Research ArticleResearch on Key Technologies of Unit-Based CNCMachine Tool Assembly Design

Zhongqi Sheng Lei Zhang Hualong Xie and Changchun Liu

College of Mechanical Engineering amp Automation Northeastern University Shenyang 110819 China

Correspondence should be addressed to Zhongqi Sheng zhongqishengoutlookcom

Received 26 May 2014 Accepted 18 August 2014 Published 8 December 2014

Academic Editor Xue Chen

Copyright copy 2014 Zhongqi Sheng et alThis 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

Assembly is the part that produces themaximumworkload and consumed time during product design andmanufacturing processCNC machine tool is the key basic equipment in manufacturing industry and research on assembly design technologies of CNCmachine tool has theoretical significance and practical value This study established a simplified ASRG for CNCmachine tool Theconnection between parts semantic information of transmission and geometric constraint informationwere quantified to assemblyconnection strength to depict the assembling difficulty level The transmissibility based on trust relationship was applied on theassembly connection strength Assembly unit partition based on assembly connection strength was conducted and interferentialassembly units were identified and revised The assembly sequence planning and optimization of parts in each assembly unit andbetween assembly units was conducted using genetic algorithmWith certain type of high speedCNC turning center as an examplethis paper explored into the assembly modeling assembly unit partition and assembly sequence planning and optimization andrealized the optimized assembly sequence of headstock of CNC machine tool

1 Introduction

CNC machine tool is key basic equipment in manufac-turing industry and a carrier of advanced manufacturingtechnology CNC machine tool manufacturing enterprisesmust improve the product quality and shorten the productdesign and manufacturing cycle in order to occupy vantageground in fierce market competition Assembly refers tothe connection and formation of a group of scattered partsfollowing a rational technological process and requirementswhich form products with specific functions Assembly is thepart that produces the maximum workload and consumedtime during product design and manufacturing processwhich directly affects the quality and reliability of final prod-ucts Efficient assembly performance is of great significanceto improve assembly efficiency reduce assembly costs andensure assembly quality of products

Assembly is the main line throughout entire prod-uct development process involving design manufacturingmaintenance recycling and other aspects of product lifecy-cle Assembly automation is always a ldquobottleneckrdquo of manu-facturing industry automation With increasing complexity

of products it is needed to develop new methods andtechniques to increase assembly efficiency and quality Thusresearch on the technologies of CNC machine tool assemblydesign is of great theoretical significance and practical value

2 Literature Review

Assembly design refers to the process from structural designand detailed design of products to final assembly modelof products according to conceptual design Research ofunit-based assembly design mainly comprises three partsassembly modeling assembly unit partition and assemblysequence planning

21 Assembly Modeling Assembly model is a key componentof product information model and a premise to realizeassembly design A complete assembly model should supportall assembly-related activities in the processes of productdesignmanufacturingmaintenance and recycling It shouldbe able to transmit the assembly information completely andcorrectly in product lifecycle In assemblymodel the integrity

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 191069 12 pageshttpdxdoiorg1011552014191069

2 Mathematical Problems in Engineering

and rationality of assembly information determine whetheran assembly sequence can be correctly generated

Blanchot and Daidie [1] introduced the adjustment ofa numerical model simulating a riveted link using differentapproaches and presented the simulation of riveting processand its influence on the riveted link behaviour Gu et al[2] represented subassemblies assembly states and assem-bly tasks as Boolean characteristic functions and proposedsymbolic OBDD (ordered binary decision diagram) schemefor all feasible assembly sequences Xu et al [3] simplifiedassembly resource established amatrix of polychromatic setsand set up a dynamic assembly model based on the theory ofpolychromatic sets Guo et al [4] explored into the layeredassembly model under the complex constraint condition andproposed a layered constraint assembly model based on theattributes of assembly objects

22 Assembly Unit Partition As the diversity and complexityof product assembly structure and process the assemblysequence planning is complex The partition of complexproducts into assembly units with fewer parts and takingassembly unit as the study object could effectively overcomethe issue of ldquocombinatorial explosionrdquo and reduce the diffi-culty of assembly sequence planning

Gottipolu and Ghosh [5] described an approach for gen-eration representation and selection of assembly sequencealternatives in which the geometric and mobility constraintsextracted directly from the CAD model of the assemblywere translated into two types of unidirectional matricesthe contact and the translational functions Ko et al [6]presented an assembly-decomposition model to improveproduct quality and used mixed-integer programming topartition the liaison graph of a product assembly with theconsideration of defect rates in components and assemblytasks Wu [7] proposed the connection intensity to expressthe tightness of assembly used the fuzzy clustering for partsclustering and established the identification of subassemblyunits and partition method Wang and Liu [8] quantifiedthe constraints of assembly units and proposed the decision-making diagram of assembly unit based on fuzzy analytichierarchy process

23 Assembly Sequence Planning The research on assemblysequence planning can be divided into two main categories(1) according to constraint condition the feasible assemblysequence is generated through reasoning and optimization(2) Using the modern optimization methods feasible assem-bly sequence is generated and simultaneously filtered

Wang et al [9] coupled the solution space generation anddisassembly solution optimization into one technical frame-work represented all assemblydisassembly sequences in theDFIG (disassembly feasibility information graph) modeland proposed an integrated methodology for mechanicalassembly planning Sinanoglu and Borklu [10] proposedan approach for the development of a process for thedetermination of geometric feasibility whose binary vectorrepresentation corresponds to assembly states Zhong et al[11] presented a determination system of assembly units forthe hull structure in order to shorten construction cycle

time and construction quality in shipbuilding and introducedthe assessment model by fuzzy synthetic evaluation Wanget al [12] suggested assembly sequences merging based onassembly unit partitioning to reduce the searching spaceof assembly sequence planning of complex products andcomprehensively took into account the assembly designconstraints and the assembly process constraints

With the research object of the headstock of HTC2550hsCNC turning center this paper conducted assembly mod-eling assembly unit partition assembly sequence planningand other tasks In Section 3 the simplified assembly seman-tic relation graph (ASRG) for CNCmachine tool was createdand the assembly relationship among parts was quanti-fied into assembly connection strength Section 4 proposedthe transmissibility of assembly connection strength Theassembly unit partition of CNC machine tool was realizedSection 5 adopted genetic algorithm for optimized assemblysequence of parts in assembly units and each assembly unitrespectively Finally the concurrent assembly sequence wasgenerated Section 6 took the headstock of HTC2550hs CNCturning center as the example and verified the technique ofconcurrent assembly sequence planning Section 7 provideda conclusion of the research

3 Assembly Modeling Based onAssembly Semantics

The assembly model of product is the foundation of assemblysequence planning The more the complete the informationincluded in a model is the higher the efficiency of assemblysequence planning is Meanwhile the assembly modelingbecomes more difficult In the field of assembly designdesigners tend to express the assembly relationship amongparts by assembly semantics The basic concept of assemblymodeling based on assembly semantics is to integrate theassembly engineering semantic analysis method with object-oriented techniques which can be used in the assemblymodeling the expression of assembly relationship and theproduct model description

31 Assembly Semantics Assembly units contain rich assem-bly design information which comprises assembly semanticsAssembly semantics are the abstract expressions of assem-bly relationship between assembly units and parts duringassembly design which embody the information such aspositioning constraints and engineering constraints betweenassembly units and parts as well as the two assembly objectscorresponding to semantics

According to the commonly existed assembly needssuch as axial restraint connection and transmission amongassembly units common assembly semantics in assemblydesign are extracted and divided into four categories connec-tion semantics transmission semantics coordinating seman-tics and user-defined semantics Connection semantics aremainly used to depict the assembly relationship among partswith connection which generally requires auxiliary toolsfor assembly Transmission semantics are mainly to depictthe assembly relationship among parts with transmissionrelation which generally does not need auxiliary tools for

Mathematical Problems in Engineering 3

assembly Coordinating semantics are mainly to depict theassembly relation related to axial restraint which does notneed auxiliary tools or only needs simple tools User-definedsemantics are mainly to realize the extension of assemblysemantics which are defined and extended by users

32 Assembly Semantic Relation Graph Model To simplifythe assembly modeling of complex product and supportassembly sequence planning this paper used assemblysemantic relation graph (ASRG) to depict the assemblystructure of products Related concepts and descriptionswereas follows

(a) function parts parts in assembly units with connec-tors removed

(b) ASRG with assembly semantics and connector infor-mation included in edges of undirected graph ASRGuses the undirected graph to describe the assemblyconnectivity of function parts in assembly units anddefines the graph as a quaternion

ASRG = ⟨119881 119864 119860119881 119860119864⟩ (1)

where 119881 = 1198751 1198752 119875

119899 is the set of nodes in the assembly

relationship diagram showing the function parts of products119899 is the number of parts 119864 = 119864

1 1198642 119864

119898 is the set of

edges in the assembly relationship diagram and each edgecorresponds to one assembly relationship 119898 is the numberof assembly relationship 119860

119881represents the attribute set of

nodes to describe the information of function parts 119860119864

represents the attribute set of edges to describe the assemblyinformation of function parts

Figure 1 shows the ASRGmodel between two parts Edgeattributes of 119875

119894and 119875

119895mainly include assembly semantics

connector and geometric constraint information ASRGseparates assembly information of products from parts infor-mation so that assembly information only contains labelinginformation of parts When attribute information of parts ischanged it does not influence original assembly relationshipwhile simplifying expressions of assembly relationship andreducing the node number in graph and modeling difficulty

33 Weights of Edges in ASRG To describe the importanceof assembly relationship and the difficulty of assembly thispaper defined assembly connection strength as the summa-tion of assembly connection strength of connector informa-tion semantic information of transmission and geometricconstraint information

According to the difficulty of assemblydisassembly con-nectors and transmission semantics parts and evaluatedassembly connection strength of geometric constraint infor-mation are shown by fuzzy hierarchical approach Thesenumerical values aim to exhibit certain difficulty Whenthere were several assembly characteristics among parts theconnection strength should be calculated respectively and

sums should be calculated Thus weights of edges of ASRGcould be calculated by the following formula

119877 (119894 119895) =

3

sum119896=1

119908119896119903119896(119894 119895) 119894 = 119895

1 119894 = 119895

(2)

where 119877(119894 119895) = 119877(119895 119894) 119894 119895 isin 1 2 119899 is weights ofedges between 119894 and 119895 among parts 119908

119896is the weight of 119896th

evaluation value of assembly connection strength 119903119896(119894 119895) is

the 119896th evaluation value of assembly connection strength

4 Assembly Unit Partition Based onConnection Strength

CNC machine tool products have more complex processof assembly sequence planning than normal mechanicalproductsThe complex product is partitioned into the assem-bly unit containing fewer parts which can effectively over-come the issue of ldquocombinatorial explosionrdquo and remarkablyreduce the difficulty of assembly sequence planning Thisresearch proposed the transmissibility of assembly connec-tion strength and realized the assembly unit partition ofCNC machine tool This paper used assembly relationshipinterference matrix for interference checking and revision ofthe produced assembly unit thereby obtaining assembly unitwith mutual noninterference in the assembly

41 Assembly Unit Partition A product is normally formedby several parts The complexity of assembly relationshipamong parts could greatly influence the assembly perfor-manceThis research quantifies the assembly difficulty amongparts to assembly connection strength and proposed thetransmissibility of assembly connection strength to realizeassembly unit partition based on connection strength

411 Number of Assembly Unit Partition Assembly unit is arelatively independent structural unit to realize one or severalfunctions Assembly unit is a prerequisite to produce simpleassembly planning tasks This paper assumed that each unitis assembled concurrently and after the assembly all units areassembled into a final product on the main assembly line Itwas also assumed that each unit does not show delaying inassembly process With the goal of reducing assembly timethe number of optimal unit partitionwas obtained by formula(3) 119873

119898was rounded and taken as the number of optimal

assembly unit partition Consider

119873119898

le radic119873119901 (3)

where119873119901is the total number of parts of a product and119873

119898is

the number of unit partition of a product

412 Basic Parts of AssemblyUnit Basic part is the importantfunction part for an assembly unit The selection of differentbasic part could reach different assembly units According tothe assembly connection matrix the connection number ofparts with other parts could be used as the basis to identify

4 Mathematical Problems in Engineering

Functionpart Pi

Functionpart Pj

Assemblysemantic Connector Geometric

constraint

Figure 1 Assembly semantic relation graph model

basic part In this paper the sum of edge weight of connectedwith part 119894 in ASRG was taken as the importance degreeof part 119894 in the assembly The importance degree could becalculated by the following formula

119882(119894) =

119899

sum119895=1

119877 (119894 119895) (4)

where 119877(119894 119895) is the edge weight of ASRGAfter the calculation of importance degree of all parts

is finished 119873119898parts with the maximum importance degree

were taken as basic parts for assembly unit partition

413 Assembly Connection Strength The assembly relation-ship graph among parts is similar with the trust network dia-gram among users in recommender systems of e-commerce[13] The transmissibility of similarity was introduced intothe assembly unit partition in this research so that assemblyconnection strength among parts could be transferred

The transmission rules of assembly connection strengthcan be defined as follows assume that there are 119898 pathsbetween part 119860 and part 119861 in ASRG and nodes on the 119894thpath between 119860 and 119861 are respectively 119873

1198941 119873

119894119896 assume

119896 is the number of nodes between 119860 and 119861 If in a paththere is basic part the assembly connection strength between119860 and 119861 was denoted by 0 if in a path there is not basicpart the assembly connection strength between 119860 and 119861 wascalculated by the following formulas

119860119868119894 (119860 119861)

= Min [119877 (1198601198731198941) 119877 (119873

1198941 1198731198942) 119877 (119873

119894119896 119861)] lowast 120573

119894

119860119868 (119860 119861) = Max [1198601198681 (119860 119861) 119860119868

2 (119860 119861) 119860119868119898 (119860 119861)]

(5)

where 119894 = 1 2 119898 119877(119860119873119894119896) is the edge weight of part 119860

and part 119873119894119896in ASRG 119860119868

119894(119860 119861) is the assembly connection

strength between part 119860 and part 119861 on 119894th path 119860119868(119860 119861) theassembly connection strength between part 119860 and part 119861 120573

119894

is an attenuation factor of assembly connection strength on119894th path which decreases with the increase of present transferpath The calculation is as follows

120573119894=

119871 minus 119879119894+ 1

119871 (6)

where 119871 is the threshold of the length of transmission pathdenoted as 5 119879

119894is the length of the 119894th transmission path

denoted as the number of edges

According to the above process assembly connectionstrength of part 119894 and each basic part were identified Nextbased on the assembly connection strength function partswere put in the unit where the basic part with the largestof assembly connection strength was in With the cyclicoperation all parts were partitioned in the unit where basicparts were in In this way the partition of assembly unit wasrealized

