Research ArticleResearch on the Reliability Allocation Method for a ProductionSystem Based on Availability

Yankun Wang 1 Binbin Xu2 Tianqi Ma1 and Ziyue Wang 1

1School of Mechanical and Aerospace Engineering Jilin University Changchun 130025 China2Sino-German College of Intelligent Manufacturing Shenzhen Technology University Shenzhen 518118 China

Correspondence should be addressed to Ziyue Wang ziyue18mailsjlueducn

Received 29 September 2019 Revised 13 December 2019 Accepted 3 January 2020 Published 22 January 2020

Academic Editor Elena Zaitseva

Copyright copy 2020 YankunWang et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper involves a production system which is composed of units (workstations) and buffers e buffer is used to storesemifinished and finished products in the production process to reduce the impacts of bad equipment in the production systemon the entire system performance Considering the characteristics of the large number of components and the state of the buffersin the production system this paper considers the influences of buffer states on upstream and downstream units When using theavailability as the allocation index and combining it with Markov theory the production unit (workstation) and upstream anddownstream buffers are regarded as an equivalent unit (workstation) with multiple output states We establish the relationshipbetween the availability of each equivalent unit (workstation) and the production system availability and determine a scalingfactor for the availability of the equivalent unit to account for the system availability e expected availability goal of theproduction system is allocated to each equivalent unit (workstation) by the scaling factor then the availability of each equivalentunit (workstation) is assigned to each unit Finally the Plant Simulation software is used to simulate and analyze the productionsystem to verify the correctness of the allocationmethod and realize the reliability allocation from a complex production system toa unit

1 Introduction

A production system is a complex repairable system withbuffers e buffer is used to store semifinished and finishedproducts to reduce the influences of bad equipment in theproduction system on the entire system performance andensure continuous production which is related to the overallefficiency and success of the enterprise Reliability allocationrefers to assigning the reliability index specified in the designspecification to each product component in the productdesign stage [1] For the production system the reliabilityallocation is mainly oriented to the initial design stageAccording to the purpose of the entire system the relevantreliability index is set and allocated to each relevant unitReliability allocation is one of the most critical tasks insystem reliability design Whether the allocation result isreasonable directly affects the realization of the reliabilityindex and the design cycle and life cycle cost of the product

and the result directly influences a productrsquos quality and therobustness of the system e reliability allocation of aproduction system must consider the influences of thebuffers the polymorphism of the state of buffers and thelarge number of components of the production system so itsreliability allocation algorithm is complex and difficult to use[2ndash5]

Various reliability allocation methods have been widelydiscussed and developed over the last several decades Inresponse to the reliability allocation problem of productionsystems Jilin University CNC Equipment ReliabilityTechnology Innovation Research Team conducted researchon a car crankshaft production line and an engine blockproduction line [6 7] established a fuzzy comprehensiveevaluation model for production line reliability allocation[8] and proposed a seminumerical simulation method forthe availability evaluation of discrete production lines [9]Based on the availability of equipment the availability of

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 6159462 9 pageshttpsdoiorg10115520206159462

production lines and the cost of the three indicators theyproposed a production line reliability assessment method[10] Availability is an important indicator to evaluate andanalyze the reliability of production lines e researchmethods for production line reliability assessment includethe Petri net model [11] the fuzzy Bayesian method [12] theMarkov model [13 14] and the semi-Markov model[15 16] Heungseob and Pansoo [14] developed a model fornonrepairable systems with heterogeneous components withphase-type time-to-failure distributions by using a struc-tured continuous-time Markov chain (CTMC) Loganathanet al [15] used the semi-Markov model to evaluate theavailability of a manufacturing system that consideredvariable failure rates or maintenance rates

At present there are many studies on reliability allo-cation methods but few of them are applicable to pro-duction systems e reliability allocation of a productionsystem usually allocates the system equipment levels anddoes not consider the buffer area e buffer area can im-prove the reliability index of the production system Hu andMeerkov [17] have shown that the equipment obeys theBernoulli reliability model and proposed an analysis methodto select the lean buffer for serial production lines Demiret al [18] used the decomposition method to obtain thethroughput evaluation model for asynchronous productionlines and used an adaptive tabu search algorithm to de-termine the buffer capacity of the production line

A production system is a multistate system which oftenconsists of multiple processing units and there may beparallel units In addition there is a buffer between pro-cessing units ese factors make the working state of theproduction system appear polymorphic For a multistatesystem the Markov model [19] a Bayesian network[20 21] a fuzzy mathematical method [8 22] and othermethods are currently used e Markov model is used toconstruct the reliability model by dynamically describingthe system state and the transition of the state from twoaspects of the system is method can comprehensivelyanalyze the reliability of the system However when thesystem levels increase the number of system states sharplyincreases and the analysis results exponentially increasee algorithm is complex and difficult to solve eBayesian network can describe the polymorphism of thesystem and the uncertainty of the logical relationship be-tween events perform two-way reasoning and find thesource of system failure in the reliability analysis eseadvantages make the Bayesian network to be widely used insystem reliability analysis However for complex multistatesystems when the number of variables is large or the rangeof the variables is large the scale and complexity of the localconditional probability table will increase as an exponentialfunction which makes it difficult to learn conditionalprobability parameters and affects the practicality of theentire network model e fuzzy mathematical method issuitable for reliability allocation under the condition ofuncertain parameters ere are many uncertain fuzzyfactors to be faced in the allocation process It is precise touse the fuzzy mathematical method to address these typesof inaccurate parameters which will have better results

However the research on fuzzy reliability allocation ismostly limited to the system reliability allocation problemwith a simple structure and many factors must be con-sidered in reliability allocation e quantitative expressionof each factor generally requires the participation of ex-perts and the results given by experts are often highlysubjective which increases the fuzziness of the reliabilityallocation

In this paper buffer areas are introduced into the reli-ability allocation of the production system By analyzing thecalculation process of the availability of the production sys-tem the quantitative relationship between the availability ofequivalent units (workstations) including the buffers andunits and the availability of the system is calculated econcept of the scale factor is proposed A method to allocatethe system availability according to the availability factor ofeach equivalent unit to the system availability ratio is pro-posed and the expected availability target of the productionsystem is allocated to each constituent unit Finally theproduction system is simulated and analyzed by the PlantSimulation software to verify the correctness of the allocationmethod e method realizes the reliability allocation fromcomplex production systems to units which is easy to im-plement in engineering and has strong feasibility It can alsosolve the reliability allocation problem of the multistageproduction system and provide a basis for the designtransformation and upgrade of the production system

According to the composition of the repairable systemthis paper divides the system into two categories an un-buffered system of rigid connections between every twoconstituent unit and a nonrigid connection system thatconsiders a buffer between constituent units e remainderof this paper is organized as follows Section 2 establishes themathematical relationship between the unbuffered systemavailability and the unit availability of the series and series-parallel systems respectively Section 3 first establishes themathematical relationship between the buffer systemavailability and the unit availability of the series and series-parallel hybrid systems en a method of buffer inventorycapacity allocation is introduced Section 4 establishes theavailability allocation method for the unbuffered system andbuffer system Section 5 introduces the three stages of thePlant Simulation software in the simulation process InSection 6 a case is presented and simulated by the softwareto illustrate the rationality of the proposed allocationmethod Section 7 concludes this paper

2 Establishing a MathematicalRelationship between Unbuffered SystemAvailability and Unit Availability

21 Establishing a Mathematical Relationship between Un-buffered Series System Availability and Unit Availabilitye system S consists of i(i 1 2 n) series units eachunit is recorded as mi e reliability block diagram is shownin Figure 1

Assuming that the failure rate and maintenance rate ofmi are λi and μi and the lifetime and maintenance time obey

2 Mathematical Problems in Engineering

the exponential distributions with parameters λi and μi thesteady-state availability of the series unit is

ai μi

λi + μi

(1)

e steady-state availability of the system is

A 1

1 + 1113936ni1λiμi

(2)

22 Establishing a Mathematical Relationship between Un-buffered Parallel-Series System Availability and UnitAvailability e parallel-series system consists of parallelunits that are put in series For the parallel-series systemwithrigid connections between units in this case the productionsystem must be simplified to a standard serial system andthe parallel units are converted into an equivalent unitwhich facilitates the next analysis and improvement

In this paper two identical units miL and miR arearranged in parallel on the layout as an example and thesubsystem containing parallel units is replaced by theequivalent unit m

pari e reliability block diagram is shown

in Figure 2Assuming that the failure rate and repair rate of miL and

miR are λi and μi respectively the steady-state availability ofthe equivalent unit m

pari is

apari

μ2i + 2λiμi

μ2i + 2λiμi + 2λ2i (3)

Expansion of equation (2) yields

A 1

1 + λ1μ1( 1113857( 1113857 + 1 + λ2μ2( 1113857( 1113857 + middot middot middot + 1 + λnμn( 1113857( 1113857 minus (n minus 1)

(4)

where 1 + (λiμi) (1ai) the relationship between systemavailability and unit availability can be expressed as

A 1

1a1( 1113857 + 1a2( 1113857 + middot middot middot + 1an( 1113857 minus (n minus 1) (5)

We further formulate equation (5) as1a1

minus 11113888 1113889 +1a2

minus 11113888 1113889 +1a3

minus 11113888 1113889 + middot middot middot +1an

minus 11113888 1113889 1A

minus 11113874 1113875

(6)

Dividing both sides of equation (6) by ((1A) minus 1) yieldsthe following formula

c1 + c2 + c3 + middot middot middot + cn 1 (7)

where ci ((1ai) minus 1)((1A) minus 1) and equation (7) ex-presses the relationship between unit availability and systemavailability

3 Establishing a MathematicalRelationship between Buffer SystemAvailability and Unit Availability

31 Establishing a Mathematical Relationship between BufferSeries System Availability and Unit Availability e seriessystem connects units mi(i 1 2 n) in series andtransfers the workpiece to the next-level unit mi+1 throughthe buffer Bi e reliability block diagram is shown inFigure 3

Taking mi as an example productivity refers to thenumber of products that mi can produce per unit timestarvation refers to the forced waiting caused by the lack ofworkpiece to provide to mi after mi processes which releasesa workpiece and the capacity kiminus 1 of the upstream buffer Biminus 1is zero (empty) and blocking refers to the forced waitingcaused by the capacity ki of the downstream buffer Bi being n

(full) which makes the workpiece unable to put into thebuffer

We assume that the lifetime and maintenance time of mi

obey the exponential distributions with parameters λi and μithe first-level unit is not starved and the last buffer is notblocked When the buffer capacity ki of the buffer Bi is n

(full) the previous unit mi is down and the next-level unitmi+1 continues working When the buffer capacity ki of thebuffer Bi is zero (empty) the next-level unit mi+1 is downIncreasing the buffer can effectively improve the unitavailability When the unit fails the spare parts in the buffercan maintain production for a period of time and strive formaintenance time References [5 23] have analyzed anddeduced the state of the buffer in detail

Taking Bi as an example inventory-free refers to thecapacity of Bi being ki 0 inventory refers to the capacity ofBi being ki＞0 vacancy-free refers to the capacity of Bi beingki n and vacancy refers to the capacity of Bi being ki＜nAssume that the probability of inventory-free is P0i theprobability of inventory is P0i the probability of vacancy-free is Pki

and the probability of vacancy is Pki then the

calculation equation for the buffer availability is as follows

ABi P0(iminus 1)

Pki

(8)

where

P0i ρi middot 1 minus ρki

i1113872 1113873

1 minus ρki+1i1113872 1113873

(9)

Pki

1 minus ρki

i1113872 1113873

1 minus ρki+1i1113872 1113873

(10)

ρi ωi

ωi+1 (11)

Each unit and its upstream and downstream buffers areequivalent to the unit mi

prime with multiple output states Weestablish the state transition probability equation of the i-thequivalent unit and obtain the availability of mi

prime [23]

m1 m2 mi mi+1 mn

Figure 1 Reliability block diagram of an unbuffered series system

Mathematical Problems in Engineering 3

Ai μiABi

μi + λiABi( 1113857

μiP0(iminus 1)P

ki

μi + λiP0(iminus 1)P

ki1113872 1113873

μi ρiminus 1 middot 1 minus ρk(iminus 1)

iminus 11113872 1113873 1 minus ρk(iminus 1)+1iminus 11113872 11138731113872 1113873

μi + λi ρiminus 1 middot 1 minus ρk(iminus 1)iminus 11113872 1113873 1 minus ρk(iminus 1)+1

iminus 11113872 11138731113872 1113873 1 minus ρkii( 1113857 1 minus ρki+1

i( 1113857( 11138571113960 1113961 (12)

According to equation (12) the availability of theequivalent unit is directly related to the failure rate main-tenance rate productivity and buffer inventory capacity ofthe unit If the availability index of each unit is determinedequation (12) can be used to guide the selection of pro-duction equipment and determine the buffer inventorycapacity

32 Establishing a Mathematical Relationship between BufferParallel-Series Hybrid System Availability and UnitAvailability e buffer parallel-series hybrid system con-sists of two or more units arranged in parallel on the layoutto form a workstation Mi(i 1 2 n) and workstationsMi and Mi+1 are connected in series through the bufferBi(i 1 2 n minus 1) In this paper the parallel distributionof two units miL and miR in the layout is taken as an ex-ample and the reliability block diagram is shown in Figure 4

When each workstation Mi in the parallel-series hybridsystem is composed of two units in parallel there are threestates of the workstation (1) both units are normal and Mi

works normally (2) one unit fails and Mi reduces pro-duction and (3) both units fail and Mi fails Combined withthe state of the upstream and downstream buffers Mi hasnine working states

(1) e probability of Mi working normally isPaiprime P0(iminus 1)

PaiP

ki

(2) Mi is trouble-free and the input shortage causesdiscontinuation e probability is P0(iminus 1)Pai

Pki

(3) Mi is trouble-free and the output blocking causesdiscontinuation e probability is P0(iminus 1)

PaiPki

(4) Mi is trouble-free the input shortage and output

blocking cause discontinuation e probability isP0(iminus 1)Pai

Pki

(5) e probability of Mi reducing production isPciprime P0(iminus 1)

PciP

ki

(6) Mi reduces production and the input shortagecauses discontinuation e probability isP0(iminus 1)Pci

Pki

(7) Mi reduces production and the output blockingcauses discontinuation e probability isP0(iminus 1)

PciPki

(8) Mi reduces production the input shortage andoutput blocking cause discontinuation e proba-bility is P0(iminus 1)Pci

Pki

(9) e probability of Mi stopping production due tomalfunction is Pbi

By adding the above probabilities

PaiP0(iminus 1)

Pki

+ P0(iminus 1)Pki+ P0(iminus 1)

Pki+ P0(iminus 1)Pki

1113874 1113875

+ PciP0(iminus 1)

Pki

+ P0(iminus 1)Pki+ P0(iminus 1)

Pki+ P0(iminus 1)Pki

1113874 1113875

+ Pbi 1

(13)

We can prove that the four probabilities in brackets ofequation (13) add up to 1 so that

Pai+ Pci

+ Pbi 1 (14)

e state transition probability equation is

_Paiprime minus 2λiPai

prime + μiPciprime

_Pciprime 2λiPai

prime minus λi + μi( 1113857Pciprime + μiPbi

_Pbi λiPciprime minus μiPbi

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(15)

We solve (15) and obtain

Pciprime

2λiμiABi

μ2i + 2λ2i ABi+ 2λiμi

Paiprime

μ2i ABi

μ2i + 2λ2i ABi+ 2λiμi

Pbi

2λ2i ABi

μ2i + 2λ2i ABi+ 2λiμi

(16)

Considering the influence of the buffer availability on thesystem availability the buffer available state is that its up-stream buffer is not starved and its downstream buffer is notblocked Combining the two states with the workstation asan equivalent workstation Mi

prime for analysis we obtain thesteady-state availability of Mi

prime

m1L m2L

m2Rm1R

mnL

mnR

m1par m2

par mnpar

Figure 2 Reliability block diagram of an unbuffered parallel-series system

mim1 B1 B2m2 Bi mi+1 mn

Figure 3 Reliability block diagram of a buffer serial system

4 Mathematical Problems in Engineering

Aiprime

2λiμi + μ2i( 1113857ABi

μ2i + 2λ2i ABi+ 2λiμi

(17)

e steady-state unavailability of the system is equal tothe product of the unavailability of each equivalent work-station ie

A 1113945n

i11 minus Aiprime( 1113857 (18)

us the steady-state availability of the system is

A 1 minus A 1 minus 1113945n

i11 minus Aiprime( 1113857 (19)

Formula (19) is expanded and transformed into

1 minus A1prime( 1113857 1 minus A2prime( 1113857 1 minus A3prime( 1113857 middot middot middot 1 minus Anminus 2prime( 1113857 1 minus Anminus 1prime( 1113857 1 minus Anprime( 1113857

1 minus A

(20)

Simultaneously taking the logarithm of both sides yields

ln 1 minus A1prime( 1113857 + ln 1 minus A2prime( 1113857 + ln 1 minus A3prime( 1113857 + middot middot middot

+ ln 1 minus Anminus 2prime( 1113857 + ln 1 minus Anminus 1prime( 1113857 + ln 1 minus Anprime( 1113857 ln(1 minus A)

(21)

Dividing both sides of equation (21) by ln(1 minus A) yieldsthe following formula

c1prime + c2prime + c3prime + middot middot middot + cnminus 2prime + cnminus 1prime + cnprime 1 (22)

where ciprime (ln(1 minus Ai

prime) ln(1 minus A))Increasing the buffer improves the workstation avail-

ability In this case the equivalent workstation availabilityafter combining the upstream and downstream buffers willbe higher than the availability of the unit itself us theseries-parallel hybrid system can be simplified to a seriessystem that consists of equivalent workstations that containbuffers

33 Buffer Inventory Capacity Allocation MethodFigure 5 is a structural diagram of a serial two-level pro-duction system composed of workstations Mi and Mi+1 andthe buffer Bi

