteristics of an item Itscorresponding value vector is denoted by V (u1

u2 uM) en a maintainability attribute value vectorset A V1 V2 VN1113864 1113865 is the set of maintainability at-tribute value vectors for items in a maintenance task

Definition 4 (attributed item sequence) An attributed itemsequence M langS G Arang is a three-tuple representing amaintenance task

For example the maintenance task ldquoRepairinter-change--replace the transceiverrdquo for the troubleshooting ofthe airplane HF transceiver failure includes the followingprocedures [24]

(1) Unscrew the nut(2) Lower the nut(3) Pull the HF transceiver from the shelf and dis-

connect the electrical plug(4) Dismantle the transceiver(5) Place cap on the electrical plug(6) Clean the interface and adjacent area(7) Check the interface and adjacent area(8) Remove the cap from the electrical plug(9) Check the cleanliness and condition of the electrical

plug(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug(12) Screw the nut

en according to the above definitions the repre-sentation model for the task is constructed as shown inFigure 2 (assuming the maintainability attribute set includessix attributes)

213 Identification and Formulation of MaintainabilityAttributes As numerous complex factors can affect themaintenance process researchers have used a variety ofattributes or indicators to reflect maintainability charac-teristics Several researchers have made use of a compre-hensive evaluation method [2 25ndash30] to incorporate a rangeof attributes e maintainability attribute sets in the aboveresearch are shown in Figure 3

Determine candidate maintenance tasks

Identify and formulate maintainability attributes

1

Construction of the maintenance task representation models

Calculate item weights

Identify maintenance itemsIdentify maintenance operations

Specify sequence mapping cost threshold value

Perform sequence matching

Maintenance task similarity analysis

2

Construct prior distribution

3

Elicitation of the prior distribution

Calculate sequence mapping cost

Clustering of similar candidate maintenance tasks

Figure 1 Process of prior distribution elicitation based on amaintenance task similarity analysis

Mathematical Problems in Engineering 3

Although there are differences between the above lists ofmaintainability attributes they all provide a comprehensiveoverview of the attributes required to understand main-tainability In practice one of the above maintainabilityattribute sets can be chosen as required or according toexpert opinion

22 Maintenance Task Similarity Analysis After the con-struction of maintenance task representation models asimilarity analysis can be performed on these models In thissection we propose a similarity computation algorithm formaintenance tasks After that the clusters of similarmaintenance tasks can then be obtained

221 Literature Review Similarity search methods have beenused in a wide variety of applications areas such as data mining[31] face recognition [32ndash34] image classification [35] medicalengineering [36] and human behavior analysis [37] Mainte-nance tasks are usually performed by maintenance staff sosimilarity searches for maintenance tasks fall under the remit ofhuman behavior analysis Human behavior can be representedfrom many perspectives from a low level eg individualmotions to an abstract level eg business processes Zhanget al [37] proposed an extended semantic distance calculationmethod called linked data semantic distance (LDSD) forsimilarity searches in relation to human behavior is methodis based on a multilayered process model (MLPM) whichdecomposes behaviors into three layers a processtask layer anactivity layer and an action layer However it is difficult toemploy this model for maintenance tasks because of the dif-ficulty of obtaining enough detail regarding human behaviorNeumuth et al [38] proposed using surgical process models

(SPMs) to represent surgical interventions and introduced fivesimilarity metrics for comparing SPMs ese metrics relate tothe granularity content temporal aspects transitional featuresand frequency of transitions However no clear instructions aregiven as to how to combine these five metrics into a singlesimilarity value Obweger et al [39] proposed a generic simi-larity model for time-stamped sequences in complex businessevents is model calculates similarity on the basis of devia-tions between a query pattern and its representation in acandidate event Additionally thismodel assesses dissimilaritiesat the level of single events their order their timing and theabsence of events However the single-event similarity is de-rived from the semantic distance between mapped eventswhich is not suitable for maintenance task similarity analysis

In this section the similarity measurement betweenmaintenance tasks is converted to a sequence matchingproblem e difference between two item sequences isquantified from the perspective of maintenance time andmaintainability characteristics To allow some tolerance thatcan recognize acceptable differences mapping cost functionsbetween the attributed item sequences are introduced into thematching process After that similar maintenance tasks canbe obtained by specifying the mapping cost threshold value

222 Problem Definition To ensure the clarity of this idearsquosexpression some definitions are given below

Definition 5 (similar items) For a given two items they aredefined to be similar if and only if their entity attribute setsare exactly the same that is they have the same type andoperation

Two attributed item sequences M1 langS1 G1 A1rang andM2 langS2 G2 A2rang are given assuming they both have five

V12 = (u121 u122 u123 u124 u125 u126)

V11 = (u111 u112 u113 u114 u115 u116)

V10 = (u101 u102 u103 u104 u105 u106)

V9 = (u91 u92 u93 u94 u95 u96)

V8 = (u81 u82 u83 u84 u85 u86)

V7 = (u71 u72 u73 u74 u75 u76)

V6 = (u61 u62 u63 u64 u65 u66)

V5 = (u51 u52 u53 u54 u55 u56)

V4 = (u41 u42 u43 u44 u45 u46)

V3 = (u31 u32 u33 u34 u35 u36)

V2 = (u21 u22 u23 u24 u25 u26)

V1 = (u11 u12 u13 u14 u15 u16)

E12 = lang(type nut) (operation screw)rang

E11 = lang(type transceiver) (operation press)rang

E10 = lang(type transceiver) (operation install)rang

E9 = lang(type electrical plug) (operation check)rang

E8 = lang(type cap) (operation dismantle)rang

E7 = lang(type interface) (operation check)rang

E6 = lang(type interface) (operation clean)rang

E5 = lang(type cap) (operation place)rang

E4 = lang(type transceiver) (operation dismantle)rang

E3 = lang(type transceiver) (operation pull)rang

E2 = lang(type nut) (operation lower)rang

E1 = lang(type nut) (operation unscrew)rang

M = langS G Arang whereS = I1 I2 hellip I12G = E1 E2 hellip E12A = V1 V2 hellip V12

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 t

Figure 2 Representation model for the task ldquoRepairinterchange--replace the transceiverrdquo

4 Mathematical Problems in Engineering

items and every item in one sequence has a similar item inanother sequence en to quantify the difference betweentwo sequences a one-to-one correspondence is assigned tosimilar items between S1 I11 I12 I13 I14 I151113864 1113865 andS2 I21 I22 I23 I24 I251113864 1113865 as shown in Figure 4 (the items withthe same color are similar)

Definition 6 (virtual item) A virtual item is a nonexistentitem that is used for full-sequence matching when theexisting items of the two sequences cannot all establish aone-to-one correspondence For example there are differentitems (different types or operations) or redundant itemsbetween two sequences

e given two attributed item sequencesM1 langS1 G1 A1rang and M2 langS2 G2 A2rang which have threesimilar items a different item I21 and a redundant item I15are shown in Figure 5

en in order to quantify the impact of I15 and I21 on thesimilarity analysis two virtual items I11 and I25 are added intothe sequences to enable full-sequencematching (see Figure 6)

1 General(i) Standardization

(ii) Components functionally grouped(iii) Console layout(iv) Complexity(v) Self-test

(vi) Maximum-time-to-repair(vii) Auxiliary tools and test equipment

(viii) Labeling(ix) Weight(x) Calibration requirements

(xi) Repairreplace philosophy(xii) Maintenance procedures

(xiii) Personnel requirements(xiv) Trade-off studies

2 Handling3 Equipment racks-general4 Packaging5 Accessibility6 Fasteners7 Cables8 Connectors9 Servicing and lubrication

10 Panel displays and controls11 Test points12 Adjustments13 Parts and components14 Environment15 Safety16 Reliability17 Software

Blanchard et al [2]

1 Inherent attributesConnection modeVisibilityStandardizationEntity reachabilityModularizationSecurityMaterial selectionProcessing technologyProcessing convenience

2 External factorsWorking environmentQuality of consumablesThe technical level of operatorsMaintenance cycleMaintenance positionMaintenance actionMaintenance spaceWorkerrsquos wagesRaw material costStorage securityThe technical level of maintainers

Jian et al [26]

1 Design configurationAccessibilityErgonomic factorsAutomation and mechanizationNormalization and interchangeability

2 Maintenance supportOrganization locality personnel and trainingProvision of spares facilities test equipmentEnvironmental conditions

Tarelko [28]

1 DesignAccessibilityDisassemblyassemblyStandardizationSimplicityIdentificationDiagnosabilityModularizationTribo-concepts

2 PersonnelPersonnel including ergonomicsSystem environment

3 Logistic supportTools and test equipmentDocumentation

Wani and Gandhi [30]

1 DesignStandardizationModularizationSimplicityDiagnosabilityIdentificationAccessibilityAssemblibilityServiceabilityTestabilityPartscomponentsReliability

2 PersonnelAnthropologyHuman sensoryPhysiologicalPsychological

3 Logistic supportSpares procurementTools amp test equipmentDocumentationSoftware

4 Operation contextSafetySystem environmentOperationmission type

Tjiparuro and Thompson [29]

1 General attributesSimplicityIdentificationModularityTribologyErgonomicsStandardizationFailure watchRelation with the manufacturer

2 Specific attributesAccessibilityAssemblydisassemblyTrainingPersonnel organizationEnvironmentSpare partsMaintenance tools and equipmentsInterdepartmental co-ordinationDocumentation

Leon et al [27]

1 Physical designSimplicityAccessibilityAssemblydisassemblyStandardizationModularizationTest points layout

2 Logistics supportTest equipmentAssemblydisassembly tool or maintenance toolDocumentation

3 ErgonomicsFault and operation indicatorsSkills of maintenance personnelMaintenance environmentOther ergonomics factors

Chen and Cai [25]

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)

(i)(ii)

(iii)

(iv)(v)

(vi)(vii)

(viii)(ix)

(i)(ii)

(iii)(iv)(v)

(vi)

(i)(ii)

(iii)

(i)(ii)

(iii)

(i)(ii)

(iii)(iv)

(iv)(v)

(vi)

(vii)(viii)

(ix)(x)

(xi)

(i)(ii)

(iii)(iv)(v)

(vi)(vii)

(viii)

(i)(ii)

(i)(ii)

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)

(i)(ii)

(iii)

(iv)

(v)(vi)

(vii)(viii) (i)

(ii)(iii)(iv)(v)

(vi)(vii)

(viii)

(i)(ii)

(iii)(iv)(v)

(vi)(vii)

(viii)(ix)

(ix)(x)

(xi)

Figure 3 Summary of the existing sets of maintainability attributes in the literature

t

tI11 I21 I31 I41 I51

I12 I22 I32 I42 I52

M1

M2

Figure 4 e one-to-one correspondence between sequences M1and M2

t

t

Different (redundant) items

I21 I31 I41 I51

I12 I22 I32 I42

M1

M2

Figure 5 Two sequences with different (redundant) items

Mathematical Problems in Engineering 5

e sequence of virtual items added is called the extendedattributed item sequence

Definition 7 (cosine similarity) Cosine similarity is ameasure of the similarity between two high-dimensionalvectors at is given two vectors X and Y

cos(X Y) langX Yrang

XY (1)

where ldquolangrangrdquo indicates the inner product of two vectors andldquordquo indicates the L2 norm of the vector

Definition 8 (item mapping cost IMC) For a given twoextended attributed item sequences M1 and M2 under theone-to-one correspondence the item mapping cost fi be-tween two similar items is

fi wi 1 minus cos V1i V

2i1113872 11138731113872 1113873 (2)

and the item mapping cost between one item and its cor-responding virtual item is

fi wi (3)

where wi is the item weight which represents the relativelength of the mean maintenance time spent on that type ofitem V1

i and V2i are the maintainability attribute value

vectors of the two items 1 minus cos(V1i V2

i ) represents thedifference between the maintainability characteristics of twosimilar items

Equations (2) and (3) show that the greater the differencebetween the maintainability characteristics of two similaritems or the more maintenance time the item costs thegreater the impact of the difference on the similarity analysis

Definition 9 (sequence mapping cost SMC) e sequencemapping cost H between the sequence M1 and M2 is

H12 1113944N

i1fi (4)

where N is the number of items in each sequencee SMC reflects the difference between two mainte-

nance tasks based on the representation models In generalthe larger the value of H is the larger the difference betweenthe two maintenance tasks is

Definition 10 (reference maintenance task) When theequipment to be MTTR demonstrated is specified themaintenance tasks for this equipment are defined as the

reference maintenance tasks denoted by Pri (i 1 2 K)

where K represents the number of task types

Definition 11 (candidate maintenance task set) A candidatemaintenance task P is a task that is compared to the ref-erence task e candidate maintenance task set denoted byOi Pi1 Pi2 PiNi

1113966 1113967(i 1 2 K) is the task set for thesimilarity search of the reference task Pr

i where Ni repre-sents the number of tasks Possible sources of candidate tasksinclude maintenance tasks relating to equipment or com-ponents in the same system that have similar functions orthat take place in a similar location

Definition 12 (similarity calculation) For a given referencemaintenance task Pr with a corresponding candidatemaintenance task set O and a user-specified SMC thresholdof ε a similarity search will retrieve all maintenance tasksPj isin O such that

Hj le ε j 1 2 Ni (5)

where Hj is the SMC between maintenance tasks Pj and PrIf equation (5) holds it can be stated that Pr and Pj aresimilar to the ε boundary We can then obtain the cluster ofsimilar candidate tasks for the reference task which isdenoted as C e sequence mapping cost (SMC) threshold εis a user-specified value and it is obvious that the larger the εthe more candidate maintenance tasks will be determined tobe similar to the reference maintenance task and then moredata will be available for constructing prior distributionHowever a larger ε will make some candidate maintenancetasks that are less similar to the reference task similar enoughfor a prior distribution elicitation which in turn makes theobtained prior distribution unreliable Hence it is importantto achieve a balance between the quantity and quality of datawhen specifying the SMC threshold value e SMCthreshold ε value can be determined through discussion withexperts based on the SMC calculation result to obtain datafrom the equipment or components as similar as possibleunder the precondition of having enough data for con-structing a prior distribution

223 Calculation of the Item Weights In this study theexpert experience is used to estimate the weight coefficientw As human judgments can be vague or ill-defined a fuzzyanalytic hierarchy process (FAHP) is used to calculate theweight coefficient of each item is method is mature andeasy to use in engineering practice and can make the weightsmore scientific when combined with the fuzzy judgment of

Extended

t

t

Different (redundant) items

I21 I31 I41 I51

I12 I22 I32 I42

M1

M2

Virtual items

t

tI11 I21 I31 I41 I51

I12 I22 I32 I42 I52

M1

M2

Figure 6 Extended sequences for full-sequence matching

6 Mathematical Problems in Engineering

experts based on their experience e implementation ofthis procedure is described below [40]

First a priority matrix Q (qij)ntimesn needs to be con-structed where the value of qij can be acquired through thepriority matrix scale method shown in Table 1

According to the results of the comparison betweendifferent items a priority matrix for each item can beconstructed as shown in Table 2

en the overall priority matrix Q is given by

Q qij1113872 1113873ntimesn

q11 middot middot middot q1n

middot middot middot

middot middot middot

middot middot middot

qn1 middot middot middot qnn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

Now a fuzzy consistent matrix can be constructedwhere R (rij)ntimesn

First the fuzzy complementary matrix is summed line byline where Q (qij)ntimesn

qi 1113944n

j1qij i 1 2 n (7)

en the following transformation is implemented toconstruct the fuzzy consistent matrix R (rij)ntimesn

rij qi minus qj

2n+ 05 (8)

e next set of calculations begins with the weight vectorof R is is given by the following

gi 1113945n

j1rij

⎛⎝ ⎞⎠

1n

(9)

e weight vector gi is now normalized

gi gi

1113936ni1 gi

i 1 2 n (10)

Finally the weight vector w can be constructed asfollows

w g1 g2 middot middot middot gn( 1113857T i 1 2 n (11)

e similarity computation algorithm based on theabove definitions is shown in Algorithm 1

To illustrate the method the maintenance task ldquoFaultisolationrdquo for the troubleshooting of the HF transceiverfailure is taken as an example After referring to the trou-bleshooting manual the chosen candidate tasks and theirprocedures are shown in Table 3

Assume that the maintainability attribute set includesentity reachability visibility maintenance space toolstechnical level of the maintainers maintenance position andsecurity e indicators are scored with a number from 0 to10 e higher the score is the better the maintainability isen the representation models for the reference andcandidate maintenance tasks are constructed as shown inTable 4

ere are two types of items in the sequences circuitbreaker and pin Using fuzzy AHP the priority matrices forthe two items are

Q 05 0

1 051113890 1113891 (12)

en according to equations (7)sim(11) the item weightsare obtained as

w1 0366

w2 0634(13)

e sequence matching between the reference and twocandidate maintenance tasks is shown in Figure 7

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks is ascertained (the resulthas been multiplied by 1000 for better comparison)

H1 asymp 655

H2 asymp 1272(14)

After discussions with the experts the SMC threshold isspecified as ε 800 en because H1 lt 800 H2 gt 800 themaintenance task for the wiring between the transceiver pinand the ground terminal is determined to be similar to thereference task

23 Elicitation of the Prior Distribution Commonly usedmethods for constructing a prior distribution include eli-cited priors conjugate priors and noninformative priors[41] As the similar candidate tasks in each cluster onlycontain the maintenance time data for the correspondingreference task not the whole maintenance action we use anoptimistic and pessimistic value method to estimate theparameters of the prior distribution A normal distributionis the commonly used form of the prior distribution inMTTR Bayesian demonstrations [14 16 20 42] so in thisstudy we also assumed a normal prior distribution for theparameter of interest

Let X sim LogN(μ σ2) denote the maintenance actiontime distribution of a specified product e variance σ2will either be known from prior information or a reasonablyprecise estimate can be obtained e prior distribution of μ

Table 1 Priority matrix scale method

Scale Definition Illustration1 More time Ii consumes more time than Ij

05 Equal time Ii and Ij consume equal time0 Less time Ii consumes less time than Ij

Table 2 Priority matrix of each item

Q Q1 Qn

Q1 q11 qn1 Qn q1n qnn

Mathematical Problems in Engineering 7

is denoted as N(μπ σ2π) According to the properties of thelognormal distribution

θ eμ+σ22

(15)

where θ is the mean of the maintenance time distributionen μ can be calculated as follows

μ ln θ minusσ2

2 (16)

If Xi(i 1 2 K) denotes the time spent on eachmaintenance task and xi(i 1 2 K) denotes the cor-responding maintenance task time data set then

X 1113944K

i1Xi (17)

Two predictions of the mean of the maintenance actiontimemdashthe lower or optimistic value θL and the upper orpessimistic value θUmdashcan be obtained as follows

1113954θL 1113944K

i1xi(min) (18)

1113954θU 1113944

K

i1xi(max) (19)

where xi(min) and xi(max) are the minimum and maximumvalues respectively for the time data set corresponding tocluster Ci

According to equation (16) the two possible predictionsof μ are

Input Pri Oi εi

Output Ci

(1) for each Pri do

(2) Ci⟵empty(3) Construct representation models for Pr

i and maintenance tasks in Oi(4) for each Pij isin Oi do(5) Perform sequence matching between Pr

i and Pij(6) Calculate item weights(7) Calculate SMC Hij(8) if Hij lt εi then(9) Ci⟵Pij(10) end if(11) end for(12) end for(13) Return Ci

ALGORITHM 1 Similarity computation algorithm based on maintenance task representation models

Table 3 Reference and candidate maintenance task procedures

Reference task Candidate task

HF transceiverPr

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC4 5 and 6 of the transceiver

e wiring between the transceiverpin and ground terminal P1

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC45 and 6 of the HF transceiver

(3) Do a check for a ground signal at pinAC8 of the HF transceiver

VHF transceiver P2

(1) Do a check of the circuit breakerstatus

(2) Do a check for 28DC at pin AC2 ofthe VHF transceiver

t

tI1r I2r I3r I4r I5r

I1 I2 I3 I4 I5

Mr

M1

Mr

M2 t

tI1r I2r I3r I4r

I1 I2 I3 I4

Figure 7 Sequence matching between the reference and candidate maintenance tasks

8 Mathematical Problems in Engineering

1113954μL ln 1113954θL minusσ2

2 (20)

1113954μU ln 1113954θU minusσ2

2 (21)

It can then be assumed that the range (1113954μU minus 1113954μL) en-compasses 100 times (1 minus p) percent of the total possible valuesof μ and that the best estimate is at the midpoint of the rangeerefore the following prior distribution estimates can beused

μπ 1113954μU + 1113954μL

2 (22)

σ2π 1113954μU minus 1113954μL( 1113857

2

4 times Z2p2

(23)

3 Case Study

In this section the implementation of an MTTR demon-stration for an HF transceiver is once again used to illustrateour method

31 Selection of Candidate Maintenance Tasks An HFtransceiver is part of the HF system and is installed at thefront of the electronics rack in a plane After referring to thetroubleshooting manual and the aircraft maintenancemanual [24 43] we established candidate tasks for eachreference task ese relate to other components in the HFsystem or other equipment at the front of the electronicsrack A breakdown of the tasks is shown in Table 5

32 Identification and Formulation of the MaintainabilityAttribute Set and Evaluation Rules e maintainability at-tribute set developed by Jian et al [26] was used for thesimilarity analysis of the maintenance tasks e main-tainability attributes were tailored to the characteristics ofthe different tasks as shown in Table 6 e correspondingevaluation rules are shown in Table 7

33 Similarity Analysis between the Maintenance Tasks

331 Construction of the Maintenance Task RepresentationModels After referring to the maintenance manuals and theexpertsrsquo experience representation models for the referenceand candidate maintenance tasks were established as shownin Table 8

332 Calculation of the Item Weights On the basis of therepresentation models an item list for each type of main-tenance task was established as shown in Table 9

Using fuzzy AHP the priority matrices for the variousitems in each type of maintenance task were thenobtained

Q1

05 05 05

05 05 05

05 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2

05 0 0

1 05 0

1 1 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q3

05 1 1 1 1 0 05 1 1 05 1 05 1 1

0 05 0 0 05 0 0 05 0 0 05 0 0 0

0 1 05 0 1 0 0 05 0 0 05 0 05 0

0 1 1 05 1 05 1 1 1 0 1 05 1 1

0 05 0 0 05 0 0 05 0 0 0 0 0 0

1 1 1 05 1 05 1 1 1 05 1 05 1 1

05 1 1 0 1 0 05 1 05 0 05 0 1 1

0 05 05 0 05 0 0 05 0 0 05 0 0 0

0 1 1 0 1 0 05 1 05 0 05 0 05 05

05 1 1 1 1 05 1 1 1 05 1 05 1 1

0 05 05 0 1 0 05 05 05 0 05 0 05 05

05 1 1 05 1 05 1 1 1 05 1 05 1 1

0 1 05 0 1 0 0 1 05 0 05 0 05 05

0 1 1 0 1 0 0 1 05 0 05 0 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

After this equations (7)sim(11) were used to establish theweight vectors for the items in each type of maintenancetask as follows

w1 (0333 0333 0333)

w2 (0211 0335 0454)

w3 (0094 0044 0055 0091 0040 0099 0077

0046 0069 0099 0061 0096 0063 0066)