42 Interference Checking and Assembly Unit CorrectionSince the factors like assembly precedence constraint andgeometric constraints were not considered in assembly unitpartition interference may happen during assembling Thusin order to improve the accuracy of assembly unit partitionthis paper generated assembly unit interference judgmentmatrix through assembly relationship interference matrix forinterference checking and correction of partitioned assemblyunit

421 Assembly Relationship Interference Matrix Let 119864 =

1198901 1198902 119890

119899 be the set of assembly relationship If assembly

relationship 119890119894after the preferential assembly interfere with

the assembly relationship 119890119895 it is denoted that assembly rela-

tionship 119890119894interferes assembly relationship 119890

119895 119890119894is called the

assembly relationship that produces assembly interference 119890119895

is called the interfered assembly relationship [14] Assemblyrelationship interference exhibits the priority of two assemblyrelationships Assembly relationship interference matrix isused to express assembly interference denoted as 119868(119877) In thematrix if 119903

119894119895= 1 119890

119894after the priority assembly interferes 119890

119895

if 119903119894119895

= 0 no assembly interference occurs between 119890119894and 119890119895

if 119903119894119895

= 0 and 119903119895119894

= 1 119890119894is prior to 119890

119895 If the row of 119890

119894has

all values as zero 119890119894can be assembled prior to other assembly

relationshipsIn the assembly process the composition parts of the

product were only assembled with adjacent parts rather withall rest parts Therefore most interferences of assembly rela-tionship exist in adjacent assembly relationships Howeverdue to the complexity of product structure and assemblyprocess some nonadjacent assembly relationships might beinterfered The interference information could be obtainedthrough the interference extension of assembly relationship

422 Assembly Unit Interference Judgment Matrix With theassembly unit interference judgment matrix for interferencechecking and correction of each assembly unit assemblyunits with no interference could be obtained Through thefollowing 3 steps the assembly unit interference judgmentmatrix can be obtained

Mathematical Problems in Engineering 5

(1) 119898 assembly relationships 119864119880 = 1198901 1198902 119890

119898 in unit

119880 were extracted(2) each line of elements corresponding to 119890

1 1198902 119890

119898

was extracted from the assembly relationship interfer-ence matrix to form the intermediary matrix 119872119880

(3) each line of elements corresponding to 1198901 1198902 119890

119898

in 119872119880 was taken as zero to obtain the assembly unitinterference judgment matrix 119862119880

If the corresponding line of assembly relationship 119890119894(1 le

119894 le 119898) in assembly unit interference judgment matrix 119862119880

has 1 it indicates that assembly relationship 119890119894will interrupt

the assembly of other units that is there is interferencebetween assembly unit 119880 and other units The assemblyrelationship that produced interference was dissembled Thenewly formed assembly unit did not create interference in theprocess of assembly Disassembled parts could be adjusted toother units or partitioned into new units if needed

5 Assembly Sequence Planning Based onGenetic Algorithm

The solutions of assembly sequence planning could bedivided into two categories serial assembly and concur-rent assembly Concurrent assembly could increase the par-allelism of assembly improve assembly efficiency reduceassembly costs and could not trigger ldquocombinatorial explo-sionrdquoThus it fits the assembly sequence planning of complexproducts Genetic algorithm generates satisfactory effectsin the research of combinatorial optimization Assemblysequence planning is a typical combinatorial optimizationissue Therefore this paper used genetic algorithm in opti-mized assembly sequence of parts in assembly units andeach assembly unit and then generated concurrent assemblysequence

51 Chromosome Coding of Assembly Sequence The assemblysequence planning is under the umbrella of combinatorialoptimization and suitable for the coding approach of com-binatorial optimization Therefore this research used order-sensitive codes for the coding of parts For assembly sequenceplanning of 119873 parts the chromosome was divided into 119873

segments and each segment is the corresponding code ofparts In order to facilitate the coding part number in theassembly unit was arranged from small to large and then wascoded

52 Ways to Generate Initial Population Generally speakingunits in the initial population are randomly generated Inthis paper the automatic generation was used to generateinitial population 119910 = Randperm (119899) in MATLAB wasused for population initialization to obtain a nonrepeatedrandom structure including integers from 1 to 119899 as a validchromosome

53 Fitness Function Design of Assembly Sequence Accordingto the feasibility of assembly sequence and the incidence ofthe strengths and weaknesses indicators with the maximuminfluence and easy access to information were used to estab-lish fitness function This research mainly constructed thefitness functions from three aspects including the geometricfeasibility of assembly sequence stability of assembly unitsand the position of basic parts in assembly sequence

(1) Geometric Feasibility Geometric feasibility requires nointerference during disassembly This research used inte-grated interference matrix [15] to derive the feasible disas-sembly direction of any part in the assembly unit It wasassumed that an assembly unit 119860 = 1 2 119899 comprised119899 parts and its corresponding integrated interference matrixwas a matrix 119868

119860with 119899 rows and 6119899 columns as shown in the

following formula

119868119860

=

[[[[[[[[

[

11986811119909

11986811119910

11986811119911

119868111199091015840119868111199101015840119868111199111015840 11986812119909

11986812119910

11986812119911

119868121199091015840119868121199101015840119868121199111015840 sdot sdot sdot 119868

11198991199091198681119899119910

1198681119899119911

119868111989911990910158401198681119899119910101584011986811198991199111015840

11986821119909

11986821119910

11986821119911

119868211199091015840119868211199101015840119868211199111015840 11986822119909

11986822119910

11986822119911

119868221199091015840119868221199101015840119868221199111015840 sdot sdot sdot 119868

21198991199091198682119899119910

1198682119899119911

119868211989911990910158401198682119899119910101584011986821198991199111015840

sdot sdot sdot

1198681198991119909

1198681198991119910

1198681198991119911

119868119899111990910158401198681198991119910101584011986811989911199111015840 1198681198992119909

1198681198992119910

1198681198992119911

119868119899211990910158401198681198992119910101584011986811989921199111015840 sdot sdot sdot 119868

119898119909119868119898119910

119868119898119911

119868119898119909101584011986811989811991010158401198681198981199111015840

]]]]]]]]

]

(7)

If at one direction 119889 part 119894 is not interfered by anypart during disassembly then 119868

119894119889= 0 which indicated

that part 119894 met the geometric feasibility of disassembly andthe disassembly was activated If at one direction 119889 part119894 is interfered by any part during disassembly then 119868

119894119889=

1 which exhibited that parts did not meet the geometricfeasibility and the disassembly was not feasible When part119894 was successfully disassembled all elements at the 119894th row inintegrated interference matrix 119868

119860were 0 indicating that part

119894 did not produce interference with other parts formula (8)could evaluate the geometric feasibility of assembly sequence

1198911=

119873 (119873 minus 1)

2minus 119872 (8)

where 119873 is the total number of parts in assembly unit and119872 is the times of an assembly sequence that generatedinterferences during assembly

(2) Stability of Assembly Unit The stability connection matrix119862 was used to quantify the stability of feasible assemblysequence 119862

119894119895represented the connection relation between

part 119894 and part 119895 When a stable connection relation betweenpart 119894 and part 119895 was observed let 119862

119894119895= 2 when a contact

6 Mathematical Problems in Engineering

connection relation between part 119894 and part 119895 was observedlet 119862119894119895

= 1 when there was no connection relation betweenpart 119894 and part 119895 let 119862

119894119895= 0 The stable connection in

this research refers to connection relations with mandatoryconstraints such as screw joint interference fit and pinnedconnection Hence formula (9) was used to evaluate thestability of assembly sequence

1198912=

119873minus1

sum119894=1

119870119894 (9)

where 119873 is the total number of parts in an assembly unitand 119870

119894is a quantified connection relation between part 119894 and

part 119894 + 1 When a stable connection existed 119870119894= 2 when

a contact connection existed 119870119894

= 1 when no connectionrelation existed 119870

119894= 0

(3) Position of Basic Parts The basic part is the key functionpart in assembly unit and should be firstly assembled duringthe assembly process In initial population number 1 partin each assembly sequence is not necessarily a basic partTherefore the assembly sequence with the basic part in thevery first place should be distinguished from that with thebasic part not in the first place In this research formula (10)was used to evaluate the position of basic parts in assemblysequence

1198913= 119873 minus 119875 (10)

where 119873 is the total number of parts of assembly unit and 119875

is the position of basic parts in assembly sequence

(4) Fitness Function According to the analysis on the aboveevaluation indexes a fitness function of genetic algorithmwas designed for assembly sequence planning as shown inthe following formula

119865 (119878) = 12059611198911+ 12059621198912+ 12059631198913 (11)

where 119878 is the assembly sequence in the population 1205961is the

weight coefficient of geometric feasibility 1205962is the weight

coefficient of assembly unit stability and 1205963is the weight

coefficient of positions of basic parts

54 Genetic Manipulation Genetic algorithm simulates theevolutionary mechanism of ldquosurvival of the fittestrdquo in theliving nature through genetic manipulation The missionof genetic manipulation is to apply certain operations onindividuals in the population according to their fitness (fit-ness evaluation) to environment thereby realizing the evo-lutionary process of survival of the fittest Common geneticmanipulation approaches include selection intersection andmutation

Selection operation refers to the selection of individualswith better fitness to produce new population For assemblysequence planning this research provided three approachesroulette selection linear ranking selection and tournamentselection Crossover operation is used to simulate geneticrecombination in the genetic evolution In the assembly

sequence planning of this paper single-point crossover two-point andmultipoint crossover as well as uniform crossoverwere applied Mutation is used to mimic the variation ofbiological heredity in the evolutionary process In this paperswitched mutation and inserted mutation were provided

55 Control Parameters and Termination Criterion of GeneticAlgorithm In order to ensure that genetic algorithm couldreach the optimal solution along the optimal research trackthe control parameter should be rationally selected whichmainly included the population size 119873 encoding length119871 crossover probability 119875

119888 mutation probability 119875

119898 and

termination conditionFor the assembly sequence planning in this research

encoding length 119871 is the number of parts in the assemblyunit The population size 119873 doubled the number of partsin assembly unit that is 119873 = 2119871 Crossover probability119875119888controls the utilization frequency of crossover operation

Generally 119875119888= 06sim10 Mutation probability 119875

119898controls the

utilization frequency of mutation Generally 119875119898

= 0005sim001 The termination condition of genetic algorithm in thispaper is when a specific algebrawas run on genetic algorithmthe calculation was terminated Besides optimal individualsin the current populationwere taken as the optimal solutions

6 Case Study

With the example of headstock of HTC2550hs CNC turningcenter this paper illustrated the technologies of concurrentassembly sequence planning

61 Assembly Modeling of Headstock of CNC Machine ToolAfter the removal of connectors in headstock there were 43function parts The assembly relationship among functionparts of the headstock was analyzed to establish ASRG ofCNC machine tool Parts were expressed by numbers asshown in Figure 2

As the headstock had a large number of function partsandASRGwas complex this paper usedGraphviz to establishASRG The function parts of headstock were expressed bynumbers The established ASRG is shown in Figure 3 Theassembly relationship information among function partsof headstock was analyzed The analytic hierarchy process(AHP) was used to generate weights of connection semanticsinformation semantic information of transmission and geo-metric constraint information The corresponding weightswere respectively 119908

1= 1199082

= 043 and 1199083

= 014 Nextedge weights in ASRG were calculated according to formula(2)

In Figure 3 as function parts were distinguished fromconnectors there were only 43 nodes in this model Inaddition the assembly relationship information was added inedges Edge information in ASRG could be revised accordingto needs so that ASRG has good maintainability

62 Assembly Unit Partition of Headstock of CNCMachine Tool

621 Selection of Basic Parts in Assembly Unit The totalnumber of function parts of this headstock was 43 According

Mathematical Problems in Engineering 7

Brak

e obj

ects6

Brak

e obj

ects29

Fric

tion

bloc

k41

Fric

tion

bloc

k43

Pisto

n ro

d20

Pisto

n ro

d35

Lid31

Back

-up

bloc

k27

Tank

cove

r28

Sign

al co

nver

ter3

6

Pisto

n ro

d30

Pisto

n ro

d7

Lid9

Lid32

Bend

ing

plat

e10

Bend

ing

plat

e34

Lid33

Fric

tion

bloc

k42

Fric

tion

bloc

k8

4 B

elt p

ulle

y37

Exp

ansio

n sle

eve

38

Exp

ansio

n sle

eve

39

Bra

ke d

isk

12

Rub

ber s

heet

40

Pro

tect

ive b

ox3 L

oop

2 H

eel c

ap1 B

ack

glan

d26

Bac

k be

arin

g ho

usin

g13

Hou

sing

14

For

mer

spac

er15

Hou

sing

21

Spa

cer

25

Spi

ndle

box

22 Locating pin

5 S

pind

le

19

Gla

nd17

Ext

erna

l gla

nd18

Lab

yrin

th ca

sing

23 Special bolts16

Hou

sing

24 Adjustingblock

11

Blin

d fr

ange

Figure 2 Explosive view of headstock

to formula (3) 119873119898

le 656 so the number of optimal unitpartition was 119873

119898= 6 According to edge weight in ASRG

formula (4) was used to calculate the importance degree ofeach part in the assembly The result was shown in Table 1

6 parts with the largest importance degrees were taken asthe basic parts of assembly unit partition In Table 1 part 2529 6 5 40 4 had the largest importance degree so that theywere selected as the basic parts of assembly unit

622 Assembly Unit Partition The assembly connectionstrength between part 119894 and other basic parts was calculated

and part 119894 was put in the unit containing basic part withthe largest connection strength to accomplish the assemblyunit partition Result of assembly unit partition was shown inTable 2 and Figure 4

623 Interference Checking and Assembly Unit CorrectionInterference checking based on the above partition resultswas carried out The interference judgment matrix of assem-bly unit 1 was shown as follows

1198621198801=

11989011198902119890311989041198905

sdot sdot sdot sdot sdot sdot sdot sdot sdot 119890511198905211989053

1198901

1198902

1198904

11989039

11989040

11989041

11989042

11989045

11989046

11989047

[[[[[[[[[[[[[[

[

00000

00000

00000

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00001

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00100

00100

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000

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000

000

000

000

000

]]]]]]]]]]]]]]

]

(12)

By the analysis of interference judgment matrix in eachassembly unit it was found that part 2 17 18 and 19 wouldtrigger assembly interference among units After the disas-sembly the assembly unit without interference was obtainedIn addition part 39 shared an assembly relationship with

part 2 When part 2 was disassembled part 39 was alsodisassembled

Elements of the assembly relationship between part 19and part 5 in the interference matrix were 0 The assemblywas prioritized and it was placed in the unit where the spindle5 was Part 2 and part 39 had bolted connections with the

8 Mathematical Problems in Engineering

Table 1 The importance degree of each part in the assembly

Partnumber Importance degree Part number Importance degree Part number Importance degree Part number Importance degree