Assume that the buffer is completely reliable during theplanning period T e failure rate and maintenance rate areλB and μB respectively Bi is separately treated as a unit andthe steady-state availability of Bi is

AB μB

μB + λB

(23)

Considering the occasional malfunction of Bi in actualproduction the correction factor ε is introduced in thispaper and ε (1AB) During the planning period time Tto ensure the continuous production of the system theminimum buffer inventory capacity kimin is [24]

kimin ε middot max ki1 ki21113966 1113967

ki1 ku

1μB

1 minus eminus μBT

1113872 1113873 minus Teminus μBT

1113890 1113891 + 1

ki2 2ku

1μB

1 minus eminus μBT

1113872 1113873 minus Teminus μBT

1113890 1113891

minus kp minus ku1113872 11138731λB

1 minus eminus λBT

1113872 1113873 minus Teminus λBT

1113890 1113891 + 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

4 Availability Allocation Method

41 Proportional Allocation Method e proportional al-location method is based on the failure rate of each unit inthe original system and the failure rate is proportionallydistributed to each unit of the new system according to thereliability prediction of the new system e mathematicalexpression is [25]

λin λio

λso

λsn (25)

where λsn is the failure rate of the new system λin is thefailure rate allocated to the unit i in the new system λso is thefailure rate of the old system and λio is the failure rate of theunit i in the old system

Similarly this paper uses the system availability as theallocation index which is proportionally allocated to eachunit (workstation) according to the scale factor to determinethe unit (workstation) availability index value

Workstation M1 Workstation M2 Workstation Mn

m1L m2L

m2Rm1R

mnL

mnR

B1 B2 Bnndash1

Figure 4 Reliability block diagram of a buffer parallel-series hybrid system

Mi Bi Mi+1

Figure 5 Structural diagram of a serial two-level productionsystem

Mathematical Problems in Engineering 5

42 Availability AllocationMethod for an Unbuffered SystemAssume that the production system availability is A and theunit availability is ai A and ai satisfy formula (6) and theinfluence factor is ci If the system target availability is Am

and the availability assigned to each unit is Aim then1

Aim

minus 11113888 1113889 ci

1Am

minus 11113888 1113889 (26)

e unit availability Aim after sorting is

Aim 1

ci 1Am( 1113857 minus 1( 1113857 + 11113858 1113859 (27)

43 Availability Allocation Method for a Buffer System Ifki＜n and ki+1＜n the system target availability is Am

prime theavailability assigned to each workstation is Aci

prime and the scalefactor is ci

prime then

ciprime

ln 1 minus Aciprime1113872 1113873

ln 1 minus Amprime( 1113857

(28)

Aciprime 1 minus e

ciprime ln 1minus Am

prime( ) (29)

Aciprime is assigned to each unit by an equal reliability allo-

cation method of the parallel system Assume that theavailability assigned to each unit is Ami

prime then the allocationmethod formula is

Amiprime

Aciprime

1113969

1 minus Aciprime

1113969

eciprime ln 1minus Am

prime( )

1113969

(30)

Amiprime 1 minus

Amiprime

1113969

1 minus

eciprime ln 1minus Am

prime( )

1113969

(31)

5 Simulation Analysis for theProduction System

e simulation establishes a model for the system and usesthe model instead of the real system to perform variousexperiments to study the performance In this paper thePlant Simulation software is used as the simulation platformAfter obtaining the availability allocation results the systemis simulated and analyzed to verify the correctness of theresults

e simulation of the production system corresponds tothree stages

(1) Establishing a simulation model using the ldquologis-ticsrdquo the production equipment and buffer can beobtained and the basic system framework model canbe established For each unit the processing capacityand availability are input for each buffer the buffercapacity type and availability are set

(2) Performing a simulation experiment using the eventcontrol unit the simulation time is set to 30 days 90days 180 days 360 days and 720 days is time isset to absolute time which is convenient for ob-serving and recording simulation events e

corresponding time jump speed is 10000 times thereal-time value and the remaining options are de-fault values e parameters of relevant productionunits are set and the software to simulate the entireline is run

(3) Analyzing the simulation data after the simulationtest is completed the intrinsic availability totalthroughput hourly throughput and dailythroughput related data are collated and comparedwith the given expected availability According to thecomparison result it is determined whether the unitreliability index can achieve the expected goal andwhether the reliability allocation method is correct

6 Examples

Assume that a buffer series-parallel hybrid productionsystem consists of a 4-level workstation Each level work-station consists of two identical processing units connectedin parallel e failure rate maintenance rate and pro-ductivity of each unit are λ1λ2λ3λ4 μ1μ2μ3μ4 and ω1ω2ω3ω4respectively which are listed in Table 1

Assume that the failure rate λBi and maintenance rate μBi

of each buffer are 0002 and 006 the planning period time Tis 10 minutes and kp and ku are 1 minute per pieceSubstituting parameters into equations (23) and (24) weobtain AB1 AB2 AB3 09677 and k1 k2 k3 5

It is also assumed that the first-level workstation is notstarved and the last-level workstation is not blocked LetP00 1 and P4 1 e buffer availability can be calculatedby formulas (8)ndash(11) e specific values are shown inTable 2

e steady-state availability of each equivalent work-station can be obtained by using equation (17) and issummarized in Table 3

By substituting the obtained steady-state availability into(19) the stable availability of the system isA 1 minus (1113937

4i1(1 minus Ai

prime)) 09925 If the expected availabilityis increased to 09999 the estimated availability assigned toeach workstation is calculated according to equations (28)and (29) and the specific values are shown in Table 4

e estimated availability of each workstation is redis-tributed to each unit according to formulas (30) and (31)and the expected unit availability is calculated and shown inTable 5

After obtaining the estimated availability allocation re-sults of each processing unit the Plant Simulation software isused to simulate the production system online to determinewhether the allocation results are correct and reasonableUsing the ldquologisticsrdquo element in the software elementtoolbox the production unit and buffer can be obtainedesimulation model of the production system is shown inFigure 6

We input the corresponding parameters for each unitand buffer and use the event control unit to set the simu-lation time to 30 days 90 days 180 days 360 days and 720days and the corresponding time jump speed is 10000 timesthe real-time value

6 Mathematical Problems in Engineering

Table 5 Expected availability of each unit

unit m1L m1R m2L m2R m3L m3R m4L m4R

Amiprime 06873 06873 06761 06761 06406 06406 07253 07253

Table 1 Failure rate maintenance rate and productivity of each unit

Parameter m1L m1R m2L m2R m3L m3R m4L m4R

λi 00032 00032 00027 00027 00026 00026 0003 0003μi 002 002 003 003 002 002 001 001ωi 5 5 4 4 35 35 3 3

Table 2 Buffer availability

Bi B1 B2 B3 B4

ABi07289 07045 06745 08146

Table 3 Steady-state availability of each equivalent workstation

Miprime M1prime M2prime M3prime M4prime

Aiprime 07089 06978 06625 07462

Table 4 Estimated availability of each workstation

Mi M1 M2 M3 M4

Aciprime 09022 08951 08708 09245

Event controller

m1L

B1 B2 B3DrainSource

m2L m3L m4L

m1R m2R m3R m4R

Figure 6 Simulation model of the production system

Entity

Event controller

m1L

ErEntityB1

ErEntityB2

ErEntityB3

DrainSource

Entitym2L

Entitym3L

Entitym4L

Entitym1R

Entitym2R

Entitym3R

Entitym4R

Figure 7 Operation simulation model of the production system

Mathematical Problems in Engineering 7

We set the parameters of the relevant production unitsand run the software to simulate the whole line e op-eration simulation model is shown in Figure 7

en we complete the simulation test sort out therelevant data and obtain the reliability and performanceindex values e specific values are shown in Table 6

e simulation results show that the reliability allocationresults obtained in this paper satisfy the requirements of thegiven expected availability erefore the reliability indexallocated to each unit can achieve the desired goal and thereliability allocation method is correct

7 Conclusions

(1) Aiming at the reliability allocation problem of theproduction system this paper proposes a reliabilityallocation method that is easy to implement in en-gineering By constructing the proportional rela-tionship between unit (workstation) availability andsystem availability the system availability can beproportionally allocated according to the scale fac-tors is allocation method is simple and easy toimplement and it can solve the reliability allocationproblem of multistage production systems

(2) e availability allocation results can help guide theselection of production units determine the numberof spare parts in the buffer comprehensively con-sider factors such as the failure rate maintenancerate inventory capacity and productivity and ra-tionally select the reliability index of the processingunit

(3) Using the Plant Simulation software to simulate andanalyze the production system one can verify thecorrectness of the allocation method and realize thereliability allocation from the complex productionsystem to the unit

(4) e reliability allocation process without consideringthe influence of the buffer can be extended to generalseries and series-parallel hybrid repairable systemssuch as CNCmachine tools and serve as the basis forcomponent selection or reliability improvement inthe product design stage

Notations

mi Unit of a series system (i 1 2 n)

λi Failure rate of a unitμi Maintenance rate of a unitωi Productivity of a unit

ρi Productivity ratio of a two-level unitai Steady-state availability of mi

A Steady-state availability of a systemmiL andmiR

Units of a hybrid system (i 1 2 n)

mpari Equivalent unit combining miL and miR

apari Steady-state availability of m

pari

ci Scale factor in an unbuffered hybrid systemBi Buffers of a systemki Inventory capacity of Bi(ki 1 2 n)

kimin Minimum inventory capacity of Bi

λB Failure rate of a bufferμB Maintenance rate of a bufferP0i Probability of inventory-freeP0i Probability of inventoryPki

Probability of vacancy-freeP

ki Probability of vacancy

ABi Buffer availability

miprime Equivalent unit combining mi Bi and mi+1 in

the buffer series systemAi Steady-state availability of mi

primeMi Workstation of a buffer hybrid system

(i 1 2 n)

Pai Trouble-free probability of Mi

Pbi Discontinuation probability of Mi

Pci Production-reduction probability of Mi

Miprime Equivalent workstation combining Mi Bi and

Mi+1 in the buffer hybrid systemAiprime Steady-state availability of Mi

primeciprime Scale factor in the buffer hybrid system

kp and ku Output per unit time of M1 and M2Am Target availability of an unbuffered systemAim Assigned availability to each unit in the

unbuffered systemAmprime Target availability of a buffer system

Aciprime and

Aciprime

Assigned availability and unavailability to theequivalent workstation

Amiprime and

Amiprime

Assigned availability and unavailability to theunit in the buffer system

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Table 6 Simulation data of the production system

Time Intrinsic availability () Total throughput Hourly throughput Daily throughput30 days 9999 86386 120 288090 days 9999 259290 121 2881180 days 9999 518760 120 2882360 days 9999 1037520 120 2882720 days 9999 2075040 120 2882

8 Mathematical Problems in Engineering

Acknowledgments

e research in this paper was supported by the NationalScience and Technology Major Project of China (Grant no2014ZX04015031) Science and Technology DevelopmentPlan Project of Jilin Province (Grant no 20180520068JH)and Jilin Province Education Departmentrsquos irteenth Five-Year Plan Science and Technology Project (Grant noJJKH20180079KJ)

References

[1] Z J Yang Q B Hao C Fei et al ldquoA comprehensive fuzzyreliability allocation method of NC machine tools based oninterval analysisrdquo Journal of Beijing University of Technologyvol 37 no 3 pp 321ndash329 2011

[2] P-C Chang ldquoReliability estimation for a stochastic pro-duction system with finite buffer storage by a simulationapproachrdquo Annals of Operations Research vol 277 no 1pp 119ndash133 2019

[3] G Liberopoulos ldquoPerformance evaluation of a productionline operated under an echelon buffer policyrdquo IISE Trans-actions vol 50 no 3 pp 161ndash177 2018

[4] S Weiss J A Schwarz and R Stolletz ldquoe buffer allocationproblem in production lines formulations solution methodsand instancesrdquo IISE Transactions vol 51 no 5 pp 456ndash4852019

[5] J Duan A P Li X Nan et al ldquoMulti-state reliabilitymodeling and analysis of reconfigurable manufacturing sys-temsrdquo Journal of Mechanical Engineering vol 47 no 17pp 104ndash111 2011

[6] J L Li Y H Jia F Xu F Chen Z Yang and X Li ldquoAnimprovement scheme for the overall line effectiveness of aproduction line a case studyrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Singapore April 2018

[7] G F Li Y Li X G Zang et al ldquoDevelopment of a preventivemaintenance strategy for an automatic production line basedon group maintenance methodrdquo Applied Sciences vol 8no 10 p 1781 2018

[8] G F Li C Hou G F Liu Y Jia C Liu and J DongldquoReliability allocation method of production line based onfuzzy comprehensive evaluationrdquo in Proceedings of the In-ternational Conference on Materials EngineeringManufacturing Technology and Control (ICMEMTC)Taiyuan China April 2016

[9] Y H Jia Z J Yang G F Li et al ldquoAn efficient semi-analyticalsimulation for availability evaluation of discrete productionlines with unreliable machinesrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Paris France November 2016

[10] F Chen B Liu and B B Xu ldquoResearch on ReliabilityEvaluation of Engine Cylinder Flexible Production Line Basedon AHP Fuzzy Comprehensive Evaluation Methodrdquo ChinaMechanical Engineering Society Changzhou China August2015

[11] F Long P Zeiler and B Bertsche ldquoModelling the flexibility ofproduction systems in industry 40 for analysing their pro-ductivity and availability with high-level Petri netsrdquo IFAC-PapersOnLine vol 50 no 1 pp 5680ndash5687 2017

[12] L Gorkemli and S Kapan Ulusoy ldquoFuzzy bayesian reliabilityand availability analysis of production systemsrdquo Computers ampIndustrial Engineering vol 59 no 4 pp 690ndash696 2010

[13] S Rebello H Yu and L Ma ldquoAn integrated approach forsystem functional reliability assessment using dynamicbayesian network and hidden Markov modelrdquo ReliabilityEngineering amp System Safety vol 180 no 12 pp 124ndash1352018

[14] K Heungseob and K Pansoo ldquoReliability models for anonrepairable systemwith heterogeneous components havinga phase-type time-to-failure distributionrdquo Reliability Engi-neering and System Safety vol 159 no 3 pp 37ndash46 2017

[15] M K Loganathan G Kumar and O P Gandhi ldquoAvailabilityevaluation of manufacturing systems using semi-markovmodelrdquo International Journal of Computer IntegratedManufacturing vol 29 no 7 pp 720ndash735 2016

[16] X-Y Li H-Z Huang and Y-F Li ldquoReliability analysis ofphased mission system with non-exponential and partiallyrepairable componentsrdquo Reliability Engineering amp SystemSafety vol 175 no 7 pp 119ndash127 2018

[17] A B Hu and S M Meerkov ldquoLean buffering in serial pro-duction lines with Bernoulli machinesrdquo MathematicalProblems in Engineering vol 2006 Article ID 17105 24 pages2006

[18] L Demir S Tunalı and D T Eliiyi ldquoAn adaptive tabu searchapproach for buffer allocation problem in unreliable non-homogenous production linesrdquo Computers amp OperationsResearch vol 39 no 7 pp 1477ndash1486 2012

[19] R Li X Liu and N Huang ldquoAvailability allocation of net-worked systems using Markov model and heuristics algo-rithmrdquo Mathematical Problems in Engineering vol 2014Article ID 315385 9 pages 2014

[20] W X Qian X W Yin and L Y Xie ldquoSystem reliabilityallocation based on bayesian networkrdquo Applied Mathematicsamp Information Sciences vol 6 no 3 pp 681ndash687 2012

[21] M Xiao and Y P Zhang ldquoParameters learning of bayesiannetworks for multistate system with small samplerdquo ComputerScience vol 42 no 4 pp 253ndash257 2015

[22] V Sriramdas S K Chaturvedi and H Gargama ldquoFuzzyarithmetic based reliability allocation approach during earlydesign and developmentrdquo Expert Systems with Applicationsvol 41 no 7 pp 3444ndash3449 2014

[23] S G Shu ldquoe manufacturing system (CIMS) with buffersand a study of the system reliabilityrdquo Acta Automatica Sinicavol 18 no 1 pp 15ndash20 1992

[24] H R Zhou S H Wang L Zhang et al ldquoModeling of in-process inventory in continuous production based on reli-abilityrdquo Machine Tool amp Hydraulics vol 39 no 9 pp 111ndash113 2011

[25] M Y You and J C Zheng ldquoA synthetic approach to reliabilityallocation of electronic systems based on AGREE method andthe extended proportional combination methodrdquo ElectronicProduct Reliability and Environmental Testing vol 29 no 4pp 1ndash6 2011

Mathematical Problems in Engineering 9

the exponential distributions with parameters λi and μi thesteady-state availability of the series unit is

ai μi

λi + μi

(1)

e steady-state availability of the system is

A 1

1 + 1113936ni1λiμi

(2)

22 Establishing a Mathematical Relationship between Un-buffered Parallel-Series System Availability and UnitAvailability e parallel-series system consists of parallelunits that are put in series For the parallel-series systemwithrigid connections between units in this case the productionsystem must be simplified to a standard serial system andthe parallel units are converted into an equivalent unitwhich facilitates the next analysis and improvement

In this paper two identical units miL and miR arearranged in parallel on the layout as an example and thesubsystem containing parallel units is replaced by theequivalent unit m

pari e reliability block diagram is shown

in Figure 2Assuming that the failure rate and repair rate of miL and

miR are λi and μi respectively the steady-state availability ofthe equivalent unit m

pari is

apari

μ2i + 2λiμi

μ2i + 2λiμi + 2λ2i (3)

Expansion of equation (2) yields

A 1

1 + λ1μ1( 1113857( 1113857 + 1 + λ2μ2( 1113857( 1113857 + middot middot middot + 1 + λnμn( 1113857( 1113857 minus (n minus 1)

(4)

where 1 + (λiμi) (1ai) the relationship between systemavailability and unit availability can be expressed as