(25)

333 Clustering of the Candidate Maintenance Taskse sequence matchings between the reference and candi-date maintenance tasks are shown in Figure 8

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks was ascertained (the resultwas multiplied by 1000 for better comparison)

H11 269

H12 198

H13 599

H21 34640

H22 80325

H23 67239

H31 41494

H32 038

H33 142

(26)

Based on the SMC calculation result the thresholds werespecified as

Mathematical Problems in Engineering 9

Tabl

e4

Representatio

nmod

elsforthereferenceandcand

idatemaintenance

tasks

Referencetask

Candidate

task

Mr

I 1rI 2r

I 3rI 4r

t

Mr

lang

SrG

rA

rrangwhere

Sr

Ir 1

Ir 2Ir 3

Ir 41113864

1113865

Gr

E

r 1E

r 2E

r 3E

r 41113864

1113865

Ar

V

r 1V

r 2V

r 3V

r 41113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Er 1

lang(type

circuitbreaker)

(op

erationcheck)

rangV

r 1

(77865910

)

Er 2

lang(type

pin

)(op

erationcheck)

rangV

r 2

(8873899)

Er 3

lang(type

pin

)(op

erationcheck)

rangV

r 3

(78738910

)

Er 4

lang(type

pin

)(op

erationcheck)

rangV

r 4

(8973898)

M1

I 1I 2

I 3I 4

I 5t

M1

lang

S1

G1

A1rang

where

S1

I 1

I2

I 3I

4I 5

11138641113865

G1

E1

E2

E3

E4

E5

11138641113865

A1

V

1V

2V

3V

4V

51113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

eration

check)

rangV

1

(98747710

)

E2

lang(type

pin

)(op

erationcheck)

rangV

2

(88728810

)

E3

lang(type

pin

)(op

erationcheck)

rangV

3

(89728710

)

E4

lang(type

pin

)(op

erationcheck)

rangV

4

(8962879)

E5

lang(type

pin

)(op

erationcheck)

rangV

5

(9872879)

M2

I 1I 2

t

M2

lang

S2

G2

A2rang

where

S2

I 1

I2

11138641113865

G2

E1

E2

11138641113865

A2

V

1V

21113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

erationcheck)

rang

E2

lang(type

pin

)(op

erationcheck)

rang

V1

(87868910

)

V2

(7873899)

lowastTo

quantifytheim

pact

ofdifferent

numbers

ofchecks

atpins

ontheSM

Ccalculation

echecks

atpins

AC4A

C5A

C6and

AC8

aretreatedseparately

10 Mathematical Problems in Engineering

Table 5 Candidate maintenance tasks for each reference task

Reference task Componentequipment Candidate task

Fault confirmationcheckout Pr

1

HF antenna coupler P11 (1) Press the row key near the HF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menuHF antenna P12

Very high-frequency (VHF) transceiver P13

(1) Press the row key near the VHF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menu

Fault isolation Pr2

e wiring between the transceiver pin AC8and the ground terminal P21

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver

P22

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver(4) Do a check of the wiring from the circuit breaker to the

HF transceiver pins AC4 5 and 6

VHF transceiver P23(1) Do a check of the circuit breaker status

(2) Do a check for 28 DC at pin AC2 of the VHF transceiver

Repairinterchange Pr3

HF antenna coupler P31

(1) Disconnect the electrical plug(2) Place cap on the electrical plug

(3) Unscrew the nut(4) Lower the nut

(5) Dismantle the antenna coupler(6) Place cap on the electrical plug

(7) Clean the interface and adjacent area(8) Check the interface and adjacent area

(9) Dismantle the cap from the electrical plug(10) Check the cleanliness and condition of the electrical

plug(11) Install the antenna coupler on the shelf

(12) Screw the nut(13) Dismantle the cap from electrical plug

(14) Check the cleanliness and condition of the electricalplug

(15) Connect the electrical plug to the antenna coupler

Audio management unit (AMU) P32

(1) Unscrew the nut(2) Lower the nut

(3) Pull the AMU from the shelf and disconnect the electricalplug

(4) Dismantle the AMU(5) Place cap on the electrical cap

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the AMU on the shelf(11) Press the AMU to connect the electrical plug

(12) Screw the nut

VHF transceiver P33

(1) Unscrew the nut(2) Lower the nut

(3) Pull the transceiver from the shelf and disconnect theelectrical plug

(4) Dismantle the transceiver(5) Place cap on the electrical plug

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug

(12) Screw the nut

Mathematical Problems in Engineering 11

ε1 6

ε2 420

ε3 3

(27)

en according to equation (5) the clusters for similarcandidate tasks for each reference task were established

C1 P11 P12 P131113864 1113865

C2 P211113864 1113865

C3 P32 P331113864 1113865

(28)

e results showed that the candidate tasks for faultconfirmationcheckout were all similar to the reference taskbecause they had the same items as the reference task de-spite having slightly different maintainability characteristicsIn the case of fault isolation three candidate tasks had

different items to the reference task Task P22 had two ad-ditional items (I5 and I6) and P23 had two missing items (Ir

3and Ir

4) P21 however had only one additional item (I5)making it more similar to the reference task than the othertwo tasks In the case of repairinterchange task P31 hadseven different items (I1 I2 Ir

3 Ir11 I13 I14 and I15) while

the other two tasks all had the same items as the referencetask us task P31 had the most differences and was theleast similar to the reference task

34 HF Transceiver MTTR Demonstration Based onBayesian eory

341 MTTR Demonstration Methods Based on Bayesianeory e Bayesian maintainability demonstrationmethod proposed by Balaban [44] is used in our paper for

Table 6 Maintainability attributes tailored to maintenance tasks

Maintainability attributes Fault confirmation Fault isolation Repairinterchange CheckoutEntity reachability Visibility Maintenance space Tools Technical level of the maintainers Maintenance position Security

Table 7 Evaluation rules

Maintainability attribute Evaluation rules

Entity reachability (U1)

Good can observe the maintenance component comfortably and there is a wide observation angle(7ndash10)

General can generally see the outline of the maintenance component though it can easily cause eye andbody fatigue (4ndash6)

Poor prone to fatigue in this pose (0ndash3)

Visibility (U2)Good clear line of sight and there is enough light (7ndash10)

General the line of sight is blocked or the light is dark (4ndash6)Poor the line of sight is seriously blocked or the light is insufficient (0ndash3)

Maintenance space (U3)

Good no restrictions on the maintenance space for operating posture (7ndash10)General maintenance space for the body is basically enough but the operatorrsquos posture is abnormal

(4ndash6)Poor there is not enough maintenance space for a body (0ndash3)

Tools (U4)Good do not need the aid of auxiliary tools (7ndash10)

General need auxiliary tools sometimes (4ndash6)Poor dependent on auxiliary tools (0ndash3)

Technical level of themaintainers (U5)

Good familiar with the relevant technical knowledge and can quickly determine the operation processand solve the problem (7ndash10)

General is familiar with the relevant knowledge and can solve problems by referring to the operationmanual (4ndash6)

Poor only a general understanding of the situation and how to carry out the relevant work (0ndash3)

Maintenance position (U6)Good ground (7ndash10)

General have to climb to the machine (4ndash6)Poor need to stand outside the machine or in a similar position to the machine (0ndash3)

Security (U7)

Good no danger of being injured by heavy objects and there are no sharp edges that may scratch ordanger of electrical shock (7ndash10)

General there is a certain security threat sometimes there are sharp edges that may cause bumps orscratches (4ndash6)

Poor there is a security threat (0ndash3)

12 Mathematical Problems in Engineering

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

Although there are differences between the above lists ofmaintainability attributes they all provide a comprehensiveoverview of the attributes required to understand main-tainability In practice one of the above maintainabilityattribute sets can be chosen as required or according toexpert opinion

22 Maintenance Task Similarity Analysis After the con-struction of maintenance task representation models asimilarity analysis can be performed on these models In thissection we propose a similarity computation algorithm formaintenance tasks After that the clusters of similarmaintenance tasks can then be obtained

221 Literature Review Similarity search methods have beenused in a wide variety of applications areas such as data mining[31] face recognition [32ndash34] image classification [35] medicalengineering [36] and human behavior analysis [37] Mainte-nance tasks are usually performed by maintenance staff sosimilarity searches for maintenance tasks fall under the remit ofhuman behavior analysis Human behavior can be representedfrom many perspectives from a low level eg individualmotions to an abstract level eg business processes Zhanget al [37] proposed an extended semantic distance calculationmethod called linked data semantic distance (LDSD) forsimilarity searches in relation to human behavior is methodis based on a multilayered process model (MLPM) whichdecomposes behaviors into three layers a processtask layer anactivity layer and an action layer However it is difficult toemploy this model for maintenance tasks because of the dif-ficulty of obtaining enough detail regarding human behaviorNeumuth et al [38] proposed using surgical process models

(SPMs) to represent surgical interventions and introduced fivesimilarity metrics for comparing SPMs ese metrics relate tothe granularity content temporal aspects transitional featuresand frequency of transitions However no clear instructions aregiven as to how to combine these five metrics into a singlesimilarity value Obweger et al [39] proposed a generic simi-larity model for time-stamped sequences in complex businessevents is model calculates similarity on the basis of devia-tions between a query pattern and its representation in acandidate event Additionally thismodel assesses dissimilaritiesat the level of single events their order their timing and theabsence of events However the single-event similarity is de-rived from the semantic distance between mapped eventswhich is not suitable for maintenance task similarity analysis

In this section the similarity measurement betweenmaintenance tasks is converted to a sequence matchingproblem e difference between two item sequences isquantified from the perspective of maintenance time andmaintainability characteristics To allow some tolerance thatcan recognize acceptable differences mapping cost functionsbetween the attributed item sequences are introduced into thematching process After that similar maintenance tasks canbe obtained by specifying the mapping cost threshold value

222 Problem Definition To ensure the clarity of this idearsquosexpression some definitions are given below

Definition 5 (similar items) For a given two items they aredefined to be similar if and only if their entity attribute setsare exactly the same that is they have the same type andoperation

Two attributed item sequences M1 langS1 G1 A1rang andM2 langS2 G2 A2rang are given assuming they both have five

V12 = (u121 u122 u123 u124 u125 u126)

V11 = (u111 u112 u113 u114 u115 u116)

V10 = (u101 u102 u103 u104 u105 u106)

V9 = (u91 u92 u93 u94 u95 u96)

V8 = (u81 u82 u83 u84 u85 u86)

V7 = (u71 u72 u73 u74 u75 u76)

V6 = (u61 u62 u63 u64 u65 u66)

V5 = (u51 u52 u53 u54 u55 u56)

V4 = (u41 u42 u43 u44 u45 u46)

V3 = (u31 u32 u33 u34 u35 u36)

V2 = (u21 u22 u23 u24 u25 u26)

V1 = (u11 u12 u13 u14 u15 u16)

E12 = lang(type nut) (operation screw)rang

E11 = lang(type transceiver) (operation press)rang

E10 = lang(type transceiver) (operation install)rang

E9 = lang(type electrical plug) (operation check)rang

E8 = lang(type cap) (operation dismantle)rang

E7 = lang(type interface) (operation check)rang

E6 = lang(type interface) (operation clean)rang

E5 = lang(type cap) (operation place)rang

E4 = lang(type transceiver) (operation dismantle)rang

E3 = lang(type transceiver) (operation pull)rang

E2 = lang(type nut) (operation lower)rang

E1 = lang(type nut) (operation unscrew)rang

M = langS G Arang whereS = I1 I2 hellip I12G = E1 E2 hellip E12A = V1 V2 hellip V12

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 t

Figure 2 Representation model for the task ldquoRepairinterchange--replace the transceiverrdquo

4 Mathematical Problems in Engineering

items and every item in one sequence has a similar item inanother sequence en to quantify the difference betweentwo sequences a one-to-one correspondence is assigned tosimilar items between S1 I11 I12 I13 I14 I151113864 1113865 andS2 I21 I22 I23 I24 I251113864 1113865 as shown in Figure 4 (the items withthe same color are similar)

Definition 6 (virtual item) A virtual item is a nonexistentitem that is used for full-sequence matching when theexisting items of the two sequences cannot all establish aone-to-one correspondence For example there are differentitems (different types or operations) or redundant itemsbetween two sequences

e given two attributed item sequencesM1 langS1 G1 A1rang and M2 langS2 G2 A2rang which have threesimilar items a different item I21 and a redundant item I15are shown in Figure 5

en in order to quantify the impact of I15 and I21 on thesimilarity analysis two virtual items I11 and I25 are added intothe sequences to enable full-sequencematching (see Figure 6)

1 General(i) Standardization

(ii) Components functionally grouped(iii) Console layout(iv) Complexity(v) Self-test

(vi) Maximum-time-to-repair(vii) Auxiliary tools and test equipment

(viii) Labeling(ix) Weight(x) Calibration requirements

(xi) Repairreplace philosophy(xii) Maintenance procedures

(xiii) Personnel requirements(xiv) Trade-off studies

2 Handling3 Equipment racks-general4 Packaging5 Accessibility6 Fasteners7 Cables8 Connectors9 Servicing and lubrication

10 Panel displays and controls11 Test points12 Adjustments13 Parts and components14 Environment15 Safety16 Reliability17 Software

Blanchard et al [2]

1 Inherent attributesConnection modeVisibilityStandardizationEntity reachabilityModularizationSecurityMaterial selectionProcessing technologyProcessing convenience

2 External factorsWorking environmentQuality of consumablesThe technical level of operatorsMaintenance cycleMaintenance positionMaintenance actionMaintenance spaceWorkerrsquos wagesRaw material costStorage securityThe technical level of maintainers

Jian et al [26]

1 Design configurationAccessibilityErgonomic factorsAutomation and mechanizationNormalization and interchangeability

2 Maintenance supportOrganization locality personnel and trainingProvision of spares facilities test equipmentEnvironmental conditions

Tarelko [28]

1 DesignAccessibilityDisassemblyassemblyStandardizationSimplicityIdentificationDiagnosabilityModularizationTribo-concepts

2 PersonnelPersonnel including ergonomicsSystem environment

3 Logistic supportTools and test equipmentDocumentation

Wani and Gandhi [30]

1 DesignStandardizationModularizationSimplicityDiagnosabilityIdentificationAccessibilityAssemblibilityServiceabilityTestabilityPartscomponentsReliability

2 PersonnelAnthropologyHuman sensoryPhysiologicalPsychological

3 Logistic supportSpares procurementTools amp test equipmentDocumentationSoftware

4 Operation contextSafetySystem environmentOperationmission type

Tjiparuro and Thompson [29]

1 General attributesSimplicityIdentificationModularityTribologyErgonomicsStandardizationFailure watchRelation with the manufacturer

2 Specific attributesAccessibilityAssemblydisassemblyTrainingPersonnel organizationEnvironmentSpare partsMaintenance tools and equipmentsInterdepartmental co-ordinationDocumentation

Leon et al [27]

1 Physical designSimplicityAccessibilityAssemblydisassemblyStandardizationModularizationTest points layout

2 Logistics supportTest equipmentAssemblydisassembly tool or maintenance toolDocumentation

3 ErgonomicsFault and operation indicatorsSkills of maintenance personnelMaintenance environmentOther ergonomics factors

Chen and Cai [25]

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)

(i)(ii)

(iii)

(iv)(v)

(vi)(vii)

(viii)(ix)

(i)(ii)

(iii)(iv)(v)

(vi)

(i)(ii)

(iii)

(i)(ii)

(iii)

(i)(ii)

(iii)(iv)

(iv)(v)

(vi)

(vii)(viii)

(ix)(x)

(xi)

(i)(ii)

(iii)(iv)(v)

(vi)(vii)

(viii)

(i)(ii)

(i)(ii)

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)

(i)(ii)

(iii)

(iv)

(v)(vi)

(vii)(viii) (i)

(ii)(iii)(iv)(v)

(vi)(vii)

(viii)

(i)(ii)

(iii)(iv)(v)

(vi)(vii)

(viii)(ix)

(ix)(x)

(xi)

Figure 3 Summary of the existing sets of maintainability attributes in the literature

t

tI11 I21 I31 I41 I51

I12 I22 I32 I42 I52

M1

M2

Figure 4 e one-to-one correspondence between sequences M1and M2

t

t

Different (redundant) items

I21 I31 I41 I51

I12 I22 I32 I42

M1

M2

Figure 5 Two sequences with different (redundant) items

Mathematical Problems in Engineering 5

e sequence of virtual items added is called the extendedattributed item sequence

Definition 7 (cosine similarity) Cosine similarity is ameasure of the similarity between two high-dimensionalvectors at is given two vectors X and Y

cos(X Y) langX Yrang

XY (1)

where ldquolangrangrdquo indicates the inner product of two vectors andldquordquo indicates the L2 norm of the vector

Definition 8 (item mapping cost IMC) For a given twoextended attributed item sequences M1 and M2 under theone-to-one correspondence the item mapping cost fi be-tween two similar items is

fi wi 1 minus cos V1i V

2i1113872 11138731113872 1113873 (2)

and the item mapping cost between one item and its cor-responding virtual item is

fi wi (3)

where wi is the item weight which represents the relativelength of the mean maintenance time spent on that type ofitem V1

i and V2i are the maintainability attribute value

vectors of the two items 1 minus cos(V1i V2

i ) represents thedifference between the maintainability characteristics of twosimilar items

Equations (2) and (3) show that the greater the differencebetween the maintainability characteristics of two similaritems or the more maintenance time the item costs thegreater the impact of the difference on the similarity analysis

Definition 9 (sequence mapping cost SMC) e sequencemapping cost H between the sequence M1 and M2 is

H12 1113944N

i1fi (4)

where N is the number of items in each sequencee SMC reflects the difference between two mainte-

nance tasks based on the representation models In generalthe larger the value of H is the larger the difference betweenthe two maintenance tasks is

Definition 10 (reference maintenance task) When theequipment to be MTTR demonstrated is specified themaintenance tasks for this equipment are defined as the

reference maintenance tasks denoted by Pri (i 1 2 K)

where K represents the number of task types

Definition 11 (candidate maintenance task set) A candidatemaintenance task P is a task that is compared to the ref-erence task e candidate maintenance task set denoted byOi Pi1 Pi2 PiNi

1113966 1113967(i 1 2 K) is the task set for thesimilarity search of the reference task Pr

i where Ni repre-sents the number of tasks Possible sources of candidate tasksinclude maintenance tasks relating to equipment or com-ponents in the same system that have similar functions orthat take place in a similar location

Definition 12 (similarity calculation) For a given referencemaintenance task Pr with a corresponding candidatemaintenance task set O and a user-specified SMC thresholdof ε a similarity search will retrieve all maintenance tasksPj isin O such that

Hj le ε j 1 2 Ni (5)

where Hj is the SMC between maintenance tasks Pj and PrIf equation (5) holds it can be stated that Pr and Pj aresimilar to the ε boundary We can then obtain the cluster ofsimilar candidate tasks for the reference task which isdenoted as C e sequence mapping cost (SMC) threshold εis a user-specified value and it is obvious that the larger the εthe more candidate maintenance tasks will be determined tobe similar to the reference maintenance task and then moredata will be available for constructing prior distributionHowever a larger ε will make some candidate maintenancetasks that are less similar to the reference task similar enoughfor a prior distribution elicitation which in turn makes theobtained prior distribution unreliable Hence it is importantto achieve a balance between the quantity and quality of datawhen specifying the SMC threshold value e SMCthreshold ε value can be determined through discussion withexperts based on the SMC calculation result to obtain datafrom the equipment or components as similar as possibleunder the precondition of having enough data for con-structing a prior distribution

223 Calculation of the Item Weights In this study theexpert experience is used to estimate the weight coefficientw As human judgments can be vague or ill-defined a fuzzyanalytic hierarchy process (FAHP) is used to calculate theweight coefficient of each item is method is mature andeasy to use in engineering practice and can make the weightsmore scientific when combined with the fuzzy judgment of

Extended

t

t

Different (redundant) items

I21 I31 I41 I51

I12 I22 I32 I42

M1

M2

Virtual items

t

tI11 I21 I31 I41 I51

I12 I22 I32 I42 I52

M1

M2

Figure 6 Extended sequences for full-sequence matching

6 Mathematical Problems in Engineering

experts based on their experience e implementation ofthis procedure is described below [40]

First a priority matrix Q (qij)ntimesn needs to be con-structed where the value of qij can be acquired through thepriority matrix scale method shown in Table 1

According to the results of the comparison betweendifferent items a priority matrix for each item can beconstructed as shown in Table 2

en the overall priority matrix Q is given by

Q qij1113872 1113873ntimesn

q11 middot middot middot q1n

middot middot middot

middot middot middot

middot middot middot

qn1 middot middot middot qnn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

Now a fuzzy consistent matrix can be constructedwhere R (rij)ntimesn

First the fuzzy complementary matrix is summed line byline where Q (qij)ntimesn

qi 1113944n

j1qij i 1 2 n (7)

en the following transformation is implemented toconstruct the fuzzy consistent matrix R (rij)ntimesn

rij qi minus qj

2n+ 05 (8)

e next set of calculations begins with the weight vectorof R is is given by the following

gi 1113945n

j1rij

⎛⎝ ⎞⎠

1n

(9)

e weight vector gi is now normalized

gi gi

1113936ni1 gi

i 1 2 n (10)

Finally the weight vector w can be constructed asfollows

w g1 g2 middot middot middot gn( 1113857T i 1 2 n (11)

e similarity computation algorithm based on theabove definitions is shown in Algorithm 1

To illustrate the method the maintenance task ldquoFaultisolationrdquo for the troubleshooting of the HF transceiverfailure is taken as an example After referring to the trou-bleshooting manual the chosen candidate tasks and theirprocedures are shown in Table 3

Assume that the maintainability attribute set includesentity reachability visibility maintenance space toolstechnical level of the maintainers maintenance position andsecurity e indicators are scored with a number from 0 to10 e higher the score is the better the maintainability isen the representation models for the reference andcandidate maintenance tasks are constructed as shown inTable 4

ere are two types of items in the sequences circuitbreaker and pin Using fuzzy AHP the priority matrices forthe two items are

Q 05 0

1 051113890 1113891 (12)

en according to equations (7)sim(11) the item weightsare obtained as

w1 0366

w2 0634(13)

e sequence matching between the reference and twocandidate maintenance tasks is shown in Figure 7

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks is ascertained (the resulthas been multiplied by 1000 for better comparison)

H1 asymp 655

H2 asymp 1272(14)

After discussions with the experts the SMC threshold isspecified as ε 800 en because H1 lt 800 H2 gt 800 themaintenance task for the wiring between the transceiver pinand the ground terminal is determined to be similar to thereference task