1 0796 12 088 23 0426 34 04262 0622 13 014 24 0852 35 05663 0734 14 014 25 4566 36 03984 1182 15 0112 26 1052 37 06225 1862 16 0112 27 088 38 06226 2208 17 0566 28 0482 39 04827 0566 18 0482 29 2606 40 14468 0482 19 0824 30 0566 41 04829 051 20 0566 31 051 42 048210 0426 21 0168 32 051 43 048211 088 22 0426 33 051

13 14 15 21 16 18 2 36 31 9 10

17 5 39 1 6

28 19 26 4 20 7

25 37 41 8

22 24

23

27

40 38 3

424311

12 35 30 34 33 32

29

e7 e6 e52 e10 e9 e8 e11e12 e1 e31 e30

e32

e41

e27e26e2

e13

e33

e53e43 e3

e5

e40

e45 e46

e49

e50

e51

e48

e42 e4 e15

e14

e17

e16

e18 e19

e22

e21

e24e25

e29 e28 e37 e36 e35

e20

e23

e23

e39

e34

e38

Figure 3 ASRG of headstock

Mathematical Problems in Engineering 9

Table 2 Results of assembly unit partition

Unitnumber Basic part number Results of assembly unit partition

1 25 1 17 19 22 23 24 26 27 28 362 29 30 32 33 34 35 42 433 6 7 9 8 10 20 31 414 5 2 13 14 15 16 18 21 395 40 11 126 4 3 37 38

relatively intensive connection strength Elements of assem-bly relationship in the interference matrix were 0 whichcould prioritize assembly Therefore an independent unitwas formed As the assembly connection strength between

part 17 part 18 and basic parts in other unit was smallthe partition was not conducted for independent assemblyResults of modified unit partition were shown in Table 3 andFigure 5

63 Assembly Sequence Planning and Optimization for CNCMachine Tool

631 Case of Optimized Assembly Sequence Assembly unit 3was the braking part of headstock and contained 8 functionparts The corresponding part number was 6 7 8 9 10 2031 41 as shown in Figure 6

The interference relationship between parts in an assem-bly unit at directions (plusmn119883 plusmn119884 plusmn119885) during disassembly andother parts was analyzed An integrated interference matrix119868119860was constructed as follows

119868119860

=

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

000000

111011

010000

111011

100000

011111

011111

010000

011111

000000

000100

111011

100000

000100

000100

000100

000010

100000

000000

000000

100000

000100

000100

000100

011111

011111

000000

000000

100000

001000

000100

000100

000100

010100

000100

000100

000000

000100

000100

000100

111011

100000

100000

100000

100000

000000

011111

100000

111011

100000

000000

100000

100000

111011

000000

000000

000010

100000

100000

100000

100000

000100

000100

000000

]]]]]]]]]]

]

(13)

The stability of the connectivity among parts in theassembly unit was analyzed and a stability connection matrix119862 was constructed as follows

119862 =

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

0

1

0

2

2

1

2

0

1

0

2

1

0

0

0

0

0

2

0

0

0

0

0

0

2

1

0

0

0

0

0

0

2

0

0

0

0

0

0

0

1

0

0

0

0

0

1

2

2

0

0

0

0

1

0

0

0

0

0

0

0

2

0

0

]]]]]]]]]]

]

(14)

The initial population was produced by the randomapproach and the control parameter of genetic algorithmwasset as follows encoding length 119871 = 8 population size119873 = 16crossover probability 119875

119888= 08 mutation probability 119875

119898=

001 generation gen = 100 and genetic manipulation wasrespectively selected by roulette This study conducted two-point crossover switched mutation and reverse operationPart 6 of assembly unit 3 was a basic part Weights ofgeometric feasibility assembly unit stability and positionsof basic parts in the fitness calculation formula (11) wererespectively 120596

1= 04 120596

2= 02 and 120596

3= 04

Run the program in MATLAB to solve the optimalassembly sequence of assembly unit 3 Figure 7 was thefitness variation diagram of genetic algorithm optimizationAccording to the calculation result the last generation of

maximum fitness was 152 and the corresponding optimalassembly sequence was as follows

6 997888rarr 8 997888rarr 7 997888rarr 9 997888rarr 41 997888rarr 20 997888rarr 31 997888rarr 10 (15)

During the assembly process basic parts should be firstassembled and plate cover parts should be assembled aftershaft parts According to the optimized assembly sequenceresults in assembly unit 3 braking part 6 was a basic part andwas firstly assembled Cover 9 and cover 31 were assembledafter the piston rodThus it could be found that the assemblysequence solved from genetic algorithm was rational andfeasible

For assembly units with the number of parts larger than5 the genetic algorithm was used for optimized assem-bly sequence and assembly sequences of other units wereobtained according to experiences The result was shown inTable 4

Similar to the optimized assembly sequencing of parts inthe assembly unit integrated interferencematrix and stabilityconnection matrix among assembly units were constructedAssembly unit 1 was taken as the basic unit so that theassembly sequence of each assembly unit could be obtainedas shown in the following order

Unit 1 997888rarr Unit 4 997888rarr Unit 7 997888rarr Unit 5 997888rarr Unit 6

997888rarr Unit 8 997888rarr Unit 9 997888rarr Unit 2 997888rarr Unit 3(16)

10 Mathematical Problems in Engineering

Assembly unit 1

Assembly unit 2

Assembly unit 6

Assembly unit 3

Assembly unit 4

Assembly unit 5

Figure 4 Result of assembly unit partition

Assembly unit 1

Assembly unit 2

Assembly unit 8

Assembly unit 7Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 3

Assembly unit 9

Figure 5 Result of modified assembly unit partition

Table 3 Results of modified unit partition

Unit number Result of modified assembly unit partition1 1 22 23 24 25 26 27 28 362 29 30 32 33 34 35 42 433 6 7 8 9 10 20 31 414 5 13 14 15 16 19 215 11 12 406 3 4 37 387 2 398 179 18

632 Expression of Assembly Sequence As a fishbone dia-gram can simply directly and flexibly express the assemblysequence product hierarchies and other information thispaper used the fishbone diagram to express concurrentassembly sequence Among them fish head demonstrated thefinal product of assembly Big bones connected with the fishspine represented each assembly unit Small bones connectedwith big bones represented parts Positions of big bones and

Brake object 6Bending plate 10

7 Piston rod 31 Lid41 Friction block

20 Piston rod9 Lid 8 Friction block

Figure 6 Section view of braking part

7

8

9

10

11

12

13

14

15

16

10 20 30 40 50 60 70 80 90 100

Col

ony

adap

tatio

n de

gree

Maximum colony adaptation degreeEvolutionary process

Average colony adaptation degree

Genetic algebra

Figure 7 Fitness variation diagram

Table 4 Assembly sequence planning of each unit

Assembly unitnumber Assembly sequence

1 25rarr 27rarr 26rarr 1rarr 36rarr 28rarr 24rarr 22rarr 232 29rarr 42rarr 30rarr 32rarr 43rarr 35rarr 33rarr 343 6rarr 8rarr 7rarr 9rarr 41rarr 20rarr 31rarr 104 5rarr 19rarr 21rarr 16rarr 15rarr 14rarr 135 40rarr 12rarr 116 4rarr 3rarr 38rarr 377 39rarr 28 179 18

small bones in the figure reflected the assembly sequenceThe concurrent assembly sequence of headstock was shownin Figure 8

Mathematical Problems in Engineering 11

5

19

21

16

15

14

13

40

12

11 17

29

42

30

32

43

35

33

34

25

27

26

1

36

28

24

22

23

39

2

4

3

38

37 18 10

31

20

41

9

7

8

6

Assembly unit 4

Assembly unit 5

Assembly unit 2

Assembly unit 3

Assembly unit 9

Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 8

Spindle box

Figure 8 Expression of headstock assembly sequence

7 Conclusions

In an era of global economic integration to occupy avantage ground in fierce market competition enterprisesmust enhance their product innovation capability improveproduct quality and performance also shorten the productdesign andmanufacturing cycle and reduce costs Inmodernmanufacturing industry the workload of product in theassembling stage is very heavy producing high costs andhard to realize assembly automationTherefore newmethodsand technologies are in urgent need to support the productassembly design On the basis of related research of assemblydesign this paper explored into assemblymodeling assemblyunit partition and optimized assembly sequence of CNCmachine tool In this way the assembly design efficiency ofproduct was improved This research reached the followingconclusions

(1) By distinguishing connectors from function parts asimplified ASRG was established and semantic informationof assembly was added in the edges of assembly model Thestudy on assembly modeling cases of headstock of CNCmachine tool showed that the model efficiently expressed theassembly relationship among parts

(2)According to the transmissibility of assembly connec-tion strength the assembly connection strength among partswas calculated so that the information of nonadjacent partscould be utilized The assembly unit partition based on theconnection strength changed the previous partition methodsby experiences so as to quantify the partition of assembly unitIn addition through interference checking and correction ofassembly unit the rationality and accuracy of assembly unitpartition were improved

(3) According to needs of assembly sequence planninggenerations of coding approach fitness function genetic

operator control parameters and initial population of geneticalgorithm were designedThrough case analysis of headstockof CNC machine tool the effectiveness of using geneticalgorithm for assembly sequence planning and optimizationwas verified

This research actively explored into the assembly mod-eling assembly unit partition assembly sequence planningand optimization and other aspects of CNC machine toolrealized the optimized assembly sequence of CNC machinetool and provided theoretical basis and technical supports forthe improvement of assembly quality and efficiency of CNCmachine tool

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (N120403003) the Scienceand Technology Funds of Liaoning Province (20130200522011216010) and the Research Funds of State ldquoTwelve FiverdquoSupport Program (2012BAF12B08)

References

[1] V Blanchot and A Daidie ldquoRiveted assembly modelling Studyand numerical characterisation of a riveting processrdquo Journal ofMaterials Processing Technology vol 180 no 1ndash3 pp 201ndash2092006

[2] T Gu Z Xu and Z Yang ldquoSymbolic OBDD representations formechanical assembly sequencesrdquo Computer Aided Design vol40 no 4 pp 411ndash421 2008

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

and rationality of assembly information determine whetheran assembly sequence can be correctly generated

Blanchot and Daidie [1] introduced the adjustment ofa numerical model simulating a riveted link using differentapproaches and presented the simulation of riveting processand its influence on the riveted link behaviour Gu et al[2] represented subassemblies assembly states and assem-bly tasks as Boolean characteristic functions and proposedsymbolic OBDD (ordered binary decision diagram) schemefor all feasible assembly sequences Xu et al [3] simplifiedassembly resource established amatrix of polychromatic setsand set up a dynamic assembly model based on the theory ofpolychromatic sets Guo et al [4] explored into the layeredassembly model under the complex constraint condition andproposed a layered constraint assembly model based on theattributes of assembly objects

22 Assembly Unit Partition As the diversity and complexityof product assembly structure and process the assemblysequence planning is complex The partition of complexproducts into assembly units with fewer parts and takingassembly unit as the study object could effectively overcomethe issue of ldquocombinatorial explosionrdquo and reduce the diffi-culty of assembly sequence planning

Gottipolu and Ghosh [5] described an approach for gen-eration representation and selection of assembly sequencealternatives in which the geometric and mobility constraintsextracted directly from the CAD model of the assemblywere translated into two types of unidirectional matricesthe contact and the translational functions Ko et al [6]presented an assembly-decomposition model to improveproduct quality and used mixed-integer programming topartition the liaison graph of a product assembly with theconsideration of defect rates in components and assemblytasks Wu [7] proposed the connection intensity to expressthe tightness of assembly used the fuzzy clustering for partsclustering and established the identification of subassemblyunits and partition method Wang and Liu [8] quantifiedthe constraints of assembly units and proposed the decision-making diagram of assembly unit based on fuzzy analytichierarchy process

23 Assembly Sequence Planning The research on assemblysequence planning can be divided into two main categories(1) according to constraint condition the feasible assemblysequence is generated through reasoning and optimization(2) Using the modern optimization methods feasible assem-bly sequence is generated and simultaneously filtered

Wang et al [9] coupled the solution space generation anddisassembly solution optimization into one technical frame-work represented all assemblydisassembly sequences in theDFIG (disassembly feasibility information graph) modeland proposed an integrated methodology for mechanicalassembly planning Sinanoglu and Borklu [10] proposedan approach for the development of a process for thedetermination of geometric feasibility whose binary vectorrepresentation corresponds to assembly states Zhong et al[11] presented a determination system of assembly units forthe hull structure in order to shorten construction cycle

time and construction quality in shipbuilding and introducedthe assessment model by fuzzy synthetic evaluation Wanget al [12] suggested assembly sequences merging based onassembly unit partitioning to reduce the searching spaceof assembly sequence planning of complex products andcomprehensively took into account the assembly designconstraints and the assembly process constraints

With the research object of the headstock of HTC2550hsCNC turning center this paper conducted assembly mod-eling assembly unit partition assembly sequence planningand other tasks In Section 3 the simplified assembly seman-tic relation graph (ASRG) for CNCmachine tool was createdand the assembly relationship among parts was quanti-fied into assembly connection strength Section 4 proposedthe transmissibility of assembly connection strength Theassembly unit partition of CNC machine tool was realizedSection 5 adopted genetic algorithm for optimized assemblysequence of parts in assembly units and each assembly unitrespectively Finally the concurrent assembly sequence wasgenerated Section 6 took the headstock of HTC2550hs CNCturning center as the example and verified the technique ofconcurrent assembly sequence planning Section 7 provideda conclusion of the research

3 Assembly Modeling Based onAssembly Semantics

The assembly model of product is the foundation of assemblysequence planning The more the complete the informationincluded in a model is the higher the efficiency of assemblysequence planning is Meanwhile the assembly modelingbecomes more difficult In the field of assembly designdesigners tend to express the assembly relationship amongparts by assembly semantics The basic concept of assemblymodeling based on assembly semantics is to integrate theassembly engineering semantic analysis method with object-oriented techniques which can be used in the assemblymodeling the expression of assembly relationship and theproduct model description

31 Assembly Semantics Assembly units contain rich assem-bly design information which comprises assembly semanticsAssembly semantics are the abstract expressions of assem-bly relationship between assembly units and parts duringassembly design which embody the information such aspositioning constraints and engineering constraints betweenassembly units and parts as well as the two assembly objectscorresponding to semantics

According to the commonly existed assembly needssuch as axial restraint connection and transmission amongassembly units common assembly semantics in assemblydesign are extracted and divided into four categories connec-tion semantics transmission semantics coordinating seman-tics and user-defined semantics Connection semantics aremainly used to depict the assembly relationship among partswith connection which generally requires auxiliary toolsfor assembly Transmission semantics are mainly to depictthe assembly relationship among parts with transmissionrelation which generally does not need auxiliary tools for

Mathematical Problems in Engineering 3

assembly Coordinating semantics are mainly to depict theassembly relation related to axial restraint which does notneed auxiliary tools or only needs simple tools User-definedsemantics are mainly to realize the extension of assemblysemantics which are defined and extended by users

32 Assembly Semantic Relation Graph Model To simplifythe assembly modeling of complex product and supportassembly sequence planning this paper used assemblysemantic relation graph (ASRG) to depict the assemblystructure of products Related concepts and descriptionswereas follows