A 1

1a1( 1113857 + 1a2( 1113857 + middot middot middot + 1an( 1113857 minus (n minus 1) (5)

We further formulate equation (5) as1a1

minus 11113888 1113889 +1a2

minus 11113888 1113889 +1a3

minus 11113888 1113889 + middot middot middot +1an

minus 11113888 1113889 1A

minus 11113874 1113875

(6)

Dividing both sides of equation (6) by ((1A) minus 1) yieldsthe following formula

c1 + c2 + c3 + middot middot middot + cn 1 (7)

where ci ((1ai) minus 1)((1A) minus 1) and equation (7) ex-presses the relationship between unit availability and systemavailability

3 Establishing a MathematicalRelationship between Buffer SystemAvailability and Unit Availability

31 Establishing a Mathematical Relationship between BufferSeries System Availability and Unit Availability e seriessystem connects units mi(i 1 2 n) in series andtransfers the workpiece to the next-level unit mi+1 throughthe buffer Bi e reliability block diagram is shown inFigure 3

Taking mi as an example productivity refers to thenumber of products that mi can produce per unit timestarvation refers to the forced waiting caused by the lack ofworkpiece to provide to mi after mi processes which releasesa workpiece and the capacity kiminus 1 of the upstream buffer Biminus 1is zero (empty) and blocking refers to the forced waitingcaused by the capacity ki of the downstream buffer Bi being n

(full) which makes the workpiece unable to put into thebuffer

We assume that the lifetime and maintenance time of mi

obey the exponential distributions with parameters λi and μithe first-level unit is not starved and the last buffer is notblocked When the buffer capacity ki of the buffer Bi is n

(full) the previous unit mi is down and the next-level unitmi+1 continues working When the buffer capacity ki of thebuffer Bi is zero (empty) the next-level unit mi+1 is downIncreasing the buffer can effectively improve the unitavailability When the unit fails the spare parts in the buffercan maintain production for a period of time and strive formaintenance time References [5 23] have analyzed anddeduced the state of the buffer in detail

Taking Bi as an example inventory-free refers to thecapacity of Bi being ki 0 inventory refers to the capacity ofBi being ki＞0 vacancy-free refers to the capacity of Bi beingki n and vacancy refers to the capacity of Bi being ki＜nAssume that the probability of inventory-free is P0i theprobability of inventory is P0i the probability of vacancy-free is Pki

and the probability of vacancy is Pki then the

calculation equation for the buffer availability is as follows

ABi P0(iminus 1)

Pki

(8)

where

P0i ρi middot 1 minus ρki

i1113872 1113873

1 minus ρki+1i1113872 1113873

(9)

Pki

1 minus ρki

i1113872 1113873

1 minus ρki+1i1113872 1113873

(10)

ρi ωi

ωi+1 (11)

Each unit and its upstream and downstream buffers areequivalent to the unit mi

prime with multiple output states Weestablish the state transition probability equation of the i-thequivalent unit and obtain the availability of mi

prime [23]

m1 m2 mi mi+1 mn

Figure 1 Reliability block diagram of an unbuffered series system

Mathematical Problems in Engineering 3

Ai μiABi

μi + λiABi( 1113857

μiP0(iminus 1)P

ki

μi + λiP0(iminus 1)P

ki1113872 1113873

μi ρiminus 1 middot 1 minus ρk(iminus 1)

iminus 11113872 1113873 1 minus ρk(iminus 1)+1iminus 11113872 11138731113872 1113873

μi + λi ρiminus 1 middot 1 minus ρk(iminus 1)iminus 11113872 1113873 1 minus ρk(iminus 1)+1

iminus 11113872 11138731113872 1113873 1 minus ρkii( 1113857 1 minus ρki+1

i( 1113857( 11138571113960 1113961 (12)

According to equation (12) the availability of theequivalent unit is directly related to the failure rate main-tenance rate productivity and buffer inventory capacity ofthe unit If the availability index of each unit is determinedequation (12) can be used to guide the selection of pro-duction equipment and determine the buffer inventorycapacity

32 Establishing a Mathematical Relationship between BufferParallel-Series Hybrid System Availability and UnitAvailability e buffer parallel-series hybrid system con-sists of two or more units arranged in parallel on the layoutto form a workstation Mi(i 1 2 n) and workstationsMi and Mi+1 are connected in series through the bufferBi(i 1 2 n minus 1) In this paper the parallel distributionof two units miL and miR in the layout is taken as an ex-ample and the reliability block diagram is shown in Figure 4

When each workstation Mi in the parallel-series hybridsystem is composed of two units in parallel there are threestates of the workstation (1) both units are normal and Mi

works normally (2) one unit fails and Mi reduces pro-duction and (3) both units fail and Mi fails Combined withthe state of the upstream and downstream buffers Mi hasnine working states

(1) e probability of Mi working normally isPaiprime P0(iminus 1)

PaiP

ki

(2) Mi is trouble-free and the input shortage causesdiscontinuation e probability is P0(iminus 1)Pai

Pki

(3) Mi is trouble-free and the output blocking causesdiscontinuation e probability is P0(iminus 1)

PaiPki

(4) Mi is trouble-free the input shortage and output

blocking cause discontinuation e probability isP0(iminus 1)Pai

Pki

(5) e probability of Mi reducing production isPciprime P0(iminus 1)

PciP

ki

(6) Mi reduces production and the input shortagecauses discontinuation e probability isP0(iminus 1)Pci

Pki

(7) Mi reduces production and the output blockingcauses discontinuation e probability isP0(iminus 1)

PciPki

(8) Mi reduces production the input shortage andoutput blocking cause discontinuation e proba-bility is P0(iminus 1)Pci

Pki

(9) e probability of Mi stopping production due tomalfunction is Pbi

By adding the above probabilities

PaiP0(iminus 1)

Pki

+ P0(iminus 1)Pki+ P0(iminus 1)

Pki+ P0(iminus 1)Pki

1113874 1113875

+ PciP0(iminus 1)

Pki

+ P0(iminus 1)Pki+ P0(iminus 1)

Pki+ P0(iminus 1)Pki

1113874 1113875

+ Pbi 1

(13)

We can prove that the four probabilities in brackets ofequation (13) add up to 1 so that

Pai+ Pci

+ Pbi 1 (14)

e state transition probability equation is

_Paiprime minus 2λiPai

prime + μiPciprime

_Pciprime 2λiPai

prime minus λi + μi( 1113857Pciprime + μiPbi

_Pbi λiPciprime minus μiPbi

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(15)

We solve (15) and obtain

Pciprime

2λiμiABi

μ2i + 2λ2i ABi+ 2λiμi

Paiprime

μ2i ABi

μ2i + 2λ2i ABi+ 2λiμi

Pbi

2λ2i ABi

μ2i + 2λ2i ABi+ 2λiμi

(16)

Considering the influence of the buffer availability on thesystem availability the buffer available state is that its up-stream buffer is not starved and its downstream buffer is notblocked Combining the two states with the workstation asan equivalent workstation Mi

prime for analysis we obtain thesteady-state availability of Mi

prime

m1L m2L

m2Rm1R

mnL

mnR

m1par m2

par mnpar

Figure 2 Reliability block diagram of an unbuffered parallel-series system

mim1 B1 B2m2 Bi mi+1 mn

Figure 3 Reliability block diagram of a buffer serial system

4 Mathematical Problems in Engineering

Aiprime

2λiμi + μ2i( 1113857ABi

μ2i + 2λ2i ABi+ 2λiμi

(17)

e steady-state unavailability of the system is equal tothe product of the unavailability of each equivalent work-station ie

A 1113945n

i11 minus Aiprime( 1113857 (18)

us the steady-state availability of the system is

A 1 minus A 1 minus 1113945n

i11 minus Aiprime( 1113857 (19)

Formula (19) is expanded and transformed into

1 minus A1prime( 1113857 1 minus A2prime( 1113857 1 minus A3prime( 1113857 middot middot middot 1 minus Anminus 2prime( 1113857 1 minus Anminus 1prime( 1113857 1 minus Anprime( 1113857

1 minus A

(20)

Simultaneously taking the logarithm of both sides yields

ln 1 minus A1prime( 1113857 + ln 1 minus A2prime( 1113857 + ln 1 minus A3prime( 1113857 + middot middot middot

+ ln 1 minus Anminus 2prime( 1113857 + ln 1 minus Anminus 1prime( 1113857 + ln 1 minus Anprime( 1113857 ln(1 minus A)

(21)

Dividing both sides of equation (21) by ln(1 minus A) yieldsthe following formula

c1prime + c2prime + c3prime + middot middot middot + cnminus 2prime + cnminus 1prime + cnprime 1 (22)

where ciprime (ln(1 minus Ai

prime) ln(1 minus A))Increasing the buffer improves the workstation avail-

ability In this case the equivalent workstation availabilityafter combining the upstream and downstream buffers willbe higher than the availability of the unit itself us theseries-parallel hybrid system can be simplified to a seriessystem that consists of equivalent workstations that containbuffers

33 Buffer Inventory Capacity Allocation MethodFigure 5 is a structural diagram of a serial two-level pro-duction system composed of workstations Mi and Mi+1 andthe buffer Bi

Assume that the buffer is completely reliable during theplanning period T e failure rate and maintenance rate areλB and μB respectively Bi is separately treated as a unit andthe steady-state availability of Bi is

AB μB

μB + λB

(23)

Considering the occasional malfunction of Bi in actualproduction the correction factor ε is introduced in thispaper and ε (1AB) During the planning period time Tto ensure the continuous production of the system theminimum buffer inventory capacity kimin is [24]

kimin ε middot max ki1 ki21113966 1113967

ki1 ku

1μB

1 minus eminus μBT

1113872 1113873 minus Teminus μBT

1113890 1113891 + 1

ki2 2ku

1μB

1 minus eminus μBT

1113872 1113873 minus Teminus μBT

1113890 1113891

minus kp minus ku1113872 11138731λB

1 minus eminus λBT

1113872 1113873 minus Teminus λBT

1113890 1113891 + 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

4 Availability Allocation Method

41 Proportional Allocation Method e proportional al-location method is based on the failure rate of each unit inthe original system and the failure rate is proportionallydistributed to each unit of the new system according to thereliability prediction of the new system e mathematicalexpression is [25]

λin λio

λso

λsn (25)

where λsn is the failure rate of the new system λin is thefailure rate allocated to the unit i in the new system λso is thefailure rate of the old system and λio is the failure rate of theunit i in the old system

Similarly this paper uses the system availability as theallocation index which is proportionally allocated to eachunit (workstation) according to the scale factor to determinethe unit (workstation) availability index value

Workstation M1 Workstation M2 Workstation Mn

m1L m2L

m2Rm1R

mnL

mnR

B1 B2 Bnndash1

Figure 4 Reliability block diagram of a buffer parallel-series hybrid system

Mi Bi Mi+1

Figure 5 Structural diagram of a serial two-level productionsystem

Mathematical Problems in Engineering 5

42 Availability AllocationMethod for an Unbuffered SystemAssume that the production system availability is A and theunit availability is ai A and ai satisfy formula (6) and theinfluence factor is ci If the system target availability is Am

and the availability assigned to each unit is Aim then1

Aim

minus 11113888 1113889 ci

1Am

minus 11113888 1113889 (26)

e unit availability Aim after sorting is

Aim 1

ci 1Am( 1113857 minus 1( 1113857 + 11113858 1113859 (27)

43 Availability Allocation Method for a Buffer System Ifki＜n and ki+1＜n the system target availability is Am

prime theavailability assigned to each workstation is Aci

prime and the scalefactor is ci

prime then

ciprime

ln 1 minus Aciprime1113872 1113873

ln 1 minus Amprime( 1113857

(28)

Aciprime 1 minus e

ciprime ln 1minus Am

prime( ) (29)

Aciprime is assigned to each unit by an equal reliability allo-

cation method of the parallel system Assume that theavailability assigned to each unit is Ami

prime then the allocationmethod formula is

Amiprime

Aciprime

1113969

1 minus Aciprime

1113969

eciprime ln 1minus Am

prime( )

1113969

(30)

Amiprime 1 minus

Amiprime

1113969

1 minus

eciprime ln 1minus Am

prime( )

1113969

(31)

5 Simulation Analysis for theProduction System

e simulation establishes a model for the system and usesthe model instead of the real system to perform variousexperiments to study the performance In this paper thePlant Simulation software is used as the simulation platformAfter obtaining the availability allocation results the systemis simulated and analyzed to verify the correctness of theresults

e simulation of the production system corresponds tothree stages

(1) Establishing a simulation model using the ldquologis-ticsrdquo the production equipment and buffer can beobtained and the basic system framework model canbe established For each unit the processing capacityand availability are input for each buffer the buffercapacity type and availability are set

(2) Performing a simulation experiment using the eventcontrol unit the simulation time is set to 30 days 90days 180 days 360 days and 720 days is time isset to absolute time which is convenient for ob-serving and recording simulation events e

corresponding time jump speed is 10000 times thereal-time value and the remaining options are de-fault values e parameters of relevant productionunits are set and the software to simulate the entireline is run

(3) Analyzing the simulation data after the simulationtest is completed the intrinsic availability totalthroughput hourly throughput and dailythroughput related data are collated and comparedwith the given expected availability According to thecomparison result it is determined whether the unitreliability index can achieve the expected goal andwhether the reliability allocation method is correct

6 Examples

Assume that a buffer series-parallel hybrid productionsystem consists of a 4-level workstation Each level work-station consists of two identical processing units connectedin parallel e failure rate maintenance rate and pro-ductivity of each unit are λ1λ2λ3λ4 μ1μ2μ3μ4 and ω1ω2ω3ω4respectively which are listed in Table 1

Assume that the failure rate λBi and maintenance rate μBi

of each buffer are 0002 and 006 the planning period time Tis 10 minutes and kp and ku are 1 minute per pieceSubstituting parameters into equations (23) and (24) weobtain AB1 AB2 AB3 09677 and k1 k2 k3 5

It is also assumed that the first-level workstation is notstarved and the last-level workstation is not blocked LetP00 1 and P4 1 e buffer availability can be calculatedby formulas (8)ndash(11) e specific values are shown inTable 2

e steady-state availability of each equivalent work-station can be obtained by using equation (17) and issummarized in Table 3

By substituting the obtained steady-state availability into(19) the stable availability of the system isA 1 minus (1113937

4i1(1 minus Ai

prime)) 09925 If the expected availabilityis increased to 09999 the estimated availability assigned toeach workstation is calculated according to equations (28)and (29) and the specific values are shown in Table 4

e estimated availability of each workstation is redis-tributed to each unit according to formulas (30) and (31)and the expected unit availability is calculated and shown inTable 5

After obtaining the estimated availability allocation re-sults of each processing unit the Plant Simulation software isused to simulate the production system online to determinewhether the allocation results are correct and reasonableUsing the ldquologisticsrdquo element in the software elementtoolbox the production unit and buffer can be obtainedesimulation model of the production system is shown inFigure 6

We input the corresponding parameters for each unitand buffer and use the event control unit to set the simu-lation time to 30 days 90 days 180 days 360 days and 720days and the corresponding time jump speed is 10000 timesthe real-time value

6 Mathematical Problems in Engineering

Table 5 Expected availability of each unit

unit m1L m1R m2L m2R m3L m3R m4L m4R

Amiprime 06873 06873 06761 06761 06406 06406 07253 07253

Table 1 Failure rate maintenance rate and productivity of each unit

Parameter m1L m1R m2L m2R m3L m3R m4L m4R

λi 00032 00032 00027 00027 00026 00026 0003 0003μi 002 002 003 003 002 002 001 001ωi 5 5 4 4 35 35 3 3

Table 2 Buffer availability

Bi B1 B2 B3 B4

ABi07289 07045 06745 08146

Table 3 Steady-state availability of each equivalent workstation

Miprime M1prime M2prime M3prime M4prime

Aiprime 07089 06978 06625 07462

Table 4 Estimated availability of each workstation

Mi M1 M2 M3 M4

Aciprime 09022 08951 08708 09245

Event controller

m1L

B1 B2 B3DrainSource

m2L m3L m4L

m1R m2R m3R m4R

Figure 6 Simulation model of the production system

Entity

Event controller

m1L

ErEntityB1

ErEntityB2

ErEntityB3

DrainSource

Entitym2L

Entitym3L

Entitym4L

Entitym1R

Entitym2R

Entitym3R

Entitym4R

Figure 7 Operation simulation model of the production system

Mathematical Problems in Engineering 7

We set the parameters of the relevant production unitsand run the software to simulate the whole line e op-eration simulation model is shown in Figure 7

en we complete the simulation test sort out therelevant data and obtain the reliability and performanceindex values e specific values are shown in Table 6

e simulation results show that the reliability allocationresults obtained in this paper satisfy the requirements of thegiven expected availability erefore the reliability indexallocated to each unit can achieve the desired goal and thereliability allocation method is correct

7 Conclusions

(1) Aiming at the reliability allocation problem of theproduction system this paper proposes a reliabilityallocation method that is easy to implement in en-gineering By constructing the proportional rela-tionship between unit (workstation) availability andsystem availability the system availability can beproportionally allocated according to the scale fac-tors is allocation method is simple and easy toimplement and it can solve the reliability allocationproblem of multistage production systems

(2) e availability allocation results can help guide theselection of production units determine the numberof spare parts in the buffer comprehensively con-sider factors such as the failure rate maintenancerate inventory capacity and productivity and ra-tionally select the reliability index of the processingunit

(3) Using the Plant Simulation software to simulate andanalyze the production system one can verify thecorrectness of the allocation method and realize thereliability allocation from the complex productionsystem to the unit

(4) e reliability allocation process without consideringthe influence of the buffer can be extended to generalseries and series-parallel hybrid repairable systemssuch as CNCmachine tools and serve as the basis forcomponent selection or reliability improvement inthe product design stage

Notations

mi Unit of a series system (i 1 2 n)