23 Elicitation of the Prior Distribution Commonly usedmethods for constructing a prior distribution include eli-cited priors conjugate priors and noninformative priors[41] As the similar candidate tasks in each cluster onlycontain the maintenance time data for the correspondingreference task not the whole maintenance action we use anoptimistic and pessimistic value method to estimate theparameters of the prior distribution A normal distributionis the commonly used form of the prior distribution inMTTR Bayesian demonstrations [14 16 20 42] so in thisstudy we also assumed a normal prior distribution for theparameter of interest

Let X sim LogN(μ σ2) denote the maintenance actiontime distribution of a specified product e variance σ2will either be known from prior information or a reasonablyprecise estimate can be obtained e prior distribution of μ

Table 1 Priority matrix scale method

Scale Definition Illustration1 More time Ii consumes more time than Ij

05 Equal time Ii and Ij consume equal time0 Less time Ii consumes less time than Ij

Table 2 Priority matrix of each item

Q Q1 Qn

Q1 q11 qn1 Qn q1n qnn

Mathematical Problems in Engineering 7

is denoted as N(μπ σ2π) According to the properties of thelognormal distribution

θ eμ+σ22

(15)

where θ is the mean of the maintenance time distributionen μ can be calculated as follows

μ ln θ minusσ2

2 (16)

If Xi(i 1 2 K) denotes the time spent on eachmaintenance task and xi(i 1 2 K) denotes the cor-responding maintenance task time data set then

X 1113944K

i1Xi (17)

Two predictions of the mean of the maintenance actiontimemdashthe lower or optimistic value θL and the upper orpessimistic value θUmdashcan be obtained as follows

1113954θL 1113944K

i1xi(min) (18)

1113954θU 1113944

K

i1xi(max) (19)

where xi(min) and xi(max) are the minimum and maximumvalues respectively for the time data set corresponding tocluster Ci

According to equation (16) the two possible predictionsof μ are

Input Pri Oi εi

Output Ci

(1) for each Pri do

(2) Ci⟵empty(3) Construct representation models for Pr

i and maintenance tasks in Oi(4) for each Pij isin Oi do(5) Perform sequence matching between Pr

i and Pij(6) Calculate item weights(7) Calculate SMC Hij(8) if Hij lt εi then(9) Ci⟵Pij(10) end if(11) end for(12) end for(13) Return Ci

ALGORITHM 1 Similarity computation algorithm based on maintenance task representation models

Table 3 Reference and candidate maintenance task procedures

Reference task Candidate task

HF transceiverPr

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC4 5 and 6 of the transceiver

e wiring between the transceiverpin and ground terminal P1

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC45 and 6 of the HF transceiver

(3) Do a check for a ground signal at pinAC8 of the HF transceiver

VHF transceiver P2

(1) Do a check of the circuit breakerstatus

(2) Do a check for 28DC at pin AC2 ofthe VHF transceiver

t

tI1r I2r I3r I4r I5r

I1 I2 I3 I4 I5

Mr

M1

Mr

M2 t

tI1r I2r I3r I4r

I1 I2 I3 I4

Figure 7 Sequence matching between the reference and candidate maintenance tasks

8 Mathematical Problems in Engineering

1113954μL ln 1113954θL minusσ2

2 (20)

1113954μU ln 1113954θU minusσ2

2 (21)

It can then be assumed that the range (1113954μU minus 1113954μL) en-compasses 100 times (1 minus p) percent of the total possible valuesof μ and that the best estimate is at the midpoint of the rangeerefore the following prior distribution estimates can beused

μπ 1113954μU + 1113954μL

2 (22)

σ2π 1113954μU minus 1113954μL( 1113857

2

4 times Z2p2

(23)

3 Case Study

In this section the implementation of an MTTR demon-stration for an HF transceiver is once again used to illustrateour method

31 Selection of Candidate Maintenance Tasks An HFtransceiver is part of the HF system and is installed at thefront of the electronics rack in a plane After referring to thetroubleshooting manual and the aircraft maintenancemanual [24 43] we established candidate tasks for eachreference task ese relate to other components in the HFsystem or other equipment at the front of the electronicsrack A breakdown of the tasks is shown in Table 5

32 Identification and Formulation of the MaintainabilityAttribute Set and Evaluation Rules e maintainability at-tribute set developed by Jian et al [26] was used for thesimilarity analysis of the maintenance tasks e main-tainability attributes were tailored to the characteristics ofthe different tasks as shown in Table 6 e correspondingevaluation rules are shown in Table 7

33 Similarity Analysis between the Maintenance Tasks

331 Construction of the Maintenance Task RepresentationModels After referring to the maintenance manuals and theexpertsrsquo experience representation models for the referenceand candidate maintenance tasks were established as shownin Table 8

332 Calculation of the Item Weights On the basis of therepresentation models an item list for each type of main-tenance task was established as shown in Table 9

Using fuzzy AHP the priority matrices for the variousitems in each type of maintenance task were thenobtained

Q1

05 05 05

05 05 05

05 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2

05 0 0

1 05 0

1 1 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q3

05 1 1 1 1 0 05 1 1 05 1 05 1 1

0 05 0 0 05 0 0 05 0 0 05 0 0 0

0 1 05 0 1 0 0 05 0 0 05 0 05 0

0 1 1 05 1 05 1 1 1 0 1 05 1 1

0 05 0 0 05 0 0 05 0 0 0 0 0 0

1 1 1 05 1 05 1 1 1 05 1 05 1 1

05 1 1 0 1 0 05 1 05 0 05 0 1 1

0 05 05 0 05 0 0 05 0 0 05 0 0 0

0 1 1 0 1 0 05 1 05 0 05 0 05 05

05 1 1 1 1 05 1 1 1 05 1 05 1 1

0 05 05 0 1 0 05 05 05 0 05 0 05 05

05 1 1 05 1 05 1 1 1 05 1 05 1 1

0 1 05 0 1 0 0 1 05 0 05 0 05 05

0 1 1 0 1 0 0 1 05 0 05 0 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

After this equations (7)sim(11) were used to establish theweight vectors for the items in each type of maintenancetask as follows

w1 (0333 0333 0333)

w2 (0211 0335 0454)

w3 (0094 0044 0055 0091 0040 0099 0077

0046 0069 0099 0061 0096 0063 0066)

(25)

333 Clustering of the Candidate Maintenance Taskse sequence matchings between the reference and candi-date maintenance tasks are shown in Figure 8

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks was ascertained (the resultwas multiplied by 1000 for better comparison)

H11 269

H12 198

H13 599

H21 34640

H22 80325

H23 67239

H31 41494

H32 038

H33 142

(26)

Based on the SMC calculation result the thresholds werespecified as

Mathematical Problems in Engineering 9

Tabl

e4

Representatio

nmod

elsforthereferenceandcand

idatemaintenance

tasks

Referencetask

Candidate

task

Mr

I 1rI 2r

I 3rI 4r

t

Mr

lang

SrG

rA

rrangwhere

Sr

Ir 1

Ir 2Ir 3

Ir 41113864

1113865

Gr

E

r 1E

r 2E

r 3E

r 41113864

1113865

Ar

V

r 1V

r 2V

r 3V

r 41113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Er 1

lang(type

circuitbreaker)

(op

erationcheck)

rangV

r 1

(77865910

)

Er 2

lang(type

pin

)(op

erationcheck)

rangV

r 2

(8873899)

Er 3

lang(type

pin

)(op

erationcheck)

rangV

r 3

(78738910

)

Er 4

lang(type

pin

)(op

erationcheck)

rangV

r 4

(8973898)

M1

I 1I 2

I 3I 4

I 5t

M1

lang

S1

G1

A1rang

where

S1

I 1

I2

I 3I

4I 5

11138641113865

G1

E1

E2

E3

E4

E5

11138641113865

A1

V

1V

2V

3V

4V

51113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

eration

check)

rangV

1

(98747710

)

E2

lang(type

pin

)(op

erationcheck)

rangV

2

(88728810

)

E3

lang(type

pin

)(op

erationcheck)

rangV

3

(89728710

)

E4

lang(type

pin

)(op

erationcheck)

rangV

4

(8962879)

E5

lang(type

pin

)(op

erationcheck)

rangV

5

(9872879)

M2

I 1I 2

t

M2

lang

S2

G2

A2rang

where

S2

I 1

I2

11138641113865

G2

E1

E2

11138641113865

A2

V

1V

21113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

erationcheck)

rang

E2

lang(type

pin

)(op

erationcheck)

rang

V1

(87868910

)

V2

(7873899)

lowastTo

quantifytheim

pact

ofdifferent

numbers

ofchecks

atpins

ontheSM

Ccalculation

echecks

atpins

AC4A

C5A

C6and

AC8

aretreatedseparately

10 Mathematical Problems in Engineering

Table 5 Candidate maintenance tasks for each reference task

Reference task Componentequipment Candidate task

Fault confirmationcheckout Pr

1

HF antenna coupler P11 (1) Press the row key near the HF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menuHF antenna P12

Very high-frequency (VHF) transceiver P13

(1) Press the row key near the VHF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menu

Fault isolation Pr2

e wiring between the transceiver pin AC8and the ground terminal P21

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver

P22

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver(4) Do a check of the wiring from the circuit breaker to the

HF transceiver pins AC4 5 and 6

VHF transceiver P23(1) Do a check of the circuit breaker status

(2) Do a check for 28 DC at pin AC2 of the VHF transceiver

Repairinterchange Pr3

HF antenna coupler P31

(1) Disconnect the electrical plug(2) Place cap on the electrical plug

(3) Unscrew the nut(4) Lower the nut

(5) Dismantle the antenna coupler(6) Place cap on the electrical plug

(7) Clean the interface and adjacent area(8) Check the interface and adjacent area

(9) Dismantle the cap from the electrical plug(10) Check the cleanliness and condition of the electrical

plug(11) Install the antenna coupler on the shelf

(12) Screw the nut(13) Dismantle the cap from electrical plug

(14) Check the cleanliness and condition of the electricalplug

(15) Connect the electrical plug to the antenna coupler

Audio management unit (AMU) P32

(1) Unscrew the nut(2) Lower the nut

(3) Pull the AMU from the shelf and disconnect the electricalplug

(4) Dismantle the AMU(5) Place cap on the electrical cap

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the AMU on the shelf(11) Press the AMU to connect the electrical plug

(12) Screw the nut

VHF transceiver P33

(1) Unscrew the nut(2) Lower the nut

(3) Pull the transceiver from the shelf and disconnect theelectrical plug

(4) Dismantle the transceiver(5) Place cap on the electrical plug

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug

(12) Screw the nut

Mathematical Problems in Engineering 11

ε1 6

ε2 420

ε3 3

(27)

en according to equation (5) the clusters for similarcandidate tasks for each reference task were established

C1 P11 P12 P131113864 1113865

C2 P211113864 1113865

C3 P32 P331113864 1113865

(28)

e results showed that the candidate tasks for faultconfirmationcheckout were all similar to the reference taskbecause they had the same items as the reference task de-spite having slightly different maintainability characteristicsIn the case of fault isolation three candidate tasks had

different items to the reference task Task P22 had two ad-ditional items (I5 and I6) and P23 had two missing items (Ir

3and Ir

4) P21 however had only one additional item (I5)making it more similar to the reference task than the othertwo tasks In the case of repairinterchange task P31 hadseven different items (I1 I2 Ir

3 Ir11 I13 I14 and I15) while

the other two tasks all had the same items as the referencetask us task P31 had the most differences and was theleast similar to the reference task

34 HF Transceiver MTTR Demonstration Based onBayesian eory

341 MTTR Demonstration Methods Based on Bayesianeory e Bayesian maintainability demonstrationmethod proposed by Balaban [44] is used in our paper for

Table 6 Maintainability attributes tailored to maintenance tasks

Maintainability attributes Fault confirmation Fault isolation Repairinterchange CheckoutEntity reachability Visibility Maintenance space Tools Technical level of the maintainers Maintenance position Security

Table 7 Evaluation rules

Maintainability attribute Evaluation rules

Entity reachability (U1)

Good can observe the maintenance component comfortably and there is a wide observation angle(7ndash10)

General can generally see the outline of the maintenance component though it can easily cause eye andbody fatigue (4ndash6)

Poor prone to fatigue in this pose (0ndash3)

Visibility (U2)Good clear line of sight and there is enough light (7ndash10)

General the line of sight is blocked or the light is dark (4ndash6)Poor the line of sight is seriously blocked or the light is insufficient (0ndash3)

Maintenance space (U3)

Good no restrictions on the maintenance space for operating posture (7ndash10)General maintenance space for the body is basically enough but the operatorrsquos posture is abnormal

(4ndash6)Poor there is not enough maintenance space for a body (0ndash3)

Tools (U4)Good do not need the aid of auxiliary tools (7ndash10)

General need auxiliary tools sometimes (4ndash6)Poor dependent on auxiliary tools (0ndash3)

Technical level of themaintainers (U5)

Good familiar with the relevant technical knowledge and can quickly determine the operation processand solve the problem (7ndash10)

General is familiar with the relevant knowledge and can solve problems by referring to the operationmanual (4ndash6)

Poor only a general understanding of the situation and how to carry out the relevant work (0ndash3)

Maintenance position (U6)Good ground (7ndash10)

General have to climb to the machine (4ndash6)Poor need to stand outside the machine or in a similar position to the machine (0ndash3)

Security (U7)

Good no danger of being injured by heavy objects and there are no sharp edges that may scratch ordanger of electrical shock (7ndash10)

General there is a certain security threat sometimes there are sharp edges that may cause bumps orscratches (4ndash6)

Poor there is a security threat (0ndash3)

12 Mathematical Problems in Engineering

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

items and every item in one sequence has a similar item inanother sequence en to quantify the difference betweentwo sequences a one-to-one correspondence is assigned tosimilar items between S1 I11 I12 I13 I14 I151113864 1113865 andS2 I21 I22 I23 I24 I251113864 1113865 as shown in Figure 4 (the items withthe same color are similar)

Definition 6 (virtual item) A virtual item is a nonexistentitem that is used for full-sequence matching when theexisting items of the two sequences cannot all establish aone-to-one correspondence For example there are differentitems (different types or operations) or redundant itemsbetween two sequences

e given two attributed item sequencesM1 langS1 G1 A1rang and M2 langS2 G2 A2rang which have threesimilar items a different item I21 and a redundant item I15are shown in Figure 5

en in order to quantify the impact of I15 and I21 on thesimilarity analysis two virtual items I11 and I25 are added intothe sequences to enable full-sequencematching (see Figure 6)

1 General(i) Standardization

(ii) Components functionally grouped(iii) Console layout(iv) Complexity(v) Self-test

(vi) Maximum-time-to-repair(vii) Auxiliary tools and test equipment

(viii) Labeling(ix) Weight(x) Calibration requirements

(xi) Repairreplace philosophy(xii) Maintenance procedures

(xiii) Personnel requirements(xiv) Trade-off studies

2 Handling3 Equipment racks-general4 Packaging5 Accessibility6 Fasteners7 Cables8 Connectors9 Servicing and lubrication

10 Panel displays and controls11 Test points12 Adjustments13 Parts and components14 Environment15 Safety16 Reliability17 Software

Blanchard et al [2]

1 Inherent attributesConnection modeVisibilityStandardizationEntity reachabilityModularizationSecurityMaterial selectionProcessing technologyProcessing convenience

2 External factorsWorking environmentQuality of consumablesThe technical level of operatorsMaintenance cycleMaintenance positionMaintenance actionMaintenance spaceWorkerrsquos wagesRaw material costStorage securityThe technical level of maintainers

Jian et al [26]

1 Design configurationAccessibilityErgonomic factorsAutomation and mechanizationNormalization and interchangeability

2 Maintenance supportOrganization locality personnel and trainingProvision of spares facilities test equipmentEnvironmental conditions

Tarelko [28]

1 DesignAccessibilityDisassemblyassemblyStandardizationSimplicityIdentificationDiagnosabilityModularizationTribo-concepts

2 PersonnelPersonnel including ergonomicsSystem environment

3 Logistic supportTools and test equipmentDocumentation

Wani and Gandhi [30]

1 DesignStandardizationModularizationSimplicityDiagnosabilityIdentificationAccessibilityAssemblibilityServiceabilityTestabilityPartscomponentsReliability

2 PersonnelAnthropologyHuman sensoryPhysiologicalPsychological

3 Logistic supportSpares procurementTools amp test equipmentDocumentationSoftware

4 Operation contextSafetySystem environmentOperationmission type

Tjiparuro and Thompson [29]

1 General attributesSimplicityIdentificationModularityTribologyErgonomicsStandardizationFailure watchRelation with the manufacturer

2 Specific attributesAccessibilityAssemblydisassemblyTrainingPersonnel organizationEnvironmentSpare partsMaintenance tools and equipmentsInterdepartmental co-ordinationDocumentation

Leon et al [27]

1 Physical designSimplicityAccessibilityAssemblydisassemblyStandardizationModularizationTest points layout

2 Logistics supportTest equipmentAssemblydisassembly tool or maintenance toolDocumentation

3 ErgonomicsFault and operation indicatorsSkills of maintenance personnelMaintenance environmentOther ergonomics factors

Chen and Cai [25]

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)

(i)(ii)

(iii)

(iv)(v)

(vi)(vii)

(viii)(ix)

(i)(ii)

(iii)(iv)(v)

(vi)

(i)(ii)

(iii)

(i)(ii)

(iii)

(i)(ii)

(iii)(iv)

(iv)(v)

(vi)

(vii)(viii)

(ix)(x)

(xi)

(i)(ii)

(iii)(iv)(v)

(vi)(vii)

(viii)

(i)(ii)

(i)(ii)

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)(iv)

(i)(ii)

(iii)

(i)(ii)

(iii)

(iv)

(v)(vi)

(vii)(viii) (i)

(ii)(iii)(iv)(v)

(vi)(vii)

(viii)

(i)(ii)

(iii)(iv)(v)

(vi)(vii)

(viii)(ix)

(ix)(x)

(xi)

Figure 3 Summary of the existing sets of maintainability attributes in the literature

t

tI11 I21 I31 I41 I51

I12 I22 I32 I42 I52

M1

M2

Figure 4 e one-to-one correspondence between sequences M1and M2

t

t

Different (redundant) items

I21 I31 I41 I51

I12 I22 I32 I42

M1

M2

Figure 5 Two sequences with different (redundant) items

Mathematical Problems in Engineering 5

e sequence of virtual items added is called the extendedattributed item sequence

Definition 7 (cosine similarity) Cosine similarity is ameasure of the similarity between two high-dimensionalvectors at is given two vectors X and Y

cos(X Y) langX Yrang

XY (1)

where ldquolangrangrdquo indicates the inner product of two vectors andldquordquo indicates the L2 norm of the vector

Definition 8 (item mapping cost IMC) For a given twoextended attributed item sequences M1 and M2 under theone-to-one correspondence the item mapping cost fi be-tween two similar items is

fi wi 1 minus cos V1i V

2i1113872 11138731113872 1113873 (2)

and the item mapping cost between one item and its cor-responding virtual item is

fi wi (3)

where wi is the item weight which represents the relativelength of the mean maintenance time spent on that type ofitem V1

i and V2i are the maintainability attribute value

vectors of the two items 1 minus cos(V1i V2

i ) represents thedifference between the maintainability characteristics of twosimilar items

Equations (2) and (3) show that the greater the differencebetween the maintainability characteristics of two similaritems or the more maintenance time the item costs thegreater the impact of the difference on the similarity analysis

Definition 9 (sequence mapping cost SMC) e sequencemapping cost H between the sequence M1 and M2 is

H12 1113944N

i1fi (4)

where N is the number of items in each sequencee SMC reflects the difference between two mainte-

nance tasks based on the representation models In generalthe larger the value of H is the larger the difference betweenthe two maintenance tasks is

Definition 10 (reference maintenance task) When theequipment to be MTTR demonstrated is specified themaintenance tasks for this equipment are defined as the

reference maintenance tasks denoted by Pri (i 1 2 K)

where K represents the number of task types

Definition 11 (candidate maintenance task set) A candidatemaintenance task P is a task that is compared to the ref-erence task e candidate maintenance task set denoted byOi Pi1 Pi2 PiNi

1113966 1113967(i 1 2 K) is the task set for thesimilarity search of the reference task Pr

i where Ni repre-sents the number of tasks Possible sources of candidate tasksinclude maintenance tasks relating to equipment or com-ponents in the same system that have similar functions orthat take place in a similar location

Definition 12 (similarity calculation) For a given referencemaintenance task Pr with a corresponding candidatemaintenance task set O and a user-specified SMC thresholdof ε a similarity search will retrieve all maintenance tasksPj isin O such that

Hj le ε j 1 2 Ni (5)

where Hj is the SMC between maintenance tasks Pj and PrIf equation (5) holds it can be stated that Pr and Pj aresimilar to the ε boundary We can then obtain the cluster ofsimilar candidate tasks for the reference task which isdenoted as C e sequence mapping cost (SMC) threshold εis a user-specified value and it is obvious that the larger the εthe more candidate maintenance tasks will be determined tobe similar to the reference maintenance task and then moredata will be available for constructing prior distributionHowever a larger ε will make some candidate maintenancetasks that are less similar to the reference task similar enoughfor a prior distribution elicitation which in turn makes theobtained prior distribution unreliable Hence it is importantto achieve a balance between the quantity and quality of datawhen specifying the SMC threshold value e SMCthreshold ε value can be determined through discussion withexperts based on the SMC calculation result to obtain datafrom the equipment or components as similar as possibleunder the precondition of having enough data for con-structing a prior distribution

223 Calculation of the Item Weights In this study theexpert experience is used to estimate the weight coefficientw As human judgments can be vague or ill-defined a fuzzyanalytic hierarchy process (FAHP) is used to calculate theweight coefficient of each item is method is mature andeasy to use in engineering practice and can make the weightsmore scientific when combined with the fuzzy judgment of

Extended

t

t

Different (redundant) items

I21 I31 I41 I51

I12 I22 I32 I42

M1

M2

Virtual items

t

tI11 I21 I31 I41 I51

I12 I22 I32 I42 I52

M1

M2

Figure 6 Extended sequences for full-sequence matching

6 Mathematical Problems in Engineering

experts based on their experience e implementation ofthis procedure is described below [40]

First a priority matrix Q (qij)ntimesn needs to be con-structed where the value of qij can be acquired through thepriority matrix scale method shown in Table 1

According to the results of the comparison betweendifferent items a priority matrix for each item can beconstructed as shown in Table 2

en the overall priority matrix Q is given by

Q qij1113872 1113873ntimesn

q11 middot middot middot q1n

middot middot middot

middot middot middot

middot middot middot

qn1 middot middot middot qnn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

Now a fuzzy consistent matrix can be constructedwhere R (rij)ntimesn

First the fuzzy complementary matrix is summed line byline where Q (qij)ntimesn

qi 1113944n

j1qij i 1 2 n (7)

en the following transformation is implemented toconstruct the fuzzy consistent matrix R (rij)ntimesn

rij qi minus qj

2n+ 05 (8)

e next set of calculations begins with the weight vectorof R is is given by the following

gi 1113945n

j1rij

⎛⎝ ⎞⎠

1n

(9)

e weight vector gi is now normalized

gi gi

1113936ni1 gi

i 1 2 n (10)