(a) function parts parts in assembly units with connec-tors removed

(b) ASRG with assembly semantics and connector infor-mation included in edges of undirected graph ASRGuses the undirected graph to describe the assemblyconnectivity of function parts in assembly units anddefines the graph as a quaternion

ASRG = ⟨119881 119864 119860119881 119860119864⟩ (1)

where 119881 = 1198751 1198752 119875

119899 is the set of nodes in the assembly

relationship diagram showing the function parts of products119899 is the number of parts 119864 = 119864

1 1198642 119864

119898 is the set of

edges in the assembly relationship diagram and each edgecorresponds to one assembly relationship 119898 is the numberof assembly relationship 119860

119881represents the attribute set of

nodes to describe the information of function parts 119860119864

represents the attribute set of edges to describe the assemblyinformation of function parts

Figure 1 shows the ASRGmodel between two parts Edgeattributes of 119875

119894and 119875

119895mainly include assembly semantics

connector and geometric constraint information ASRGseparates assembly information of products from parts infor-mation so that assembly information only contains labelinginformation of parts When attribute information of parts ischanged it does not influence original assembly relationshipwhile simplifying expressions of assembly relationship andreducing the node number in graph and modeling difficulty

33 Weights of Edges in ASRG To describe the importanceof assembly relationship and the difficulty of assembly thispaper defined assembly connection strength as the summa-tion of assembly connection strength of connector informa-tion semantic information of transmission and geometricconstraint information

According to the difficulty of assemblydisassembly con-nectors and transmission semantics parts and evaluatedassembly connection strength of geometric constraint infor-mation are shown by fuzzy hierarchical approach Thesenumerical values aim to exhibit certain difficulty Whenthere were several assembly characteristics among parts theconnection strength should be calculated respectively and

sums should be calculated Thus weights of edges of ASRGcould be calculated by the following formula

119877 (119894 119895) =

3

sum119896=1

119908119896119903119896(119894 119895) 119894 = 119895

1 119894 = 119895

(2)

where 119877(119894 119895) = 119877(119895 119894) 119894 119895 isin 1 2 119899 is weights ofedges between 119894 and 119895 among parts 119908

119896is the weight of 119896th

evaluation value of assembly connection strength 119903119896(119894 119895) is

the 119896th evaluation value of assembly connection strength

4 Assembly Unit Partition Based onConnection Strength

CNC machine tool products have more complex processof assembly sequence planning than normal mechanicalproductsThe complex product is partitioned into the assem-bly unit containing fewer parts which can effectively over-come the issue of ldquocombinatorial explosionrdquo and remarkablyreduce the difficulty of assembly sequence planning Thisresearch proposed the transmissibility of assembly connec-tion strength and realized the assembly unit partition ofCNC machine tool This paper used assembly relationshipinterference matrix for interference checking and revision ofthe produced assembly unit thereby obtaining assembly unitwith mutual noninterference in the assembly

41 Assembly Unit Partition A product is normally formedby several parts The complexity of assembly relationshipamong parts could greatly influence the assembly perfor-manceThis research quantifies the assembly difficulty amongparts to assembly connection strength and proposed thetransmissibility of assembly connection strength to realizeassembly unit partition based on connection strength

411 Number of Assembly Unit Partition Assembly unit is arelatively independent structural unit to realize one or severalfunctions Assembly unit is a prerequisite to produce simpleassembly planning tasks This paper assumed that each unitis assembled concurrently and after the assembly all units areassembled into a final product on the main assembly line Itwas also assumed that each unit does not show delaying inassembly process With the goal of reducing assembly timethe number of optimal unit partitionwas obtained by formula(3) 119873

119898was rounded and taken as the number of optimal

assembly unit partition Consider

119873119898

le radic119873119901 (3)

where119873119901is the total number of parts of a product and119873

119898is

the number of unit partition of a product

412 Basic Parts of AssemblyUnit Basic part is the importantfunction part for an assembly unit The selection of differentbasic part could reach different assembly units According tothe assembly connection matrix the connection number ofparts with other parts could be used as the basis to identify

4 Mathematical Problems in Engineering

Functionpart Pi

Functionpart Pj

Assemblysemantic Connector Geometric

constraint

Figure 1 Assembly semantic relation graph model

basic part In this paper the sum of edge weight of connectedwith part 119894 in ASRG was taken as the importance degreeof part 119894 in the assembly The importance degree could becalculated by the following formula

119882(119894) =

119899

sum119895=1

119877 (119894 119895) (4)

where 119877(119894 119895) is the edge weight of ASRGAfter the calculation of importance degree of all parts

is finished 119873119898parts with the maximum importance degree

were taken as basic parts for assembly unit partition

413 Assembly Connection Strength The assembly relation-ship graph among parts is similar with the trust network dia-gram among users in recommender systems of e-commerce[13] The transmissibility of similarity was introduced intothe assembly unit partition in this research so that assemblyconnection strength among parts could be transferred

The transmission rules of assembly connection strengthcan be defined as follows assume that there are 119898 pathsbetween part 119860 and part 119861 in ASRG and nodes on the 119894thpath between 119860 and 119861 are respectively 119873

1198941 119873

119894119896 assume

119896 is the number of nodes between 119860 and 119861 If in a paththere is basic part the assembly connection strength between119860 and 119861 was denoted by 0 if in a path there is not basicpart the assembly connection strength between 119860 and 119861 wascalculated by the following formulas

119860119868119894 (119860 119861)

= Min [119877 (1198601198731198941) 119877 (119873

1198941 1198731198942) 119877 (119873

119894119896 119861)] lowast 120573

119894

119860119868 (119860 119861) = Max [1198601198681 (119860 119861) 119860119868

2 (119860 119861) 119860119868119898 (119860 119861)]

(5)

where 119894 = 1 2 119898 119877(119860119873119894119896) is the edge weight of part 119860

and part 119873119894119896in ASRG 119860119868

119894(119860 119861) is the assembly connection

strength between part 119860 and part 119861 on 119894th path 119860119868(119860 119861) theassembly connection strength between part 119860 and part 119861 120573

119894

is an attenuation factor of assembly connection strength on119894th path which decreases with the increase of present transferpath The calculation is as follows

120573119894=

119871 minus 119879119894+ 1

119871 (6)

where 119871 is the threshold of the length of transmission pathdenoted as 5 119879

119894is the length of the 119894th transmission path

denoted as the number of edges

According to the above process assembly connectionstrength of part 119894 and each basic part were identified Nextbased on the assembly connection strength function partswere put in the unit where the basic part with the largestof assembly connection strength was in With the cyclicoperation all parts were partitioned in the unit where basicparts were in In this way the partition of assembly unit wasrealized

42 Interference Checking and Assembly Unit CorrectionSince the factors like assembly precedence constraint andgeometric constraints were not considered in assembly unitpartition interference may happen during assembling Thusin order to improve the accuracy of assembly unit partitionthis paper generated assembly unit interference judgmentmatrix through assembly relationship interference matrix forinterference checking and correction of partitioned assemblyunit

421 Assembly Relationship Interference Matrix Let 119864 =

1198901 1198902 119890

119899 be the set of assembly relationship If assembly

relationship 119890119894after the preferential assembly interfere with

the assembly relationship 119890119895 it is denoted that assembly rela-

tionship 119890119894interferes assembly relationship 119890

119895 119890119894is called the

assembly relationship that produces assembly interference 119890119895

is called the interfered assembly relationship [14] Assemblyrelationship interference exhibits the priority of two assemblyrelationships Assembly relationship interference matrix isused to express assembly interference denoted as 119868(119877) In thematrix if 119903

119894119895= 1 119890

119894after the priority assembly interferes 119890

119895

if 119903119894119895

= 0 no assembly interference occurs between 119890119894and 119890119895

if 119903119894119895

= 0 and 119903119895119894

= 1 119890119894is prior to 119890

119895 If the row of 119890

119894has

all values as zero 119890119894can be assembled prior to other assembly

relationshipsIn the assembly process the composition parts of the

product were only assembled with adjacent parts rather withall rest parts Therefore most interferences of assembly rela-tionship exist in adjacent assembly relationships Howeverdue to the complexity of product structure and assemblyprocess some nonadjacent assembly relationships might beinterfered The interference information could be obtainedthrough the interference extension of assembly relationship

422 Assembly Unit Interference Judgment Matrix With theassembly unit interference judgment matrix for interferencechecking and correction of each assembly unit assemblyunits with no interference could be obtained Through thefollowing 3 steps the assembly unit interference judgmentmatrix can be obtained

Mathematical Problems in Engineering 5

(1) 119898 assembly relationships 119864119880 = 1198901 1198902 119890

119898 in unit

119880 were extracted(2) each line of elements corresponding to 119890

1 1198902 119890

119898

was extracted from the assembly relationship interfer-ence matrix to form the intermediary matrix 119872119880

(3) each line of elements corresponding to 1198901 1198902 119890

119898

in 119872119880 was taken as zero to obtain the assembly unitinterference judgment matrix 119862119880

If the corresponding line of assembly relationship 119890119894(1 le

119894 le 119898) in assembly unit interference judgment matrix 119862119880

has 1 it indicates that assembly relationship 119890119894will interrupt

the assembly of other units that is there is interferencebetween assembly unit 119880 and other units The assemblyrelationship that produced interference was dissembled Thenewly formed assembly unit did not create interference in theprocess of assembly Disassembled parts could be adjusted toother units or partitioned into new units if needed

5 Assembly Sequence Planning Based onGenetic Algorithm

The solutions of assembly sequence planning could bedivided into two categories serial assembly and concur-rent assembly Concurrent assembly could increase the par-allelism of assembly improve assembly efficiency reduceassembly costs and could not trigger ldquocombinatorial explo-sionrdquoThus it fits the assembly sequence planning of complexproducts Genetic algorithm generates satisfactory effectsin the research of combinatorial optimization Assemblysequence planning is a typical combinatorial optimizationissue Therefore this paper used genetic algorithm in opti-mized assembly sequence of parts in assembly units andeach assembly unit and then generated concurrent assemblysequence

51 Chromosome Coding of Assembly Sequence The assemblysequence planning is under the umbrella of combinatorialoptimization and suitable for the coding approach of com-binatorial optimization Therefore this research used order-sensitive codes for the coding of parts For assembly sequenceplanning of 119873 parts the chromosome was divided into 119873

segments and each segment is the corresponding code ofparts In order to facilitate the coding part number in theassembly unit was arranged from small to large and then wascoded

52 Ways to Generate Initial Population Generally speakingunits in the initial population are randomly generated Inthis paper the automatic generation was used to generateinitial population 119910 = Randperm (119899) in MATLAB wasused for population initialization to obtain a nonrepeatedrandom structure including integers from 1 to 119899 as a validchromosome

53 Fitness Function Design of Assembly Sequence Accordingto the feasibility of assembly sequence and the incidence ofthe strengths and weaknesses indicators with the maximuminfluence and easy access to information were used to estab-lish fitness function This research mainly constructed thefitness functions from three aspects including the geometricfeasibility of assembly sequence stability of assembly unitsand the position of basic parts in assembly sequence

(1) Geometric Feasibility Geometric feasibility requires nointerference during disassembly This research used inte-grated interference matrix [15] to derive the feasible disas-sembly direction of any part in the assembly unit It wasassumed that an assembly unit 119860 = 1 2 119899 comprised119899 parts and its corresponding integrated interference matrixwas a matrix 119868

119860with 119899 rows and 6119899 columns as shown in the

following formula

119868119860

=

[[[[[[[[

[

11986811119909

11986811119910

11986811119911

119868111199091015840119868111199101015840119868111199111015840 11986812119909

11986812119910

11986812119911

119868121199091015840119868121199101015840119868121199111015840 sdot sdot sdot 119868

11198991199091198681119899119910

1198681119899119911

119868111989911990910158401198681119899119910101584011986811198991199111015840

11986821119909

11986821119910

11986821119911

119868211199091015840119868211199101015840119868211199111015840 11986822119909

11986822119910

11986822119911

119868221199091015840119868221199101015840119868221199111015840 sdot sdot sdot 119868

21198991199091198682119899119910

1198682119899119911

119868211989911990910158401198682119899119910101584011986821198991199111015840

sdot sdot sdot

1198681198991119909

1198681198991119910

1198681198991119911

119868119899111990910158401198681198991119910101584011986811989911199111015840 1198681198992119909

1198681198992119910

1198681198992119911

119868119899211990910158401198681198992119910101584011986811989921199111015840 sdot sdot sdot 119868

119898119909119868119898119910

119868119898119911

119868119898119909101584011986811989811991010158401198681198981199111015840

]]]]]]]]

]

(7)

If at one direction 119889 part 119894 is not interfered by anypart during disassembly then 119868

119894119889= 0 which indicated

that part 119894 met the geometric feasibility of disassembly andthe disassembly was activated If at one direction 119889 part119894 is interfered by any part during disassembly then 119868

119894119889=

1 which exhibited that parts did not meet the geometricfeasibility and the disassembly was not feasible When part119894 was successfully disassembled all elements at the 119894th row inintegrated interference matrix 119868

119860were 0 indicating that part

119894 did not produce interference with other parts formula (8)could evaluate the geometric feasibility of assembly sequence

1198911=

119873 (119873 minus 1)

2minus 119872 (8)

where 119873 is the total number of parts in assembly unit and119872 is the times of an assembly sequence that generatedinterferences during assembly

(2) Stability of Assembly Unit The stability connection matrix119862 was used to quantify the stability of feasible assemblysequence 119862

119894119895represented the connection relation between

part 119894 and part 119895 When a stable connection relation betweenpart 119894 and part 119895 was observed let 119862

119894119895= 2 when a contact

6 Mathematical Problems in Engineering

connection relation between part 119894 and part 119895 was observedlet 119862119894119895

= 1 when there was no connection relation betweenpart 119894 and part 119895 let 119862

119894119895= 0 The stable connection in

this research refers to connection relations with mandatoryconstraints such as screw joint interference fit and pinnedconnection Hence formula (9) was used to evaluate thestability of assembly sequence

1198912=

119873minus1

sum119894=1

119870119894 (9)

where 119873 is the total number of parts in an assembly unitand 119870

119894is a quantified connection relation between part 119894 and

part 119894 + 1 When a stable connection existed 119870119894= 2 when

a contact connection existed 119870119894

= 1 when no connectionrelation existed 119870

119894= 0

(3) Position of Basic Parts The basic part is the key functionpart in assembly unit and should be firstly assembled duringthe assembly process In initial population number 1 partin each assembly sequence is not necessarily a basic partTherefore the assembly sequence with the basic part in thevery first place should be distinguished from that with thebasic part not in the first place In this research formula (10)was used to evaluate the position of basic parts in assemblysequence