λi Failure rate of a unitμi Maintenance rate of a unitωi Productivity of a unit

ρi Productivity ratio of a two-level unitai Steady-state availability of mi

A Steady-state availability of a systemmiL andmiR

Units of a hybrid system (i 1 2 n)

mpari Equivalent unit combining miL and miR

apari Steady-state availability of m

pari

ci Scale factor in an unbuffered hybrid systemBi Buffers of a systemki Inventory capacity of Bi(ki 1 2 n)

kimin Minimum inventory capacity of Bi

λB Failure rate of a bufferμB Maintenance rate of a bufferP0i Probability of inventory-freeP0i Probability of inventoryPki

Probability of vacancy-freeP

ki Probability of vacancy

ABi Buffer availability

miprime Equivalent unit combining mi Bi and mi+1 in

the buffer series systemAi Steady-state availability of mi

primeMi Workstation of a buffer hybrid system

(i 1 2 n)

Pai Trouble-free probability of Mi

Pbi Discontinuation probability of Mi

Pci Production-reduction probability of Mi

Miprime Equivalent workstation combining Mi Bi and

Mi+1 in the buffer hybrid systemAiprime Steady-state availability of Mi

primeciprime Scale factor in the buffer hybrid system

kp and ku Output per unit time of M1 and M2Am Target availability of an unbuffered systemAim Assigned availability to each unit in the

unbuffered systemAmprime Target availability of a buffer system

Aciprime and

Aciprime

Assigned availability and unavailability to theequivalent workstation

Amiprime and

Amiprime

Assigned availability and unavailability to theunit in the buffer system

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Table 6 Simulation data of the production system

Time Intrinsic availability () Total throughput Hourly throughput Daily throughput30 days 9999 86386 120 288090 days 9999 259290 121 2881180 days 9999 518760 120 2882360 days 9999 1037520 120 2882720 days 9999 2075040 120 2882

8 Mathematical Problems in Engineering

Acknowledgments

e research in this paper was supported by the NationalScience and Technology Major Project of China (Grant no2014ZX04015031) Science and Technology DevelopmentPlan Project of Jilin Province (Grant no 20180520068JH)and Jilin Province Education Departmentrsquos irteenth Five-Year Plan Science and Technology Project (Grant noJJKH20180079KJ)

References

[1] Z J Yang Q B Hao C Fei et al ldquoA comprehensive fuzzyreliability allocation method of NC machine tools based oninterval analysisrdquo Journal of Beijing University of Technologyvol 37 no 3 pp 321ndash329 2011

[2] P-C Chang ldquoReliability estimation for a stochastic pro-duction system with finite buffer storage by a simulationapproachrdquo Annals of Operations Research vol 277 no 1pp 119ndash133 2019

[3] G Liberopoulos ldquoPerformance evaluation of a productionline operated under an echelon buffer policyrdquo IISE Trans-actions vol 50 no 3 pp 161ndash177 2018

[4] S Weiss J A Schwarz and R Stolletz ldquoe buffer allocationproblem in production lines formulations solution methodsand instancesrdquo IISE Transactions vol 51 no 5 pp 456ndash4852019

[5] J Duan A P Li X Nan et al ldquoMulti-state reliabilitymodeling and analysis of reconfigurable manufacturing sys-temsrdquo Journal of Mechanical Engineering vol 47 no 17pp 104ndash111 2011

[6] J L Li Y H Jia F Xu F Chen Z Yang and X Li ldquoAnimprovement scheme for the overall line effectiveness of aproduction line a case studyrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Singapore April 2018

[7] G F Li Y Li X G Zang et al ldquoDevelopment of a preventivemaintenance strategy for an automatic production line basedon group maintenance methodrdquo Applied Sciences vol 8no 10 p 1781 2018

[8] G F Li C Hou G F Liu Y Jia C Liu and J DongldquoReliability allocation method of production line based onfuzzy comprehensive evaluationrdquo in Proceedings of the In-ternational Conference on Materials EngineeringManufacturing Technology and Control (ICMEMTC)Taiyuan China April 2016

[9] Y H Jia Z J Yang G F Li et al ldquoAn efficient semi-analyticalsimulation for availability evaluation of discrete productionlines with unreliable machinesrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Paris France November 2016

[10] F Chen B Liu and B B Xu ldquoResearch on ReliabilityEvaluation of Engine Cylinder Flexible Production Line Basedon AHP Fuzzy Comprehensive Evaluation Methodrdquo ChinaMechanical Engineering Society Changzhou China August2015

[11] F Long P Zeiler and B Bertsche ldquoModelling the flexibility ofproduction systems in industry 40 for analysing their pro-ductivity and availability with high-level Petri netsrdquo IFAC-PapersOnLine vol 50 no 1 pp 5680ndash5687 2017

[12] L Gorkemli and S Kapan Ulusoy ldquoFuzzy bayesian reliabilityand availability analysis of production systemsrdquo Computers ampIndustrial Engineering vol 59 no 4 pp 690ndash696 2010

[13] S Rebello H Yu and L Ma ldquoAn integrated approach forsystem functional reliability assessment using dynamicbayesian network and hidden Markov modelrdquo ReliabilityEngineering amp System Safety vol 180 no 12 pp 124ndash1352018

[14] K Heungseob and K Pansoo ldquoReliability models for anonrepairable systemwith heterogeneous components havinga phase-type time-to-failure distributionrdquo Reliability Engi-neering and System Safety vol 159 no 3 pp 37ndash46 2017

[15] M K Loganathan G Kumar and O P Gandhi ldquoAvailabilityevaluation of manufacturing systems using semi-markovmodelrdquo International Journal of Computer IntegratedManufacturing vol 29 no 7 pp 720ndash735 2016

[16] X-Y Li H-Z Huang and Y-F Li ldquoReliability analysis ofphased mission system with non-exponential and partiallyrepairable componentsrdquo Reliability Engineering amp SystemSafety vol 175 no 7 pp 119ndash127 2018

[17] A B Hu and S M Meerkov ldquoLean buffering in serial pro-duction lines with Bernoulli machinesrdquo MathematicalProblems in Engineering vol 2006 Article ID 17105 24 pages2006

[18] L Demir S Tunalı and D T Eliiyi ldquoAn adaptive tabu searchapproach for buffer allocation problem in unreliable non-homogenous production linesrdquo Computers amp OperationsResearch vol 39 no 7 pp 1477ndash1486 2012

[19] R Li X Liu and N Huang ldquoAvailability allocation of net-worked systems using Markov model and heuristics algo-rithmrdquo Mathematical Problems in Engineering vol 2014Article ID 315385 9 pages 2014

[20] W X Qian X W Yin and L Y Xie ldquoSystem reliabilityallocation based on bayesian networkrdquo Applied Mathematicsamp Information Sciences vol 6 no 3 pp 681ndash687 2012

[21] M Xiao and Y P Zhang ldquoParameters learning of bayesiannetworks for multistate system with small samplerdquo ComputerScience vol 42 no 4 pp 253ndash257 2015

[22] V Sriramdas S K Chaturvedi and H Gargama ldquoFuzzyarithmetic based reliability allocation approach during earlydesign and developmentrdquo Expert Systems with Applicationsvol 41 no 7 pp 3444ndash3449 2014

[23] S G Shu ldquoe manufacturing system (CIMS) with buffersand a study of the system reliabilityrdquo Acta Automatica Sinicavol 18 no 1 pp 15ndash20 1992

[24] H R Zhou S H Wang L Zhang et al ldquoModeling of in-process inventory in continuous production based on reli-abilityrdquo Machine Tool amp Hydraulics vol 39 no 9 pp 111ndash113 2011

[25] M Y You and J C Zheng ldquoA synthetic approach to reliabilityallocation of electronic systems based on AGREE method andthe extended proportional combination methodrdquo ElectronicProduct Reliability and Environmental Testing vol 29 no 4pp 1ndash6 2011

Mathematical Problems in Engineering 9

Ai μiABi

μi + λiABi( 1113857

μiP0(iminus 1)P

ki

μi + λiP0(iminus 1)P

ki1113872 1113873

μi ρiminus 1 middot 1 minus ρk(iminus 1)

iminus 11113872 1113873 1 minus ρk(iminus 1)+1iminus 11113872 11138731113872 1113873

μi + λi ρiminus 1 middot 1 minus ρk(iminus 1)iminus 11113872 1113873 1 minus ρk(iminus 1)+1

iminus 11113872 11138731113872 1113873 1 minus ρkii( 1113857 1 minus ρki+1

i( 1113857( 11138571113960 1113961 (12)

According to equation (12) the availability of theequivalent unit is directly related to the failure rate main-tenance rate productivity and buffer inventory capacity ofthe unit If the availability index of each unit is determinedequation (12) can be used to guide the selection of pro-duction equipment and determine the buffer inventorycapacity

32 Establishing a Mathematical Relationship between BufferParallel-Series Hybrid System Availability and UnitAvailability e buffer parallel-series hybrid system con-sists of two or more units arranged in parallel on the layoutto form a workstation Mi(i 1 2 n) and workstationsMi and Mi+1 are connected in series through the bufferBi(i 1 2 n minus 1) In this paper the parallel distributionof two units miL and miR in the layout is taken as an ex-ample and the reliability block diagram is shown in Figure 4

When each workstation Mi in the parallel-series hybridsystem is composed of two units in parallel there are threestates of the workstation (1) both units are normal and Mi

works normally (2) one unit fails and Mi reduces pro-duction and (3) both units fail and Mi fails Combined withthe state of the upstream and downstream buffers Mi hasnine working states

(1) e probability of Mi working normally isPaiprime P0(iminus 1)

PaiP

ki

(2) Mi is trouble-free and the input shortage causesdiscontinuation e probability is P0(iminus 1)Pai

Pki

(3) Mi is trouble-free and the output blocking causesdiscontinuation e probability is P0(iminus 1)

PaiPki

(4) Mi is trouble-free the input shortage and output

blocking cause discontinuation e probability isP0(iminus 1)Pai

Pki

(5) e probability of Mi reducing production isPciprime P0(iminus 1)

PciP

ki

(6) Mi reduces production and the input shortagecauses discontinuation e probability isP0(iminus 1)Pci

Pki

(7) Mi reduces production and the output blockingcauses discontinuation e probability isP0(iminus 1)

PciPki

(8) Mi reduces production the input shortage andoutput blocking cause discontinuation e proba-bility is P0(iminus 1)Pci

Pki

(9) e probability of Mi stopping production due tomalfunction is Pbi

By adding the above probabilities

PaiP0(iminus 1)

Pki

+ P0(iminus 1)Pki+ P0(iminus 1)

Pki+ P0(iminus 1)Pki

1113874 1113875

+ PciP0(iminus 1)

Pki

+ P0(iminus 1)Pki+ P0(iminus 1)

Pki+ P0(iminus 1)Pki

1113874 1113875

+ Pbi 1

(13)

We can prove that the four probabilities in brackets ofequation (13) add up to 1 so that

Pai+ Pci

+ Pbi 1 (14)

e state transition probability equation is

_Paiprime minus 2λiPai

prime + μiPciprime

_Pciprime 2λiPai

prime minus λi + μi( 1113857Pciprime + μiPbi

_Pbi λiPciprime minus μiPbi

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(15)

We solve (15) and obtain

Pciprime

2λiμiABi

μ2i + 2λ2i ABi+ 2λiμi

Paiprime

μ2i ABi

μ2i + 2λ2i ABi+ 2λiμi

Pbi

2λ2i ABi

μ2i + 2λ2i ABi+ 2λiμi

(16)

Considering the influence of the buffer availability on thesystem availability the buffer available state is that its up-stream buffer is not starved and its downstream buffer is notblocked Combining the two states with the workstation asan equivalent workstation Mi

prime for analysis we obtain thesteady-state availability of Mi

prime

m1L m2L

m2Rm1R

mnL

mnR

m1par m2

par mnpar

Figure 2 Reliability block diagram of an unbuffered parallel-series system

mim1 B1 B2m2 Bi mi+1 mn

Figure 3 Reliability block diagram of a buffer serial system

4 Mathematical Problems in Engineering

Aiprime

2λiμi + μ2i( 1113857ABi

μ2i + 2λ2i ABi+ 2λiμi

(17)

e steady-state unavailability of the system is equal tothe product of the unavailability of each equivalent work-station ie

A 1113945n

i11 minus Aiprime( 1113857 (18)

us the steady-state availability of the system is

A 1 minus A 1 minus 1113945n

i11 minus Aiprime( 1113857 (19)

Formula (19) is expanded and transformed into

1 minus A1prime( 1113857 1 minus A2prime( 1113857 1 minus A3prime( 1113857 middot middot middot 1 minus Anminus 2prime( 1113857 1 minus Anminus 1prime( 1113857 1 minus Anprime( 1113857

1 minus A

(20)

Simultaneously taking the logarithm of both sides yields

ln 1 minus A1prime( 1113857 + ln 1 minus A2prime( 1113857 + ln 1 minus A3prime( 1113857 + middot middot middot

+ ln 1 minus Anminus 2prime( 1113857 + ln 1 minus Anminus 1prime( 1113857 + ln 1 minus Anprime( 1113857 ln(1 minus A)

(21)

Dividing both sides of equation (21) by ln(1 minus A) yieldsthe following formula

c1prime + c2prime + c3prime + middot middot middot + cnminus 2prime + cnminus 1prime + cnprime 1 (22)

where ciprime (ln(1 minus Ai

prime) ln(1 minus A))Increasing the buffer improves the workstation avail-

ability In this case the equivalent workstation availabilityafter combining the upstream and downstream buffers willbe higher than the availability of the unit itself us theseries-parallel hybrid system can be simplified to a seriessystem that consists of equivalent workstations that containbuffers

33 Buffer Inventory Capacity Allocation MethodFigure 5 is a structural diagram of a serial two-level pro-duction system composed of workstations Mi and Mi+1 andthe buffer Bi

Assume that the buffer is completely reliable during theplanning period T e failure rate and maintenance rate areλB and μB respectively Bi is separately treated as a unit andthe steady-state availability of Bi is

AB μB

μB + λB

(23)

Considering the occasional malfunction of Bi in actualproduction the correction factor ε is introduced in thispaper and ε (1AB) During the planning period time Tto ensure the continuous production of the system theminimum buffer inventory capacity kimin is [24]

kimin ε middot max ki1 ki21113966 1113967

ki1 ku

1μB

1 minus eminus μBT

1113872 1113873 minus Teminus μBT

1113890 1113891 + 1

ki2 2ku

1μB

1 minus eminus μBT

1113872 1113873 minus Teminus μBT

1113890 1113891

minus kp minus ku1113872 11138731λB

1 minus eminus λBT

1113872 1113873 minus Teminus λBT

1113890 1113891 + 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

4 Availability Allocation Method

41 Proportional Allocation Method e proportional al-location method is based on the failure rate of each unit inthe original system and the failure rate is proportionallydistributed to each unit of the new system according to thereliability prediction of the new system e mathematicalexpression is [25]

λin λio

λso

λsn (25)

where λsn is the failure rate of the new system λin is thefailure rate allocated to the unit i in the new system λso is thefailure rate of the old system and λio is the failure rate of theunit i in the old system

Similarly this paper uses the system availability as theallocation index which is proportionally allocated to eachunit (workstation) according to the scale factor to determinethe unit (workstation) availability index value

Workstation M1 Workstation M2 Workstation Mn

m1L m2L

m2Rm1R

mnL

mnR

B1 B2 Bnndash1

Figure 4 Reliability block diagram of a buffer parallel-series hybrid system

Mi Bi Mi+1

Figure 5 Structural diagram of a serial two-level productionsystem

Mathematical Problems in Engineering 5

42 Availability AllocationMethod for an Unbuffered SystemAssume that the production system availability is A and theunit availability is ai A and ai satisfy formula (6) and theinfluence factor is ci If the system target availability is Am

and the availability assigned to each unit is Aim then1

Aim

minus 11113888 1113889 ci

1Am

minus 11113888 1113889 (26)

e unit availability Aim after sorting is

Aim 1

ci 1Am( 1113857 minus 1( 1113857 + 11113858 1113859 (27)

43 Availability Allocation Method for a Buffer System Ifki＜n and ki+1＜n the system target availability is Am

prime theavailability assigned to each workstation is Aci

prime and the scalefactor is ci

prime then

ciprime

ln 1 minus Aciprime1113872 1113873

ln 1 minus Amprime( 1113857

(28)

Aciprime 1 minus e

ciprime ln 1minus Am

prime( ) (29)

Aciprime is assigned to each unit by an equal reliability allo-

cation method of the parallel system Assume that theavailability assigned to each unit is Ami

prime then the allocationmethod formula is

Amiprime

Aciprime

1113969

1 minus Aciprime

1113969

eciprime ln 1minus Am

prime( )

1113969

(30)

Amiprime 1 minus

Amiprime

1113969

1 minus

eciprime ln 1minus Am

prime( )

1113969

(31)

5 Simulation Analysis for theProduction System

e simulation establishes a model for the system and usesthe model instead of the real system to perform variousexperiments to study the performance In this paper thePlant Simulation software is used as the simulation platformAfter obtaining the availability allocation results the systemis simulated and analyzed to verify the correctness of theresults

e simulation of the production system corresponds tothree stages

(1) Establishing a simulation model using the ldquologis-ticsrdquo the production equipment and buffer can beobtained and the basic system framework model canbe established For each unit the processing capacityand availability are input for each buffer the buffercapacity type and availability are set

(2) Performing a simulation experiment using the eventcontrol unit the simulation time is set to 30 days 90days 180 days 360 days and 720 days is time isset to absolute time which is convenient for ob-serving and recording simulation events e

corresponding time jump speed is 10000 times thereal-time value and the remaining options are de-fault values e parameters of relevant productionunits are set and the software to simulate the entireline is run