Finally the weight vector w can be constructed asfollows

w g1 g2 middot middot middot gn( 1113857T i 1 2 n (11)

e similarity computation algorithm based on theabove definitions is shown in Algorithm 1

To illustrate the method the maintenance task ldquoFaultisolationrdquo for the troubleshooting of the HF transceiverfailure is taken as an example After referring to the trou-bleshooting manual the chosen candidate tasks and theirprocedures are shown in Table 3

Assume that the maintainability attribute set includesentity reachability visibility maintenance space toolstechnical level of the maintainers maintenance position andsecurity e indicators are scored with a number from 0 to10 e higher the score is the better the maintainability isen the representation models for the reference andcandidate maintenance tasks are constructed as shown inTable 4

ere are two types of items in the sequences circuitbreaker and pin Using fuzzy AHP the priority matrices forthe two items are

Q 05 0

1 051113890 1113891 (12)

en according to equations (7)sim(11) the item weightsare obtained as

w1 0366

w2 0634(13)

e sequence matching between the reference and twocandidate maintenance tasks is shown in Figure 7

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks is ascertained (the resulthas been multiplied by 1000 for better comparison)

H1 asymp 655

H2 asymp 1272(14)

After discussions with the experts the SMC threshold isspecified as ε 800 en because H1 lt 800 H2 gt 800 themaintenance task for the wiring between the transceiver pinand the ground terminal is determined to be similar to thereference task

23 Elicitation of the Prior Distribution Commonly usedmethods for constructing a prior distribution include eli-cited priors conjugate priors and noninformative priors[41] As the similar candidate tasks in each cluster onlycontain the maintenance time data for the correspondingreference task not the whole maintenance action we use anoptimistic and pessimistic value method to estimate theparameters of the prior distribution A normal distributionis the commonly used form of the prior distribution inMTTR Bayesian demonstrations [14 16 20 42] so in thisstudy we also assumed a normal prior distribution for theparameter of interest

Let X sim LogN(μ σ2) denote the maintenance actiontime distribution of a specified product e variance σ2will either be known from prior information or a reasonablyprecise estimate can be obtained e prior distribution of μ

Table 1 Priority matrix scale method

Scale Definition Illustration1 More time Ii consumes more time than Ij

05 Equal time Ii and Ij consume equal time0 Less time Ii consumes less time than Ij

Table 2 Priority matrix of each item

Q Q1 Qn

Q1 q11 qn1 Qn q1n qnn

Mathematical Problems in Engineering 7

is denoted as N(μπ σ2π) According to the properties of thelognormal distribution

θ eμ+σ22

(15)

where θ is the mean of the maintenance time distributionen μ can be calculated as follows

μ ln θ minusσ2

2 (16)

If Xi(i 1 2 K) denotes the time spent on eachmaintenance task and xi(i 1 2 K) denotes the cor-responding maintenance task time data set then

X 1113944K

i1Xi (17)

Two predictions of the mean of the maintenance actiontimemdashthe lower or optimistic value θL and the upper orpessimistic value θUmdashcan be obtained as follows

1113954θL 1113944K

i1xi(min) (18)

1113954θU 1113944

K

i1xi(max) (19)

where xi(min) and xi(max) are the minimum and maximumvalues respectively for the time data set corresponding tocluster Ci

According to equation (16) the two possible predictionsof μ are

Input Pri Oi εi

Output Ci

(1) for each Pri do

(2) Ci⟵empty(3) Construct representation models for Pr

i and maintenance tasks in Oi(4) for each Pij isin Oi do(5) Perform sequence matching between Pr

i and Pij(6) Calculate item weights(7) Calculate SMC Hij(8) if Hij lt εi then(9) Ci⟵Pij(10) end if(11) end for(12) end for(13) Return Ci

ALGORITHM 1 Similarity computation algorithm based on maintenance task representation models

Table 3 Reference and candidate maintenance task procedures

Reference task Candidate task

HF transceiverPr

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC4 5 and 6 of the transceiver

e wiring between the transceiverpin and ground terminal P1

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC45 and 6 of the HF transceiver

(3) Do a check for a ground signal at pinAC8 of the HF transceiver

VHF transceiver P2

(1) Do a check of the circuit breakerstatus

(2) Do a check for 28DC at pin AC2 ofthe VHF transceiver

t

tI1r I2r I3r I4r I5r

I1 I2 I3 I4 I5

Mr

M1

Mr

M2 t

tI1r I2r I3r I4r

I1 I2 I3 I4

Figure 7 Sequence matching between the reference and candidate maintenance tasks

8 Mathematical Problems in Engineering

1113954μL ln 1113954θL minusσ2

2 (20)

1113954μU ln 1113954θU minusσ2

2 (21)

It can then be assumed that the range (1113954μU minus 1113954μL) en-compasses 100 times (1 minus p) percent of the total possible valuesof μ and that the best estimate is at the midpoint of the rangeerefore the following prior distribution estimates can beused

μπ 1113954μU + 1113954μL

2 (22)

σ2π 1113954μU minus 1113954μL( 1113857

2

4 times Z2p2

(23)

3 Case Study

In this section the implementation of an MTTR demon-stration for an HF transceiver is once again used to illustrateour method

31 Selection of Candidate Maintenance Tasks An HFtransceiver is part of the HF system and is installed at thefront of the electronics rack in a plane After referring to thetroubleshooting manual and the aircraft maintenancemanual [24 43] we established candidate tasks for eachreference task ese relate to other components in the HFsystem or other equipment at the front of the electronicsrack A breakdown of the tasks is shown in Table 5

32 Identification and Formulation of the MaintainabilityAttribute Set and Evaluation Rules e maintainability at-tribute set developed by Jian et al [26] was used for thesimilarity analysis of the maintenance tasks e main-tainability attributes were tailored to the characteristics ofthe different tasks as shown in Table 6 e correspondingevaluation rules are shown in Table 7

33 Similarity Analysis between the Maintenance Tasks

331 Construction of the Maintenance Task RepresentationModels After referring to the maintenance manuals and theexpertsrsquo experience representation models for the referenceand candidate maintenance tasks were established as shownin Table 8

332 Calculation of the Item Weights On the basis of therepresentation models an item list for each type of main-tenance task was established as shown in Table 9

Using fuzzy AHP the priority matrices for the variousitems in each type of maintenance task were thenobtained

Q1

05 05 05

05 05 05

05 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2

05 0 0

1 05 0

1 1 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q3

05 1 1 1 1 0 05 1 1 05 1 05 1 1

0 05 0 0 05 0 0 05 0 0 05 0 0 0

0 1 05 0 1 0 0 05 0 0 05 0 05 0

0 1 1 05 1 05 1 1 1 0 1 05 1 1

0 05 0 0 05 0 0 05 0 0 0 0 0 0

1 1 1 05 1 05 1 1 1 05 1 05 1 1

05 1 1 0 1 0 05 1 05 0 05 0 1 1

0 05 05 0 05 0 0 05 0 0 05 0 0 0

0 1 1 0 1 0 05 1 05 0 05 0 05 05

05 1 1 1 1 05 1 1 1 05 1 05 1 1

0 05 05 0 1 0 05 05 05 0 05 0 05 05

05 1 1 05 1 05 1 1 1 05 1 05 1 1

0 1 05 0 1 0 0 1 05 0 05 0 05 05

0 1 1 0 1 0 0 1 05 0 05 0 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

After this equations (7)sim(11) were used to establish theweight vectors for the items in each type of maintenancetask as follows

w1 (0333 0333 0333)

w2 (0211 0335 0454)

w3 (0094 0044 0055 0091 0040 0099 0077

0046 0069 0099 0061 0096 0063 0066)

(25)

333 Clustering of the Candidate Maintenance Taskse sequence matchings between the reference and candi-date maintenance tasks are shown in Figure 8

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks was ascertained (the resultwas multiplied by 1000 for better comparison)

H11 269

H12 198

H13 599

H21 34640

H22 80325

H23 67239

H31 41494

H32 038

H33 142

(26)

Based on the SMC calculation result the thresholds werespecified as

Mathematical Problems in Engineering 9

Tabl

e4

Representatio

nmod

elsforthereferenceandcand

idatemaintenance

tasks

Referencetask

Candidate

task

Mr

I 1rI 2r

I 3rI 4r

t

Mr

lang

SrG

rA

rrangwhere

Sr

Ir 1

Ir 2Ir 3

Ir 41113864

1113865

Gr

E

r 1E

r 2E

r 3E

r 41113864

1113865

Ar

V

r 1V

r 2V

r 3V

r 41113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Er 1

lang(type

circuitbreaker)

(op

erationcheck)

rangV

r 1

(77865910

)

Er 2

lang(type

pin

)(op

erationcheck)

rangV

r 2

(8873899)

Er 3

lang(type

pin

)(op

erationcheck)

rangV

r 3

(78738910

)

Er 4

lang(type

pin

)(op

erationcheck)

rangV

r 4

(8973898)

M1

I 1I 2

I 3I 4

I 5t

M1

lang

S1

G1

A1rang

where

S1

I 1

I2

I 3I

4I 5

11138641113865

G1

E1

E2

E3

E4

E5

11138641113865

A1

V

1V

2V

3V

4V

51113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

eration

check)

rangV

1

(98747710

)

E2

lang(type

pin

)(op

erationcheck)

rangV

2

(88728810

)

E3

lang(type

pin

)(op

erationcheck)

rangV

3

(89728710

)

E4

lang(type

pin

)(op

erationcheck)

rangV

4

(8962879)

E5

lang(type

pin

)(op

erationcheck)

rangV

5

(9872879)

M2

I 1I 2

t

M2

lang

S2

G2

A2rang

where

S2

I 1

I2

11138641113865

G2

E1

E2

11138641113865

A2

V

1V

21113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

erationcheck)

rang

E2

lang(type

pin

)(op

erationcheck)

rang

V1

(87868910

)

V2

(7873899)

lowastTo

quantifytheim

pact

ofdifferent

numbers

ofchecks

atpins

ontheSM

Ccalculation

echecks

atpins

AC4A

C5A

C6and

AC8

aretreatedseparately

10 Mathematical Problems in Engineering

Table 5 Candidate maintenance tasks for each reference task

Reference task Componentequipment Candidate task

Fault confirmationcheckout Pr

1

HF antenna coupler P11 (1) Press the row key near the HF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menuHF antenna P12

Very high-frequency (VHF) transceiver P13

(1) Press the row key near the VHF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menu

Fault isolation Pr2

e wiring between the transceiver pin AC8and the ground terminal P21

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver

P22

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver(4) Do a check of the wiring from the circuit breaker to the

HF transceiver pins AC4 5 and 6

VHF transceiver P23(1) Do a check of the circuit breaker status

(2) Do a check for 28 DC at pin AC2 of the VHF transceiver

Repairinterchange Pr3

HF antenna coupler P31

(1) Disconnect the electrical plug(2) Place cap on the electrical plug

(3) Unscrew the nut(4) Lower the nut

(5) Dismantle the antenna coupler(6) Place cap on the electrical plug

(7) Clean the interface and adjacent area(8) Check the interface and adjacent area

(9) Dismantle the cap from the electrical plug(10) Check the cleanliness and condition of the electrical

plug(11) Install the antenna coupler on the shelf

(12) Screw the nut(13) Dismantle the cap from electrical plug

(14) Check the cleanliness and condition of the electricalplug

(15) Connect the electrical plug to the antenna coupler

Audio management unit (AMU) P32

(1) Unscrew the nut(2) Lower the nut

(3) Pull the AMU from the shelf and disconnect the electricalplug

(4) Dismantle the AMU(5) Place cap on the electrical cap

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the AMU on the shelf(11) Press the AMU to connect the electrical plug

(12) Screw the nut

VHF transceiver P33

(1) Unscrew the nut(2) Lower the nut

(3) Pull the transceiver from the shelf and disconnect theelectrical plug

(4) Dismantle the transceiver(5) Place cap on the electrical plug

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug

(12) Screw the nut

Mathematical Problems in Engineering 11

ε1 6

ε2 420

ε3 3

(27)

en according to equation (5) the clusters for similarcandidate tasks for each reference task were established

C1 P11 P12 P131113864 1113865

C2 P211113864 1113865

C3 P32 P331113864 1113865

(28)

e results showed that the candidate tasks for faultconfirmationcheckout were all similar to the reference taskbecause they had the same items as the reference task de-spite having slightly different maintainability characteristicsIn the case of fault isolation three candidate tasks had

different items to the reference task Task P22 had two ad-ditional items (I5 and I6) and P23 had two missing items (Ir

3and Ir

4) P21 however had only one additional item (I5)making it more similar to the reference task than the othertwo tasks In the case of repairinterchange task P31 hadseven different items (I1 I2 Ir

3 Ir11 I13 I14 and I15) while

the other two tasks all had the same items as the referencetask us task P31 had the most differences and was theleast similar to the reference task

34 HF Transceiver MTTR Demonstration Based onBayesian eory

341 MTTR Demonstration Methods Based on Bayesianeory e Bayesian maintainability demonstrationmethod proposed by Balaban [44] is used in our paper for

Table 6 Maintainability attributes tailored to maintenance tasks

Maintainability attributes Fault confirmation Fault isolation Repairinterchange CheckoutEntity reachability Visibility Maintenance space Tools Technical level of the maintainers Maintenance position Security

Table 7 Evaluation rules

Maintainability attribute Evaluation rules

Entity reachability (U1)

Good can observe the maintenance component comfortably and there is a wide observation angle(7ndash10)

General can generally see the outline of the maintenance component though it can easily cause eye andbody fatigue (4ndash6)

Poor prone to fatigue in this pose (0ndash3)

Visibility (U2)Good clear line of sight and there is enough light (7ndash10)

General the line of sight is blocked or the light is dark (4ndash6)Poor the line of sight is seriously blocked or the light is insufficient (0ndash3)

Maintenance space (U3)

Good no restrictions on the maintenance space for operating posture (7ndash10)General maintenance space for the body is basically enough but the operatorrsquos posture is abnormal

(4ndash6)Poor there is not enough maintenance space for a body (0ndash3)

Tools (U4)Good do not need the aid of auxiliary tools (7ndash10)

General need auxiliary tools sometimes (4ndash6)Poor dependent on auxiliary tools (0ndash3)

Technical level of themaintainers (U5)

Good familiar with the relevant technical knowledge and can quickly determine the operation processand solve the problem (7ndash10)

General is familiar with the relevant knowledge and can solve problems by referring to the operationmanual (4ndash6)

Poor only a general understanding of the situation and how to carry out the relevant work (0ndash3)

Maintenance position (U6)Good ground (7ndash10)

General have to climb to the machine (4ndash6)Poor need to stand outside the machine or in a similar position to the machine (0ndash3)

Security (U7)

Good no danger of being injured by heavy objects and there are no sharp edges that may scratch ordanger of electrical shock (7ndash10)

General there is a certain security threat sometimes there are sharp edges that may cause bumps orscratches (4ndash6)

Poor there is a security threat (0ndash3)

12 Mathematical Problems in Engineering

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

e sequence of virtual items added is called the extendedattributed item sequence

Definition 7 (cosine similarity) Cosine similarity is ameasure of the similarity between two high-dimensionalvectors at is given two vectors X and Y

cos(X Y) langX Yrang

XY (1)

where ldquolangrangrdquo indicates the inner product of two vectors andldquordquo indicates the L2 norm of the vector

Definition 8 (item mapping cost IMC) For a given twoextended attributed item sequences M1 and M2 under theone-to-one correspondence the item mapping cost fi be-tween two similar items is

fi wi 1 minus cos V1i V

2i1113872 11138731113872 1113873 (2)

and the item mapping cost between one item and its cor-responding virtual item is

fi wi (3)

where wi is the item weight which represents the relativelength of the mean maintenance time spent on that type ofitem V1

i and V2i are the maintainability attribute value

vectors of the two items 1 minus cos(V1i V2

i ) represents thedifference between the maintainability characteristics of twosimilar items

Equations (2) and (3) show that the greater the differencebetween the maintainability characteristics of two similaritems or the more maintenance time the item costs thegreater the impact of the difference on the similarity analysis

Definition 9 (sequence mapping cost SMC) e sequencemapping cost H between the sequence M1 and M2 is

H12 1113944N

i1fi (4)

where N is the number of items in each sequencee SMC reflects the difference between two mainte-

nance tasks based on the representation models In generalthe larger the value of H is the larger the difference betweenthe two maintenance tasks is

Definition 10 (reference maintenance task) When theequipment to be MTTR demonstrated is specified themaintenance tasks for this equipment are defined as the

reference maintenance tasks denoted by Pri (i 1 2 K)

where K represents the number of task types

Definition 11 (candidate maintenance task set) A candidatemaintenance task P is a task that is compared to the ref-erence task e candidate maintenance task set denoted byOi Pi1 Pi2 PiNi

1113966 1113967(i 1 2 K) is the task set for thesimilarity search of the reference task Pr

i where Ni repre-sents the number of tasks Possible sources of candidate tasksinclude maintenance tasks relating to equipment or com-ponents in the same system that have similar functions orthat take place in a similar location

Definition 12 (similarity calculation) For a given referencemaintenance task Pr with a corresponding candidatemaintenance task set O and a user-specified SMC thresholdof ε a similarity search will retrieve all maintenance tasksPj isin O such that

Hj le ε j 1 2 Ni (5)

where Hj is the SMC between maintenance tasks Pj and PrIf equation (5) holds it can be stated that Pr and Pj aresimilar to the ε boundary We can then obtain the cluster ofsimilar candidate tasks for the reference task which isdenoted as C e sequence mapping cost (SMC) threshold εis a user-specified value and it is obvious that the larger the εthe more candidate maintenance tasks will be determined tobe similar to the reference maintenance task and then moredata will be available for constructing prior distributionHowever a larger ε will make some candidate maintenancetasks that are less similar to the reference task similar enoughfor a prior distribution elicitation which in turn makes theobtained prior distribution unreliable Hence it is importantto achieve a balance between the quantity and quality of datawhen specifying the SMC threshold value e SMCthreshold ε value can be determined through discussion withexperts based on the SMC calculation result to obtain datafrom the equipment or components as similar as possibleunder the precondition of having enough data for con-structing a prior distribution

223 Calculation of the Item Weights In this study theexpert experience is used to estimate the weight coefficientw As human judgments can be vague or ill-defined a fuzzyanalytic hierarchy process (FAHP) is used to calculate theweight coefficient of each item is method is mature andeasy to use in engineering practice and can make the weightsmore scientific when combined with the fuzzy judgment of

Extended

t

t

Different (redundant) items

I21 I31 I41 I51

I12 I22 I32 I42

M1

M2

Virtual items

t

tI11 I21 I31 I41 I51

I12 I22 I32 I42 I52

M1

M2

Figure 6 Extended sequences for full-sequence matching

6 Mathematical Problems in Engineering

experts based on their experience e implementation ofthis procedure is described below [40]

First a priority matrix Q (qij)ntimesn needs to be con-structed where the value of qij can be acquired through thepriority matrix scale method shown in Table 1

According to the results of the comparison betweendifferent items a priority matrix for each item can beconstructed as shown in Table 2

en the overall priority matrix Q is given by

Q qij1113872 1113873ntimesn

q11 middot middot middot q1n

middot middot middot

middot middot middot

middot middot middot

qn1 middot middot middot qnn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

Now a fuzzy consistent matrix can be constructedwhere R (rij)ntimesn

First the fuzzy complementary matrix is summed line byline where Q (qij)ntimesn

qi 1113944n

j1qij i 1 2 n (7)

en the following transformation is implemented toconstruct the fuzzy consistent matrix R (rij)ntimesn

rij qi minus qj

2n+ 05 (8)

e next set of calculations begins with the weight vectorof R is is given by the following

gi 1113945n

j1rij

⎛⎝ ⎞⎠

1n

(9)

e weight vector gi is now normalized

gi gi

1113936ni1 gi

i 1 2 n (10)

Finally the weight vector w can be constructed asfollows

w g1 g2 middot middot middot gn( 1113857T i 1 2 n (11)

e similarity computation algorithm based on theabove definitions is shown in Algorithm 1

To illustrate the method the maintenance task ldquoFaultisolationrdquo for the troubleshooting of the HF transceiverfailure is taken as an example After referring to the trou-bleshooting manual the chosen candidate tasks and theirprocedures are shown in Table 3

Assume that the maintainability attribute set includesentity reachability visibility maintenance space toolstechnical level of the maintainers maintenance position andsecurity e indicators are scored with a number from 0 to10 e higher the score is the better the maintainability isen the representation models for the reference andcandidate maintenance tasks are constructed as shown inTable 4

ere are two types of items in the sequences circuitbreaker and pin Using fuzzy AHP the priority matrices forthe two items are

Q 05 0

1 051113890 1113891 (12)

en according to equations (7)sim(11) the item weightsare obtained as

w1 0366

w2 0634(13)

e sequence matching between the reference and twocandidate maintenance tasks is shown in Figure 7

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks is ascertained (the resulthas been multiplied by 1000 for better comparison)

H1 asymp 655

H2 asymp 1272(14)

After discussions with the experts the SMC threshold isspecified as ε 800 en because H1 lt 800 H2 gt 800 themaintenance task for the wiring between the transceiver pinand the ground terminal is determined to be similar to thereference task

23 Elicitation of the Prior Distribution Commonly usedmethods for constructing a prior distribution include eli-cited priors conjugate priors and noninformative priors[41] As the similar candidate tasks in each cluster onlycontain the maintenance time data for the correspondingreference task not the whole maintenance action we use anoptimistic and pessimistic value method to estimate theparameters of the prior distribution A normal distributionis the commonly used form of the prior distribution inMTTR Bayesian demonstrations [14 16 20 42] so in thisstudy we also assumed a normal prior distribution for theparameter of interest

Let X sim LogN(μ σ2) denote the maintenance actiontime distribution of a specified product e variance σ2will either be known from prior information or a reasonablyprecise estimate can be obtained e prior distribution of μ

Table 1 Priority matrix scale method

Scale Definition Illustration1 More time Ii consumes more time than Ij

05 Equal time Ii and Ij consume equal time0 Less time Ii consumes less time than Ij

Table 2 Priority matrix of each item

Q Q1 Qn

Q1 q11 qn1 Qn q1n qnn

Mathematical Problems in Engineering 7

is denoted as N(μπ σ2π) According to the properties of thelognormal distribution

θ eμ+σ22

(15)

where θ is the mean of the maintenance time distributionen μ can be calculated as follows

μ ln θ minusσ2

2 (16)

If Xi(i 1 2 K) denotes the time spent on eachmaintenance task and xi(i 1 2 K) denotes the cor-responding maintenance task time data set then

X 1113944K

i1Xi (17)

Two predictions of the mean of the maintenance actiontimemdashthe lower or optimistic value θL and the upper orpessimistic value θUmdashcan be obtained as follows