1198913= 119873 minus 119875 (10)

where 119873 is the total number of parts of assembly unit and 119875

is the position of basic parts in assembly sequence

(4) Fitness Function According to the analysis on the aboveevaluation indexes a fitness function of genetic algorithmwas designed for assembly sequence planning as shown inthe following formula

119865 (119878) = 12059611198911+ 12059621198912+ 12059631198913 (11)

where 119878 is the assembly sequence in the population 1205961is the

weight coefficient of geometric feasibility 1205962is the weight

coefficient of assembly unit stability and 1205963is the weight

coefficient of positions of basic parts

54 Genetic Manipulation Genetic algorithm simulates theevolutionary mechanism of ldquosurvival of the fittestrdquo in theliving nature through genetic manipulation The missionof genetic manipulation is to apply certain operations onindividuals in the population according to their fitness (fit-ness evaluation) to environment thereby realizing the evo-lutionary process of survival of the fittest Common geneticmanipulation approaches include selection intersection andmutation

Selection operation refers to the selection of individualswith better fitness to produce new population For assemblysequence planning this research provided three approachesroulette selection linear ranking selection and tournamentselection Crossover operation is used to simulate geneticrecombination in the genetic evolution In the assembly

sequence planning of this paper single-point crossover two-point andmultipoint crossover as well as uniform crossoverwere applied Mutation is used to mimic the variation ofbiological heredity in the evolutionary process In this paperswitched mutation and inserted mutation were provided

55 Control Parameters and Termination Criterion of GeneticAlgorithm In order to ensure that genetic algorithm couldreach the optimal solution along the optimal research trackthe control parameter should be rationally selected whichmainly included the population size 119873 encoding length119871 crossover probability 119875

119888 mutation probability 119875

119898 and

termination conditionFor the assembly sequence planning in this research

encoding length 119871 is the number of parts in the assemblyunit The population size 119873 doubled the number of partsin assembly unit that is 119873 = 2119871 Crossover probability119875119888controls the utilization frequency of crossover operation

Generally 119875119888= 06sim10 Mutation probability 119875

119898controls the

utilization frequency of mutation Generally 119875119898

= 0005sim001 The termination condition of genetic algorithm in thispaper is when a specific algebrawas run on genetic algorithmthe calculation was terminated Besides optimal individualsin the current populationwere taken as the optimal solutions

6 Case Study

With the example of headstock of HTC2550hs CNC turningcenter this paper illustrated the technologies of concurrentassembly sequence planning

61 Assembly Modeling of Headstock of CNC Machine ToolAfter the removal of connectors in headstock there were 43function parts The assembly relationship among functionparts of the headstock was analyzed to establish ASRG ofCNC machine tool Parts were expressed by numbers asshown in Figure 2

As the headstock had a large number of function partsandASRGwas complex this paper usedGraphviz to establishASRG The function parts of headstock were expressed bynumbers The established ASRG is shown in Figure 3 Theassembly relationship information among function partsof headstock was analyzed The analytic hierarchy process(AHP) was used to generate weights of connection semanticsinformation semantic information of transmission and geo-metric constraint information The corresponding weightswere respectively 119908

1= 1199082

= 043 and 1199083

= 014 Nextedge weights in ASRG were calculated according to formula(2)

In Figure 3 as function parts were distinguished fromconnectors there were only 43 nodes in this model Inaddition the assembly relationship information was added inedges Edge information in ASRG could be revised accordingto needs so that ASRG has good maintainability

62 Assembly Unit Partition of Headstock of CNCMachine Tool

621 Selection of Basic Parts in Assembly Unit The totalnumber of function parts of this headstock was 43 According

Mathematical Problems in Engineering 7

Brak

e obj

ects6

Brak

e obj

ects29

Fric

tion

bloc

k41

Fric

tion

bloc

k43

Pisto

n ro

d20

Pisto

n ro

d35

Lid31

Back

-up

bloc

k27

Tank

cove

r28

Sign

al co

nver

ter3

6

Pisto

n ro

d30

Pisto

n ro

d7

Lid9

Lid32

Bend

ing

plat

e10

Bend

ing

plat

e34

Lid33

Fric

tion

bloc

k42

Fric

tion

bloc

k8

4 B

elt p

ulle

y37

Exp

ansio

n sle

eve

38

Exp

ansio

n sle

eve

39

Bra

ke d

isk

12

Rub

ber s

heet

40

Pro

tect

ive b

ox3 L

oop

2 H

eel c

ap1 B

ack

glan

d26

Bac

k be

arin

g ho

usin

g13

Hou

sing

14

For

mer

spac

er15

Hou

sing

21

Spa

cer

25

Spi

ndle

box

22 Locating pin

5 S

pind

le

19

Gla

nd17

Ext

erna

l gla

nd18

Lab

yrin

th ca

sing

23 Special bolts16

Hou

sing

24 Adjustingblock

11

Blin

d fr

ange

Figure 2 Explosive view of headstock

to formula (3) 119873119898

le 656 so the number of optimal unitpartition was 119873

119898= 6 According to edge weight in ASRG

formula (4) was used to calculate the importance degree ofeach part in the assembly The result was shown in Table 1

6 parts with the largest importance degrees were taken asthe basic parts of assembly unit partition In Table 1 part 2529 6 5 40 4 had the largest importance degree so that theywere selected as the basic parts of assembly unit

622 Assembly Unit Partition The assembly connectionstrength between part 119894 and other basic parts was calculated

and part 119894 was put in the unit containing basic part withthe largest connection strength to accomplish the assemblyunit partition Result of assembly unit partition was shown inTable 2 and Figure 4

623 Interference Checking and Assembly Unit CorrectionInterference checking based on the above partition resultswas carried out The interference judgment matrix of assem-bly unit 1 was shown as follows

1198621198801=

11989011198902119890311989041198905

sdot sdot sdot sdot sdot sdot sdot sdot sdot 119890511198905211989053

1198901

1198902

1198904

11989039

11989040

11989041

11989042

11989045

11989046

11989047

[[[[[[[[[[[[[[

[

00000

00000

00000

00000

00000

00000

00001

00000

00000

00000

00000

00000

00000

00000

00000

00100

00100

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

000

000

000

000

000

000

000

000

000

000

]]]]]]]]]]]]]]

]

(12)

By the analysis of interference judgment matrix in eachassembly unit it was found that part 2 17 18 and 19 wouldtrigger assembly interference among units After the disas-sembly the assembly unit without interference was obtainedIn addition part 39 shared an assembly relationship with

part 2 When part 2 was disassembled part 39 was alsodisassembled

Elements of the assembly relationship between part 19and part 5 in the interference matrix were 0 The assemblywas prioritized and it was placed in the unit where the spindle5 was Part 2 and part 39 had bolted connections with the

8 Mathematical Problems in Engineering

Table 1 The importance degree of each part in the assembly

Partnumber Importance degree Part number Importance degree Part number Importance degree Part number Importance degree

1 0796 12 088 23 0426 34 04262 0622 13 014 24 0852 35 05663 0734 14 014 25 4566 36 03984 1182 15 0112 26 1052 37 06225 1862 16 0112 27 088 38 06226 2208 17 0566 28 0482 39 04827 0566 18 0482 29 2606 40 14468 0482 19 0824 30 0566 41 04829 051 20 0566 31 051 42 048210 0426 21 0168 32 051 43 048211 088 22 0426 33 051

13 14 15 21 16 18 2 36 31 9 10

17 5 39 1 6

28 19 26 4 20 7

25 37 41 8

22 24

23

27

40 38 3

424311

12 35 30 34 33 32

29

e7 e6 e52 e10 e9 e8 e11e12 e1 e31 e30

e32

e41

e27e26e2

e13

e33

e53e43 e3

e5

e40

e45 e46

e49

e50

e51

e48

e42 e4 e15

e14

e17

e16

e18 e19

e22

e21

e24e25

e29 e28 e37 e36 e35

e20

e23

e23

e39

e34

e38

Figure 3 ASRG of headstock

Mathematical Problems in Engineering 9

Table 2 Results of assembly unit partition

Unitnumber Basic part number Results of assembly unit partition

1 25 1 17 19 22 23 24 26 27 28 362 29 30 32 33 34 35 42 433 6 7 9 8 10 20 31 414 5 2 13 14 15 16 18 21 395 40 11 126 4 3 37 38

relatively intensive connection strength Elements of assem-bly relationship in the interference matrix were 0 whichcould prioritize assembly Therefore an independent unitwas formed As the assembly connection strength between

part 17 part 18 and basic parts in other unit was smallthe partition was not conducted for independent assemblyResults of modified unit partition were shown in Table 3 andFigure 5

63 Assembly Sequence Planning and Optimization for CNCMachine Tool

631 Case of Optimized Assembly Sequence Assembly unit 3was the braking part of headstock and contained 8 functionparts The corresponding part number was 6 7 8 9 10 2031 41 as shown in Figure 6

The interference relationship between parts in an assem-bly unit at directions (plusmn119883 plusmn119884 plusmn119885) during disassembly andother parts was analyzed An integrated interference matrix119868119860was constructed as follows

119868119860

=

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

000000

111011

010000

111011

100000

011111

011111

010000

011111

000000

000100

111011

100000

000100

000100

000100

000010

100000

000000

000000

100000

000100

000100

000100

011111

011111

000000

000000

100000

001000

000100

000100

000100

010100

000100

000100

000000

000100

000100

000100

111011

100000

100000

100000

100000

000000

011111

100000

111011

100000

000000

100000

100000

111011

000000

000000

000010

100000

100000

100000

100000

000100

000100

000000

]]]]]]]]]]

]

(13)

The stability of the connectivity among parts in theassembly unit was analyzed and a stability connection matrix119862 was constructed as follows

119862 =

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

0

1

0

2

2

1

2

0

1

0

2

1

0

0

0

0

0

2

0

0

0

0

0

0

2

1

0

0

0

0

0

0

2

0

0

0

0

0

0

0

1

0

0

0

0

0

1

2

2

0

0

0

0

1

0

0

0

0

0

0

0

2

0

0

]]]]]]]]]]

]

(14)

The initial population was produced by the randomapproach and the control parameter of genetic algorithmwasset as follows encoding length 119871 = 8 population size119873 = 16crossover probability 119875

119888= 08 mutation probability 119875

119898=

001 generation gen = 100 and genetic manipulation wasrespectively selected by roulette This study conducted two-point crossover switched mutation and reverse operationPart 6 of assembly unit 3 was a basic part Weights ofgeometric feasibility assembly unit stability and positionsof basic parts in the fitness calculation formula (11) wererespectively 120596

1= 04 120596

2= 02 and 120596

3= 04

Run the program in MATLAB to solve the optimalassembly sequence of assembly unit 3 Figure 7 was thefitness variation diagram of genetic algorithm optimizationAccording to the calculation result the last generation of

maximum fitness was 152 and the corresponding optimalassembly sequence was as follows

6 997888rarr 8 997888rarr 7 997888rarr 9 997888rarr 41 997888rarr 20 997888rarr 31 997888rarr 10 (15)

During the assembly process basic parts should be firstassembled and plate cover parts should be assembled aftershaft parts According to the optimized assembly sequenceresults in assembly unit 3 braking part 6 was a basic part andwas firstly assembled Cover 9 and cover 31 were assembledafter the piston rodThus it could be found that the assemblysequence solved from genetic algorithm was rational andfeasible

For assembly units with the number of parts larger than5 the genetic algorithm was used for optimized assem-bly sequence and assembly sequences of other units wereobtained according to experiences The result was shown inTable 4

Similar to the optimized assembly sequencing of parts inthe assembly unit integrated interferencematrix and stabilityconnection matrix among assembly units were constructedAssembly unit 1 was taken as the basic unit so that theassembly sequence of each assembly unit could be obtainedas shown in the following order

Unit 1 997888rarr Unit 4 997888rarr Unit 7 997888rarr Unit 5 997888rarr Unit 6

997888rarr Unit 8 997888rarr Unit 9 997888rarr Unit 2 997888rarr Unit 3(16)

10 Mathematical Problems in Engineering

Assembly unit 1

Assembly unit 2

Assembly unit 6

Assembly unit 3

Assembly unit 4

Assembly unit 5

Figure 4 Result of assembly unit partition

Assembly unit 1

Assembly unit 2

Assembly unit 8

Assembly unit 7Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 3

Assembly unit 9

Figure 5 Result of modified assembly unit partition

Table 3 Results of modified unit partition

Unit number Result of modified assembly unit partition1 1 22 23 24 25 26 27 28 362 29 30 32 33 34 35 42 433 6 7 8 9 10 20 31 414 5 13 14 15 16 19 215 11 12 406 3 4 37 387 2 398 179 18

632 Expression of Assembly Sequence As a fishbone dia-gram can simply directly and flexibly express the assemblysequence product hierarchies and other information thispaper used the fishbone diagram to express concurrentassembly sequence Among them fish head demonstrated thefinal product of assembly Big bones connected with the fishspine represented each assembly unit Small bones connectedwith big bones represented parts Positions of big bones and

Brake object 6Bending plate 10

7 Piston rod 31 Lid41 Friction block

20 Piston rod9 Lid 8 Friction block

Figure 6 Section view of braking part

7

8

9

10

11

12

13

14

15

16

10 20 30 40 50 60 70 80 90 100

Col

ony

adap

tatio

n de

gree

Maximum colony adaptation degreeEvolutionary process

Average colony adaptation degree

Genetic algebra

Figure 7 Fitness variation diagram

Table 4 Assembly sequence planning of each unit

Assembly unitnumber Assembly sequence

1 25rarr 27rarr 26rarr 1rarr 36rarr 28rarr 24rarr 22rarr 232 29rarr 42rarr 30rarr 32rarr 43rarr 35rarr 33rarr 343 6rarr 8rarr 7rarr 9rarr 41rarr 20rarr 31rarr 104 5rarr 19rarr 21rarr 16rarr 15rarr 14rarr 135 40rarr 12rarr 116 4rarr 3rarr 38rarr 377 39rarr 28 179 18

small bones in the figure reflected the assembly sequenceThe concurrent assembly sequence of headstock was shownin Figure 8

Mathematical Problems in Engineering 11

5

19

21

16

15

14

13

40

12

11 17

29

42

30

32

43

35

33

34

25

27

26

1

36

28

24

22

23

39

2

4

3

38

37 18 10

31

20

41

9

7

8

6

Assembly unit 4

Assembly unit 5

Assembly unit 2

Assembly unit 3

Assembly unit 9

Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 8

Spindle box

Figure 8 Expression of headstock assembly sequence

7 Conclusions

In an era of global economic integration to occupy avantage ground in fierce market competition enterprisesmust enhance their product innovation capability improveproduct quality and performance also shorten the productdesign andmanufacturing cycle and reduce costs Inmodernmanufacturing industry the workload of product in theassembling stage is very heavy producing high costs andhard to realize assembly automationTherefore newmethodsand technologies are in urgent need to support the productassembly design On the basis of related research of assemblydesign this paper explored into assemblymodeling assemblyunit partition and optimized assembly sequence of CNCmachine tool In this way the assembly design efficiency ofproduct was improved This research reached the followingconclusions