(3) Analyzing the simulation data after the simulationtest is completed the intrinsic availability totalthroughput hourly throughput and dailythroughput related data are collated and comparedwith the given expected availability According to thecomparison result it is determined whether the unitreliability index can achieve the expected goal andwhether the reliability allocation method is correct

6 Examples

Assume that a buffer series-parallel hybrid productionsystem consists of a 4-level workstation Each level work-station consists of two identical processing units connectedin parallel e failure rate maintenance rate and pro-ductivity of each unit are λ1λ2λ3λ4 μ1μ2μ3μ4 and ω1ω2ω3ω4respectively which are listed in Table 1

Assume that the failure rate λBi and maintenance rate μBi

of each buffer are 0002 and 006 the planning period time Tis 10 minutes and kp and ku are 1 minute per pieceSubstituting parameters into equations (23) and (24) weobtain AB1 AB2 AB3 09677 and k1 k2 k3 5

It is also assumed that the first-level workstation is notstarved and the last-level workstation is not blocked LetP00 1 and P4 1 e buffer availability can be calculatedby formulas (8)ndash(11) e specific values are shown inTable 2

e steady-state availability of each equivalent work-station can be obtained by using equation (17) and issummarized in Table 3

By substituting the obtained steady-state availability into(19) the stable availability of the system isA 1 minus (1113937

4i1(1 minus Ai

prime)) 09925 If the expected availabilityis increased to 09999 the estimated availability assigned toeach workstation is calculated according to equations (28)and (29) and the specific values are shown in Table 4

e estimated availability of each workstation is redis-tributed to each unit according to formulas (30) and (31)and the expected unit availability is calculated and shown inTable 5

After obtaining the estimated availability allocation re-sults of each processing unit the Plant Simulation software isused to simulate the production system online to determinewhether the allocation results are correct and reasonableUsing the ldquologisticsrdquo element in the software elementtoolbox the production unit and buffer can be obtainedesimulation model of the production system is shown inFigure 6

We input the corresponding parameters for each unitand buffer and use the event control unit to set the simu-lation time to 30 days 90 days 180 days 360 days and 720days and the corresponding time jump speed is 10000 timesthe real-time value

6 Mathematical Problems in Engineering

Table 5 Expected availability of each unit

unit m1L m1R m2L m2R m3L m3R m4L m4R

Amiprime 06873 06873 06761 06761 06406 06406 07253 07253

Table 1 Failure rate maintenance rate and productivity of each unit

Parameter m1L m1R m2L m2R m3L m3R m4L m4R

λi 00032 00032 00027 00027 00026 00026 0003 0003μi 002 002 003 003 002 002 001 001ωi 5 5 4 4 35 35 3 3

Table 2 Buffer availability

Bi B1 B2 B3 B4

ABi07289 07045 06745 08146

Table 3 Steady-state availability of each equivalent workstation

Miprime M1prime M2prime M3prime M4prime

Aiprime 07089 06978 06625 07462

Table 4 Estimated availability of each workstation

Mi M1 M2 M3 M4

Aciprime 09022 08951 08708 09245

Event controller

m1L

B1 B2 B3DrainSource

m2L m3L m4L

m1R m2R m3R m4R

Figure 6 Simulation model of the production system

Entity

Event controller

m1L

ErEntityB1

ErEntityB2

ErEntityB3

DrainSource

Entitym2L

Entitym3L

Entitym4L

Entitym1R

Entitym2R

Entitym3R

Entitym4R

Figure 7 Operation simulation model of the production system

Mathematical Problems in Engineering 7

We set the parameters of the relevant production unitsand run the software to simulate the whole line e op-eration simulation model is shown in Figure 7

en we complete the simulation test sort out therelevant data and obtain the reliability and performanceindex values e specific values are shown in Table 6

e simulation results show that the reliability allocationresults obtained in this paper satisfy the requirements of thegiven expected availability erefore the reliability indexallocated to each unit can achieve the desired goal and thereliability allocation method is correct

7 Conclusions

(1) Aiming at the reliability allocation problem of theproduction system this paper proposes a reliabilityallocation method that is easy to implement in en-gineering By constructing the proportional rela-tionship between unit (workstation) availability andsystem availability the system availability can beproportionally allocated according to the scale fac-tors is allocation method is simple and easy toimplement and it can solve the reliability allocationproblem of multistage production systems

(2) e availability allocation results can help guide theselection of production units determine the numberof spare parts in the buffer comprehensively con-sider factors such as the failure rate maintenancerate inventory capacity and productivity and ra-tionally select the reliability index of the processingunit

(3) Using the Plant Simulation software to simulate andanalyze the production system one can verify thecorrectness of the allocation method and realize thereliability allocation from the complex productionsystem to the unit

(4) e reliability allocation process without consideringthe influence of the buffer can be extended to generalseries and series-parallel hybrid repairable systemssuch as CNCmachine tools and serve as the basis forcomponent selection or reliability improvement inthe product design stage

Notations

mi Unit of a series system (i 1 2 n)

λi Failure rate of a unitμi Maintenance rate of a unitωi Productivity of a unit

ρi Productivity ratio of a two-level unitai Steady-state availability of mi

A Steady-state availability of a systemmiL andmiR

Units of a hybrid system (i 1 2 n)

mpari Equivalent unit combining miL and miR

apari Steady-state availability of m

pari

ci Scale factor in an unbuffered hybrid systemBi Buffers of a systemki Inventory capacity of Bi(ki 1 2 n)

kimin Minimum inventory capacity of Bi

λB Failure rate of a bufferμB Maintenance rate of a bufferP0i Probability of inventory-freeP0i Probability of inventoryPki

Probability of vacancy-freeP

ki Probability of vacancy

ABi Buffer availability

miprime Equivalent unit combining mi Bi and mi+1 in

the buffer series systemAi Steady-state availability of mi

primeMi Workstation of a buffer hybrid system

(i 1 2 n)

Pai Trouble-free probability of Mi

Pbi Discontinuation probability of Mi

Pci Production-reduction probability of Mi

Miprime Equivalent workstation combining Mi Bi and

Mi+1 in the buffer hybrid systemAiprime Steady-state availability of Mi

primeciprime Scale factor in the buffer hybrid system

kp and ku Output per unit time of M1 and M2Am Target availability of an unbuffered systemAim Assigned availability to each unit in the

unbuffered systemAmprime Target availability of a buffer system

Aciprime and

Aciprime

Assigned availability and unavailability to theequivalent workstation

Amiprime and

Amiprime

Assigned availability and unavailability to theunit in the buffer system

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Table 6 Simulation data of the production system

Time Intrinsic availability () Total throughput Hourly throughput Daily throughput30 days 9999 86386 120 288090 days 9999 259290 121 2881180 days 9999 518760 120 2882360 days 9999 1037520 120 2882720 days 9999 2075040 120 2882

8 Mathematical Problems in Engineering

Acknowledgments

e research in this paper was supported by the NationalScience and Technology Major Project of China (Grant no2014ZX04015031) Science and Technology DevelopmentPlan Project of Jilin Province (Grant no 20180520068JH)and Jilin Province Education Departmentrsquos irteenth Five-Year Plan Science and Technology Project (Grant noJJKH20180079KJ)

References

[1] Z J Yang Q B Hao C Fei et al ldquoA comprehensive fuzzyreliability allocation method of NC machine tools based oninterval analysisrdquo Journal of Beijing University of Technologyvol 37 no 3 pp 321ndash329 2011

[2] P-C Chang ldquoReliability estimation for a stochastic pro-duction system with finite buffer storage by a simulationapproachrdquo Annals of Operations Research vol 277 no 1pp 119ndash133 2019

[3] G Liberopoulos ldquoPerformance evaluation of a productionline operated under an echelon buffer policyrdquo IISE Trans-actions vol 50 no 3 pp 161ndash177 2018

[4] S Weiss J A Schwarz and R Stolletz ldquoe buffer allocationproblem in production lines formulations solution methodsand instancesrdquo IISE Transactions vol 51 no 5 pp 456ndash4852019

[5] J Duan A P Li X Nan et al ldquoMulti-state reliabilitymodeling and analysis of reconfigurable manufacturing sys-temsrdquo Journal of Mechanical Engineering vol 47 no 17pp 104ndash111 2011

[6] J L Li Y H Jia F Xu F Chen Z Yang and X Li ldquoAnimprovement scheme for the overall line effectiveness of aproduction line a case studyrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Singapore April 2018

[7] G F Li Y Li X G Zang et al ldquoDevelopment of a preventivemaintenance strategy for an automatic production line basedon group maintenance methodrdquo Applied Sciences vol 8no 10 p 1781 2018

[8] G F Li C Hou G F Liu Y Jia C Liu and J DongldquoReliability allocation method of production line based onfuzzy comprehensive evaluationrdquo in Proceedings of the In-ternational Conference on Materials EngineeringManufacturing Technology and Control (ICMEMTC)Taiyuan China April 2016

[9] Y H Jia Z J Yang G F Li et al ldquoAn efficient semi-analyticalsimulation for availability evaluation of discrete productionlines with unreliable machinesrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Paris France November 2016

[10] F Chen B Liu and B B Xu ldquoResearch on ReliabilityEvaluation of Engine Cylinder Flexible Production Line Basedon AHP Fuzzy Comprehensive Evaluation Methodrdquo ChinaMechanical Engineering Society Changzhou China August2015

[11] F Long P Zeiler and B Bertsche ldquoModelling the flexibility ofproduction systems in industry 40 for analysing their pro-ductivity and availability with high-level Petri netsrdquo IFAC-PapersOnLine vol 50 no 1 pp 5680ndash5687 2017

[12] L Gorkemli and S Kapan Ulusoy ldquoFuzzy bayesian reliabilityand availability analysis of production systemsrdquo Computers ampIndustrial Engineering vol 59 no 4 pp 690ndash696 2010

[13] S Rebello H Yu and L Ma ldquoAn integrated approach forsystem functional reliability assessment using dynamicbayesian network and hidden Markov modelrdquo ReliabilityEngineering amp System Safety vol 180 no 12 pp 124ndash1352018

[14] K Heungseob and K Pansoo ldquoReliability models for anonrepairable systemwith heterogeneous components havinga phase-type time-to-failure distributionrdquo Reliability Engi-neering and System Safety vol 159 no 3 pp 37ndash46 2017

[15] M K Loganathan G Kumar and O P Gandhi ldquoAvailabilityevaluation of manufacturing systems using semi-markovmodelrdquo International Journal of Computer IntegratedManufacturing vol 29 no 7 pp 720ndash735 2016

[16] X-Y Li H-Z Huang and Y-F Li ldquoReliability analysis ofphased mission system with non-exponential and partiallyrepairable componentsrdquo Reliability Engineering amp SystemSafety vol 175 no 7 pp 119ndash127 2018

[17] A B Hu and S M Meerkov ldquoLean buffering in serial pro-duction lines with Bernoulli machinesrdquo MathematicalProblems in Engineering vol 2006 Article ID 17105 24 pages2006

[18] L Demir S Tunalı and D T Eliiyi ldquoAn adaptive tabu searchapproach for buffer allocation problem in unreliable non-homogenous production linesrdquo Computers amp OperationsResearch vol 39 no 7 pp 1477ndash1486 2012

[19] R Li X Liu and N Huang ldquoAvailability allocation of net-worked systems using Markov model and heuristics algo-rithmrdquo Mathematical Problems in Engineering vol 2014Article ID 315385 9 pages 2014

[20] W X Qian X W Yin and L Y Xie ldquoSystem reliabilityallocation based on bayesian networkrdquo Applied Mathematicsamp Information Sciences vol 6 no 3 pp 681ndash687 2012

[21] M Xiao and Y P Zhang ldquoParameters learning of bayesiannetworks for multistate system with small samplerdquo ComputerScience vol 42 no 4 pp 253ndash257 2015

[22] V Sriramdas S K Chaturvedi and H Gargama ldquoFuzzyarithmetic based reliability allocation approach during earlydesign and developmentrdquo Expert Systems with Applicationsvol 41 no 7 pp 3444ndash3449 2014

[23] S G Shu ldquoe manufacturing system (CIMS) with buffersand a study of the system reliabilityrdquo Acta Automatica Sinicavol 18 no 1 pp 15ndash20 1992

[24] H R Zhou S H Wang L Zhang et al ldquoModeling of in-process inventory in continuous production based on reli-abilityrdquo Machine Tool amp Hydraulics vol 39 no 9 pp 111ndash113 2011

[25] M Y You and J C Zheng ldquoA synthetic approach to reliabilityallocation of electronic systems based on AGREE method andthe extended proportional combination methodrdquo ElectronicProduct Reliability and Environmental Testing vol 29 no 4pp 1ndash6 2011

Mathematical Problems in Engineering 9

Aiprime

2λiμi + μ2i( 1113857ABi

μ2i + 2λ2i ABi+ 2λiμi

(17)

e steady-state unavailability of the system is equal tothe product of the unavailability of each equivalent work-station ie

A 1113945n

i11 minus Aiprime( 1113857 (18)

us the steady-state availability of the system is

A 1 minus A 1 minus 1113945n

i11 minus Aiprime( 1113857 (19)

Formula (19) is expanded and transformed into

1 minus A1prime( 1113857 1 minus A2prime( 1113857 1 minus A3prime( 1113857 middot middot middot 1 minus Anminus 2prime( 1113857 1 minus Anminus 1prime( 1113857 1 minus Anprime( 1113857

1 minus A

(20)

Simultaneously taking the logarithm of both sides yields

ln 1 minus A1prime( 1113857 + ln 1 minus A2prime( 1113857 + ln 1 minus A3prime( 1113857 + middot middot middot

+ ln 1 minus Anminus 2prime( 1113857 + ln 1 minus Anminus 1prime( 1113857 + ln 1 minus Anprime( 1113857 ln(1 minus A)

(21)

Dividing both sides of equation (21) by ln(1 minus A) yieldsthe following formula

c1prime + c2prime + c3prime + middot middot middot + cnminus 2prime + cnminus 1prime + cnprime 1 (22)

where ciprime (ln(1 minus Ai

prime) ln(1 minus A))Increasing the buffer improves the workstation avail-

ability In this case the equivalent workstation availabilityafter combining the upstream and downstream buffers willbe higher than the availability of the unit itself us theseries-parallel hybrid system can be simplified to a seriessystem that consists of equivalent workstations that containbuffers

33 Buffer Inventory Capacity Allocation MethodFigure 5 is a structural diagram of a serial two-level pro-duction system composed of workstations Mi and Mi+1 andthe buffer Bi

Assume that the buffer is completely reliable during theplanning period T e failure rate and maintenance rate areλB and μB respectively Bi is separately treated as a unit andthe steady-state availability of Bi is

AB μB

μB + λB

(23)

Considering the occasional malfunction of Bi in actualproduction the correction factor ε is introduced in thispaper and ε (1AB) During the planning period time Tto ensure the continuous production of the system theminimum buffer inventory capacity kimin is [24]

kimin ε middot max ki1 ki21113966 1113967

ki1 ku

1μB

1 minus eminus μBT

1113872 1113873 minus Teminus μBT

1113890 1113891 + 1

ki2 2ku

1μB

1 minus eminus μBT

1113872 1113873 minus Teminus μBT

1113890 1113891

minus kp minus ku1113872 11138731λB

1 minus eminus λBT

1113872 1113873 minus Teminus λBT

1113890 1113891 + 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

4 Availability Allocation Method

41 Proportional Allocation Method e proportional al-location method is based on the failure rate of each unit inthe original system and the failure rate is proportionallydistributed to each unit of the new system according to thereliability prediction of the new system e mathematicalexpression is [25]

λin λio

λso

λsn (25)

where λsn is the failure rate of the new system λin is thefailure rate allocated to the unit i in the new system λso is thefailure rate of the old system and λio is the failure rate of theunit i in the old system

Similarly this paper uses the system availability as theallocation index which is proportionally allocated to eachunit (workstation) according to the scale factor to determinethe unit (workstation) availability index value

Workstation M1 Workstation M2 Workstation Mn

m1L m2L

m2Rm1R

mnL

mnR

B1 B2 Bnndash1

Figure 4 Reliability block diagram of a buffer parallel-series hybrid system

Mi Bi Mi+1

Figure 5 Structural diagram of a serial two-level productionsystem

Mathematical Problems in Engineering 5

42 Availability AllocationMethod for an Unbuffered SystemAssume that the production system availability is A and theunit availability is ai A and ai satisfy formula (6) and theinfluence factor is ci If the system target availability is Am

and the availability assigned to each unit is Aim then1

Aim

minus 11113888 1113889 ci

1Am

minus 11113888 1113889 (26)

e unit availability Aim after sorting is

Aim 1

ci 1Am( 1113857 minus 1( 1113857 + 11113858 1113859 (27)

43 Availability Allocation Method for a Buffer System Ifki＜n and ki+1＜n the system target availability is Am

prime theavailability assigned to each workstation is Aci

prime and the scalefactor is ci

prime then

ciprime

ln 1 minus Aciprime1113872 1113873

ln 1 minus Amprime( 1113857

(28)

Aciprime 1 minus e

ciprime ln 1minus Am

prime( ) (29)

Aciprime is assigned to each unit by an equal reliability allo-

cation method of the parallel system Assume that theavailability assigned to each unit is Ami

prime then the allocationmethod formula is

Amiprime

Aciprime

1113969

1 minus Aciprime

1113969

eciprime ln 1minus Am

prime( )

1113969

(30)

Amiprime 1 minus

Amiprime

1113969

1 minus

eciprime ln 1minus Am

prime( )

1113969

(31)

5 Simulation Analysis for theProduction System

e simulation establishes a model for the system and usesthe model instead of the real system to perform variousexperiments to study the performance In this paper thePlant Simulation software is used as the simulation platformAfter obtaining the availability allocation results the systemis simulated and analyzed to verify the correctness of theresults

e simulation of the production system corresponds tothree stages

(1) Establishing a simulation model using the ldquologis-ticsrdquo the production equipment and buffer can beobtained and the basic system framework model canbe established For each unit the processing capacityand availability are input for each buffer the buffercapacity type and availability are set