1113954θL 1113944K

i1xi(min) (18)

1113954θU 1113944

K

i1xi(max) (19)

where xi(min) and xi(max) are the minimum and maximumvalues respectively for the time data set corresponding tocluster Ci

According to equation (16) the two possible predictionsof μ are

Input Pri Oi εi

Output Ci

(1) for each Pri do

(2) Ci⟵empty(3) Construct representation models for Pr

i and maintenance tasks in Oi(4) for each Pij isin Oi do(5) Perform sequence matching between Pr

i and Pij(6) Calculate item weights(7) Calculate SMC Hij(8) if Hij lt εi then(9) Ci⟵Pij(10) end if(11) end for(12) end for(13) Return Ci

ALGORITHM 1 Similarity computation algorithm based on maintenance task representation models

Table 3 Reference and candidate maintenance task procedures

Reference task Candidate task

HF transceiverPr

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC4 5 and 6 of the transceiver

e wiring between the transceiverpin and ground terminal P1

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC45 and 6 of the HF transceiver

(3) Do a check for a ground signal at pinAC8 of the HF transceiver

VHF transceiver P2

(1) Do a check of the circuit breakerstatus

(2) Do a check for 28DC at pin AC2 ofthe VHF transceiver

t

tI1r I2r I3r I4r I5r

I1 I2 I3 I4 I5

Mr

M1

Mr

M2 t

tI1r I2r I3r I4r

I1 I2 I3 I4

Figure 7 Sequence matching between the reference and candidate maintenance tasks

8 Mathematical Problems in Engineering

1113954μL ln 1113954θL minusσ2

2 (20)

1113954μU ln 1113954θU minusσ2

2 (21)

It can then be assumed that the range (1113954μU minus 1113954μL) en-compasses 100 times (1 minus p) percent of the total possible valuesof μ and that the best estimate is at the midpoint of the rangeerefore the following prior distribution estimates can beused

μπ 1113954μU + 1113954μL

2 (22)

σ2π 1113954μU minus 1113954μL( 1113857

2

4 times Z2p2

(23)

3 Case Study

In this section the implementation of an MTTR demon-stration for an HF transceiver is once again used to illustrateour method

31 Selection of Candidate Maintenance Tasks An HFtransceiver is part of the HF system and is installed at thefront of the electronics rack in a plane After referring to thetroubleshooting manual and the aircraft maintenancemanual [24 43] we established candidate tasks for eachreference task ese relate to other components in the HFsystem or other equipment at the front of the electronicsrack A breakdown of the tasks is shown in Table 5

32 Identification and Formulation of the MaintainabilityAttribute Set and Evaluation Rules e maintainability at-tribute set developed by Jian et al [26] was used for thesimilarity analysis of the maintenance tasks e main-tainability attributes were tailored to the characteristics ofthe different tasks as shown in Table 6 e correspondingevaluation rules are shown in Table 7

33 Similarity Analysis between the Maintenance Tasks

331 Construction of the Maintenance Task RepresentationModels After referring to the maintenance manuals and theexpertsrsquo experience representation models for the referenceand candidate maintenance tasks were established as shownin Table 8

332 Calculation of the Item Weights On the basis of therepresentation models an item list for each type of main-tenance task was established as shown in Table 9

Using fuzzy AHP the priority matrices for the variousitems in each type of maintenance task were thenobtained

Q1

05 05 05

05 05 05

05 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2

05 0 0

1 05 0

1 1 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q3

05 1 1 1 1 0 05 1 1 05 1 05 1 1

0 05 0 0 05 0 0 05 0 0 05 0 0 0

0 1 05 0 1 0 0 05 0 0 05 0 05 0

0 1 1 05 1 05 1 1 1 0 1 05 1 1

0 05 0 0 05 0 0 05 0 0 0 0 0 0

1 1 1 05 1 05 1 1 1 05 1 05 1 1

05 1 1 0 1 0 05 1 05 0 05 0 1 1

0 05 05 0 05 0 0 05 0 0 05 0 0 0

0 1 1 0 1 0 05 1 05 0 05 0 05 05

05 1 1 1 1 05 1 1 1 05 1 05 1 1

0 05 05 0 1 0 05 05 05 0 05 0 05 05

05 1 1 05 1 05 1 1 1 05 1 05 1 1

0 1 05 0 1 0 0 1 05 0 05 0 05 05

0 1 1 0 1 0 0 1 05 0 05 0 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

After this equations (7)sim(11) were used to establish theweight vectors for the items in each type of maintenancetask as follows

w1 (0333 0333 0333)

w2 (0211 0335 0454)

w3 (0094 0044 0055 0091 0040 0099 0077

0046 0069 0099 0061 0096 0063 0066)

(25)

333 Clustering of the Candidate Maintenance Taskse sequence matchings between the reference and candi-date maintenance tasks are shown in Figure 8

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks was ascertained (the resultwas multiplied by 1000 for better comparison)

H11 269

H12 198

H13 599

H21 34640

H22 80325

H23 67239

H31 41494

H32 038

H33 142

(26)

Based on the SMC calculation result the thresholds werespecified as

Mathematical Problems in Engineering 9

Tabl

e4

Representatio

nmod

elsforthereferenceandcand

idatemaintenance

tasks

Referencetask

Candidate

task

Mr

I 1rI 2r

I 3rI 4r

t

Mr

lang

SrG

rA

rrangwhere

Sr

Ir 1

Ir 2Ir 3

Ir 41113864

1113865

Gr

E

r 1E

r 2E

r 3E

r 41113864

1113865

Ar

V

r 1V

r 2V

r 3V

r 41113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Er 1

lang(type

circuitbreaker)

(op

erationcheck)

rangV

r 1

(77865910

)

Er 2

lang(type

pin

)(op

erationcheck)

rangV

r 2

(8873899)

Er 3

lang(type

pin

)(op

erationcheck)

rangV

r 3

(78738910

)

Er 4

lang(type

pin

)(op

erationcheck)

rangV

r 4

(8973898)

M1

I 1I 2

I 3I 4

I 5t

M1

lang

S1

G1

A1rang

where

S1

I 1

I2

I 3I

4I 5

11138641113865

G1

E1

E2

E3

E4

E5

11138641113865

A1

V

1V

2V

3V

4V

51113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

eration

check)

rangV

1

(98747710

)

E2

lang(type

pin

)(op

erationcheck)

rangV

2

(88728810

)

E3

lang(type

pin

)(op

erationcheck)

rangV

3

(89728710

)

E4

lang(type

pin

)(op

erationcheck)

rangV

4

(8962879)

E5

lang(type

pin

)(op

erationcheck)

rangV

5

(9872879)

M2

I 1I 2

t

M2

lang

S2

G2

A2rang

where

S2

I 1

I2

11138641113865

G2

E1

E2

11138641113865

A2

V

1V

21113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

erationcheck)

rang

E2

lang(type

pin

)(op

erationcheck)

rang

V1

(87868910

)

V2

(7873899)

lowastTo

quantifytheim

pact

ofdifferent

numbers

ofchecks

atpins

ontheSM

Ccalculation

echecks

atpins

AC4A

C5A

C6and

AC8

aretreatedseparately

10 Mathematical Problems in Engineering

Table 5 Candidate maintenance tasks for each reference task

Reference task Componentequipment Candidate task

Fault confirmationcheckout Pr

1

HF antenna coupler P11 (1) Press the row key near the HF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menuHF antenna P12

Very high-frequency (VHF) transceiver P13

(1) Press the row key near the VHF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menu

Fault isolation Pr2

e wiring between the transceiver pin AC8and the ground terminal P21

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver

P22

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver(4) Do a check of the wiring from the circuit breaker to the

HF transceiver pins AC4 5 and 6

VHF transceiver P23(1) Do a check of the circuit breaker status

(2) Do a check for 28 DC at pin AC2 of the VHF transceiver

Repairinterchange Pr3

HF antenna coupler P31

(1) Disconnect the electrical plug(2) Place cap on the electrical plug

(3) Unscrew the nut(4) Lower the nut

(5) Dismantle the antenna coupler(6) Place cap on the electrical plug

(7) Clean the interface and adjacent area(8) Check the interface and adjacent area

(9) Dismantle the cap from the electrical plug(10) Check the cleanliness and condition of the electrical

plug(11) Install the antenna coupler on the shelf

(12) Screw the nut(13) Dismantle the cap from electrical plug

(14) Check the cleanliness and condition of the electricalplug

(15) Connect the electrical plug to the antenna coupler

Audio management unit (AMU) P32

(1) Unscrew the nut(2) Lower the nut

(3) Pull the AMU from the shelf and disconnect the electricalplug

(4) Dismantle the AMU(5) Place cap on the electrical cap

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the AMU on the shelf(11) Press the AMU to connect the electrical plug

(12) Screw the nut

VHF transceiver P33

(1) Unscrew the nut(2) Lower the nut

(3) Pull the transceiver from the shelf and disconnect theelectrical plug

(4) Dismantle the transceiver(5) Place cap on the electrical plug

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug

(12) Screw the nut

Mathematical Problems in Engineering 11

ε1 6

ε2 420

ε3 3

(27)

en according to equation (5) the clusters for similarcandidate tasks for each reference task were established

C1 P11 P12 P131113864 1113865

C2 P211113864 1113865

C3 P32 P331113864 1113865

(28)

e results showed that the candidate tasks for faultconfirmationcheckout were all similar to the reference taskbecause they had the same items as the reference task de-spite having slightly different maintainability characteristicsIn the case of fault isolation three candidate tasks had

different items to the reference task Task P22 had two ad-ditional items (I5 and I6) and P23 had two missing items (Ir

3and Ir

4) P21 however had only one additional item (I5)making it more similar to the reference task than the othertwo tasks In the case of repairinterchange task P31 hadseven different items (I1 I2 Ir

3 Ir11 I13 I14 and I15) while

the other two tasks all had the same items as the referencetask us task P31 had the most differences and was theleast similar to the reference task

34 HF Transceiver MTTR Demonstration Based onBayesian eory

341 MTTR Demonstration Methods Based on Bayesianeory e Bayesian maintainability demonstrationmethod proposed by Balaban [44] is used in our paper for

Table 6 Maintainability attributes tailored to maintenance tasks

Maintainability attributes Fault confirmation Fault isolation Repairinterchange CheckoutEntity reachability Visibility Maintenance space Tools Technical level of the maintainers Maintenance position Security

Table 7 Evaluation rules

Maintainability attribute Evaluation rules

Entity reachability (U1)

Good can observe the maintenance component comfortably and there is a wide observation angle(7ndash10)

General can generally see the outline of the maintenance component though it can easily cause eye andbody fatigue (4ndash6)

Poor prone to fatigue in this pose (0ndash3)

Visibility (U2)Good clear line of sight and there is enough light (7ndash10)

General the line of sight is blocked or the light is dark (4ndash6)Poor the line of sight is seriously blocked or the light is insufficient (0ndash3)

Maintenance space (U3)

Good no restrictions on the maintenance space for operating posture (7ndash10)General maintenance space for the body is basically enough but the operatorrsquos posture is abnormal

(4ndash6)Poor there is not enough maintenance space for a body (0ndash3)

Tools (U4)Good do not need the aid of auxiliary tools (7ndash10)

General need auxiliary tools sometimes (4ndash6)Poor dependent on auxiliary tools (0ndash3)

Technical level of themaintainers (U5)

Good familiar with the relevant technical knowledge and can quickly determine the operation processand solve the problem (7ndash10)

General is familiar with the relevant knowledge and can solve problems by referring to the operationmanual (4ndash6)

Poor only a general understanding of the situation and how to carry out the relevant work (0ndash3)

Maintenance position (U6)Good ground (7ndash10)

General have to climb to the machine (4ndash6)Poor need to stand outside the machine or in a similar position to the machine (0ndash3)

Security (U7)

Good no danger of being injured by heavy objects and there are no sharp edges that may scratch ordanger of electrical shock (7ndash10)

General there is a certain security threat sometimes there are sharp edges that may cause bumps orscratches (4ndash6)

Poor there is a security threat (0ndash3)

12 Mathematical Problems in Engineering

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

experts based on their experience e implementation ofthis procedure is described below [40]

First a priority matrix Q (qij)ntimesn needs to be con-structed where the value of qij can be acquired through thepriority matrix scale method shown in Table 1

According to the results of the comparison betweendifferent items a priority matrix for each item can beconstructed as shown in Table 2

en the overall priority matrix Q is given by

Q qij1113872 1113873ntimesn

q11 middot middot middot q1n

middot middot middot

middot middot middot

middot middot middot

qn1 middot middot middot qnn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

Now a fuzzy consistent matrix can be constructedwhere R (rij)ntimesn

First the fuzzy complementary matrix is summed line byline where Q (qij)ntimesn

qi 1113944n

j1qij i 1 2 n (7)

en the following transformation is implemented toconstruct the fuzzy consistent matrix R (rij)ntimesn

rij qi minus qj

2n+ 05 (8)

e next set of calculations begins with the weight vectorof R is is given by the following

gi 1113945n

j1rij

⎛⎝ ⎞⎠

1n

(9)

e weight vector gi is now normalized

gi gi

1113936ni1 gi

i 1 2 n (10)

Finally the weight vector w can be constructed asfollows

w g1 g2 middot middot middot gn( 1113857T i 1 2 n (11)

e similarity computation algorithm based on theabove definitions is shown in Algorithm 1

To illustrate the method the maintenance task ldquoFaultisolationrdquo for the troubleshooting of the HF transceiverfailure is taken as an example After referring to the trou-bleshooting manual the chosen candidate tasks and theirprocedures are shown in Table 3

Assume that the maintainability attribute set includesentity reachability visibility maintenance space toolstechnical level of the maintainers maintenance position andsecurity e indicators are scored with a number from 0 to10 e higher the score is the better the maintainability isen the representation models for the reference andcandidate maintenance tasks are constructed as shown inTable 4

ere are two types of items in the sequences circuitbreaker and pin Using fuzzy AHP the priority matrices forthe two items are

Q 05 0

1 051113890 1113891 (12)

en according to equations (7)sim(11) the item weightsare obtained as

w1 0366

w2 0634(13)

e sequence matching between the reference and twocandidate maintenance tasks is shown in Figure 7

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks is ascertained (the resulthas been multiplied by 1000 for better comparison)

H1 asymp 655

H2 asymp 1272(14)

After discussions with the experts the SMC threshold isspecified as ε 800 en because H1 lt 800 H2 gt 800 themaintenance task for the wiring between the transceiver pinand the ground terminal is determined to be similar to thereference task

23 Elicitation of the Prior Distribution Commonly usedmethods for constructing a prior distribution include eli-cited priors conjugate priors and noninformative priors[41] As the similar candidate tasks in each cluster onlycontain the maintenance time data for the correspondingreference task not the whole maintenance action we use anoptimistic and pessimistic value method to estimate theparameters of the prior distribution A normal distributionis the commonly used form of the prior distribution inMTTR Bayesian demonstrations [14 16 20 42] so in thisstudy we also assumed a normal prior distribution for theparameter of interest

Let X sim LogN(μ σ2) denote the maintenance actiontime distribution of a specified product e variance σ2will either be known from prior information or a reasonablyprecise estimate can be obtained e prior distribution of μ

Table 1 Priority matrix scale method

Scale Definition Illustration1 More time Ii consumes more time than Ij

05 Equal time Ii and Ij consume equal time0 Less time Ii consumes less time than Ij

Table 2 Priority matrix of each item

Q Q1 Qn

Q1 q11 qn1 Qn q1n qnn

Mathematical Problems in Engineering 7

is denoted as N(μπ σ2π) According to the properties of thelognormal distribution

θ eμ+σ22

(15)

where θ is the mean of the maintenance time distributionen μ can be calculated as follows

μ ln θ minusσ2

2 (16)

If Xi(i 1 2 K) denotes the time spent on eachmaintenance task and xi(i 1 2 K) denotes the cor-responding maintenance task time data set then

X 1113944K

i1Xi (17)

Two predictions of the mean of the maintenance actiontimemdashthe lower or optimistic value θL and the upper orpessimistic value θUmdashcan be obtained as follows

1113954θL 1113944K

i1xi(min) (18)

1113954θU 1113944

K

i1xi(max) (19)

where xi(min) and xi(max) are the minimum and maximumvalues respectively for the time data set corresponding tocluster Ci

According to equation (16) the two possible predictionsof μ are

Input Pri Oi εi

Output Ci

(1) for each Pri do

(2) Ci⟵empty(3) Construct representation models for Pr

i and maintenance tasks in Oi(4) for each Pij isin Oi do(5) Perform sequence matching between Pr

i and Pij(6) Calculate item weights(7) Calculate SMC Hij(8) if Hij lt εi then(9) Ci⟵Pij(10) end if(11) end for(12) end for(13) Return Ci

ALGORITHM 1 Similarity computation algorithm based on maintenance task representation models

Table 3 Reference and candidate maintenance task procedures

Reference task Candidate task

HF transceiverPr

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC4 5 and 6 of the transceiver

e wiring between the transceiverpin and ground terminal P1

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC45 and 6 of the HF transceiver

(3) Do a check for a ground signal at pinAC8 of the HF transceiver

VHF transceiver P2

(1) Do a check of the circuit breakerstatus

(2) Do a check for 28DC at pin AC2 ofthe VHF transceiver

t

tI1r I2r I3r I4r I5r

I1 I2 I3 I4 I5

Mr

M1

Mr

M2 t

tI1r I2r I3r I4r

I1 I2 I3 I4

Figure 7 Sequence matching between the reference and candidate maintenance tasks

8 Mathematical Problems in Engineering

1113954μL ln 1113954θL minusσ2

2 (20)

1113954μU ln 1113954θU minusσ2

2 (21)

It can then be assumed that the range (1113954μU minus 1113954μL) en-compasses 100 times (1 minus p) percent of the total possible valuesof μ and that the best estimate is at the midpoint of the rangeerefore the following prior distribution estimates can beused

μπ 1113954μU + 1113954μL

2 (22)

σ2π 1113954μU minus 1113954μL( 1113857

2

4 times Z2p2

(23)

3 Case Study

In this section the implementation of an MTTR demon-stration for an HF transceiver is once again used to illustrateour method

31 Selection of Candidate Maintenance Tasks An HFtransceiver is part of the HF system and is installed at thefront of the electronics rack in a plane After referring to thetroubleshooting manual and the aircraft maintenancemanual [24 43] we established candidate tasks for eachreference task ese relate to other components in the HFsystem or other equipment at the front of the electronicsrack A breakdown of the tasks is shown in Table 5

32 Identification and Formulation of the MaintainabilityAttribute Set and Evaluation Rules e maintainability at-tribute set developed by Jian et al [26] was used for thesimilarity analysis of the maintenance tasks e main-tainability attributes were tailored to the characteristics ofthe different tasks as shown in Table 6 e correspondingevaluation rules are shown in Table 7

33 Similarity Analysis between the Maintenance Tasks

331 Construction of the Maintenance Task RepresentationModels After referring to the maintenance manuals and theexpertsrsquo experience representation models for the referenceand candidate maintenance tasks were established as shownin Table 8

332 Calculation of the Item Weights On the basis of therepresentation models an item list for each type of main-tenance task was established as shown in Table 9

Using fuzzy AHP the priority matrices for the variousitems in each type of maintenance task were thenobtained

Q1

05 05 05

05 05 05

05 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2

05 0 0

1 05 0

1 1 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q3

05 1 1 1 1 0 05 1 1 05 1 05 1 1

0 05 0 0 05 0 0 05 0 0 05 0 0 0

0 1 05 0 1 0 0 05 0 0 05 0 05 0

0 1 1 05 1 05 1 1 1 0 1 05 1 1

0 05 0 0 05 0 0 05 0 0 0 0 0 0

1 1 1 05 1 05 1 1 1 05 1 05 1 1

05 1 1 0 1 0 05 1 05 0 05 0 1 1

0 05 05 0 05 0 0 05 0 0 05 0 0 0

0 1 1 0 1 0 05 1 05 0 05 0 05 05

05 1 1 1 1 05 1 1 1 05 1 05 1 1

0 05 05 0 1 0 05 05 05 0 05 0 05 05

05 1 1 05 1 05 1 1 1 05 1 05 1 1

0 1 05 0 1 0 0 1 05 0 05 0 05 05

0 1 1 0 1 0 0 1 05 0 05 0 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

After this equations (7)sim(11) were used to establish theweight vectors for the items in each type of maintenancetask as follows

w1 (0333 0333 0333)

w2 (0211 0335 0454)

w3 (0094 0044 0055 0091 0040 0099 0077

0046 0069 0099 0061 0096 0063 0066)

(25)

333 Clustering of the Candidate Maintenance Taskse sequence matchings between the reference and candi-date maintenance tasks are shown in Figure 8

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks was ascertained (the resultwas multiplied by 1000 for better comparison)

H11 269

H12 198

H13 599

H21 34640

H22 80325

H23 67239

H31 41494

H32 038

H33 142

(26)

Based on the SMC calculation result the thresholds werespecified as

Mathematical Problems in Engineering 9

Tabl

e4

Representatio

nmod

elsforthereferenceandcand

idatemaintenance

tasks

Referencetask

Candidate

task

Mr

I 1rI 2r

I 3rI 4r

t

Mr

lang

SrG

rA

rrangwhere

Sr

Ir 1

Ir 2Ir 3

Ir 41113864

1113865

Gr

E

r 1E

r 2E

r 3E

r 41113864

1113865

Ar

V

r 1V

r 2V

r 3V

r 41113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Er 1

lang(type

circuitbreaker)

(op

erationcheck)

rangV

r 1

(77865910

)

Er 2

lang(type

pin

)(op

erationcheck)

rangV

r 2

(8873899)

Er 3

lang(type

pin

)(op

erationcheck)

rangV

r 3

(78738910

)

Er 4

lang(type

pin

)(op

erationcheck)

rangV

r 4

(8973898)

M1

I 1I 2

I 3I 4

I 5t

M1

lang

S1

G1

A1rang

where

S1

I 1

I2

I 3I

4I 5

11138641113865

G1

E1

E2

E3

E4

E5

11138641113865

A1

V

1V

2V

3V

4V

51113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

eration

check)

rangV

1

(98747710

)

E2

lang(type

pin

)(op

erationcheck)

rangV

2

(88728810

)

E3

lang(type

pin

)(op

erationcheck)

rangV

3

(89728710

)

E4

lang(type

pin

)(op

erationcheck)

rangV

4

(8962879)

E5

lang(type

pin

)(op

erationcheck)

rangV

5

(9872879)

M2

I 1I 2

t

M2

lang

S2

G2

A2rang

where

S2

I 1

I2

11138641113865

G2

E1

E2

11138641113865

A2

V

1V

21113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

erationcheck)

rang

E2

lang(type

pin

)(op

erationcheck)

rang

V1

(87868910

)