(1) By distinguishing connectors from function parts asimplified ASRG was established and semantic informationof assembly was added in the edges of assembly model Thestudy on assembly modeling cases of headstock of CNCmachine tool showed that the model efficiently expressed theassembly relationship among parts

(2)According to the transmissibility of assembly connec-tion strength the assembly connection strength among partswas calculated so that the information of nonadjacent partscould be utilized The assembly unit partition based on theconnection strength changed the previous partition methodsby experiences so as to quantify the partition of assembly unitIn addition through interference checking and correction ofassembly unit the rationality and accuracy of assembly unitpartition were improved

(3) According to needs of assembly sequence planninggenerations of coding approach fitness function genetic

operator control parameters and initial population of geneticalgorithm were designedThrough case analysis of headstockof CNC machine tool the effectiveness of using geneticalgorithm for assembly sequence planning and optimizationwas verified

This research actively explored into the assembly mod-eling assembly unit partition assembly sequence planningand optimization and other aspects of CNC machine toolrealized the optimized assembly sequence of CNC machinetool and provided theoretical basis and technical supports forthe improvement of assembly quality and efficiency of CNCmachine tool

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (N120403003) the Scienceand Technology Funds of Liaoning Province (20130200522011216010) and the Research Funds of State ldquoTwelve FiverdquoSupport Program (2012BAF12B08)

References

[1] V Blanchot and A Daidie ldquoRiveted assembly modelling Studyand numerical characterisation of a riveting processrdquo Journal ofMaterials Processing Technology vol 180 no 1ndash3 pp 201ndash2092006

[2] T Gu Z Xu and Z Yang ldquoSymbolic OBDD representations formechanical assembly sequencesrdquo Computer Aided Design vol40 no 4 pp 411ndash421 2008

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

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Mathematical Problems in Engineering 3

assembly Coordinating semantics are mainly to depict theassembly relation related to axial restraint which does notneed auxiliary tools or only needs simple tools User-definedsemantics are mainly to realize the extension of assemblysemantics which are defined and extended by users

32 Assembly Semantic Relation Graph Model To simplifythe assembly modeling of complex product and supportassembly sequence planning this paper used assemblysemantic relation graph (ASRG) to depict the assemblystructure of products Related concepts and descriptionswereas follows

(a) function parts parts in assembly units with connec-tors removed

(b) ASRG with assembly semantics and connector infor-mation included in edges of undirected graph ASRGuses the undirected graph to describe the assemblyconnectivity of function parts in assembly units anddefines the graph as a quaternion

ASRG = ⟨119881 119864 119860119881 119860119864⟩ (1)

where 119881 = 1198751 1198752 119875

119899 is the set of nodes in the assembly

relationship diagram showing the function parts of products119899 is the number of parts 119864 = 119864

1 1198642 119864

119898 is the set of

edges in the assembly relationship diagram and each edgecorresponds to one assembly relationship 119898 is the numberof assembly relationship 119860

119881represents the attribute set of

nodes to describe the information of function parts 119860119864

represents the attribute set of edges to describe the assemblyinformation of function parts

Figure 1 shows the ASRGmodel between two parts Edgeattributes of 119875

119894and 119875

119895mainly include assembly semantics

connector and geometric constraint information ASRGseparates assembly information of products from parts infor-mation so that assembly information only contains labelinginformation of parts When attribute information of parts ischanged it does not influence original assembly relationshipwhile simplifying expressions of assembly relationship andreducing the node number in graph and modeling difficulty

33 Weights of Edges in ASRG To describe the importanceof assembly relationship and the difficulty of assembly thispaper defined assembly connection strength as the summa-tion of assembly connection strength of connector informa-tion semantic information of transmission and geometricconstraint information

According to the difficulty of assemblydisassembly con-nectors and transmission semantics parts and evaluatedassembly connection strength of geometric constraint infor-mation are shown by fuzzy hierarchical approach Thesenumerical values aim to exhibit certain difficulty Whenthere were several assembly characteristics among parts theconnection strength should be calculated respectively and

sums should be calculated Thus weights of edges of ASRGcould be calculated by the following formula

119877 (119894 119895) =

3

sum119896=1

119908119896119903119896(119894 119895) 119894 = 119895

1 119894 = 119895

(2)

where 119877(119894 119895) = 119877(119895 119894) 119894 119895 isin 1 2 119899 is weights ofedges between 119894 and 119895 among parts 119908

119896is the weight of 119896th

evaluation value of assembly connection strength 119903119896(119894 119895) is

the 119896th evaluation value of assembly connection strength

4 Assembly Unit Partition Based onConnection Strength

CNC machine tool products have more complex processof assembly sequence planning than normal mechanicalproductsThe complex product is partitioned into the assem-bly unit containing fewer parts which can effectively over-come the issue of ldquocombinatorial explosionrdquo and remarkablyreduce the difficulty of assembly sequence planning Thisresearch proposed the transmissibility of assembly connec-tion strength and realized the assembly unit partition ofCNC machine tool This paper used assembly relationshipinterference matrix for interference checking and revision ofthe produced assembly unit thereby obtaining assembly unitwith mutual noninterference in the assembly

41 Assembly Unit Partition A product is normally formedby several parts The complexity of assembly relationshipamong parts could greatly influence the assembly perfor-manceThis research quantifies the assembly difficulty amongparts to assembly connection strength and proposed thetransmissibility of assembly connection strength to realizeassembly unit partition based on connection strength

411 Number of Assembly Unit Partition Assembly unit is arelatively independent structural unit to realize one or severalfunctions Assembly unit is a prerequisite to produce simpleassembly planning tasks This paper assumed that each unitis assembled concurrently and after the assembly all units areassembled into a final product on the main assembly line Itwas also assumed that each unit does not show delaying inassembly process With the goal of reducing assembly timethe number of optimal unit partitionwas obtained by formula(3) 119873

119898was rounded and taken as the number of optimal

assembly unit partition Consider

119873119898

le radic119873119901 (3)

where119873119901is the total number of parts of a product and119873

119898is

the number of unit partition of a product

412 Basic Parts of AssemblyUnit Basic part is the importantfunction part for an assembly unit The selection of differentbasic part could reach different assembly units According tothe assembly connection matrix the connection number ofparts with other parts could be used as the basis to identify

4 Mathematical Problems in Engineering

Functionpart Pi

Functionpart Pj

Assemblysemantic Connector Geometric

constraint

Figure 1 Assembly semantic relation graph model

basic part In this paper the sum of edge weight of connectedwith part 119894 in ASRG was taken as the importance degreeof part 119894 in the assembly The importance degree could becalculated by the following formula

119882(119894) =

119899

sum119895=1

119877 (119894 119895) (4)

where 119877(119894 119895) is the edge weight of ASRGAfter the calculation of importance degree of all parts

is finished 119873119898parts with the maximum importance degree

were taken as basic parts for assembly unit partition

413 Assembly Connection Strength The assembly relation-ship graph among parts is similar with the trust network dia-gram among users in recommender systems of e-commerce[13] The transmissibility of similarity was introduced intothe assembly unit partition in this research so that assemblyconnection strength among parts could be transferred

The transmission rules of assembly connection strengthcan be defined as follows assume that there are 119898 pathsbetween part 119860 and part 119861 in ASRG and nodes on the 119894thpath between 119860 and 119861 are respectively 119873

1198941 119873

119894119896 assume

119896 is the number of nodes between 119860 and 119861 If in a paththere is basic part the assembly connection strength between119860 and 119861 was denoted by 0 if in a path there is not basicpart the assembly connection strength between 119860 and 119861 wascalculated by the following formulas

119860119868119894 (119860 119861)

= Min [119877 (1198601198731198941) 119877 (119873

1198941 1198731198942) 119877 (119873

119894119896 119861)] lowast 120573

119894

119860119868 (119860 119861) = Max [1198601198681 (119860 119861) 119860119868

2 (119860 119861) 119860119868119898 (119860 119861)]

(5)

where 119894 = 1 2 119898 119877(119860119873119894119896) is the edge weight of part 119860

and part 119873119894119896in ASRG 119860119868

119894(119860 119861) is the assembly connection

strength between part 119860 and part 119861 on 119894th path 119860119868(119860 119861) theassembly connection strength between part 119860 and part 119861 120573

119894

is an attenuation factor of assembly connection strength on119894th path which decreases with the increase of present transferpath The calculation is as follows

120573119894=

119871 minus 119879119894+ 1

119871 (6)

where 119871 is the threshold of the length of transmission pathdenoted as 5 119879

119894is the length of the 119894th transmission path

denoted as the number of edges

According to the above process assembly connectionstrength of part 119894 and each basic part were identified Nextbased on the assembly connection strength function partswere put in the unit where the basic part with the largestof assembly connection strength was in With the cyclicoperation all parts were partitioned in the unit where basicparts were in In this way the partition of assembly unit wasrealized

42 Interference Checking and Assembly Unit CorrectionSince the factors like assembly precedence constraint andgeometric constraints were not considered in assembly unitpartition interference may happen during assembling Thusin order to improve the accuracy of assembly unit partitionthis paper generated assembly unit interference judgmentmatrix through assembly relationship interference matrix forinterference checking and correction of partitioned assemblyunit

421 Assembly Relationship Interference Matrix Let 119864 =

1198901 1198902 119890

119899 be the set of assembly relationship If assembly

relationship 119890119894after the preferential assembly interfere with

the assembly relationship 119890119895 it is denoted that assembly rela-

tionship 119890119894interferes assembly relationship 119890

119895 119890119894is called the

assembly relationship that produces assembly interference 119890119895

is called the interfered assembly relationship [14] Assemblyrelationship interference exhibits the priority of two assemblyrelationships Assembly relationship interference matrix isused to express assembly interference denoted as 119868(119877) In thematrix if 119903

119894119895= 1 119890

119894after the priority assembly interferes 119890

119895

if 119903119894119895

= 0 no assembly interference occurs between 119890119894and 119890119895

if 119903119894119895

= 0 and 119903119895119894

= 1 119890119894is prior to 119890

119895 If the row of 119890

119894has

all values as zero 119890119894can be assembled prior to other assembly

relationshipsIn the assembly process the composition parts of the

product were only assembled with adjacent parts rather withall rest parts Therefore most interferences of assembly rela-tionship exist in adjacent assembly relationships Howeverdue to the complexity of product structure and assemblyprocess some nonadjacent assembly relationships might beinterfered The interference information could be obtainedthrough the interference extension of assembly relationship

422 Assembly Unit Interference Judgment Matrix With theassembly unit interference judgment matrix for interferencechecking and correction of each assembly unit assemblyunits with no interference could be obtained Through thefollowing 3 steps the assembly unit interference judgmentmatrix can be obtained

Mathematical Problems in Engineering 5

(1) 119898 assembly relationships 119864119880 = 1198901 1198902 119890

119898 in unit

119880 were extracted(2) each line of elements corresponding to 119890

1 1198902 119890

119898

was extracted from the assembly relationship interfer-ence matrix to form the intermediary matrix 119872119880

(3) each line of elements corresponding to 1198901 1198902 119890

119898

in 119872119880 was taken as zero to obtain the assembly unitinterference judgment matrix 119862119880

If the corresponding line of assembly relationship 119890119894(1 le

119894 le 119898) in assembly unit interference judgment matrix 119862119880

has 1 it indicates that assembly relationship 119890119894will interrupt

the assembly of other units that is there is interferencebetween assembly unit 119880 and other units The assemblyrelationship that produced interference was dissembled Thenewly formed assembly unit did not create interference in theprocess of assembly Disassembled parts could be adjusted toother units or partitioned into new units if needed

5 Assembly Sequence Planning Based onGenetic Algorithm

The solutions of assembly sequence planning could bedivided into two categories serial assembly and concur-rent assembly Concurrent assembly could increase the par-allelism of assembly improve assembly efficiency reduceassembly costs and could not trigger ldquocombinatorial explo-sionrdquoThus it fits the assembly sequence planning of complexproducts Genetic algorithm generates satisfactory effectsin the research of combinatorial optimization Assemblysequence planning is a typical combinatorial optimizationissue Therefore this paper used genetic algorithm in opti-mized assembly sequence of parts in assembly units andeach assembly unit and then generated concurrent assemblysequence

51 Chromosome Coding of Assembly Sequence The assemblysequence planning is under the umbrella of combinatorialoptimization and suitable for the coding approach of com-binatorial optimization Therefore this research used order-sensitive codes for the coding of parts For assembly sequenceplanning of 119873 parts the chromosome was divided into 119873

segments and each segment is the corresponding code ofparts In order to facilitate the coding part number in theassembly unit was arranged from small to large and then wascoded

52 Ways to Generate Initial Population Generally speakingunits in the initial population are randomly generated Inthis paper the automatic generation was used to generateinitial population 119910 = Randperm (119899) in MATLAB wasused for population initialization to obtain a nonrepeatedrandom structure including integers from 1 to 119899 as a validchromosome

53 Fitness Function Design of Assembly Sequence Accordingto the feasibility of assembly sequence and the incidence ofthe strengths and weaknesses indicators with the maximuminfluence and easy access to information were used to estab-lish fitness function This research mainly constructed thefitness functions from three aspects including the geometricfeasibility of assembly sequence stability of assembly unitsand the position of basic parts in assembly sequence

(1) Geometric Feasibility Geometric feasibility requires nointerference during disassembly This research used inte-grated interference matrix [15] to derive the feasible disas-sembly direction of any part in the assembly unit It wasassumed that an assembly unit 119860 = 1 2 119899 comprised119899 parts and its corresponding integrated interference matrixwas a matrix 119868

119860with 119899 rows and 6119899 columns as shown in the

following formula

119868119860

=

[[[[[[[[

[

11986811119909

11986811119910

11986811119911

119868111199091015840119868111199101015840119868111199111015840 11986812119909

11986812119910

11986812119911

119868121199091015840119868121199101015840119868121199111015840 sdot sdot sdot 119868

11198991199091198681119899119910

1198681119899119911

119868111989911990910158401198681119899119910101584011986811198991199111015840

11986821119909

11986821119910

11986821119911

119868211199091015840119868211199101015840119868211199111015840 11986822119909

11986822119910

11986822119911

119868221199091015840119868221199101015840119868221199111015840 sdot sdot sdot 119868

21198991199091198682119899119910

1198682119899119911

119868211989911990910158401198682119899119910101584011986821198991199111015840

sdot sdot sdot

1198681198991119909

1198681198991119910

1198681198991119911

119868119899111990910158401198681198991119910101584011986811989911199111015840 1198681198992119909

1198681198992119910

1198681198992119911

119868119899211990910158401198681198992119910101584011986811989921199111015840 sdot sdot sdot 119868

119898119909119868119898119910

119868119898119911

119868119898119909101584011986811989811991010158401198681198981199111015840

]]]]]]]]

]

(7)