(2) Performing a simulation experiment using the eventcontrol unit the simulation time is set to 30 days 90days 180 days 360 days and 720 days is time isset to absolute time which is convenient for ob-serving and recording simulation events e

corresponding time jump speed is 10000 times thereal-time value and the remaining options are de-fault values e parameters of relevant productionunits are set and the software to simulate the entireline is run

(3) Analyzing the simulation data after the simulationtest is completed the intrinsic availability totalthroughput hourly throughput and dailythroughput related data are collated and comparedwith the given expected availability According to thecomparison result it is determined whether the unitreliability index can achieve the expected goal andwhether the reliability allocation method is correct

6 Examples

Assume that a buffer series-parallel hybrid productionsystem consists of a 4-level workstation Each level work-station consists of two identical processing units connectedin parallel e failure rate maintenance rate and pro-ductivity of each unit are λ1λ2λ3λ4 μ1μ2μ3μ4 and ω1ω2ω3ω4respectively which are listed in Table 1

Assume that the failure rate λBi and maintenance rate μBi

of each buffer are 0002 and 006 the planning period time Tis 10 minutes and kp and ku are 1 minute per pieceSubstituting parameters into equations (23) and (24) weobtain AB1 AB2 AB3 09677 and k1 k2 k3 5

It is also assumed that the first-level workstation is notstarved and the last-level workstation is not blocked LetP00 1 and P4 1 e buffer availability can be calculatedby formulas (8)ndash(11) e specific values are shown inTable 2

e steady-state availability of each equivalent work-station can be obtained by using equation (17) and issummarized in Table 3

By substituting the obtained steady-state availability into(19) the stable availability of the system isA 1 minus (1113937

4i1(1 minus Ai

prime)) 09925 If the expected availabilityis increased to 09999 the estimated availability assigned toeach workstation is calculated according to equations (28)and (29) and the specific values are shown in Table 4

e estimated availability of each workstation is redis-tributed to each unit according to formulas (30) and (31)and the expected unit availability is calculated and shown inTable 5

After obtaining the estimated availability allocation re-sults of each processing unit the Plant Simulation software isused to simulate the production system online to determinewhether the allocation results are correct and reasonableUsing the ldquologisticsrdquo element in the software elementtoolbox the production unit and buffer can be obtainedesimulation model of the production system is shown inFigure 6

We input the corresponding parameters for each unitand buffer and use the event control unit to set the simu-lation time to 30 days 90 days 180 days 360 days and 720days and the corresponding time jump speed is 10000 timesthe real-time value

6 Mathematical Problems in Engineering

Table 5 Expected availability of each unit

unit m1L m1R m2L m2R m3L m3R m4L m4R

Amiprime 06873 06873 06761 06761 06406 06406 07253 07253

Table 1 Failure rate maintenance rate and productivity of each unit

Parameter m1L m1R m2L m2R m3L m3R m4L m4R

λi 00032 00032 00027 00027 00026 00026 0003 0003μi 002 002 003 003 002 002 001 001ωi 5 5 4 4 35 35 3 3

Table 2 Buffer availability

Bi B1 B2 B3 B4

ABi07289 07045 06745 08146

Table 3 Steady-state availability of each equivalent workstation

Miprime M1prime M2prime M3prime M4prime

Aiprime 07089 06978 06625 07462

Table 4 Estimated availability of each workstation

Mi M1 M2 M3 M4

Aciprime 09022 08951 08708 09245

Event controller

m1L

B1 B2 B3DrainSource

m2L m3L m4L

m1R m2R m3R m4R

Figure 6 Simulation model of the production system

Entity

Event controller

m1L

ErEntityB1

ErEntityB2

ErEntityB3

DrainSource

Entitym2L

Entitym3L

Entitym4L

Entitym1R

Entitym2R

Entitym3R

Entitym4R

Figure 7 Operation simulation model of the production system

Mathematical Problems in Engineering 7

We set the parameters of the relevant production unitsand run the software to simulate the whole line e op-eration simulation model is shown in Figure 7

en we complete the simulation test sort out therelevant data and obtain the reliability and performanceindex values e specific values are shown in Table 6

e simulation results show that the reliability allocationresults obtained in this paper satisfy the requirements of thegiven expected availability erefore the reliability indexallocated to each unit can achieve the desired goal and thereliability allocation method is correct

7 Conclusions

(1) Aiming at the reliability allocation problem of theproduction system this paper proposes a reliabilityallocation method that is easy to implement in en-gineering By constructing the proportional rela-tionship between unit (workstation) availability andsystem availability the system availability can beproportionally allocated according to the scale fac-tors is allocation method is simple and easy toimplement and it can solve the reliability allocationproblem of multistage production systems

(2) e availability allocation results can help guide theselection of production units determine the numberof spare parts in the buffer comprehensively con-sider factors such as the failure rate maintenancerate inventory capacity and productivity and ra-tionally select the reliability index of the processingunit

(3) Using the Plant Simulation software to simulate andanalyze the production system one can verify thecorrectness of the allocation method and realize thereliability allocation from the complex productionsystem to the unit

(4) e reliability allocation process without consideringthe influence of the buffer can be extended to generalseries and series-parallel hybrid repairable systemssuch as CNCmachine tools and serve as the basis forcomponent selection or reliability improvement inthe product design stage

Notations

mi Unit of a series system (i 1 2 n)

λi Failure rate of a unitμi Maintenance rate of a unitωi Productivity of a unit

ρi Productivity ratio of a two-level unitai Steady-state availability of mi

A Steady-state availability of a systemmiL andmiR

Units of a hybrid system (i 1 2 n)

mpari Equivalent unit combining miL and miR

apari Steady-state availability of m

pari

ci Scale factor in an unbuffered hybrid systemBi Buffers of a systemki Inventory capacity of Bi(ki 1 2 n)

kimin Minimum inventory capacity of Bi

λB Failure rate of a bufferμB Maintenance rate of a bufferP0i Probability of inventory-freeP0i Probability of inventoryPki

Probability of vacancy-freeP

ki Probability of vacancy

ABi Buffer availability

miprime Equivalent unit combining mi Bi and mi+1 in

the buffer series systemAi Steady-state availability of mi

primeMi Workstation of a buffer hybrid system

(i 1 2 n)

Pai Trouble-free probability of Mi

Pbi Discontinuation probability of Mi

Pci Production-reduction probability of Mi

Miprime Equivalent workstation combining Mi Bi and

Mi+1 in the buffer hybrid systemAiprime Steady-state availability of Mi

primeciprime Scale factor in the buffer hybrid system

kp and ku Output per unit time of M1 and M2Am Target availability of an unbuffered systemAim Assigned availability to each unit in the

unbuffered systemAmprime Target availability of a buffer system

Aciprime and

Aciprime

Assigned availability and unavailability to theequivalent workstation

Amiprime and

Amiprime

Assigned availability and unavailability to theunit in the buffer system

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Table 6 Simulation data of the production system

Time Intrinsic availability () Total throughput Hourly throughput Daily throughput30 days 9999 86386 120 288090 days 9999 259290 121 2881180 days 9999 518760 120 2882360 days 9999 1037520 120 2882720 days 9999 2075040 120 2882

8 Mathematical Problems in Engineering

Acknowledgments

e research in this paper was supported by the NationalScience and Technology Major Project of China (Grant no2014ZX04015031) Science and Technology DevelopmentPlan Project of Jilin Province (Grant no 20180520068JH)and Jilin Province Education Departmentrsquos irteenth Five-Year Plan Science and Technology Project (Grant noJJKH20180079KJ)

References

[1] Z J Yang Q B Hao C Fei et al ldquoA comprehensive fuzzyreliability allocation method of NC machine tools based oninterval analysisrdquo Journal of Beijing University of Technologyvol 37 no 3 pp 321ndash329 2011

[2] P-C Chang ldquoReliability estimation for a stochastic pro-duction system with finite buffer storage by a simulationapproachrdquo Annals of Operations Research vol 277 no 1pp 119ndash133 2019

[3] G Liberopoulos ldquoPerformance evaluation of a productionline operated under an echelon buffer policyrdquo IISE Trans-actions vol 50 no 3 pp 161ndash177 2018

[4] S Weiss J A Schwarz and R Stolletz ldquoe buffer allocationproblem in production lines formulations solution methodsand instancesrdquo IISE Transactions vol 51 no 5 pp 456ndash4852019

[5] J Duan A P Li X Nan et al ldquoMulti-state reliabilitymodeling and analysis of reconfigurable manufacturing sys-temsrdquo Journal of Mechanical Engineering vol 47 no 17pp 104ndash111 2011

[6] J L Li Y H Jia F Xu F Chen Z Yang and X Li ldquoAnimprovement scheme for the overall line effectiveness of aproduction line a case studyrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Singapore April 2018

[7] G F Li Y Li X G Zang et al ldquoDevelopment of a preventivemaintenance strategy for an automatic production line basedon group maintenance methodrdquo Applied Sciences vol 8no 10 p 1781 2018

[8] G F Li C Hou G F Liu Y Jia C Liu and J DongldquoReliability allocation method of production line based onfuzzy comprehensive evaluationrdquo in Proceedings of the In-ternational Conference on Materials EngineeringManufacturing Technology and Control (ICMEMTC)Taiyuan China April 2016

[9] Y H Jia Z J Yang G F Li et al ldquoAn efficient semi-analyticalsimulation for availability evaluation of discrete productionlines with unreliable machinesrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Paris France November 2016

[10] F Chen B Liu and B B Xu ldquoResearch on ReliabilityEvaluation of Engine Cylinder Flexible Production Line Basedon AHP Fuzzy Comprehensive Evaluation Methodrdquo ChinaMechanical Engineering Society Changzhou China August2015

[11] F Long P Zeiler and B Bertsche ldquoModelling the flexibility ofproduction systems in industry 40 for analysing their pro-ductivity and availability with high-level Petri netsrdquo IFAC-PapersOnLine vol 50 no 1 pp 5680ndash5687 2017

[12] L Gorkemli and S Kapan Ulusoy ldquoFuzzy bayesian reliabilityand availability analysis of production systemsrdquo Computers ampIndustrial Engineering vol 59 no 4 pp 690ndash696 2010

[13] S Rebello H Yu and L Ma ldquoAn integrated approach forsystem functional reliability assessment using dynamicbayesian network and hidden Markov modelrdquo ReliabilityEngineering amp System Safety vol 180 no 12 pp 124ndash1352018

[14] K Heungseob and K Pansoo ldquoReliability models for anonrepairable systemwith heterogeneous components havinga phase-type time-to-failure distributionrdquo Reliability Engi-neering and System Safety vol 159 no 3 pp 37ndash46 2017

[15] M K Loganathan G Kumar and O P Gandhi ldquoAvailabilityevaluation of manufacturing systems using semi-markovmodelrdquo International Journal of Computer IntegratedManufacturing vol 29 no 7 pp 720ndash735 2016

[16] X-Y Li H-Z Huang and Y-F Li ldquoReliability analysis ofphased mission system with non-exponential and partiallyrepairable componentsrdquo Reliability Engineering amp SystemSafety vol 175 no 7 pp 119ndash127 2018

[17] A B Hu and S M Meerkov ldquoLean buffering in serial pro-duction lines with Bernoulli machinesrdquo MathematicalProblems in Engineering vol 2006 Article ID 17105 24 pages2006

[18] L Demir S Tunalı and D T Eliiyi ldquoAn adaptive tabu searchapproach for buffer allocation problem in unreliable non-homogenous production linesrdquo Computers amp OperationsResearch vol 39 no 7 pp 1477ndash1486 2012

[19] R Li X Liu and N Huang ldquoAvailability allocation of net-worked systems using Markov model and heuristics algo-rithmrdquo Mathematical Problems in Engineering vol 2014Article ID 315385 9 pages 2014

[20] W X Qian X W Yin and L Y Xie ldquoSystem reliabilityallocation based on bayesian networkrdquo Applied Mathematicsamp Information Sciences vol 6 no 3 pp 681ndash687 2012

[21] M Xiao and Y P Zhang ldquoParameters learning of bayesiannetworks for multistate system with small samplerdquo ComputerScience vol 42 no 4 pp 253ndash257 2015

[22] V Sriramdas S K Chaturvedi and H Gargama ldquoFuzzyarithmetic based reliability allocation approach during earlydesign and developmentrdquo Expert Systems with Applicationsvol 41 no 7 pp 3444ndash3449 2014

[23] S G Shu ldquoe manufacturing system (CIMS) with buffersand a study of the system reliabilityrdquo Acta Automatica Sinicavol 18 no 1 pp 15ndash20 1992

[24] H R Zhou S H Wang L Zhang et al ldquoModeling of in-process inventory in continuous production based on reli-abilityrdquo Machine Tool amp Hydraulics vol 39 no 9 pp 111ndash113 2011

[25] M Y You and J C Zheng ldquoA synthetic approach to reliabilityallocation of electronic systems based on AGREE method andthe extended proportional combination methodrdquo ElectronicProduct Reliability and Environmental Testing vol 29 no 4pp 1ndash6 2011

Mathematical Problems in Engineering 9

42 Availability AllocationMethod for an Unbuffered SystemAssume that the production system availability is A and theunit availability is ai A and ai satisfy formula (6) and theinfluence factor is ci If the system target availability is Am

and the availability assigned to each unit is Aim then1

Aim

minus 11113888 1113889 ci

1Am

minus 11113888 1113889 (26)

e unit availability Aim after sorting is

Aim 1

ci 1Am( 1113857 minus 1( 1113857 + 11113858 1113859 (27)

43 Availability Allocation Method for a Buffer System Ifki＜n and ki+1＜n the system target availability is Am

prime theavailability assigned to each workstation is Aci

prime and the scalefactor is ci

prime then

ciprime

ln 1 minus Aciprime1113872 1113873

ln 1 minus Amprime( 1113857

(28)

Aciprime 1 minus e

ciprime ln 1minus Am

prime( ) (29)

Aciprime is assigned to each unit by an equal reliability allo-

cation method of the parallel system Assume that theavailability assigned to each unit is Ami

prime then the allocationmethod formula is

Amiprime

Aciprime

1113969

1 minus Aciprime

1113969

eciprime ln 1minus Am

prime( )

1113969

(30)

Amiprime 1 minus

Amiprime

1113969

1 minus

eciprime ln 1minus Am

prime( )

1113969

(31)

5 Simulation Analysis for theProduction System

e simulation establishes a model for the system and usesthe model instead of the real system to perform variousexperiments to study the performance In this paper thePlant Simulation software is used as the simulation platformAfter obtaining the availability allocation results the systemis simulated and analyzed to verify the correctness of theresults

e simulation of the production system corresponds tothree stages

(1) Establishing a simulation model using the ldquologis-ticsrdquo the production equipment and buffer can beobtained and the basic system framework model canbe established For each unit the processing capacityand availability are input for each buffer the buffercapacity type and availability are set

(2) Performing a simulation experiment using the eventcontrol unit the simulation time is set to 30 days 90days 180 days 360 days and 720 days is time isset to absolute time which is convenient for ob-serving and recording simulation events e

corresponding time jump speed is 10000 times thereal-time value and the remaining options are de-fault values e parameters of relevant productionunits are set and the software to simulate the entireline is run

(3) Analyzing the simulation data after the simulationtest is completed the intrinsic availability totalthroughput hourly throughput and dailythroughput related data are collated and comparedwith the given expected availability According to thecomparison result it is determined whether the unitreliability index can achieve the expected goal andwhether the reliability allocation method is correct

6 Examples

Assume that a buffer series-parallel hybrid productionsystem consists of a 4-level workstation Each level work-station consists of two identical processing units connectedin parallel e failure rate maintenance rate and pro-ductivity of each unit are λ1λ2λ3λ4 μ1μ2μ3μ4 and ω1ω2ω3ω4respectively which are listed in Table 1

Assume that the failure rate λBi and maintenance rate μBi

of each buffer are 0002 and 006 the planning period time Tis 10 minutes and kp and ku are 1 minute per pieceSubstituting parameters into equations (23) and (24) weobtain AB1 AB2 AB3 09677 and k1 k2 k3 5

It is also assumed that the first-level workstation is notstarved and the last-level workstation is not blocked LetP00 1 and P4 1 e buffer availability can be calculatedby formulas (8)ndash(11) e specific values are shown inTable 2

e steady-state availability of each equivalent work-station can be obtained by using equation (17) and issummarized in Table 3

By substituting the obtained steady-state availability into(19) the stable availability of the system isA 1 minus (1113937

4i1(1 minus Ai

prime)) 09925 If the expected availabilityis increased to 09999 the estimated availability assigned toeach workstation is calculated according to equations (28)and (29) and the specific values are shown in Table 4

e estimated availability of each workstation is redis-tributed to each unit according to formulas (30) and (31)and the expected unit availability is calculated and shown inTable 5

After obtaining the estimated availability allocation re-sults of each processing unit the Plant Simulation software isused to simulate the production system online to determinewhether the allocation results are correct and reasonableUsing the ldquologisticsrdquo element in the software elementtoolbox the production unit and buffer can be obtainedesimulation model of the production system is shown inFigure 6

We input the corresponding parameters for each unitand buffer and use the event control unit to set the simu-lation time to 30 days 90 days 180 days 360 days and 720days and the corresponding time jump speed is 10000 timesthe real-time value

6 Mathematical Problems in Engineering

Table 5 Expected availability of each unit

unit m1L m1R m2L m2R m3L m3R m4L m4R

Amiprime 06873 06873 06761 06761 06406 06406 07253 07253

Table 1 Failure rate maintenance rate and productivity of each unit

Parameter m1L m1R m2L m2R m3L m3R m4L m4R

λi 00032 00032 00027 00027 00026 00026 0003 0003μi 002 002 003 003 002 002 001 001ωi 5 5 4 4 35 35 3 3