V2

(7873899)

lowastTo

quantifytheim

pact

ofdifferent

numbers

ofchecks

atpins

ontheSM

Ccalculation

echecks

atpins

AC4A

C5A

C6and

AC8

aretreatedseparately

10 Mathematical Problems in Engineering

Table 5 Candidate maintenance tasks for each reference task

Reference task Componentequipment Candidate task

Fault confirmationcheckout Pr

1

HF antenna coupler P11 (1) Press the row key near the HF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menuHF antenna P12

Very high-frequency (VHF) transceiver P13

(1) Press the row key near the VHF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menu

Fault isolation Pr2

e wiring between the transceiver pin AC8and the ground terminal P21

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver

P22

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver(4) Do a check of the wiring from the circuit breaker to the

HF transceiver pins AC4 5 and 6

VHF transceiver P23(1) Do a check of the circuit breaker status

(2) Do a check for 28 DC at pin AC2 of the VHF transceiver

Repairinterchange Pr3

HF antenna coupler P31

(1) Disconnect the electrical plug(2) Place cap on the electrical plug

(3) Unscrew the nut(4) Lower the nut

(5) Dismantle the antenna coupler(6) Place cap on the electrical plug

(7) Clean the interface and adjacent area(8) Check the interface and adjacent area

(9) Dismantle the cap from the electrical plug(10) Check the cleanliness and condition of the electrical

plug(11) Install the antenna coupler on the shelf

(12) Screw the nut(13) Dismantle the cap from electrical plug

(14) Check the cleanliness and condition of the electricalplug

(15) Connect the electrical plug to the antenna coupler

Audio management unit (AMU) P32

(1) Unscrew the nut(2) Lower the nut

(3) Pull the AMU from the shelf and disconnect the electricalplug

(4) Dismantle the AMU(5) Place cap on the electrical cap

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the AMU on the shelf(11) Press the AMU to connect the electrical plug

(12) Screw the nut

VHF transceiver P33

(1) Unscrew the nut(2) Lower the nut

(3) Pull the transceiver from the shelf and disconnect theelectrical plug

(4) Dismantle the transceiver(5) Place cap on the electrical plug

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug

(12) Screw the nut

Mathematical Problems in Engineering 11

ε1 6

ε2 420

ε3 3

(27)

en according to equation (5) the clusters for similarcandidate tasks for each reference task were established

C1 P11 P12 P131113864 1113865

C2 P211113864 1113865

C3 P32 P331113864 1113865

(28)

e results showed that the candidate tasks for faultconfirmationcheckout were all similar to the reference taskbecause they had the same items as the reference task de-spite having slightly different maintainability characteristicsIn the case of fault isolation three candidate tasks had

different items to the reference task Task P22 had two ad-ditional items (I5 and I6) and P23 had two missing items (Ir

3and Ir

4) P21 however had only one additional item (I5)making it more similar to the reference task than the othertwo tasks In the case of repairinterchange task P31 hadseven different items (I1 I2 Ir

3 Ir11 I13 I14 and I15) while

the other two tasks all had the same items as the referencetask us task P31 had the most differences and was theleast similar to the reference task

34 HF Transceiver MTTR Demonstration Based onBayesian eory

341 MTTR Demonstration Methods Based on Bayesianeory e Bayesian maintainability demonstrationmethod proposed by Balaban [44] is used in our paper for

Table 6 Maintainability attributes tailored to maintenance tasks

Maintainability attributes Fault confirmation Fault isolation Repairinterchange CheckoutEntity reachability Visibility Maintenance space Tools Technical level of the maintainers Maintenance position Security

Table 7 Evaluation rules

Maintainability attribute Evaluation rules

Entity reachability (U1)

Good can observe the maintenance component comfortably and there is a wide observation angle(7ndash10)

General can generally see the outline of the maintenance component though it can easily cause eye andbody fatigue (4ndash6)

Poor prone to fatigue in this pose (0ndash3)

Visibility (U2)Good clear line of sight and there is enough light (7ndash10)

General the line of sight is blocked or the light is dark (4ndash6)Poor the line of sight is seriously blocked or the light is insufficient (0ndash3)

Maintenance space (U3)

Good no restrictions on the maintenance space for operating posture (7ndash10)General maintenance space for the body is basically enough but the operatorrsquos posture is abnormal

(4ndash6)Poor there is not enough maintenance space for a body (0ndash3)

Tools (U4)Good do not need the aid of auxiliary tools (7ndash10)

General need auxiliary tools sometimes (4ndash6)Poor dependent on auxiliary tools (0ndash3)

Technical level of themaintainers (U5)

Good familiar with the relevant technical knowledge and can quickly determine the operation processand solve the problem (7ndash10)

General is familiar with the relevant knowledge and can solve problems by referring to the operationmanual (4ndash6)

Poor only a general understanding of the situation and how to carry out the relevant work (0ndash3)

Maintenance position (U6)Good ground (7ndash10)

General have to climb to the machine (4ndash6)Poor need to stand outside the machine or in a similar position to the machine (0ndash3)

Security (U7)

Good no danger of being injured by heavy objects and there are no sharp edges that may scratch ordanger of electrical shock (7ndash10)

General there is a certain security threat sometimes there are sharp edges that may cause bumps orscratches (4ndash6)

Poor there is a security threat (0ndash3)

12 Mathematical Problems in Engineering

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

is denoted as N(μπ σ2π) According to the properties of thelognormal distribution

θ eμ+σ22

(15)

where θ is the mean of the maintenance time distributionen μ can be calculated as follows

μ ln θ minusσ2

2 (16)

If Xi(i 1 2 K) denotes the time spent on eachmaintenance task and xi(i 1 2 K) denotes the cor-responding maintenance task time data set then

X 1113944K

i1Xi (17)

Two predictions of the mean of the maintenance actiontimemdashthe lower or optimistic value θL and the upper orpessimistic value θUmdashcan be obtained as follows

1113954θL 1113944K

i1xi(min) (18)

1113954θU 1113944

K

i1xi(max) (19)

where xi(min) and xi(max) are the minimum and maximumvalues respectively for the time data set corresponding tocluster Ci

According to equation (16) the two possible predictionsof μ are

Input Pri Oi εi

Output Ci

(1) for each Pri do

(2) Ci⟵empty(3) Construct representation models for Pr

i and maintenance tasks in Oi(4) for each Pij isin Oi do(5) Perform sequence matching between Pr

i and Pij(6) Calculate item weights(7) Calculate SMC Hij(8) if Hij lt εi then(9) Ci⟵Pij(10) end if(11) end for(12) end for(13) Return Ci

ALGORITHM 1 Similarity computation algorithm based on maintenance task representation models

Table 3 Reference and candidate maintenance task procedures

Reference task Candidate task

HF transceiverPr

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC4 5 and 6 of the transceiver

e wiring between the transceiverpin and ground terminal P1

(1) Do a check of the circuit breakerstatus

(2) Do a check for 115 VAC at pins AC45 and 6 of the HF transceiver

(3) Do a check for a ground signal at pinAC8 of the HF transceiver

VHF transceiver P2

(1) Do a check of the circuit breakerstatus

(2) Do a check for 28DC at pin AC2 ofthe VHF transceiver

t

tI1r I2r I3r I4r I5r

I1 I2 I3 I4 I5

Mr

M1

Mr

M2 t

tI1r I2r I3r I4r

I1 I2 I3 I4

Figure 7 Sequence matching between the reference and candidate maintenance tasks

8 Mathematical Problems in Engineering

1113954μL ln 1113954θL minusσ2

2 (20)

1113954μU ln 1113954θU minusσ2

2 (21)

It can then be assumed that the range (1113954μU minus 1113954μL) en-compasses 100 times (1 minus p) percent of the total possible valuesof μ and that the best estimate is at the midpoint of the rangeerefore the following prior distribution estimates can beused

μπ 1113954μU + 1113954μL

2 (22)

σ2π 1113954μU minus 1113954μL( 1113857

2

4 times Z2p2

(23)

3 Case Study

In this section the implementation of an MTTR demon-stration for an HF transceiver is once again used to illustrateour method

31 Selection of Candidate Maintenance Tasks An HFtransceiver is part of the HF system and is installed at thefront of the electronics rack in a plane After referring to thetroubleshooting manual and the aircraft maintenancemanual [24 43] we established candidate tasks for eachreference task ese relate to other components in the HFsystem or other equipment at the front of the electronicsrack A breakdown of the tasks is shown in Table 5

32 Identification and Formulation of the MaintainabilityAttribute Set and Evaluation Rules e maintainability at-tribute set developed by Jian et al [26] was used for thesimilarity analysis of the maintenance tasks e main-tainability attributes were tailored to the characteristics ofthe different tasks as shown in Table 6 e correspondingevaluation rules are shown in Table 7

33 Similarity Analysis between the Maintenance Tasks

331 Construction of the Maintenance Task RepresentationModels After referring to the maintenance manuals and theexpertsrsquo experience representation models for the referenceand candidate maintenance tasks were established as shownin Table 8

332 Calculation of the Item Weights On the basis of therepresentation models an item list for each type of main-tenance task was established as shown in Table 9

Using fuzzy AHP the priority matrices for the variousitems in each type of maintenance task were thenobtained

Q1

05 05 05

05 05 05

05 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2

05 0 0

1 05 0

1 1 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q3

05 1 1 1 1 0 05 1 1 05 1 05 1 1

0 05 0 0 05 0 0 05 0 0 05 0 0 0

0 1 05 0 1 0 0 05 0 0 05 0 05 0

0 1 1 05 1 05 1 1 1 0 1 05 1 1

0 05 0 0 05 0 0 05 0 0 0 0 0 0

1 1 1 05 1 05 1 1 1 05 1 05 1 1

05 1 1 0 1 0 05 1 05 0 05 0 1 1

0 05 05 0 05 0 0 05 0 0 05 0 0 0

0 1 1 0 1 0 05 1 05 0 05 0 05 05

05 1 1 1 1 05 1 1 1 05 1 05 1 1

0 05 05 0 1 0 05 05 05 0 05 0 05 05

05 1 1 05 1 05 1 1 1 05 1 05 1 1

0 1 05 0 1 0 0 1 05 0 05 0 05 05

0 1 1 0 1 0 0 1 05 0 05 0 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

After this equations (7)sim(11) were used to establish theweight vectors for the items in each type of maintenancetask as follows

w1 (0333 0333 0333)

w2 (0211 0335 0454)

w3 (0094 0044 0055 0091 0040 0099 0077

0046 0069 0099 0061 0096 0063 0066)

(25)

333 Clustering of the Candidate Maintenance Taskse sequence matchings between the reference and candi-date maintenance tasks are shown in Figure 8

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks was ascertained (the resultwas multiplied by 1000 for better comparison)

H11 269

H12 198

H13 599

H21 34640

H22 80325

H23 67239

H31 41494

H32 038

H33 142

(26)

Based on the SMC calculation result the thresholds werespecified as

Mathematical Problems in Engineering 9

Tabl

e4

Representatio

nmod

elsforthereferenceandcand

idatemaintenance

tasks

Referencetask

Candidate

task

Mr

I 1rI 2r

I 3rI 4r

t

Mr

lang

SrG

rA

rrangwhere

Sr

Ir 1

Ir 2Ir 3

Ir 41113864

1113865

Gr

E

r 1E

r 2E

r 3E

r 41113864

1113865

Ar

V

r 1V

r 2V

r 3V

r 41113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Er 1

lang(type

circuitbreaker)

(op

erationcheck)

rangV

r 1

(77865910

)

Er 2

lang(type

pin

)(op

erationcheck)

rangV

r 2

(8873899)

Er 3

lang(type

pin

)(op

erationcheck)

rangV

r 3

(78738910

)

Er 4

lang(type

pin

)(op

erationcheck)

rangV

r 4

(8973898)

M1

I 1I 2

I 3I 4

I 5t

M1

lang

S1

G1

A1rang

where

S1

I 1

I2

I 3I

4I 5

11138641113865

G1

E1

E2

E3

E4

E5

11138641113865

A1

V

1V

2V

3V

4V

51113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

eration

check)

rangV

1

(98747710

)

E2

lang(type

pin

)(op

erationcheck)

rangV

2

(88728810

)

E3

lang(type

pin

)(op

erationcheck)

rangV

3

(89728710

)

E4

lang(type

pin

)(op

erationcheck)

rangV

4

(8962879)

E5

lang(type

pin

)(op

erationcheck)

rangV

5

(9872879)

M2

I 1I 2

t

M2

lang

S2

G2

A2rang

where

S2

I 1

I2

11138641113865

G2

E1

E2

11138641113865

A2

V

1V

21113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

erationcheck)

rang

E2

lang(type

pin

)(op

erationcheck)

rang

V1

(87868910

)

V2

(7873899)

lowastTo

quantifytheim

pact

ofdifferent

numbers

ofchecks

atpins

ontheSM

Ccalculation

echecks

atpins

AC4A

C5A

C6and

AC8

aretreatedseparately

10 Mathematical Problems in Engineering

Table 5 Candidate maintenance tasks for each reference task

Reference task Componentequipment Candidate task

Fault confirmationcheckout Pr

1

HF antenna coupler P11 (1) Press the row key near the HF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menuHF antenna P12

Very high-frequency (VHF) transceiver P13

(1) Press the row key near the VHF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menu

Fault isolation Pr2

e wiring between the transceiver pin AC8and the ground terminal P21

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver

P22

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver(4) Do a check of the wiring from the circuit breaker to the

HF transceiver pins AC4 5 and 6

VHF transceiver P23(1) Do a check of the circuit breaker status

(2) Do a check for 28 DC at pin AC2 of the VHF transceiver

Repairinterchange Pr3

HF antenna coupler P31

(1) Disconnect the electrical plug(2) Place cap on the electrical plug

(3) Unscrew the nut(4) Lower the nut

(5) Dismantle the antenna coupler(6) Place cap on the electrical plug

(7) Clean the interface and adjacent area(8) Check the interface and adjacent area

(9) Dismantle the cap from the electrical plug(10) Check the cleanliness and condition of the electrical

plug(11) Install the antenna coupler on the shelf

(12) Screw the nut(13) Dismantle the cap from electrical plug

(14) Check the cleanliness and condition of the electricalplug

(15) Connect the electrical plug to the antenna coupler

Audio management unit (AMU) P32

(1) Unscrew the nut(2) Lower the nut

(3) Pull the AMU from the shelf and disconnect the electricalplug

(4) Dismantle the AMU(5) Place cap on the electrical cap

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the AMU on the shelf(11) Press the AMU to connect the electrical plug

(12) Screw the nut

VHF transceiver P33

(1) Unscrew the nut(2) Lower the nut

(3) Pull the transceiver from the shelf and disconnect theelectrical plug

(4) Dismantle the transceiver(5) Place cap on the electrical plug

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug

(12) Screw the nut

Mathematical Problems in Engineering 11

ε1 6

ε2 420

ε3 3

(27)

en according to equation (5) the clusters for similarcandidate tasks for each reference task were established

C1 P11 P12 P131113864 1113865

C2 P211113864 1113865

C3 P32 P331113864 1113865

(28)

e results showed that the candidate tasks for faultconfirmationcheckout were all similar to the reference taskbecause they had the same items as the reference task de-spite having slightly different maintainability characteristicsIn the case of fault isolation three candidate tasks had

different items to the reference task Task P22 had two ad-ditional items (I5 and I6) and P23 had two missing items (Ir

3and Ir

4) P21 however had only one additional item (I5)making it more similar to the reference task than the othertwo tasks In the case of repairinterchange task P31 hadseven different items (I1 I2 Ir

3 Ir11 I13 I14 and I15) while

the other two tasks all had the same items as the referencetask us task P31 had the most differences and was theleast similar to the reference task

34 HF Transceiver MTTR Demonstration Based onBayesian eory

341 MTTR Demonstration Methods Based on Bayesianeory e Bayesian maintainability demonstrationmethod proposed by Balaban [44] is used in our paper for

Table 6 Maintainability attributes tailored to maintenance tasks

Maintainability attributes Fault confirmation Fault isolation Repairinterchange CheckoutEntity reachability Visibility Maintenance space Tools Technical level of the maintainers Maintenance position Security

Table 7 Evaluation rules

Maintainability attribute Evaluation rules

Entity reachability (U1)

Good can observe the maintenance component comfortably and there is a wide observation angle(7ndash10)

General can generally see the outline of the maintenance component though it can easily cause eye andbody fatigue (4ndash6)

Poor prone to fatigue in this pose (0ndash3)

Visibility (U2)Good clear line of sight and there is enough light (7ndash10)

General the line of sight is blocked or the light is dark (4ndash6)Poor the line of sight is seriously blocked or the light is insufficient (0ndash3)

Maintenance space (U3)

Good no restrictions on the maintenance space for operating posture (7ndash10)General maintenance space for the body is basically enough but the operatorrsquos posture is abnormal

(4ndash6)Poor there is not enough maintenance space for a body (0ndash3)

Tools (U4)Good do not need the aid of auxiliary tools (7ndash10)

General need auxiliary tools sometimes (4ndash6)Poor dependent on auxiliary tools (0ndash3)

Technical level of themaintainers (U5)

Good familiar with the relevant technical knowledge and can quickly determine the operation processand solve the problem (7ndash10)

General is familiar with the relevant knowledge and can solve problems by referring to the operationmanual (4ndash6)

Poor only a general understanding of the situation and how to carry out the relevant work (0ndash3)

Maintenance position (U6)Good ground (7ndash10)

General have to climb to the machine (4ndash6)Poor need to stand outside the machine or in a similar position to the machine (0ndash3)

Security (U7)

Good no danger of being injured by heavy objects and there are no sharp edges that may scratch ordanger of electrical shock (7ndash10)

General there is a certain security threat sometimes there are sharp edges that may cause bumps orscratches (4ndash6)

Poor there is a security threat (0ndash3)

12 Mathematical Problems in Engineering

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

1113954μL ln 1113954θL minusσ2

2 (20)

1113954μU ln 1113954θU minusσ2

2 (21)

It can then be assumed that the range (1113954μU minus 1113954μL) en-compasses 100 times (1 minus p) percent of the total possible valuesof μ and that the best estimate is at the midpoint of the rangeerefore the following prior distribution estimates can beused

μπ 1113954μU + 1113954μL

2 (22)

σ2π 1113954μU minus 1113954μL( 1113857

2

4 times Z2p2

(23)

3 Case Study

In this section the implementation of an MTTR demon-stration for an HF transceiver is once again used to illustrateour method

31 Selection of Candidate Maintenance Tasks An HFtransceiver is part of the HF system and is installed at thefront of the electronics rack in a plane After referring to thetroubleshooting manual and the aircraft maintenancemanual [24 43] we established candidate tasks for eachreference task ese relate to other components in the HFsystem or other equipment at the front of the electronicsrack A breakdown of the tasks is shown in Table 5

32 Identification and Formulation of the MaintainabilityAttribute Set and Evaluation Rules e maintainability at-tribute set developed by Jian et al [26] was used for thesimilarity analysis of the maintenance tasks e main-tainability attributes were tailored to the characteristics ofthe different tasks as shown in Table 6 e correspondingevaluation rules are shown in Table 7

33 Similarity Analysis between the Maintenance Tasks

331 Construction of the Maintenance Task RepresentationModels After referring to the maintenance manuals and theexpertsrsquo experience representation models for the referenceand candidate maintenance tasks were established as shownin Table 8

332 Calculation of the Item Weights On the basis of therepresentation models an item list for each type of main-tenance task was established as shown in Table 9

Using fuzzy AHP the priority matrices for the variousitems in each type of maintenance task were thenobtained

Q1

05 05 05

05 05 05

05 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q2

05 0 0

1 05 0

1 1 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q3

05 1 1 1 1 0 05 1 1 05 1 05 1 1

0 05 0 0 05 0 0 05 0 0 05 0 0 0

0 1 05 0 1 0 0 05 0 0 05 0 05 0

0 1 1 05 1 05 1 1 1 0 1 05 1 1

0 05 0 0 05 0 0 05 0 0 0 0 0 0

1 1 1 05 1 05 1 1 1 05 1 05 1 1

05 1 1 0 1 0 05 1 05 0 05 0 1 1

0 05 05 0 05 0 0 05 0 0 05 0 0 0

0 1 1 0 1 0 05 1 05 0 05 0 05 05

05 1 1 1 1 05 1 1 1 05 1 05 1 1

0 05 05 0 1 0 05 05 05 0 05 0 05 05

05 1 1 05 1 05 1 1 1 05 1 05 1 1

0 1 05 0 1 0 0 1 05 0 05 0 05 05

0 1 1 0 1 0 0 1 05 0 05 0 05 05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

After this equations (7)sim(11) were used to establish theweight vectors for the items in each type of maintenancetask as follows

w1 (0333 0333 0333)

w2 (0211 0335 0454)

w3 (0094 0044 0055 0091 0040 0099 0077

0046 0069 0099 0061 0096 0063 0066)

(25)

333 Clustering of the Candidate Maintenance Taskse sequence matchings between the reference and candi-date maintenance tasks are shown in Figure 8

en drawing upon equations (1)sim(4) the SMC betweenthe candidate and reference tasks was ascertained (the resultwas multiplied by 1000 for better comparison)

H11 269

H12 198

H13 599

H21 34640

H22 80325

H23 67239

H31 41494

H32 038

H33 142

(26)

Based on the SMC calculation result the thresholds werespecified as

Mathematical Problems in Engineering 9

Tabl

e4

Representatio

nmod

elsforthereferenceandcand

idatemaintenance

tasks

Referencetask

Candidate

task

Mr

I 1rI 2r

I 3rI 4r

t

Mr

lang

SrG

rA

rrangwhere

Sr

Ir 1

Ir 2Ir 3

Ir 41113864

1113865

Gr

E

r 1E

r 2E

r 3E

r 41113864

1113865

Ar

V

r 1V

r 2V

r 3V

r 41113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Er 1

lang(type

circuitbreaker)

(op

erationcheck)

rangV

r 1

(77865910

)

Er 2

lang(type

pin

)(op

erationcheck)

rangV

r 2

(8873899)

Er 3

lang(type

pin

)(op

erationcheck)

rangV

r 3

(78738910

)

Er 4

lang(type

pin

)(op

erationcheck)

rangV

r 4

(8973898)

M1

I 1I 2

I 3I 4

I 5t

M1

lang

S1

G1

A1rang

where

S1

I 1

I2

I 3I

4I 5

11138641113865

G1

E1

E2

E3

E4

E5

11138641113865

A1

V

1V

2V

3V

4V

51113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

eration

check)

rangV

1

(98747710

)

E2

lang(type

pin

)(op

erationcheck)

rangV

2

(88728810

)

E3

lang(type

pin

)(op

erationcheck)

rangV

3

(89728710

)

E4

lang(type

pin

)(op

erationcheck)

rangV

4

(8962879)

E5

lang(type

pin

)(op

erationcheck)

rangV

5

(9872879)

M2

I 1I 2

t

M2

lang

S2

G2

A2rang

where

S2

I 1

I2

11138641113865

G2

E1

E2

11138641113865

A2

V

1V

21113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

erationcheck)

rang

E2

lang(type

pin

)(op

erationcheck)

rang

V1

(87868910

)