If at one direction 119889 part 119894 is not interfered by anypart during disassembly then 119868

119894119889= 0 which indicated

that part 119894 met the geometric feasibility of disassembly andthe disassembly was activated If at one direction 119889 part119894 is interfered by any part during disassembly then 119868

119894119889=

1 which exhibited that parts did not meet the geometricfeasibility and the disassembly was not feasible When part119894 was successfully disassembled all elements at the 119894th row inintegrated interference matrix 119868

119860were 0 indicating that part

119894 did not produce interference with other parts formula (8)could evaluate the geometric feasibility of assembly sequence

1198911=

119873 (119873 minus 1)

2minus 119872 (8)

where 119873 is the total number of parts in assembly unit and119872 is the times of an assembly sequence that generatedinterferences during assembly

(2) Stability of Assembly Unit The stability connection matrix119862 was used to quantify the stability of feasible assemblysequence 119862

119894119895represented the connection relation between

part 119894 and part 119895 When a stable connection relation betweenpart 119894 and part 119895 was observed let 119862

119894119895= 2 when a contact

6 Mathematical Problems in Engineering

connection relation between part 119894 and part 119895 was observedlet 119862119894119895

= 1 when there was no connection relation betweenpart 119894 and part 119895 let 119862

119894119895= 0 The stable connection in

this research refers to connection relations with mandatoryconstraints such as screw joint interference fit and pinnedconnection Hence formula (9) was used to evaluate thestability of assembly sequence

1198912=

119873minus1

sum119894=1

119870119894 (9)

where 119873 is the total number of parts in an assembly unitand 119870

119894is a quantified connection relation between part 119894 and

part 119894 + 1 When a stable connection existed 119870119894= 2 when

a contact connection existed 119870119894

= 1 when no connectionrelation existed 119870

119894= 0

(3) Position of Basic Parts The basic part is the key functionpart in assembly unit and should be firstly assembled duringthe assembly process In initial population number 1 partin each assembly sequence is not necessarily a basic partTherefore the assembly sequence with the basic part in thevery first place should be distinguished from that with thebasic part not in the first place In this research formula (10)was used to evaluate the position of basic parts in assemblysequence

1198913= 119873 minus 119875 (10)

where 119873 is the total number of parts of assembly unit and 119875

is the position of basic parts in assembly sequence

(4) Fitness Function According to the analysis on the aboveevaluation indexes a fitness function of genetic algorithmwas designed for assembly sequence planning as shown inthe following formula

119865 (119878) = 12059611198911+ 12059621198912+ 12059631198913 (11)

where 119878 is the assembly sequence in the population 1205961is the

weight coefficient of geometric feasibility 1205962is the weight

coefficient of assembly unit stability and 1205963is the weight

coefficient of positions of basic parts

54 Genetic Manipulation Genetic algorithm simulates theevolutionary mechanism of ldquosurvival of the fittestrdquo in theliving nature through genetic manipulation The missionof genetic manipulation is to apply certain operations onindividuals in the population according to their fitness (fit-ness evaluation) to environment thereby realizing the evo-lutionary process of survival of the fittest Common geneticmanipulation approaches include selection intersection andmutation

Selection operation refers to the selection of individualswith better fitness to produce new population For assemblysequence planning this research provided three approachesroulette selection linear ranking selection and tournamentselection Crossover operation is used to simulate geneticrecombination in the genetic evolution In the assembly

sequence planning of this paper single-point crossover two-point andmultipoint crossover as well as uniform crossoverwere applied Mutation is used to mimic the variation ofbiological heredity in the evolutionary process In this paperswitched mutation and inserted mutation were provided

55 Control Parameters and Termination Criterion of GeneticAlgorithm In order to ensure that genetic algorithm couldreach the optimal solution along the optimal research trackthe control parameter should be rationally selected whichmainly included the population size 119873 encoding length119871 crossover probability 119875

119888 mutation probability 119875

119898 and

termination conditionFor the assembly sequence planning in this research

encoding length 119871 is the number of parts in the assemblyunit The population size 119873 doubled the number of partsin assembly unit that is 119873 = 2119871 Crossover probability119875119888controls the utilization frequency of crossover operation

Generally 119875119888= 06sim10 Mutation probability 119875

119898controls the

utilization frequency of mutation Generally 119875119898

= 0005sim001 The termination condition of genetic algorithm in thispaper is when a specific algebrawas run on genetic algorithmthe calculation was terminated Besides optimal individualsin the current populationwere taken as the optimal solutions

6 Case Study

With the example of headstock of HTC2550hs CNC turningcenter this paper illustrated the technologies of concurrentassembly sequence planning

61 Assembly Modeling of Headstock of CNC Machine ToolAfter the removal of connectors in headstock there were 43function parts The assembly relationship among functionparts of the headstock was analyzed to establish ASRG ofCNC machine tool Parts were expressed by numbers asshown in Figure 2

As the headstock had a large number of function partsandASRGwas complex this paper usedGraphviz to establishASRG The function parts of headstock were expressed bynumbers The established ASRG is shown in Figure 3 Theassembly relationship information among function partsof headstock was analyzed The analytic hierarchy process(AHP) was used to generate weights of connection semanticsinformation semantic information of transmission and geo-metric constraint information The corresponding weightswere respectively 119908

1= 1199082

= 043 and 1199083

= 014 Nextedge weights in ASRG were calculated according to formula(2)

In Figure 3 as function parts were distinguished fromconnectors there were only 43 nodes in this model Inaddition the assembly relationship information was added inedges Edge information in ASRG could be revised accordingto needs so that ASRG has good maintainability

62 Assembly Unit Partition of Headstock of CNCMachine Tool

621 Selection of Basic Parts in Assembly Unit The totalnumber of function parts of this headstock was 43 According

Mathematical Problems in Engineering 7

Brak

e obj

ects6

Brak

e obj

ects29

Fric

tion

bloc

k41

Fric

tion

bloc

k43

Pisto

n ro

d20

Pisto

n ro

d35

Lid31

Back

-up

bloc

k27

Tank

cove

r28

Sign

al co

nver

ter3

6

Pisto

n ro

d30

Pisto

n ro

d7

Lid9

Lid32

Bend

ing

plat

e10

Bend

ing

plat

e34

Lid33

Fric

tion

bloc

k42

Fric

tion

bloc

k8

4 B

elt p

ulle

y37

Exp

ansio

n sle

eve

38

Exp

ansio

n sle

eve

39

Bra

ke d

isk

12

Rub

ber s

heet

40

Pro

tect

ive b

ox3 L

oop

2 H

eel c

ap1 B

ack

glan

d26

Bac

k be

arin

g ho

usin

g13

Hou

sing

14

For

mer

spac

er15

Hou

sing

21

Spa

cer

25

Spi

ndle

box

22 Locating pin

5 S

pind

le

19

Gla

nd17

Ext

erna

l gla

nd18

Lab

yrin

th ca

sing

23 Special bolts16

Hou

sing

24 Adjustingblock

11

Blin

d fr

ange

Figure 2 Explosive view of headstock

to formula (3) 119873119898

le 656 so the number of optimal unitpartition was 119873

119898= 6 According to edge weight in ASRG

formula (4) was used to calculate the importance degree ofeach part in the assembly The result was shown in Table 1

6 parts with the largest importance degrees were taken asthe basic parts of assembly unit partition In Table 1 part 2529 6 5 40 4 had the largest importance degree so that theywere selected as the basic parts of assembly unit

622 Assembly Unit Partition The assembly connectionstrength between part 119894 and other basic parts was calculated

and part 119894 was put in the unit containing basic part withthe largest connection strength to accomplish the assemblyunit partition Result of assembly unit partition was shown inTable 2 and Figure 4

623 Interference Checking and Assembly Unit CorrectionInterference checking based on the above partition resultswas carried out The interference judgment matrix of assem-bly unit 1 was shown as follows

1198621198801=

11989011198902119890311989041198905

sdot sdot sdot sdot sdot sdot sdot sdot sdot 119890511198905211989053

1198901

1198902

1198904

11989039

11989040

11989041

11989042

11989045

11989046

11989047

[[[[[[[[[[[[[[

[

00000

00000

00000

00000

00000

00000

00001

00000

00000

00000

00000

00000

00000

00000

00000

00100

00100

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

000

000

000

000

000

000

000

000

000

000

]]]]]]]]]]]]]]

]

(12)

By the analysis of interference judgment matrix in eachassembly unit it was found that part 2 17 18 and 19 wouldtrigger assembly interference among units After the disas-sembly the assembly unit without interference was obtainedIn addition part 39 shared an assembly relationship with

part 2 When part 2 was disassembled part 39 was alsodisassembled

Elements of the assembly relationship between part 19and part 5 in the interference matrix were 0 The assemblywas prioritized and it was placed in the unit where the spindle5 was Part 2 and part 39 had bolted connections with the

8 Mathematical Problems in Engineering

Table 1 The importance degree of each part in the assembly

Partnumber Importance degree Part number Importance degree Part number Importance degree Part number Importance degree

1 0796 12 088 23 0426 34 04262 0622 13 014 24 0852 35 05663 0734 14 014 25 4566 36 03984 1182 15 0112 26 1052 37 06225 1862 16 0112 27 088 38 06226 2208 17 0566 28 0482 39 04827 0566 18 0482 29 2606 40 14468 0482 19 0824 30 0566 41 04829 051 20 0566 31 051 42 048210 0426 21 0168 32 051 43 048211 088 22 0426 33 051

13 14 15 21 16 18 2 36 31 9 10

17 5 39 1 6

28 19 26 4 20 7

25 37 41 8

22 24

23

27

40 38 3

424311

12 35 30 34 33 32

29

e7 e6 e52 e10 e9 e8 e11e12 e1 e31 e30

e32

e41

e27e26e2

e13

e33

e53e43 e3

e5

e40

e45 e46

e49

e50

e51

e48

e42 e4 e15

e14

e17

e16

e18 e19

e22

e21

e24e25

e29 e28 e37 e36 e35

e20

e23

e23

e39

e34

e38

Figure 3 ASRG of headstock

Mathematical Problems in Engineering 9

Table 2 Results of assembly unit partition

Unitnumber Basic part number Results of assembly unit partition

1 25 1 17 19 22 23 24 26 27 28 362 29 30 32 33 34 35 42 433 6 7 9 8 10 20 31 414 5 2 13 14 15 16 18 21 395 40 11 126 4 3 37 38

relatively intensive connection strength Elements of assem-bly relationship in the interference matrix were 0 whichcould prioritize assembly Therefore an independent unitwas formed As the assembly connection strength between

part 17 part 18 and basic parts in other unit was smallthe partition was not conducted for independent assemblyResults of modified unit partition were shown in Table 3 andFigure 5

63 Assembly Sequence Planning and Optimization for CNCMachine Tool

631 Case of Optimized Assembly Sequence Assembly unit 3was the braking part of headstock and contained 8 functionparts The corresponding part number was 6 7 8 9 10 2031 41 as shown in Figure 6

The interference relationship between parts in an assem-bly unit at directions (plusmn119883 plusmn119884 plusmn119885) during disassembly andother parts was analyzed An integrated interference matrix119868119860was constructed as follows

119868119860

=

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

000000

111011

010000

111011

100000

011111

011111

010000

011111

000000

000100

111011

100000

000100

000100

000100

000010

100000

000000

000000

100000

000100

000100

000100

011111

011111

000000

000000

100000

001000

000100

000100

000100

010100

000100

000100

000000

000100

000100

000100

111011

100000

100000

100000

100000

000000

011111

100000

111011

100000

000000

100000

100000

111011

000000

000000

000010

100000

100000

100000

100000

000100

000100

000000

]]]]]]]]]]

]

(13)

The stability of the connectivity among parts in theassembly unit was analyzed and a stability connection matrix119862 was constructed as follows

119862 =

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

0

1

0

2

2

1

2

0

1

0

2

1

0

0

0

0

0

2

0

0

0

0

0

0

2

1

0

0

0

0

0

0

2

0

0

0

0

0

0

0

1

0

0

0

0

0

1

2

2

0

0

0

0

1

0

0

0

0

0

0

0

2

0

0

]]]]]]]]]]

]

(14)

The initial population was produced by the randomapproach and the control parameter of genetic algorithmwasset as follows encoding length 119871 = 8 population size119873 = 16crossover probability 119875

119888= 08 mutation probability 119875

119898=

001 generation gen = 100 and genetic manipulation wasrespectively selected by roulette This study conducted two-point crossover switched mutation and reverse operationPart 6 of assembly unit 3 was a basic part Weights ofgeometric feasibility assembly unit stability and positionsof basic parts in the fitness calculation formula (11) wererespectively 120596

1= 04 120596

2= 02 and 120596

3= 04

Run the program in MATLAB to solve the optimalassembly sequence of assembly unit 3 Figure 7 was thefitness variation diagram of genetic algorithm optimizationAccording to the calculation result the last generation of

maximum fitness was 152 and the corresponding optimalassembly sequence was as follows

6 997888rarr 8 997888rarr 7 997888rarr 9 997888rarr 41 997888rarr 20 997888rarr 31 997888rarr 10 (15)

During the assembly process basic parts should be firstassembled and plate cover parts should be assembled aftershaft parts According to the optimized assembly sequenceresults in assembly unit 3 braking part 6 was a basic part andwas firstly assembled Cover 9 and cover 31 were assembledafter the piston rodThus it could be found that the assemblysequence solved from genetic algorithm was rational andfeasible

For assembly units with the number of parts larger than5 the genetic algorithm was used for optimized assem-bly sequence and assembly sequences of other units wereobtained according to experiences The result was shown inTable 4

Similar to the optimized assembly sequencing of parts inthe assembly unit integrated interferencematrix and stabilityconnection matrix among assembly units were constructedAssembly unit 1 was taken as the basic unit so that theassembly sequence of each assembly unit could be obtainedas shown in the following order

Unit 1 997888rarr Unit 4 997888rarr Unit 7 997888rarr Unit 5 997888rarr Unit 6

997888rarr Unit 8 997888rarr Unit 9 997888rarr Unit 2 997888rarr Unit 3(16)

10 Mathematical Problems in Engineering

Assembly unit 1

Assembly unit 2

Assembly unit 6

Assembly unit 3

Assembly unit 4

Assembly unit 5

Figure 4 Result of assembly unit partition

Assembly unit 1

Assembly unit 2

Assembly unit 8

Assembly unit 7Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 3

Assembly unit 9

Figure 5 Result of modified assembly unit partition

Table 3 Results of modified unit partition

Unit number Result of modified assembly unit partition1 1 22 23 24 25 26 27 28 362 29 30 32 33 34 35 42 433 6 7 8 9 10 20 31 414 5 13 14 15 16 19 215 11 12 406 3 4 37 387 2 398 179 18

632 Expression of Assembly Sequence As a fishbone dia-gram can simply directly and flexibly express the assemblysequence product hierarchies and other information thispaper used the fishbone diagram to express concurrentassembly sequence Among them fish head demonstrated thefinal product of assembly Big bones connected with the fishspine represented each assembly unit Small bones connectedwith big bones represented parts Positions of big bones and