Table 2 Buffer availability

Bi B1 B2 B3 B4

ABi07289 07045 06745 08146

Table 3 Steady-state availability of each equivalent workstation

Miprime M1prime M2prime M3prime M4prime

Aiprime 07089 06978 06625 07462

Table 4 Estimated availability of each workstation

Mi M1 M2 M3 M4

Aciprime 09022 08951 08708 09245

Event controller

m1L

B1 B2 B3DrainSource

m2L m3L m4L

m1R m2R m3R m4R

Figure 6 Simulation model of the production system

Entity

Event controller

m1L

ErEntityB1

ErEntityB2

ErEntityB3

DrainSource

Entitym2L

Entitym3L

Entitym4L

Entitym1R

Entitym2R

Entitym3R

Entitym4R

Figure 7 Operation simulation model of the production system

Mathematical Problems in Engineering 7

We set the parameters of the relevant production unitsand run the software to simulate the whole line e op-eration simulation model is shown in Figure 7

en we complete the simulation test sort out therelevant data and obtain the reliability and performanceindex values e specific values are shown in Table 6

e simulation results show that the reliability allocationresults obtained in this paper satisfy the requirements of thegiven expected availability erefore the reliability indexallocated to each unit can achieve the desired goal and thereliability allocation method is correct

7 Conclusions

(1) Aiming at the reliability allocation problem of theproduction system this paper proposes a reliabilityallocation method that is easy to implement in en-gineering By constructing the proportional rela-tionship between unit (workstation) availability andsystem availability the system availability can beproportionally allocated according to the scale fac-tors is allocation method is simple and easy toimplement and it can solve the reliability allocationproblem of multistage production systems

(2) e availability allocation results can help guide theselection of production units determine the numberof spare parts in the buffer comprehensively con-sider factors such as the failure rate maintenancerate inventory capacity and productivity and ra-tionally select the reliability index of the processingunit

(3) Using the Plant Simulation software to simulate andanalyze the production system one can verify thecorrectness of the allocation method and realize thereliability allocation from the complex productionsystem to the unit

(4) e reliability allocation process without consideringthe influence of the buffer can be extended to generalseries and series-parallel hybrid repairable systemssuch as CNCmachine tools and serve as the basis forcomponent selection or reliability improvement inthe product design stage

Notations

mi Unit of a series system (i 1 2 n)

λi Failure rate of a unitμi Maintenance rate of a unitωi Productivity of a unit

ρi Productivity ratio of a two-level unitai Steady-state availability of mi

A Steady-state availability of a systemmiL andmiR

Units of a hybrid system (i 1 2 n)

mpari Equivalent unit combining miL and miR

apari Steady-state availability of m

pari

ci Scale factor in an unbuffered hybrid systemBi Buffers of a systemki Inventory capacity of Bi(ki 1 2 n)

kimin Minimum inventory capacity of Bi

λB Failure rate of a bufferμB Maintenance rate of a bufferP0i Probability of inventory-freeP0i Probability of inventoryPki

Probability of vacancy-freeP

ki Probability of vacancy

ABi Buffer availability

miprime Equivalent unit combining mi Bi and mi+1 in

the buffer series systemAi Steady-state availability of mi

primeMi Workstation of a buffer hybrid system

(i 1 2 n)

Pai Trouble-free probability of Mi

Pbi Discontinuation probability of Mi

Pci Production-reduction probability of Mi

Miprime Equivalent workstation combining Mi Bi and

Mi+1 in the buffer hybrid systemAiprime Steady-state availability of Mi

primeciprime Scale factor in the buffer hybrid system

kp and ku Output per unit time of M1 and M2Am Target availability of an unbuffered systemAim Assigned availability to each unit in the

unbuffered systemAmprime Target availability of a buffer system

Aciprime and

Aciprime

Assigned availability and unavailability to theequivalent workstation

Amiprime and

Amiprime

Assigned availability and unavailability to theunit in the buffer system

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Table 6 Simulation data of the production system

Time Intrinsic availability () Total throughput Hourly throughput Daily throughput30 days 9999 86386 120 288090 days 9999 259290 121 2881180 days 9999 518760 120 2882360 days 9999 1037520 120 2882720 days 9999 2075040 120 2882

8 Mathematical Problems in Engineering

Acknowledgments

e research in this paper was supported by the NationalScience and Technology Major Project of China (Grant no2014ZX04015031) Science and Technology DevelopmentPlan Project of Jilin Province (Grant no 20180520068JH)and Jilin Province Education Departmentrsquos irteenth Five-Year Plan Science and Technology Project (Grant noJJKH20180079KJ)

References

[1] Z J Yang Q B Hao C Fei et al ldquoA comprehensive fuzzyreliability allocation method of NC machine tools based oninterval analysisrdquo Journal of Beijing University of Technologyvol 37 no 3 pp 321ndash329 2011

[2] P-C Chang ldquoReliability estimation for a stochastic pro-duction system with finite buffer storage by a simulationapproachrdquo Annals of Operations Research vol 277 no 1pp 119ndash133 2019

[3] G Liberopoulos ldquoPerformance evaluation of a productionline operated under an echelon buffer policyrdquo IISE Trans-actions vol 50 no 3 pp 161ndash177 2018

[4] S Weiss J A Schwarz and R Stolletz ldquoe buffer allocationproblem in production lines formulations solution methodsand instancesrdquo IISE Transactions vol 51 no 5 pp 456ndash4852019

[5] J Duan A P Li X Nan et al ldquoMulti-state reliabilitymodeling and analysis of reconfigurable manufacturing sys-temsrdquo Journal of Mechanical Engineering vol 47 no 17pp 104ndash111 2011

[6] J L Li Y H Jia F Xu F Chen Z Yang and X Li ldquoAnimprovement scheme for the overall line effectiveness of aproduction line a case studyrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Singapore April 2018

[7] G F Li Y Li X G Zang et al ldquoDevelopment of a preventivemaintenance strategy for an automatic production line basedon group maintenance methodrdquo Applied Sciences vol 8no 10 p 1781 2018

[8] G F Li C Hou G F Liu Y Jia C Liu and J DongldquoReliability allocation method of production line based onfuzzy comprehensive evaluationrdquo in Proceedings of the In-ternational Conference on Materials EngineeringManufacturing Technology and Control (ICMEMTC)Taiyuan China April 2016

[9] Y H Jia Z J Yang G F Li et al ldquoAn efficient semi-analyticalsimulation for availability evaluation of discrete productionlines with unreliable machinesrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Paris France November 2016

[10] F Chen B Liu and B B Xu ldquoResearch on ReliabilityEvaluation of Engine Cylinder Flexible Production Line Basedon AHP Fuzzy Comprehensive Evaluation Methodrdquo ChinaMechanical Engineering Society Changzhou China August2015

[11] F Long P Zeiler and B Bertsche ldquoModelling the flexibility ofproduction systems in industry 40 for analysing their pro-ductivity and availability with high-level Petri netsrdquo IFAC-PapersOnLine vol 50 no 1 pp 5680ndash5687 2017

[12] L Gorkemli and S Kapan Ulusoy ldquoFuzzy bayesian reliabilityand availability analysis of production systemsrdquo Computers ampIndustrial Engineering vol 59 no 4 pp 690ndash696 2010

[13] S Rebello H Yu and L Ma ldquoAn integrated approach forsystem functional reliability assessment using dynamicbayesian network and hidden Markov modelrdquo ReliabilityEngineering amp System Safety vol 180 no 12 pp 124ndash1352018

[14] K Heungseob and K Pansoo ldquoReliability models for anonrepairable systemwith heterogeneous components havinga phase-type time-to-failure distributionrdquo Reliability Engi-neering and System Safety vol 159 no 3 pp 37ndash46 2017

[15] M K Loganathan G Kumar and O P Gandhi ldquoAvailabilityevaluation of manufacturing systems using semi-markovmodelrdquo International Journal of Computer IntegratedManufacturing vol 29 no 7 pp 720ndash735 2016

[16] X-Y Li H-Z Huang and Y-F Li ldquoReliability analysis ofphased mission system with non-exponential and partiallyrepairable componentsrdquo Reliability Engineering amp SystemSafety vol 175 no 7 pp 119ndash127 2018

[17] A B Hu and S M Meerkov ldquoLean buffering in serial pro-duction lines with Bernoulli machinesrdquo MathematicalProblems in Engineering vol 2006 Article ID 17105 24 pages2006

[18] L Demir S Tunalı and D T Eliiyi ldquoAn adaptive tabu searchapproach for buffer allocation problem in unreliable non-homogenous production linesrdquo Computers amp OperationsResearch vol 39 no 7 pp 1477ndash1486 2012

[19] R Li X Liu and N Huang ldquoAvailability allocation of net-worked systems using Markov model and heuristics algo-rithmrdquo Mathematical Problems in Engineering vol 2014Article ID 315385 9 pages 2014

[20] W X Qian X W Yin and L Y Xie ldquoSystem reliabilityallocation based on bayesian networkrdquo Applied Mathematicsamp Information Sciences vol 6 no 3 pp 681ndash687 2012

[21] M Xiao and Y P Zhang ldquoParameters learning of bayesiannetworks for multistate system with small samplerdquo ComputerScience vol 42 no 4 pp 253ndash257 2015

[22] V Sriramdas S K Chaturvedi and H Gargama ldquoFuzzyarithmetic based reliability allocation approach during earlydesign and developmentrdquo Expert Systems with Applicationsvol 41 no 7 pp 3444ndash3449 2014

[23] S G Shu ldquoe manufacturing system (CIMS) with buffersand a study of the system reliabilityrdquo Acta Automatica Sinicavol 18 no 1 pp 15ndash20 1992

[24] H R Zhou S H Wang L Zhang et al ldquoModeling of in-process inventory in continuous production based on reli-abilityrdquo Machine Tool amp Hydraulics vol 39 no 9 pp 111ndash113 2011

[25] M Y You and J C Zheng ldquoA synthetic approach to reliabilityallocation of electronic systems based on AGREE method andthe extended proportional combination methodrdquo ElectronicProduct Reliability and Environmental Testing vol 29 no 4pp 1ndash6 2011

Mathematical Problems in Engineering 9

Table 5 Expected availability of each unit

unit m1L m1R m2L m2R m3L m3R m4L m4R

Amiprime 06873 06873 06761 06761 06406 06406 07253 07253

Table 1 Failure rate maintenance rate and productivity of each unit

Parameter m1L m1R m2L m2R m3L m3R m4L m4R

λi 00032 00032 00027 00027 00026 00026 0003 0003μi 002 002 003 003 002 002 001 001ωi 5 5 4 4 35 35 3 3

Table 2 Buffer availability

Bi B1 B2 B3 B4

ABi07289 07045 06745 08146

Table 3 Steady-state availability of each equivalent workstation

Miprime M1prime M2prime M3prime M4prime

Aiprime 07089 06978 06625 07462

Table 4 Estimated availability of each workstation

Mi M1 M2 M3 M4

Aciprime 09022 08951 08708 09245

Event controller

m1L

B1 B2 B3DrainSource

m2L m3L m4L

m1R m2R m3R m4R

Figure 6 Simulation model of the production system

Entity

Event controller

m1L

ErEntityB1

ErEntityB2

ErEntityB3

DrainSource

Entitym2L

Entitym3L

Entitym4L

Entitym1R

Entitym2R

Entitym3R

Entitym4R

Figure 7 Operation simulation model of the production system

Mathematical Problems in Engineering 7

We set the parameters of the relevant production unitsand run the software to simulate the whole line e op-eration simulation model is shown in Figure 7

en we complete the simulation test sort out therelevant data and obtain the reliability and performanceindex values e specific values are shown in Table 6

e simulation results show that the reliability allocationresults obtained in this paper satisfy the requirements of thegiven expected availability erefore the reliability indexallocated to each unit can achieve the desired goal and thereliability allocation method is correct

7 Conclusions

(1) Aiming at the reliability allocation problem of theproduction system this paper proposes a reliabilityallocation method that is easy to implement in en-gineering By constructing the proportional rela-tionship between unit (workstation) availability andsystem availability the system availability can beproportionally allocated according to the scale fac-tors is allocation method is simple and easy toimplement and it can solve the reliability allocationproblem of multistage production systems

(2) e availability allocation results can help guide theselection of production units determine the numberof spare parts in the buffer comprehensively con-sider factors such as the failure rate maintenancerate inventory capacity and productivity and ra-tionally select the reliability index of the processingunit

(3) Using the Plant Simulation software to simulate andanalyze the production system one can verify thecorrectness of the allocation method and realize thereliability allocation from the complex productionsystem to the unit

(4) e reliability allocation process without consideringthe influence of the buffer can be extended to generalseries and series-parallel hybrid repairable systemssuch as CNCmachine tools and serve as the basis forcomponent selection or reliability improvement inthe product design stage

Notations

mi Unit of a series system (i 1 2 n)

λi Failure rate of a unitμi Maintenance rate of a unitωi Productivity of a unit

ρi Productivity ratio of a two-level unitai Steady-state availability of mi

A Steady-state availability of a systemmiL andmiR

Units of a hybrid system (i 1 2 n)

mpari Equivalent unit combining miL and miR

apari Steady-state availability of m

pari

ci Scale factor in an unbuffered hybrid systemBi Buffers of a systemki Inventory capacity of Bi(ki 1 2 n)

kimin Minimum inventory capacity of Bi

λB Failure rate of a bufferμB Maintenance rate of a bufferP0i Probability of inventory-freeP0i Probability of inventoryPki

Probability of vacancy-freeP

ki Probability of vacancy

ABi Buffer availability

miprime Equivalent unit combining mi Bi and mi+1 in

the buffer series systemAi Steady-state availability of mi

primeMi Workstation of a buffer hybrid system

(i 1 2 n)

Pai Trouble-free probability of Mi

Pbi Discontinuation probability of Mi

Pci Production-reduction probability of Mi

Miprime Equivalent workstation combining Mi Bi and

Mi+1 in the buffer hybrid systemAiprime Steady-state availability of Mi

primeciprime Scale factor in the buffer hybrid system

kp and ku Output per unit time of M1 and M2Am Target availability of an unbuffered systemAim Assigned availability to each unit in the

unbuffered systemAmprime Target availability of a buffer system

Aciprime and

Aciprime

Assigned availability and unavailability to theequivalent workstation

Amiprime and

Amiprime

Assigned availability and unavailability to theunit in the buffer system

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Table 6 Simulation data of the production system

Time Intrinsic availability () Total throughput Hourly throughput Daily throughput30 days 9999 86386 120 288090 days 9999 259290 121 2881180 days 9999 518760 120 2882360 days 9999 1037520 120 2882720 days 9999 2075040 120 2882

8 Mathematical Problems in Engineering

Acknowledgments

e research in this paper was supported by the NationalScience and Technology Major Project of China (Grant no2014ZX04015031) Science and Technology DevelopmentPlan Project of Jilin Province (Grant no 20180520068JH)and Jilin Province Education Departmentrsquos irteenth Five-Year Plan Science and Technology Project (Grant noJJKH20180079KJ)

References

[1] Z J Yang Q B Hao C Fei et al ldquoA comprehensive fuzzyreliability allocation method of NC machine tools based oninterval analysisrdquo Journal of Beijing University of Technologyvol 37 no 3 pp 321ndash329 2011

[2] P-C Chang ldquoReliability estimation for a stochastic pro-duction system with finite buffer storage by a simulationapproachrdquo Annals of Operations Research vol 277 no 1pp 119ndash133 2019

[3] G Liberopoulos ldquoPerformance evaluation of a productionline operated under an echelon buffer policyrdquo IISE Trans-actions vol 50 no 3 pp 161ndash177 2018

[4] S Weiss J A Schwarz and R Stolletz ldquoe buffer allocationproblem in production lines formulations solution methodsand instancesrdquo IISE Transactions vol 51 no 5 pp 456ndash4852019

[5] J Duan A P Li X Nan et al ldquoMulti-state reliabilitymodeling and analysis of reconfigurable manufacturing sys-temsrdquo Journal of Mechanical Engineering vol 47 no 17pp 104ndash111 2011

[6] J L Li Y H Jia F Xu F Chen Z Yang and X Li ldquoAnimprovement scheme for the overall line effectiveness of aproduction line a case studyrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Singapore April 2018

[7] G F Li Y Li X G Zang et al ldquoDevelopment of a preventivemaintenance strategy for an automatic production line basedon group maintenance methodrdquo Applied Sciences vol 8no 10 p 1781 2018

[8] G F Li C Hou G F Liu Y Jia C Liu and J DongldquoReliability allocation method of production line based onfuzzy comprehensive evaluationrdquo in Proceedings of the In-ternational Conference on Materials EngineeringManufacturing Technology and Control (ICMEMTC)Taiyuan China April 2016

[9] Y H Jia Z J Yang G F Li et al ldquoAn efficient semi-analyticalsimulation for availability evaluation of discrete productionlines with unreliable machinesrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Paris France November 2016

[10] F Chen B Liu and B B Xu ldquoResearch on ReliabilityEvaluation of Engine Cylinder Flexible Production Line Basedon AHP Fuzzy Comprehensive Evaluation Methodrdquo ChinaMechanical Engineering Society Changzhou China August2015

[11] F Long P Zeiler and B Bertsche ldquoModelling the flexibility ofproduction systems in industry 40 for analysing their pro-ductivity and availability with high-level Petri netsrdquo IFAC-PapersOnLine vol 50 no 1 pp 5680ndash5687 2017

[12] L Gorkemli and S Kapan Ulusoy ldquoFuzzy bayesian reliabilityand availability analysis of production systemsrdquo Computers ampIndustrial Engineering vol 59 no 4 pp 690ndash696 2010

[13] S Rebello H Yu and L Ma ldquoAn integrated approach forsystem functional reliability assessment using dynamicbayesian network and hidden Markov modelrdquo ReliabilityEngineering amp System Safety vol 180 no 12 pp 124ndash1352018