V2

(7873899)

lowastTo

quantifytheim

pact

ofdifferent

numbers

ofchecks

atpins

ontheSM

Ccalculation

echecks

atpins

AC4A

C5A

C6and

AC8

aretreatedseparately

10 Mathematical Problems in Engineering

Table 5 Candidate maintenance tasks for each reference task

Reference task Componentequipment Candidate task

Fault confirmationcheckout Pr

1

HF antenna coupler P11 (1) Press the row key near the HF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menuHF antenna P12

Very high-frequency (VHF) transceiver P13

(1) Press the row key near the VHF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menu

Fault isolation Pr2

e wiring between the transceiver pin AC8and the ground terminal P21

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver

P22

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver(4) Do a check of the wiring from the circuit breaker to the

HF transceiver pins AC4 5 and 6

VHF transceiver P23(1) Do a check of the circuit breaker status

(2) Do a check for 28 DC at pin AC2 of the VHF transceiver

Repairinterchange Pr3

HF antenna coupler P31

(1) Disconnect the electrical plug(2) Place cap on the electrical plug

(3) Unscrew the nut(4) Lower the nut

(5) Dismantle the antenna coupler(6) Place cap on the electrical plug

(7) Clean the interface and adjacent area(8) Check the interface and adjacent area

(9) Dismantle the cap from the electrical plug(10) Check the cleanliness and condition of the electrical

plug(11) Install the antenna coupler on the shelf

(12) Screw the nut(13) Dismantle the cap from electrical plug

(14) Check the cleanliness and condition of the electricalplug

(15) Connect the electrical plug to the antenna coupler

Audio management unit (AMU) P32

(1) Unscrew the nut(2) Lower the nut

(3) Pull the AMU from the shelf and disconnect the electricalplug

(4) Dismantle the AMU(5) Place cap on the electrical cap

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the AMU on the shelf(11) Press the AMU to connect the electrical plug

(12) Screw the nut

VHF transceiver P33

(1) Unscrew the nut(2) Lower the nut

(3) Pull the transceiver from the shelf and disconnect theelectrical plug

(4) Dismantle the transceiver(5) Place cap on the electrical plug

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug

(12) Screw the nut

Mathematical Problems in Engineering 11

ε1 6

ε2 420

ε3 3

(27)

en according to equation (5) the clusters for similarcandidate tasks for each reference task were established

C1 P11 P12 P131113864 1113865

C2 P211113864 1113865

C3 P32 P331113864 1113865

(28)

e results showed that the candidate tasks for faultconfirmationcheckout were all similar to the reference taskbecause they had the same items as the reference task de-spite having slightly different maintainability characteristicsIn the case of fault isolation three candidate tasks had

different items to the reference task Task P22 had two ad-ditional items (I5 and I6) and P23 had two missing items (Ir

3and Ir

4) P21 however had only one additional item (I5)making it more similar to the reference task than the othertwo tasks In the case of repairinterchange task P31 hadseven different items (I1 I2 Ir

3 Ir11 I13 I14 and I15) while

the other two tasks all had the same items as the referencetask us task P31 had the most differences and was theleast similar to the reference task

34 HF Transceiver MTTR Demonstration Based onBayesian eory

341 MTTR Demonstration Methods Based on Bayesianeory e Bayesian maintainability demonstrationmethod proposed by Balaban [44] is used in our paper for

Table 6 Maintainability attributes tailored to maintenance tasks

Maintainability attributes Fault confirmation Fault isolation Repairinterchange CheckoutEntity reachability Visibility Maintenance space Tools Technical level of the maintainers Maintenance position Security

Table 7 Evaluation rules

Maintainability attribute Evaluation rules

Entity reachability (U1)

Good can observe the maintenance component comfortably and there is a wide observation angle(7ndash10)

General can generally see the outline of the maintenance component though it can easily cause eye andbody fatigue (4ndash6)

Poor prone to fatigue in this pose (0ndash3)

Visibility (U2)Good clear line of sight and there is enough light (7ndash10)

General the line of sight is blocked or the light is dark (4ndash6)Poor the line of sight is seriously blocked or the light is insufficient (0ndash3)

Maintenance space (U3)

Good no restrictions on the maintenance space for operating posture (7ndash10)General maintenance space for the body is basically enough but the operatorrsquos posture is abnormal

(4ndash6)Poor there is not enough maintenance space for a body (0ndash3)

Tools (U4)Good do not need the aid of auxiliary tools (7ndash10)

General need auxiliary tools sometimes (4ndash6)Poor dependent on auxiliary tools (0ndash3)

Technical level of themaintainers (U5)

Good familiar with the relevant technical knowledge and can quickly determine the operation processand solve the problem (7ndash10)

General is familiar with the relevant knowledge and can solve problems by referring to the operationmanual (4ndash6)

Poor only a general understanding of the situation and how to carry out the relevant work (0ndash3)

Maintenance position (U6)Good ground (7ndash10)

General have to climb to the machine (4ndash6)Poor need to stand outside the machine or in a similar position to the machine (0ndash3)

Security (U7)

Good no danger of being injured by heavy objects and there are no sharp edges that may scratch ordanger of electrical shock (7ndash10)

General there is a certain security threat sometimes there are sharp edges that may cause bumps orscratches (4ndash6)

Poor there is a security threat (0ndash3)

12 Mathematical Problems in Engineering

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

Tabl

e4

Representatio

nmod

elsforthereferenceandcand

idatemaintenance

tasks

Referencetask

Candidate

task

Mr

I 1rI 2r

I 3rI 4r

t

Mr

lang

SrG

rA

rrangwhere

Sr

Ir 1

Ir 2Ir 3

Ir 41113864

1113865

Gr

E

r 1E

r 2E

r 3E

r 41113864

1113865

Ar

V

r 1V

r 2V

r 3V

r 41113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Er 1

lang(type

circuitbreaker)

(op

erationcheck)

rangV

r 1

(77865910

)

Er 2

lang(type

pin

)(op

erationcheck)

rangV

r 2

(8873899)

Er 3

lang(type

pin

)(op

erationcheck)

rangV

r 3

(78738910

)

Er 4

lang(type

pin

)(op

erationcheck)

rangV

r 4

(8973898)

M1

I 1I 2

I 3I 4

I 5t

M1

lang

S1

G1

A1rang

where

S1

I 1

I2

I 3I

4I 5

11138641113865

G1

E1

E2

E3

E4

E5

11138641113865

A1

V

1V

2V

3V

4V

51113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

eration

check)

rangV

1

(98747710

)

E2

lang(type

pin

)(op

erationcheck)

rangV

2

(88728810

)

E3

lang(type

pin

)(op

erationcheck)

rangV

3

(89728710

)

E4

lang(type

pin

)(op

erationcheck)

rangV

4

(8962879)

E5

lang(type

pin

)(op

erationcheck)

rangV

5

(9872879)

M2

I 1I 2

t

M2

lang

S2

G2

A2rang

where

S2

I 1

I2

11138641113865

G2

E1

E2

11138641113865

A2

V

1V

21113864

1113865

⎧⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E1

lang(type

circuitbreaker)

(op

erationcheck)

rang

E2

lang(type

pin

)(op

erationcheck)

rang

V1

(87868910

)

V2

(7873899)

lowastTo

quantifytheim

pact

ofdifferent

numbers

ofchecks

atpins

ontheSM

Ccalculation

echecks

atpins

AC4A

C5A

C6and

AC8

aretreatedseparately

10 Mathematical Problems in Engineering

Table 5 Candidate maintenance tasks for each reference task

Reference task Componentequipment Candidate task

Fault confirmationcheckout Pr

1

HF antenna coupler P11 (1) Press the row key near the HF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menuHF antenna P12

Very high-frequency (VHF) transceiver P13

(1) Press the row key near the VHF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menu

Fault isolation Pr2

e wiring between the transceiver pin AC8and the ground terminal P21

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver

P22

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver(4) Do a check of the wiring from the circuit breaker to the

HF transceiver pins AC4 5 and 6

VHF transceiver P23(1) Do a check of the circuit breaker status

(2) Do a check for 28 DC at pin AC2 of the VHF transceiver

Repairinterchange Pr3

HF antenna coupler P31

(1) Disconnect the electrical plug(2) Place cap on the electrical plug

(3) Unscrew the nut(4) Lower the nut

(5) Dismantle the antenna coupler(6) Place cap on the electrical plug

(7) Clean the interface and adjacent area(8) Check the interface and adjacent area

(9) Dismantle the cap from the electrical plug(10) Check the cleanliness and condition of the electrical

plug(11) Install the antenna coupler on the shelf

(12) Screw the nut(13) Dismantle the cap from electrical plug

(14) Check the cleanliness and condition of the electricalplug

(15) Connect the electrical plug to the antenna coupler

Audio management unit (AMU) P32

(1) Unscrew the nut(2) Lower the nut

(3) Pull the AMU from the shelf and disconnect the electricalplug

(4) Dismantle the AMU(5) Place cap on the electrical cap

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the AMU on the shelf(11) Press the AMU to connect the electrical plug

(12) Screw the nut

VHF transceiver P33

(1) Unscrew the nut(2) Lower the nut

(3) Pull the transceiver from the shelf and disconnect theelectrical plug

(4) Dismantle the transceiver(5) Place cap on the electrical plug

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug

(12) Screw the nut

Mathematical Problems in Engineering 11

ε1 6

ε2 420

ε3 3

(27)

en according to equation (5) the clusters for similarcandidate tasks for each reference task were established

C1 P11 P12 P131113864 1113865

C2 P211113864 1113865

C3 P32 P331113864 1113865

(28)

e results showed that the candidate tasks for faultconfirmationcheckout were all similar to the reference taskbecause they had the same items as the reference task de-spite having slightly different maintainability characteristicsIn the case of fault isolation three candidate tasks had

different items to the reference task Task P22 had two ad-ditional items (I5 and I6) and P23 had two missing items (Ir

3and Ir

4) P21 however had only one additional item (I5)making it more similar to the reference task than the othertwo tasks In the case of repairinterchange task P31 hadseven different items (I1 I2 Ir

3 Ir11 I13 I14 and I15) while

the other two tasks all had the same items as the referencetask us task P31 had the most differences and was theleast similar to the reference task

34 HF Transceiver MTTR Demonstration Based onBayesian eory

341 MTTR Demonstration Methods Based on Bayesianeory e Bayesian maintainability demonstrationmethod proposed by Balaban [44] is used in our paper for

Table 6 Maintainability attributes tailored to maintenance tasks

Maintainability attributes Fault confirmation Fault isolation Repairinterchange CheckoutEntity reachability Visibility Maintenance space Tools Technical level of the maintainers Maintenance position Security

Table 7 Evaluation rules

Maintainability attribute Evaluation rules

Entity reachability (U1)

Good can observe the maintenance component comfortably and there is a wide observation angle(7ndash10)

General can generally see the outline of the maintenance component though it can easily cause eye andbody fatigue (4ndash6)

Poor prone to fatigue in this pose (0ndash3)

Visibility (U2)Good clear line of sight and there is enough light (7ndash10)

General the line of sight is blocked or the light is dark (4ndash6)Poor the line of sight is seriously blocked or the light is insufficient (0ndash3)

Maintenance space (U3)

Good no restrictions on the maintenance space for operating posture (7ndash10)General maintenance space for the body is basically enough but the operatorrsquos posture is abnormal

(4ndash6)Poor there is not enough maintenance space for a body (0ndash3)

Tools (U4)Good do not need the aid of auxiliary tools (7ndash10)

General need auxiliary tools sometimes (4ndash6)Poor dependent on auxiliary tools (0ndash3)

Technical level of themaintainers (U5)

Good familiar with the relevant technical knowledge and can quickly determine the operation processand solve the problem (7ndash10)

General is familiar with the relevant knowledge and can solve problems by referring to the operationmanual (4ndash6)

Poor only a general understanding of the situation and how to carry out the relevant work (0ndash3)

Maintenance position (U6)Good ground (7ndash10)

General have to climb to the machine (4ndash6)Poor need to stand outside the machine or in a similar position to the machine (0ndash3)

Security (U7)

Good no danger of being injured by heavy objects and there are no sharp edges that may scratch ordanger of electrical shock (7ndash10)

General there is a certain security threat sometimes there are sharp edges that may cause bumps orscratches (4ndash6)

Poor there is a security threat (0ndash3)

12 Mathematical Problems in Engineering

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

Table 5 Candidate maintenance tasks for each reference task

Reference task Componentequipment Candidate task

Fault confirmationcheckout Pr

1

HF antenna coupler P11 (1) Press the row key near the HF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menuHF antenna P12

Very high-frequency (VHF) transceiver P13

(1) Press the row key near the VHF indicator(2) Press the column key near the test indicator(3) Press the mode key on the MCDU menu

Fault isolation Pr2

e wiring between the transceiver pin AC8and the ground terminal P21

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver

P22

(1) Do a check of the circuit breaker status(2) Do a check for 115 VAC at pins AC4 5 and 6 of the HF

transceiver(3) Do a check for a ground signal at pin AC8 of the HF

transceiver(4) Do a check of the wiring from the circuit breaker to the

HF transceiver pins AC4 5 and 6

VHF transceiver P23(1) Do a check of the circuit breaker status

(2) Do a check for 28 DC at pin AC2 of the VHF transceiver

Repairinterchange Pr3

HF antenna coupler P31

(1) Disconnect the electrical plug(2) Place cap on the electrical plug

(3) Unscrew the nut(4) Lower the nut

(5) Dismantle the antenna coupler(6) Place cap on the electrical plug

(7) Clean the interface and adjacent area(8) Check the interface and adjacent area

(9) Dismantle the cap from the electrical plug(10) Check the cleanliness and condition of the electrical

plug(11) Install the antenna coupler on the shelf

(12) Screw the nut(13) Dismantle the cap from electrical plug

(14) Check the cleanliness and condition of the electricalplug

(15) Connect the electrical plug to the antenna coupler

Audio management unit (AMU) P32

(1) Unscrew the nut(2) Lower the nut

(3) Pull the AMU from the shelf and disconnect the electricalplug

(4) Dismantle the AMU(5) Place cap on the electrical cap

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the AMU on the shelf(11) Press the AMU to connect the electrical plug

(12) Screw the nut

VHF transceiver P33

(1) Unscrew the nut(2) Lower the nut

(3) Pull the transceiver from the shelf and disconnect theelectrical plug

(4) Dismantle the transceiver(5) Place cap on the electrical plug

(6) Clean the interface and adjacent area(7) Check the interface and adjacent area

(8) Dismantle the cap from the electrical plug(9) Check the cleanliness and condition of the electrical plug

(10) Install the transceiver on the shelf(11) Press the transceiver to connect the electrical plug

(12) Screw the nut

Mathematical Problems in Engineering 11

ε1 6

ε2 420

ε3 3

(27)

en according to equation (5) the clusters for similarcandidate tasks for each reference task were established

C1 P11 P12 P131113864 1113865

C2 P211113864 1113865

C3 P32 P331113864 1113865

(28)

e results showed that the candidate tasks for faultconfirmationcheckout were all similar to the reference taskbecause they had the same items as the reference task de-spite having slightly different maintainability characteristicsIn the case of fault isolation three candidate tasks had

different items to the reference task Task P22 had two ad-ditional items (I5 and I6) and P23 had two missing items (Ir

3and Ir

4) P21 however had only one additional item (I5)making it more similar to the reference task than the othertwo tasks In the case of repairinterchange task P31 hadseven different items (I1 I2 Ir

3 Ir11 I13 I14 and I15) while

the other two tasks all had the same items as the referencetask us task P31 had the most differences and was theleast similar to the reference task

34 HF Transceiver MTTR Demonstration Based onBayesian eory

341 MTTR Demonstration Methods Based on Bayesianeory e Bayesian maintainability demonstrationmethod proposed by Balaban [44] is used in our paper for

Table 6 Maintainability attributes tailored to maintenance tasks

Maintainability attributes Fault confirmation Fault isolation Repairinterchange CheckoutEntity reachability Visibility Maintenance space Tools Technical level of the maintainers Maintenance position Security

Table 7 Evaluation rules

Maintainability attribute Evaluation rules

Entity reachability (U1)

Good can observe the maintenance component comfortably and there is a wide observation angle(7ndash10)

General can generally see the outline of the maintenance component though it can easily cause eye andbody fatigue (4ndash6)

Poor prone to fatigue in this pose (0ndash3)

Visibility (U2)Good clear line of sight and there is enough light (7ndash10)

General the line of sight is blocked or the light is dark (4ndash6)Poor the line of sight is seriously blocked or the light is insufficient (0ndash3)

Maintenance space (U3)

Good no restrictions on the maintenance space for operating posture (7ndash10)General maintenance space for the body is basically enough but the operatorrsquos posture is abnormal

(4ndash6)Poor there is not enough maintenance space for a body (0ndash3)

Tools (U4)Good do not need the aid of auxiliary tools (7ndash10)

General need auxiliary tools sometimes (4ndash6)Poor dependent on auxiliary tools (0ndash3)

Technical level of themaintainers (U5)

Good familiar with the relevant technical knowledge and can quickly determine the operation processand solve the problem (7ndash10)

General is familiar with the relevant knowledge and can solve problems by referring to the operationmanual (4ndash6)

Poor only a general understanding of the situation and how to carry out the relevant work (0ndash3)

Maintenance position (U6)Good ground (7ndash10)

General have to climb to the machine (4ndash6)Poor need to stand outside the machine or in a similar position to the machine (0ndash3)

Security (U7)

Good no danger of being injured by heavy objects and there are no sharp edges that may scratch ordanger of electrical shock (7ndash10)

General there is a certain security threat sometimes there are sharp edges that may cause bumps orscratches (4ndash6)

Poor there is a security threat (0ndash3)

12 Mathematical Problems in Engineering

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

ε1 6

ε2 420

ε3 3

(27)

en according to equation (5) the clusters for similarcandidate tasks for each reference task were established

C1 P11 P12 P131113864 1113865

C2 P211113864 1113865

C3 P32 P331113864 1113865

(28)

e results showed that the candidate tasks for faultconfirmationcheckout were all similar to the reference taskbecause they had the same items as the reference task de-spite having slightly different maintainability characteristicsIn the case of fault isolation three candidate tasks had

different items to the reference task Task P22 had two ad-ditional items (I5 and I6) and P23 had two missing items (Ir

3and Ir

4) P21 however had only one additional item (I5)making it more similar to the reference task than the othertwo tasks In the case of repairinterchange task P31 hadseven different items (I1 I2 Ir

3 Ir11 I13 I14 and I15) while

the other two tasks all had the same items as the referencetask us task P31 had the most differences and was theleast similar to the reference task

34 HF Transceiver MTTR Demonstration Based onBayesian eory

341 MTTR Demonstration Methods Based on Bayesianeory e Bayesian maintainability demonstrationmethod proposed by Balaban [44] is used in our paper for

Table 6 Maintainability attributes tailored to maintenance tasks

Maintainability attributes Fault confirmation Fault isolation Repairinterchange CheckoutEntity reachability Visibility Maintenance space Tools Technical level of the maintainers Maintenance position Security

Table 7 Evaluation rules

Maintainability attribute Evaluation rules

Entity reachability (U1)

Good can observe the maintenance component comfortably and there is a wide observation angle(7ndash10)

General can generally see the outline of the maintenance component though it can easily cause eye andbody fatigue (4ndash6)

Poor prone to fatigue in this pose (0ndash3)

Visibility (U2)Good clear line of sight and there is enough light (7ndash10)

General the line of sight is blocked or the light is dark (4ndash6)Poor the line of sight is seriously blocked or the light is insufficient (0ndash3)

Maintenance space (U3)

Good no restrictions on the maintenance space for operating posture (7ndash10)General maintenance space for the body is basically enough but the operatorrsquos posture is abnormal

(4ndash6)Poor there is not enough maintenance space for a body (0ndash3)

Tools (U4)Good do not need the aid of auxiliary tools (7ndash10)

General need auxiliary tools sometimes (4ndash6)Poor dependent on auxiliary tools (0ndash3)

Technical level of themaintainers (U5)

Good familiar with the relevant technical knowledge and can quickly determine the operation processand solve the problem (7ndash10)

General is familiar with the relevant knowledge and can solve problems by referring to the operationmanual (4ndash6)

Poor only a general understanding of the situation and how to carry out the relevant work (0ndash3)

Maintenance position (U6)Good ground (7ndash10)

General have to climb to the machine (4ndash6)Poor need to stand outside the machine or in a similar position to the machine (0ndash3)

Security (U7)

Good no danger of being injured by heavy objects and there are no sharp edges that may scratch ordanger of electrical shock (7ndash10)

General there is a certain security threat sometimes there are sharp edges that may cause bumps orscratches (4ndash6)

Poor there is a security threat (0ndash3)

12 Mathematical Problems in Engineering

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

Table 8 Representation models for the reference and candidate tasks

Reference task No Candidate task(a) Fault confirmationcheckout

tI1r I2r I3r

Mr1 langSr

1 Gr1 Ar

1rangwhere

Sr1 Ir

1 Ir2 Ir

31113864 1113865

Gr1 Er

1 Er2 Er

31113864 1113865

Ar1 Vr

1 Vr2 Vr

31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type row key) (operation press)rang

Er2 lang(type columnkey) (operation press)rang

Er3 lang(typemode key) (operation press)rang

Vr1 (8 10 5 10)

Vr2 (9 10 5 10)

Vr3 (9 10 5 10)

(1)

I1 I2 I3 t

M11 langS11 G11 A11rangwhere

S11 I1 I2 I31113864 1113865

G11 E1 E2 E31113864 1113865

A11 V1 V2 V31113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type row key) (operation press)rangE2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (7 10 5 10)

(2)

I1 I2 I3 t

M12 langS12 G12 A12rangwhere

S12 I1 I2 I31113864 1113865

G12 E1 E2 E31113864 1113865

A12 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 6 10)

V2 (9 9 5 10)

V3 (8 10 6 10)

(3)

I1 I2 I3 t

M13 langS13 G13 A13rangwhere

S13 I1 I2 I31113864 1113865

G13 E1 E2 E31113864 1113865

A13 V1 V2 V31113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type row key) (operation press)rang

E2 lang(type column key) (operation press)rang

E3 lang(typemode key) (operation press)rang

V1 (8 10 8 10)

V2 (9 9 5 10)

V3 (8 9 6 10)

(b) Fault isolation

I1r I2r I3r I4r t

Mr2 langSr

2 Gr2 Ar

2rangwhere

Sr2 Ir

1 Ir2 Ir

3 Ir41113864 1113865

Gr2 Er

1 Er2 Er

3 Er41113864 1113865

Ar2 Vr

1 Vr2 Vr

3 Vr41113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(type circuit breaker) (operation check)rang Vr

1 (7 7 8 6 5 9 10)

Er2 lang(type pin) (operation check)rang Vr

2 (8 8 7 3 8 9 9)

Er3 lang(type pin) (operation check)rang Vr

3 (7 8 7 3 8 9 10)