[14] K Heungseob and K Pansoo ldquoReliability models for anonrepairable systemwith heterogeneous components havinga phase-type time-to-failure distributionrdquo Reliability Engi-neering and System Safety vol 159 no 3 pp 37ndash46 2017

[15] M K Loganathan G Kumar and O P Gandhi ldquoAvailabilityevaluation of manufacturing systems using semi-markovmodelrdquo International Journal of Computer IntegratedManufacturing vol 29 no 7 pp 720ndash735 2016

[16] X-Y Li H-Z Huang and Y-F Li ldquoReliability analysis ofphased mission system with non-exponential and partiallyrepairable componentsrdquo Reliability Engineering amp SystemSafety vol 175 no 7 pp 119ndash127 2018

[17] A B Hu and S M Meerkov ldquoLean buffering in serial pro-duction lines with Bernoulli machinesrdquo MathematicalProblems in Engineering vol 2006 Article ID 17105 24 pages2006

[18] L Demir S Tunalı and D T Eliiyi ldquoAn adaptive tabu searchapproach for buffer allocation problem in unreliable non-homogenous production linesrdquo Computers amp OperationsResearch vol 39 no 7 pp 1477ndash1486 2012

[19] R Li X Liu and N Huang ldquoAvailability allocation of net-worked systems using Markov model and heuristics algo-rithmrdquo Mathematical Problems in Engineering vol 2014Article ID 315385 9 pages 2014

[20] W X Qian X W Yin and L Y Xie ldquoSystem reliabilityallocation based on bayesian networkrdquo Applied Mathematicsamp Information Sciences vol 6 no 3 pp 681ndash687 2012

[21] M Xiao and Y P Zhang ldquoParameters learning of bayesiannetworks for multistate system with small samplerdquo ComputerScience vol 42 no 4 pp 253ndash257 2015

[22] V Sriramdas S K Chaturvedi and H Gargama ldquoFuzzyarithmetic based reliability allocation approach during earlydesign and developmentrdquo Expert Systems with Applicationsvol 41 no 7 pp 3444ndash3449 2014

[23] S G Shu ldquoe manufacturing system (CIMS) with buffersand a study of the system reliabilityrdquo Acta Automatica Sinicavol 18 no 1 pp 15ndash20 1992

[24] H R Zhou S H Wang L Zhang et al ldquoModeling of in-process inventory in continuous production based on reli-abilityrdquo Machine Tool amp Hydraulics vol 39 no 9 pp 111ndash113 2011

[25] M Y You and J C Zheng ldquoA synthetic approach to reliabilityallocation of electronic systems based on AGREE method andthe extended proportional combination methodrdquo ElectronicProduct Reliability and Environmental Testing vol 29 no 4pp 1ndash6 2011

Mathematical Problems in Engineering 9

We set the parameters of the relevant production unitsand run the software to simulate the whole line e op-eration simulation model is shown in Figure 7

en we complete the simulation test sort out therelevant data and obtain the reliability and performanceindex values e specific values are shown in Table 6

e simulation results show that the reliability allocationresults obtained in this paper satisfy the requirements of thegiven expected availability erefore the reliability indexallocated to each unit can achieve the desired goal and thereliability allocation method is correct

7 Conclusions

(1) Aiming at the reliability allocation problem of theproduction system this paper proposes a reliabilityallocation method that is easy to implement in en-gineering By constructing the proportional rela-tionship between unit (workstation) availability andsystem availability the system availability can beproportionally allocated according to the scale fac-tors is allocation method is simple and easy toimplement and it can solve the reliability allocationproblem of multistage production systems

(2) e availability allocation results can help guide theselection of production units determine the numberof spare parts in the buffer comprehensively con-sider factors such as the failure rate maintenancerate inventory capacity and productivity and ra-tionally select the reliability index of the processingunit

(3) Using the Plant Simulation software to simulate andanalyze the production system one can verify thecorrectness of the allocation method and realize thereliability allocation from the complex productionsystem to the unit

(4) e reliability allocation process without consideringthe influence of the buffer can be extended to generalseries and series-parallel hybrid repairable systemssuch as CNCmachine tools and serve as the basis forcomponent selection or reliability improvement inthe product design stage

Notations

mi Unit of a series system (i 1 2 n)

λi Failure rate of a unitμi Maintenance rate of a unitωi Productivity of a unit

ρi Productivity ratio of a two-level unitai Steady-state availability of mi

A Steady-state availability of a systemmiL andmiR

Units of a hybrid system (i 1 2 n)

mpari Equivalent unit combining miL and miR

apari Steady-state availability of m

pari

ci Scale factor in an unbuffered hybrid systemBi Buffers of a systemki Inventory capacity of Bi(ki 1 2 n)

kimin Minimum inventory capacity of Bi

λB Failure rate of a bufferμB Maintenance rate of a bufferP0i Probability of inventory-freeP0i Probability of inventoryPki

Probability of vacancy-freeP

ki Probability of vacancy

ABi Buffer availability

miprime Equivalent unit combining mi Bi and mi+1 in

the buffer series systemAi Steady-state availability of mi

primeMi Workstation of a buffer hybrid system

(i 1 2 n)

Pai Trouble-free probability of Mi

Pbi Discontinuation probability of Mi

Pci Production-reduction probability of Mi

Miprime Equivalent workstation combining Mi Bi and

Mi+1 in the buffer hybrid systemAiprime Steady-state availability of Mi

primeciprime Scale factor in the buffer hybrid system

kp and ku Output per unit time of M1 and M2Am Target availability of an unbuffered systemAim Assigned availability to each unit in the

unbuffered systemAmprime Target availability of a buffer system

Aciprime and

Aciprime

Assigned availability and unavailability to theequivalent workstation

Amiprime and

Amiprime

Assigned availability and unavailability to theunit in the buffer system

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Table 6 Simulation data of the production system

Time Intrinsic availability () Total throughput Hourly throughput Daily throughput30 days 9999 86386 120 288090 days 9999 259290 121 2881180 days 9999 518760 120 2882360 days 9999 1037520 120 2882720 days 9999 2075040 120 2882

8 Mathematical Problems in Engineering

Acknowledgments

e research in this paper was supported by the NationalScience and Technology Major Project of China (Grant no2014ZX04015031) Science and Technology DevelopmentPlan Project of Jilin Province (Grant no 20180520068JH)and Jilin Province Education Departmentrsquos irteenth Five-Year Plan Science and Technology Project (Grant noJJKH20180079KJ)

References

[1] Z J Yang Q B Hao C Fei et al ldquoA comprehensive fuzzyreliability allocation method of NC machine tools based oninterval analysisrdquo Journal of Beijing University of Technologyvol 37 no 3 pp 321ndash329 2011

[2] P-C Chang ldquoReliability estimation for a stochastic pro-duction system with finite buffer storage by a simulationapproachrdquo Annals of Operations Research vol 277 no 1pp 119ndash133 2019

[3] G Liberopoulos ldquoPerformance evaluation of a productionline operated under an echelon buffer policyrdquo IISE Trans-actions vol 50 no 3 pp 161ndash177 2018

[4] S Weiss J A Schwarz and R Stolletz ldquoe buffer allocationproblem in production lines formulations solution methodsand instancesrdquo IISE Transactions vol 51 no 5 pp 456ndash4852019

[5] J Duan A P Li X Nan et al ldquoMulti-state reliabilitymodeling and analysis of reconfigurable manufacturing sys-temsrdquo Journal of Mechanical Engineering vol 47 no 17pp 104ndash111 2011

[6] J L Li Y H Jia F Xu F Chen Z Yang and X Li ldquoAnimprovement scheme for the overall line effectiveness of aproduction line a case studyrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Singapore April 2018

[7] G F Li Y Li X G Zang et al ldquoDevelopment of a preventivemaintenance strategy for an automatic production line basedon group maintenance methodrdquo Applied Sciences vol 8no 10 p 1781 2018

[8] G F Li C Hou G F Liu Y Jia C Liu and J DongldquoReliability allocation method of production line based onfuzzy comprehensive evaluationrdquo in Proceedings of the In-ternational Conference on Materials EngineeringManufacturing Technology and Control (ICMEMTC)Taiyuan China April 2016

[9] Y H Jia Z J Yang G F Li et al ldquoAn efficient semi-analyticalsimulation for availability evaluation of discrete productionlines with unreliable machinesrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Paris France November 2016

[10] F Chen B Liu and B B Xu ldquoResearch on ReliabilityEvaluation of Engine Cylinder Flexible Production Line Basedon AHP Fuzzy Comprehensive Evaluation Methodrdquo ChinaMechanical Engineering Society Changzhou China August2015

[11] F Long P Zeiler and B Bertsche ldquoModelling the flexibility ofproduction systems in industry 40 for analysing their pro-ductivity and availability with high-level Petri netsrdquo IFAC-PapersOnLine vol 50 no 1 pp 5680ndash5687 2017

[12] L Gorkemli and S Kapan Ulusoy ldquoFuzzy bayesian reliabilityand availability analysis of production systemsrdquo Computers ampIndustrial Engineering vol 59 no 4 pp 690ndash696 2010

[13] S Rebello H Yu and L Ma ldquoAn integrated approach forsystem functional reliability assessment using dynamicbayesian network and hidden Markov modelrdquo ReliabilityEngineering amp System Safety vol 180 no 12 pp 124ndash1352018

[14] K Heungseob and K Pansoo ldquoReliability models for anonrepairable systemwith heterogeneous components havinga phase-type time-to-failure distributionrdquo Reliability Engi-neering and System Safety vol 159 no 3 pp 37ndash46 2017

[15] M K Loganathan G Kumar and O P Gandhi ldquoAvailabilityevaluation of manufacturing systems using semi-markovmodelrdquo International Journal of Computer IntegratedManufacturing vol 29 no 7 pp 720ndash735 2016

[16] X-Y Li H-Z Huang and Y-F Li ldquoReliability analysis ofphased mission system with non-exponential and partiallyrepairable componentsrdquo Reliability Engineering amp SystemSafety vol 175 no 7 pp 119ndash127 2018

[17] A B Hu and S M Meerkov ldquoLean buffering in serial pro-duction lines with Bernoulli machinesrdquo MathematicalProblems in Engineering vol 2006 Article ID 17105 24 pages2006

[18] L Demir S Tunalı and D T Eliiyi ldquoAn adaptive tabu searchapproach for buffer allocation problem in unreliable non-homogenous production linesrdquo Computers amp OperationsResearch vol 39 no 7 pp 1477ndash1486 2012

[19] R Li X Liu and N Huang ldquoAvailability allocation of net-worked systems using Markov model and heuristics algo-rithmrdquo Mathematical Problems in Engineering vol 2014Article ID 315385 9 pages 2014

[20] W X Qian X W Yin and L Y Xie ldquoSystem reliabilityallocation based on bayesian networkrdquo Applied Mathematicsamp Information Sciences vol 6 no 3 pp 681ndash687 2012

[21] M Xiao and Y P Zhang ldquoParameters learning of bayesiannetworks for multistate system with small samplerdquo ComputerScience vol 42 no 4 pp 253ndash257 2015

[22] V Sriramdas S K Chaturvedi and H Gargama ldquoFuzzyarithmetic based reliability allocation approach during earlydesign and developmentrdquo Expert Systems with Applicationsvol 41 no 7 pp 3444ndash3449 2014

[23] S G Shu ldquoe manufacturing system (CIMS) with buffersand a study of the system reliabilityrdquo Acta Automatica Sinicavol 18 no 1 pp 15ndash20 1992

[24] H R Zhou S H Wang L Zhang et al ldquoModeling of in-process inventory in continuous production based on reli-abilityrdquo Machine Tool amp Hydraulics vol 39 no 9 pp 111ndash113 2011

[25] M Y You and J C Zheng ldquoA synthetic approach to reliabilityallocation of electronic systems based on AGREE method andthe extended proportional combination methodrdquo ElectronicProduct Reliability and Environmental Testing vol 29 no 4pp 1ndash6 2011

Mathematical Problems in Engineering 9

Acknowledgments

e research in this paper was supported by the NationalScience and Technology Major Project of China (Grant no2014ZX04015031) Science and Technology DevelopmentPlan Project of Jilin Province (Grant no 20180520068JH)and Jilin Province Education Departmentrsquos irteenth Five-Year Plan Science and Technology Project (Grant noJJKH20180079KJ)

References

[1] Z J Yang Q B Hao C Fei et al ldquoA comprehensive fuzzyreliability allocation method of NC machine tools based oninterval analysisrdquo Journal of Beijing University of Technologyvol 37 no 3 pp 321ndash329 2011

[2] P-C Chang ldquoReliability estimation for a stochastic pro-duction system with finite buffer storage by a simulationapproachrdquo Annals of Operations Research vol 277 no 1pp 119ndash133 2019

[3] G Liberopoulos ldquoPerformance evaluation of a productionline operated under an echelon buffer policyrdquo IISE Trans-actions vol 50 no 3 pp 161ndash177 2018

[4] S Weiss J A Schwarz and R Stolletz ldquoe buffer allocationproblem in production lines formulations solution methodsand instancesrdquo IISE Transactions vol 51 no 5 pp 456ndash4852019

[5] J Duan A P Li X Nan et al ldquoMulti-state reliabilitymodeling and analysis of reconfigurable manufacturing sys-temsrdquo Journal of Mechanical Engineering vol 47 no 17pp 104ndash111 2011

[6] J L Li Y H Jia F Xu F Chen Z Yang and X Li ldquoAnimprovement scheme for the overall line effectiveness of aproduction line a case studyrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Singapore April 2018

[7] G F Li Y Li X G Zang et al ldquoDevelopment of a preventivemaintenance strategy for an automatic production line basedon group maintenance methodrdquo Applied Sciences vol 8no 10 p 1781 2018

[8] G F Li C Hou G F Liu Y Jia C Liu and J DongldquoReliability allocation method of production line based onfuzzy comprehensive evaluationrdquo in Proceedings of the In-ternational Conference on Materials EngineeringManufacturing Technology and Control (ICMEMTC)Taiyuan China April 2016

[9] Y H Jia Z J Yang G F Li et al ldquoAn efficient semi-analyticalsimulation for availability evaluation of discrete productionlines with unreliable machinesrdquo in Proceedings of the Inter-national Conference on System Reliability and Science (ICSRS)Paris France November 2016

[10] F Chen B Liu and B B Xu ldquoResearch on ReliabilityEvaluation of Engine Cylinder Flexible Production Line Basedon AHP Fuzzy Comprehensive Evaluation Methodrdquo ChinaMechanical Engineering Society Changzhou China August2015

[11] F Long P Zeiler and B Bertsche ldquoModelling the flexibility ofproduction systems in industry 40 for analysing their pro-ductivity and availability with high-level Petri netsrdquo IFAC-PapersOnLine vol 50 no 1 pp 5680ndash5687 2017

[12] L Gorkemli and S Kapan Ulusoy ldquoFuzzy bayesian reliabilityand availability analysis of production systemsrdquo Computers ampIndustrial Engineering vol 59 no 4 pp 690ndash696 2010

[13] S Rebello H Yu and L Ma ldquoAn integrated approach forsystem functional reliability assessment using dynamicbayesian network and hidden Markov modelrdquo ReliabilityEngineering amp System Safety vol 180 no 12 pp 124ndash1352018

[14] K Heungseob and K Pansoo ldquoReliability models for anonrepairable systemwith heterogeneous components havinga phase-type time-to-failure distributionrdquo Reliability Engi-neering and System Safety vol 159 no 3 pp 37ndash46 2017

[15] M K Loganathan G Kumar and O P Gandhi ldquoAvailabilityevaluation of manufacturing systems using semi-markovmodelrdquo International Journal of Computer IntegratedManufacturing vol 29 no 7 pp 720ndash735 2016

[16] X-Y Li H-Z Huang and Y-F Li ldquoReliability analysis ofphased mission system with non-exponential and partiallyrepairable componentsrdquo Reliability Engineering amp SystemSafety vol 175 no 7 pp 119ndash127 2018

[17] A B Hu and S M Meerkov ldquoLean buffering in serial pro-duction lines with Bernoulli machinesrdquo MathematicalProblems in Engineering vol 2006 Article ID 17105 24 pages2006

[18] L Demir S Tunalı and D T Eliiyi ldquoAn adaptive tabu searchapproach for buffer allocation problem in unreliable non-homogenous production linesrdquo Computers amp OperationsResearch vol 39 no 7 pp 1477ndash1486 2012

[19] R Li X Liu and N Huang ldquoAvailability allocation of net-worked systems using Markov model and heuristics algo-rithmrdquo Mathematical Problems in Engineering vol 2014Article ID 315385 9 pages 2014

[20] W X Qian X W Yin and L Y Xie ldquoSystem reliabilityallocation based on bayesian networkrdquo Applied Mathematicsamp Information Sciences vol 6 no 3 pp 681ndash687 2012

[21] M Xiao and Y P Zhang ldquoParameters learning of bayesiannetworks for multistate system with small samplerdquo ComputerScience vol 42 no 4 pp 253ndash257 2015

[22] V Sriramdas S K Chaturvedi and H Gargama ldquoFuzzyarithmetic based reliability allocation approach during earlydesign and developmentrdquo Expert Systems with Applicationsvol 41 no 7 pp 3444ndash3449 2014

[23] S G Shu ldquoe manufacturing system (CIMS) with buffersand a study of the system reliabilityrdquo Acta Automatica Sinicavol 18 no 1 pp 15ndash20 1992

[24] H R Zhou S H Wang L Zhang et al ldquoModeling of in-process inventory in continuous production based on reli-abilityrdquo Machine Tool amp Hydraulics vol 39 no 9 pp 111ndash113 2011

[25] M Y You and J C Zheng ldquoA synthetic approach to reliabilityallocation of electronic systems based on AGREE method andthe extended proportional combination methodrdquo ElectronicProduct Reliability and Environmental Testing vol 29 no 4pp 1ndash6 2011

Mathematical Problems in Engineering 9