Er4 lang(type pin) (operation check)rang Vr

4 (8 9 7 3 8 9 8)

(1)

I1 I2 I3 I4 I5 t

M21 langS21 G21 A21rangwhere

S21 I1 I2 I3 I4 I51113864 1113865

G21 E1 E2 E3 E4 E51113864 1113865

A21 V1 V2 V3 V4 V51113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 8 7 4 7 7 10)

E2 lang(type pin) (operation check)rang V2 (8 8 7 2 8 8 10)

E3 lang(type pin) (operation check)rang V3 (8 9 7 2 8 7 10)

E4 lang(type pin) (operation check)rang V4 (8 9 6 2 8 7 9)

E5 lang(type pin) (operation check)rang V5 (9 8 7 2 8 7 9)

(2)

I1 I2 I3 I4 I5 I6 t

M22 langS22 G22 A22rangwhereS22 I1 I2 I3 I4 I5 I61113864 1113865

G22 E1 E2 E3 E4 E5 E61113864 1113865

A22 V1 V2 V3 V4 V5 V61113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang V1 (9 9 9 4 8 9 10)

E2 lang(type pin) (operation check)rang V2 (7 7 10 2 9 10 10)

E3 lang(type pin) (operation check)rang V3 (8 8 9 2 9 10 10)

E4 lang(type pin) (operation check)rang V4 (8 7 9 2 9 10 9)

E5 lang(type pin) (operation check)rang V5 (8 8 9 2 8 7 10)

E6 lang(typewiring) (operation check)rang V6 (6 9 8 3 6 9 9)

(3)

I1 I2 t

M23 langS23 G23 A23rangwhere

S23 I1 I21113864 1113865

G23 E1 E21113864 1113865

A23 V1 V21113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(type circuit breaker) (operation check)rang

E2 lang(type pin) (operation check)rang

V1 (8 7 8 6 8 9 10)

V2 (7 8 7 3 8 9 9)

Mathematical Problems in Engineering 13

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

Table 8 Continued

Reference task No Candidate task(c) Repairinterchange

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 t

Mr3 langSr

3 Gr3 Ar

3rangwhere

Sr3 Ir

1 Ir2 Ir

121113864 1113865

Gr3 Er

1 Er2 Er

121113864 1113865

Ar3 Vr

1 Vr2 Vr

121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Er1 lang(typenut)(operationunscrew)rang Vr

1 (67729910)

Er2 lang(typenut)(operation lower)rang Vr

2 (67889910)

Er3 lang(type faileddevice)(operationpull)rang Vr

3 (991099107)

Er4 lang(type faileddevice)(operationdismantle)rang Vr

4 (9982896)

Er5 lang(typecap)(operationplace)rang Vr

5 (9991091010)

Er6 lang(type interface)(operationclean)rang Vr

6 (7764799)

Er7 lang(type interface)(operationcheck)rang Vr

7 (9910107910)

Er8 lang(typecap)(operationdismantle)rang Vr

8 (9991091010)

Er9 lang(typeelectricalplug)(operationcheck)rang Vr

9 (889991010)

Er10 lang(type faileddevice)(operation install)rang Vr

10 (77867106)

Er11 lang(type faileddevice)(operationpress)rang Vr

11 (78897108)

Er12 lang(typenut)(operationscrew)rang Vr

12 (67729910)

lowast ldquofailed devicerdquo indicates HF transceiver

(1)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15

M31 langS31G31A31rangwhere

S31 I1 I2 middot middot middot I151113864 1113865

G31 E1 E2 middot middot middot E151113864 1113865

A31 V1 V2 middot middot middot V151113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typeelectricalplug)(operationdisconnect)rang V1 (67798910)

E2 lang(typecap)(operationplace)rang V2 (67898910)

E3 lang(typenut)(operationunscrew)rang V3 (67728910)

E4 lang(typenut)(operation lower)rang V4 (67728910)

E5 lang(type faileddevice)(operationdismantle)rang V5 (8876895)

E6 lang(typecap)(operationplace)rang V6 (678108910)

E7 lang(type interface)(operationclean)rang V7 (7663889)

E8 lang(type interface)(operationcheck)rang V8 (66677910)

E9 lang(typecap)(operationdismantle)rang V9 (678108910)

E10 lang(typeelectricalplug)(operationcheck)rang V10 (678981010)

E11 lang(type faileddevice)(operation install)rang V11 (57678106)

E12 lang(typenut)(operationscrew)rang V12 (67628910)

E13 lang(typecap)(operationdismantle)rang V13 (778108910)

E14 lang(typeelectricalplug)(operationcheck)rang V14 (678981010)

E15 lang(typeelectricalplug)(operationconnect)rang V15 (6779899)

lowast ldquofailed devicerdquo indicates HF antenna coupler

(2)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M32 langS32 G32 A32rangwhere

S32 I1 I2 middot middot middot I121113864 1113865

G32 E1 E2 middot middot middot E121113864 1113865

A32 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (77729910)

E2 lang(typenut)(operation lower)rang V2 (77889910)

E3 lang(type faileddevice)(operationpull)rang V3 (981099107)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9982896)

E5 lang(typecap)(operationplace)rang V5 (9891091010)

E6 lang(type interface)(operationclean)rang V6 (87647910)

E7 lang(type interface)(operationcheck)rang V7 (9910107910)

E8 lang(typecap)(operationdismantle)rang V8 (9991091010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889991010)

E10 lang(type faileddevice)(operation install)rang V10 (77867106)

E11 lang(type faileddevice)(operationpress)rang V11 (78897108)

E12 lang(typenut)(operationscrew)rang V12 (67729910)

lowast ldquofailed devicerdquo indicates AMU

(3)

tI1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12

M33 langS33 G33 A33rangwhereS33 I1 I2 middot middot middot I121113864 1113865

G33 E1 E2 middot middot middot E121113864 1113865

A33 V1 V2 middot middot middot V121113864 1113865

⎧⎪⎪⎨

⎪⎪⎩

E1 lang(typenut)(operationunscrew)rang V1 (78728910)

E2 lang(typenut)(operation lower)rang V2 (78888910)

E3 lang(type faileddevice)(operationpull)rang V3 (981098108)

E4 lang(type faileddevice)(operationdismantle)rang V4 (9882896)

E5 lang(typecap)(operationplace)rang V5 (9891081010)

E6 lang(type interface)(operationclean)rang V6 (87648910)

E7 lang(type interface)(operationcheck)rang V7 (8910108910)

E8 lang(typecap)(operationdismantle)rang V8 (9991081010)

E9 lang(typeelectricalplug)(operationcheck)rang V9 (889981010)

E10 lang(type faileddevice)(operation install)rang V10 (77868106)

E11 lang(type faileddevice)(operationpress)rang V11 (78898108)

E12 lang(typenut)(operationscrew)rang V12 (67728910)

lowast ldquofailed devicerdquo indicates VHF transceiver

14 Mathematical Problems in Engineering

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

Tabl

e9

Item

listfor

each

type

ofmaintenance

task

No

Faultc

onfirmationchecko

utFaultisolatio

nRe

pairin

terchang

e1

lang(ty

pe

row

key

)(

op

era

tion

pre

ss)rang

lang(ty

pe

circ

uit

bre

ak

er)

(op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

un

scre

w)rang

2lang(

typ

eco

lum

nk

ey)

(op

era

tion

pre

ss)rang

lang(ty

pe

pin

)(

op

era

tion

ch

eck

)ranglang(

typ

en

ut)

(op

era

tion

low

er)rang

3lang(

typ

em

od

ekey)

(op

era

tion

pre

ss)rang

lang(ty

pe

wir

ing

)(

op

era

tion

ch

eck

)ranglang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pu

ll)rang

4mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

dis

ma

ntl

e)rang

5mdash

mdashlang(

typ

eca

p)

(op

era

tion

pla

ce)rang

6mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

cle

an

)rang

7mdash

mdashlang(

typ

ein

terf

ace

)(

op

era

tion

ch

eck

)rang

8mdash

mdashlang(

typ

eca

p)

(op

era

tion

dis

ma

ntl

e)rang

9mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

ch

eck

)rang

10mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

in

sta

ll)rang

11mdash

mdashlang(

typ

ef

ail

ure

de

vice

)(

op

era

tion

pre

ss)rang

12mdash

mdashlang(

typ

en

ut)

(op

era

tion

scr

ew)rang

13mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

dis

con

nec

t)rang

14mdash

mdashlang(

typ

eel

ectr

ica

lplu

g)

(op

era

tion

con

nec

t)rang

Mathematical Problems in Engineering 15

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

the MTTR demonstration In outline the method can bedescribed as follows

Let X sim LogN(μ σ2) denote the maintenance timedistribution of specified equipment e variance σ2 isknown from prior information or a reasonably preciseestimate can be obtained Xi(i 1 2 n) is a randomsample collected from field data

e following is the hypothesis to be tested

H0 θ(MTTR) θ0

H1 θ(MTTR) θ1(29)

where θ0 is the desired value and θ1 is the minimum ac-ceptable value

According to the properties of the lognormal distribu-tion the statistical inference concerning θ can be simplifiedby referring to the statistical inference concerning μ which is

H0 μ μ0

H1 μ μ1(30)

where μ0 ln θ0 minus σ22 and μ1 ln θ1 minus σ22If the mean maintenance time is θ0 then the probability

of the productrsquos acceptance will be 1 minus α If the product isaccepted then the probability that its mean maintenancetime will be greater than θ1 will be βb e above two re-quirements can be written as

P Tn leTlowast μ μ0

11138681113868111386811138681113960 1113961 1 minus α

P μgt μ1 Tn leTlowast11138681113868111386811138681113960 1113961 βb

(31)

where Tn 1113936ni1 lnXin and Tlowast is the preselected critical

value for the decision such that H0 will be accepted ifTn leTlowast and H1 will be accepted if Tn gtTlowast

e parameter μ is assumed to have a normal priordistribution N(μπ σ2π) so

Y μπ minus μ1σπ

(32)

Z μ1 minus μ0σπ

(33)

e sample size is

n Kσ2

μ1 minus μ0( 11138572 (34)

where the values ofK for all combinations of α βb Y andZ canbe determined according to the tables presented in [44] K 0signifies that the prior distribution already meets the Bayesianrisk requirement thus obviating the need for testing After thisgiven the sample size n Tlowast can be obtained as follows

Tlowast

μ0 + Z1minusασn

radic (35)

342 HF Transceiver MTTR Demonstration LetX sim LogN(μ σ2) denote the maintenance time distributionof the HF transceiver We will assume an estimation pro-cedure has produced an estimate of 03 for σ2 e priordistribution of μ is denoted as N(μπ σ2π) Table 10 shows the

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

I1r I2r I3rM1r I1r I2r I3rM1

r I1r I2r I3rM1r

I1 I2 I3M11 I1 I2 I3M12 I1 I2 I3M13

I1r I2r I3r I4r I5rM2r

I1r I2r I3r I4rM2r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12 Ir13 Ir14 Ir15 Ir16 Ir17M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6r I7r I8r I9r Ir10 Ir11 Ir12M3r

I1r I2r I3r I4r I5r I6rM2r

I1 I2 I3 I4 I5M21

I1 I2 I3 I4M23

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14 I15 I16 I17M31

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M32

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12M33

I1 I2 I3 I4 I5 I6M22

Figure 8 Sequence matching between reference and candidate maintenance tasks

16 Mathematical Problems in Engineering

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

maintenance time data of similar candidate tasks fromhistorical records

en according to equations (18) and (19) the estimatedvalues of θL and θU for the maintenance time distributionwill be as follows

1113954θL 734

1113954θU 1046(36)

Drawing upon equations (20) and (21) the two equiv-alent predictions of μ are

1113954μL 415

1113954μU 450(37)

We will assume that the range (1113954μU minus 1113954μL) encompasses90 of the total possible values of μ en according toequations (22) and (23) we obtain

μπ 4323

σ2π 0012(38)

Assume that the hypothesis test is

H0 θ(MTTR) 75mins

H1 θ(MTTR) 95mins(39)

is relation can be simplified according to equation (30)to give

H0 μ 4167

H1 μ 4404(40)

e risks for the producer and the consumer areα β 01 From equations (32) and (33) we obtain

Y minus0751

Z 2195(41)

To be conservative the next highest tabular entries Y

minus05 andZ 25 are used giving the result K 3835enfrom equation (34) the sample size can be obtained asfollows

n 2059 asymp 21 (42)

e critical value will then be

Tlowast

4321 (43)

After taking a random sample of 21 maintenance actionsfor the HF transceiver the sample mean can be obtained asfollows

Tn 1113936

21i1lnXi

21 (44)

If Tn le 4321 then the null hypothesis will be acceptedotherwise it will be rejected

It can be seen from the result that with the introductionof prior information into the MTTR demonstration by usingthe Bayesian method the test requires fewer samples thantraditional methods that require no less than 30 samples Inaddition because each candidate task contains too fewsamples (not more than five) analyzing the prior infor-mation accuracy from the perspective of data consistency isunreliable Our method provides a better solution for priorinformation accuracy analysis and prior distribution elici-tation in this situation

4 Conclusion

In this article we have proposed a novel prior distributionelicitation method for MTTR Bayesian demonstrationismethod enables a maintenance task similarity analysis to beundertaken using task representation models that canensure the accuracy of the prior information Taking theMTTR demonstration for an HF transceiver in a civilairplane as an example we elicited the prior distributionfrom the maintenance time data for equipment andcomponents on the same rack or in the same systemDuring the elicitation process candidate tasks having anunacceptable difference from the reference tasks wereexcluded After that a demonstration method based onBayesian theory was employed for the actual MTTRdemonstration Compared with previous research whichmainly depends on maintenance time data analysis ourmethod provides a novel way of analyzing the accuracy ofprior information from the perspective of maintenanceaction is approach is a better way to proceed when theamount of field data available is limited e disadvantagesof using this method are that it relies heavily on expertexperience and can be time consuming if performedmanually However with the help of a computer and anexpert system the procedure can be performed automat-ically making it much more straightforward to use inpractice

Table 10 Maintenance time records for similar candidate tasks (min)

NoFault confirmationcheckout (X1X4) Fault isolation (X2)

Repairinterchange(X3)

P11 P12 P13 P21 P32 P33

1 47 57 30 189 647 5182 43 73 48 202 661 6553 53 62 191 528 5564 58 239 6485 63 156 613

Mathematical Problems in Engineering 17

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

In our future work we intend to establish a maintain-ability evaluation index that is better suited to similarityanalysis than the currently available indexes In addition weaim to develop a method for combining a maintenance tasksimilarity analysis with a maintenance time data analysis toobtain a more comprehensive result

Data Availability

e related data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

References

[1] Department of Defense USA Designing and DevelopingMaintainable Products and Systems Washington DC USA1997

[2] B S Blanchard D Verma and E L Peterson Maintain-ability A Key to Effective Serviceability and MaintenanceManagement Wiley New York NY USA 1995

[3] B P Cai Y Liu and M Xie ldquoA dynamic-Bayesian-network-based fault diagnosis methodology considering transient andintermittent faultsrdquo IEEE Transactions on Automation Scienceand Engineering vol 14 no 1 pp 276ndash285 2016

[4] B P Cai X Y Shao Y H Liu et al ldquoRemaining useful lifeestimation of structure systems under the influence of mul-tiple causes subsea pipelines as a case studyrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 7 pp 5737ndash57472019

[5] B P Cai Y B Zhao H L Liu and M Xie ldquoA data-drivenfault diagnosis methodology in three-phase inverters forPMSM drive systemsrdquo IEEE Transactions on Power Elec-tronics vol 32 no 7 pp 5590ndash5600 2016

[6] X Yuan B Cai Y Ma et al ldquoReliability evaluation meth-odology of complex systems based on dynamic object-ori-ented Bayesian networksrdquo IEEE Access vol 6pp 11289ndash11300 2018

[7] J Guo C Wang J Cabrera and E A Elsayed ldquoImprovedinverse Gaussian process and bootstrap degradation andreliability metricsrdquo Reliability Engineering amp System Safetyvol 178 pp 269ndash277 2018

[8] D-Q Li X-S Tang and K-K Phoon ldquoBootstrap method forcharacterizing the effect of uncertainty in shear strengthparameters on slope reliabilityrdquo Reliability Engineering ampSystem Safety vol 140 pp 99ndash106 2015

[9] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018

[10] Q Miao L Liu Y Feng and M Pecht ldquoComplex systemmaintainability verification with limited samplesrdquo Micro-electronics Reliability vol 51 no 2 pp 294ndash299 2011

[11] H Yang S G Hassan L Wang and D Li ldquoFault diagnosismethod for water quality monitoring and control equipmentin aquaculture based on multiple SVM combined with D-Sevidence theoryrdquo Computers and Electronics in Agriculturevol 141 pp 96ndash108 2017

[12] D R Insua F Ruggeri R Soyer and S Wilson ldquoAdvances inBayesian decision making in reliabilityrdquo European Journal ofOperational Research In press 2019

[13] Z M Wang and H Zhou ldquoA general method of prior elic-itation in bayesian reliability analysisrdquo in Proceedings of the8th International Conference on Reliability Maintainabilityand Safety Chengdu China July 2009

[14] Z Zhang ldquoResearch on method which confirmed smallsample maintainability verification test plan based on fittingerror simulationrdquo MS thesis National University of DefenseTechnology Changsha China 2009

[15] H M Zhang ldquoe key partsrsquo maintainability evaluation ofpanzer equipment based on similar information fusionrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[16] Z Zhu ldquoResearch on system maintainability integrationmethod based on Bayesian theory for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2008

[17] X P Huang ldquoMaintainability test data processing anddemonstration methods based on small sample for armouredequipmentrdquo MS thesis National University of DefenseTechnology Changsha China 2008

[18] H L Liu ldquoStudy on maintainability demonstration andevaluation for xx-canon based on small sample theoryrdquo MSthesis National University of Defense Technology ChangshaChina 2010

[19] J Wang ldquoe research on maintainability demonstrationmethod based on small sample for panzer equipmentrdquo MSthesis National University of Defense Technology ChangshaChina 2007

[20] Z Z Chen H Z Huang Y Liu L P He and Z L WangldquoMaintainability verification for airplanes with small samplesbased on similarity degreerdquo in Proceedings of the InternationalConference on Quality Reliability Risk Maintenance andSafety Engineering Xirsquoan China June 2011

[21] D I Agu ldquoAutomated analysis of product disassembly todetermine environmental impactrdquo MS thesis e Universityof Texas at Austin Austin TX USA 2009

[22] H Srinivasan and R Gadh ldquoEfficient geometric disassemblyof multiple components from an assembly using wavepropagationrdquo Journal of Mechanical Design vol 122 no 2pp 179ndash184 2000

[23] L S Homem de Mello and A C Sanderson ldquoANDOR graphrepresentation of assembly plansrdquo IEEE Transactions onRobotics and Automation vol 6 no 2 pp 188ndash199 1990

[24] S A S Airbus Aircraft Maintenance Manual OccitanieFrance 2016

[25] L Chen and J Cai ldquoUsing vector projection method toevaluate maintainability of mechanical system in design re-viewrdquo Reliability Engineering amp System Safety vol 81 no 2pp 147ndash154 2003

[26] X Jian S Cai and Q Chen ldquoA study on the evaluation ofproduct maintainability based on the life cycle theoryrdquoJournal of Cleaner Production vol 141 pp 481ndash491 2017

[27] P M D Leon V G P Dıaz L B Martınez andA C Marquez ldquoA practical method for the maintainabilityassessment in industrial devices using indicators and specificattributesrdquo Reliability Engineering amp System Safety vol 100pp 84ndash92 2012

[28] W Tarelko ldquoControl model of maintainability levelrdquoReliabilityEngineering amp System Safety vol 47 no 2 pp 85ndash91 1995

[29] Z Tjiparuro and G ompson ldquoReview of maintainabilitydesign principles and their application to conceptual designrdquo

18 Mathematical Problems in Engineering

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19

Proceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 218 no 2pp 103ndash113 2004

[30] M F Wani and O P Gandhi ldquoDevelopment of maintain-ability index for mechanical systemsrdquo Reliability Engineeringamp System Safety vol 65 no 3 pp 259ndash270 1999

[31] XWu V Kumar J Ross Quinlan et al ldquoTop 10 algorithms indata miningrdquo Knowledge and Information Systems vol 14no 1 pp 1ndash37 2008

[32] X Xu J Liang C Chen and Z Hou ldquoWeighted similarityand distance metric learning for unconstrained face verifi-cation with 3D frontalisationrdquo IET Image Processing vol 13no 2 pp 399ndash408 2019

[33] J Gou J Song W Ou S Zeng Y Yuan and L DuldquoRepresentation-based classification methods with enhancedlinear reconstruction measures for face recognitionrdquo Com-puters amp Electrical Engineering vol 79 Article ID 1064512019

[34] J Gou Y Xu D Zhang Q Mao L Du and Y Zhan ldquoTwo-phase linear reconstruction measure-based classification forface recognitionrdquo Information Sciences vol 433-434pp 17ndash36 2018

[35] J Gou L Wang B Hou J Lv Y Yuan and Q Mao ldquoTwo-phase probabilistic collaborative representation-based clas-sificationrdquo Expert Systems with Applications vol 133 pp 9ndash20 2019

[36] M Mannino J Fredrickson F Banaei-Kashani I Linck andR A Raghda ldquoDevelopment and evaluation of a similaritymeasure for medical event sequencesrdquo ACM Transactions onManagement Information Systems vol 8 no 2-3 pp 8ndash342017

[37] Z Zhang H H Huang and K Kawagoe ldquoSimilarity search ofhuman behavior processes using extended linked data se-mantic distancerdquo in Proceedings of the 25th InternationalWorkshop on Database and Expert Systems ApplicationsMunich Germany September 2014

[38] T Neumuth F Loebe and P Jannin ldquoSimilarity metrics forsurgical process modelsrdquo Artificial Intelligence in Medicinevol 54 no 1 pp 15ndash27 2012

[39] H Obweger M Suntinger J Schiefer and G Raidl ldquoSimi-larity searching in sequences of complex eventsrdquo in Pro-ceedings of the International Conference on ResearchChallenges in Information Science (RCIS) Nice France May2010

[40] L Wang ldquoResearch for sustainability assessment method andapplication at design phase of auto productsrdquo MS thesisShanghai Jiaotong University Shanghai China 2015

[41] B P Carlin and T A Louis Bayesian Methods for DataAnalysis Chapman amp Hall New York NY USA 2009

[42] B Guo P Jiang and Y Y Xing ldquoA censored sequentialposterior odd test (SPOT) method for verification of the meantime to repairrdquo IEEE Transactions on Reliability vol 57 no 2pp 243ndash247 2008

[43] S A S Airbus Trouble Shooting Manual Occitanie France2006

[44] H Balaban Maintainability Prediction and DemonstrationTechniques Volume II Maintainability DemonstrationARINC Research Corporation Annapolis USA 1970

Mathematical Problems in Engineering 19