Brake object 6Bending plate 10

7 Piston rod 31 Lid41 Friction block

20 Piston rod9 Lid 8 Friction block

Figure 6 Section view of braking part

7

8

9

10

11

12

13

14

15

16

10 20 30 40 50 60 70 80 90 100

Col

ony

adap

tatio

n de

gree

Maximum colony adaptation degreeEvolutionary process

Average colony adaptation degree

Genetic algebra

Figure 7 Fitness variation diagram

Table 4 Assembly sequence planning of each unit

Assembly unitnumber Assembly sequence

1 25rarr 27rarr 26rarr 1rarr 36rarr 28rarr 24rarr 22rarr 232 29rarr 42rarr 30rarr 32rarr 43rarr 35rarr 33rarr 343 6rarr 8rarr 7rarr 9rarr 41rarr 20rarr 31rarr 104 5rarr 19rarr 21rarr 16rarr 15rarr 14rarr 135 40rarr 12rarr 116 4rarr 3rarr 38rarr 377 39rarr 28 179 18

small bones in the figure reflected the assembly sequenceThe concurrent assembly sequence of headstock was shownin Figure 8

Mathematical Problems in Engineering 11

5

19

21

16

15

14

13

40

12

11 17

29

42

30

32

43

35

33

34

25

27

26

1

36

28

24

22

23

39

2

4

3

38

37 18 10

31

20

41

9

7

8

6

Assembly unit 4

Assembly unit 5

Assembly unit 2

Assembly unit 3

Assembly unit 9

Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 8

Spindle box

Figure 8 Expression of headstock assembly sequence

7 Conclusions

In an era of global economic integration to occupy avantage ground in fierce market competition enterprisesmust enhance their product innovation capability improveproduct quality and performance also shorten the productdesign andmanufacturing cycle and reduce costs Inmodernmanufacturing industry the workload of product in theassembling stage is very heavy producing high costs andhard to realize assembly automationTherefore newmethodsand technologies are in urgent need to support the productassembly design On the basis of related research of assemblydesign this paper explored into assemblymodeling assemblyunit partition and optimized assembly sequence of CNCmachine tool In this way the assembly design efficiency ofproduct was improved This research reached the followingconclusions

(1) By distinguishing connectors from function parts asimplified ASRG was established and semantic informationof assembly was added in the edges of assembly model Thestudy on assembly modeling cases of headstock of CNCmachine tool showed that the model efficiently expressed theassembly relationship among parts

(2)According to the transmissibility of assembly connec-tion strength the assembly connection strength among partswas calculated so that the information of nonadjacent partscould be utilized The assembly unit partition based on theconnection strength changed the previous partition methodsby experiences so as to quantify the partition of assembly unitIn addition through interference checking and correction ofassembly unit the rationality and accuracy of assembly unitpartition were improved

(3) According to needs of assembly sequence planninggenerations of coding approach fitness function genetic

operator control parameters and initial population of geneticalgorithm were designedThrough case analysis of headstockof CNC machine tool the effectiveness of using geneticalgorithm for assembly sequence planning and optimizationwas verified

This research actively explored into the assembly mod-eling assembly unit partition assembly sequence planningand optimization and other aspects of CNC machine toolrealized the optimized assembly sequence of CNC machinetool and provided theoretical basis and technical supports forthe improvement of assembly quality and efficiency of CNCmachine tool

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (N120403003) the Scienceand Technology Funds of Liaoning Province (20130200522011216010) and the Research Funds of State ldquoTwelve FiverdquoSupport Program (2012BAF12B08)

References

[1] V Blanchot and A Daidie ldquoRiveted assembly modelling Studyand numerical characterisation of a riveting processrdquo Journal ofMaterials Processing Technology vol 180 no 1ndash3 pp 201ndash2092006

[2] T Gu Z Xu and Z Yang ldquoSymbolic OBDD representations formechanical assembly sequencesrdquo Computer Aided Design vol40 no 4 pp 411ndash421 2008

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Functionpart Pi

Functionpart Pj

Assemblysemantic Connector Geometric

constraint

Figure 1 Assembly semantic relation graph model

basic part In this paper the sum of edge weight of connectedwith part 119894 in ASRG was taken as the importance degreeof part 119894 in the assembly The importance degree could becalculated by the following formula

119882(119894) =

119899

sum119895=1

119877 (119894 119895) (4)

where 119877(119894 119895) is the edge weight of ASRGAfter the calculation of importance degree of all parts

is finished 119873119898parts with the maximum importance degree

were taken as basic parts for assembly unit partition

413 Assembly Connection Strength The assembly relation-ship graph among parts is similar with the trust network dia-gram among users in recommender systems of e-commerce[13] The transmissibility of similarity was introduced intothe assembly unit partition in this research so that assemblyconnection strength among parts could be transferred

The transmission rules of assembly connection strengthcan be defined as follows assume that there are 119898 pathsbetween part 119860 and part 119861 in ASRG and nodes on the 119894thpath between 119860 and 119861 are respectively 119873

1198941 119873

119894119896 assume

119896 is the number of nodes between 119860 and 119861 If in a paththere is basic part the assembly connection strength between119860 and 119861 was denoted by 0 if in a path there is not basicpart the assembly connection strength between 119860 and 119861 wascalculated by the following formulas

119860119868119894 (119860 119861)

= Min [119877 (1198601198731198941) 119877 (119873

1198941 1198731198942) 119877 (119873

119894119896 119861)] lowast 120573

119894

119860119868 (119860 119861) = Max [1198601198681 (119860 119861) 119860119868

2 (119860 119861) 119860119868119898 (119860 119861)]

(5)

where 119894 = 1 2 119898 119877(119860119873119894119896) is the edge weight of part 119860

and part 119873119894119896in ASRG 119860119868

119894(119860 119861) is the assembly connection

strength between part 119860 and part 119861 on 119894th path 119860119868(119860 119861) theassembly connection strength between part 119860 and part 119861 120573

119894

is an attenuation factor of assembly connection strength on119894th path which decreases with the increase of present transferpath The calculation is as follows

120573119894=

119871 minus 119879119894+ 1

119871 (6)

where 119871 is the threshold of the length of transmission pathdenoted as 5 119879

119894is the length of the 119894th transmission path

denoted as the number of edges

According to the above process assembly connectionstrength of part 119894 and each basic part were identified Nextbased on the assembly connection strength function partswere put in the unit where the basic part with the largestof assembly connection strength was in With the cyclicoperation all parts were partitioned in the unit where basicparts were in In this way the partition of assembly unit wasrealized

42 Interference Checking and Assembly Unit CorrectionSince the factors like assembly precedence constraint andgeometric constraints were not considered in assembly unitpartition interference may happen during assembling Thusin order to improve the accuracy of assembly unit partitionthis paper generated assembly unit interference judgmentmatrix through assembly relationship interference matrix forinterference checking and correction of partitioned assemblyunit

421 Assembly Relationship Interference Matrix Let 119864 =

1198901 1198902 119890

119899 be the set of assembly relationship If assembly

relationship 119890119894after the preferential assembly interfere with

the assembly relationship 119890119895 it is denoted that assembly rela-

tionship 119890119894interferes assembly relationship 119890

119895 119890119894is called the

assembly relationship that produces assembly interference 119890119895

is called the interfered assembly relationship [14] Assemblyrelationship interference exhibits the priority of two assemblyrelationships Assembly relationship interference matrix isused to express assembly interference denoted as 119868(119877) In thematrix if 119903

119894119895= 1 119890

119894after the priority assembly interferes 119890

119895

if 119903119894119895

= 0 no assembly interference occurs between 119890119894and 119890119895

if 119903119894119895

= 0 and 119903119895119894

= 1 119890119894is prior to 119890

119895 If the row of 119890

119894has

all values as zero 119890119894can be assembled prior to other assembly

relationshipsIn the assembly process the composition parts of the

product were only assembled with adjacent parts rather withall rest parts Therefore most interferences of assembly rela-tionship exist in adjacent assembly relationships Howeverdue to the complexity of product structure and assemblyprocess some nonadjacent assembly relationships might beinterfered The interference information could be obtainedthrough the interference extension of assembly relationship

422 Assembly Unit Interference Judgment Matrix With theassembly unit interference judgment matrix for interferencechecking and correction of each assembly unit assemblyunits with no interference could be obtained Through thefollowing 3 steps the assembly unit interference judgmentmatrix can be obtained

Mathematical Problems in Engineering 5

(1) 119898 assembly relationships 119864119880 = 1198901 1198902 119890

119898 in unit

119880 were extracted(2) each line of elements corresponding to 119890

1 1198902 119890

119898

was extracted from the assembly relationship interfer-ence matrix to form the intermediary matrix 119872119880

(3) each line of elements corresponding to 1198901 1198902 119890

119898

in 119872119880 was taken as zero to obtain the assembly unitinterference judgment matrix 119862119880

If the corresponding line of assembly relationship 119890119894(1 le

119894 le 119898) in assembly unit interference judgment matrix 119862119880

has 1 it indicates that assembly relationship 119890119894will interrupt

the assembly of other units that is there is interferencebetween assembly unit 119880 and other units The assemblyrelationship that produced interference was dissembled Thenewly formed assembly unit did not create interference in theprocess of assembly Disassembled parts could be adjusted toother units or partitioned into new units if needed

5 Assembly Sequence Planning Based onGenetic Algorithm

The solutions of assembly sequence planning could bedivided into two categories serial assembly and concur-rent assembly Concurrent assembly could increase the par-allelism of assembly improve assembly efficiency reduceassembly costs and could not trigger ldquocombinatorial explo-sionrdquoThus it fits the assembly sequence planning of complexproducts Genetic algorithm generates satisfactory effectsin the research of combinatorial optimization Assemblysequence planning is a typical combinatorial optimizationissue Therefore this paper used genetic algorithm in opti-mized assembly sequence of parts in assembly units andeach assembly unit and then generated concurrent assemblysequence

51 Chromosome Coding of Assembly Sequence The assemblysequence planning is under the umbrella of combinatorialoptimization and suitable for the coding approach of com-binatorial optimization Therefore this research used order-sensitive codes for the coding of parts For assembly sequenceplanning of 119873 parts the chromosome was divided into 119873

segments and each segment is the corresponding code ofparts In order to facilitate the coding part number in theassembly unit was arranged from small to large and then wascoded

52 Ways to Generate Initial Population Generally speakingunits in the initial population are randomly generated Inthis paper the automatic generation was used to generateinitial population 119910 = Randperm (119899) in MATLAB wasused for population initialization to obtain a nonrepeatedrandom structure including integers from 1 to 119899 as a validchromosome

53 Fitness Function Design of Assembly Sequence Accordingto the feasibility of assembly sequence and the incidence ofthe strengths and weaknesses indicators with the maximuminfluence and easy access to information were used to estab-lish fitness function This research mainly constructed thefitness functions from three aspects including the geometricfeasibility of assembly sequence stability of assembly unitsand the position of basic parts in assembly sequence

(1) Geometric Feasibility Geometric feasibility requires nointerference during disassembly This research used inte-grated interference matrix [15] to derive the feasible disas-sembly direction of any part in the assembly unit It wasassumed that an assembly unit 119860 = 1 2 119899 comprised119899 parts and its corresponding integrated interference matrixwas a matrix 119868

119860with 119899 rows and 6119899 columns as shown in the

following formula

119868119860

=

[[[[[[[[

[

11986811119909

11986811119910

11986811119911

119868111199091015840119868111199101015840119868111199111015840 11986812119909

11986812119910

11986812119911

119868121199091015840119868121199101015840119868121199111015840 sdot sdot sdot 119868

11198991199091198681119899119910

1198681119899119911

119868111989911990910158401198681119899119910101584011986811198991199111015840

11986821119909

11986821119910

11986821119911

119868211199091015840119868211199101015840119868211199111015840 11986822119909

11986822119910

11986822119911

119868221199091015840119868221199101015840119868221199111015840 sdot sdot sdot 119868

21198991199091198682119899119910

1198682119899119911

119868211989911990910158401198682119899119910101584011986821198991199111015840

sdot sdot sdot

1198681198991119909

1198681198991119910

1198681198991119911

119868119899111990910158401198681198991119910101584011986811989911199111015840 1198681198992119909

1198681198992119910

1198681198992119911

119868119899211990910158401198681198992119910101584011986811989921199111015840 sdot sdot sdot 119868

119898119909119868119898119910

119868119898119911

119868119898119909101584011986811989811991010158401198681198981199111015840

]]]]]]]]

]

(7)

If at one direction 119889 part 119894 is not interfered by anypart during disassembly then 119868

119894119889= 0 which indicated

that part 119894 met the geometric feasibility of disassembly andthe disassembly was activated If at one direction 119889 part119894 is interfered by any part during disassembly then 119868

119894119889=

1 which exhibited that parts did not meet the geometricfeasibility and the disassembly was not feasible When part119894 was successfully disassembled all elements at the 119894th row inintegrated interference matrix 119868

119860were 0 indicating that part

119894 did not produce interference with other parts formula (8)could evaluate the geometric feasibility of assembly sequence

1198911=

119873 (119873 minus 1)

2minus 119872 (8)

where 119873 is the total number of parts in assembly unit and119872 is the times of an assembly sequence that generatedinterferences during assembly

(2) Stability of Assembly Unit The stability connection matrix119862 was used to quantify the stability of feasible assemblysequence 119862

119894119895represented the connection relation between

part 119894 and part 119895 When a stable connection relation betweenpart 119894 and part 119895 was observed let 119862

119894119895= 2 when a contact

6 Mathematical Problems in Engineering

connection relation between part 119894 and part 119895 was observedlet 119862119894119895

= 1 when there was no connection relation betweenpart 119894 and part 119895 let 119862

119894119895= 0 The stable connection in

this research refers to connection relations with mandatoryconstraints such as screw joint interference fit and pinnedconnection Hence formula (9) was used to evaluate thestability of assembly sequence

1198912=

119873minus1

sum119894=1

119870119894 (9)

where 119873 is the total number of parts in an assembly unitand 119870

119894is a quantified connection relation between part 119894 and

part 119894 + 1 When a stable connection existed 119870119894= 2 when

a contact connection existed 119870119894

= 1 when no connectionrelation existed 119870

119894= 0

(3) Position of Basic Parts The basic part is the key functionpart in assembly unit and should be firstly assembled duringthe assembly process In initial population number 1 partin each assembly sequence is not necessarily a basic partTherefore the assembly sequence with the basic part in thevery first place should be distinguished from that with thebasic part not in the first place In this research formula (10)was used to evaluate the position of basic parts in assemblysequence

1198913= 119873 minus 119875 (10)

where 119873 is the total number of parts of assembly unit and 119875

is the position of basic parts in assembly sequence

(4) Fitness Function According to the analysis on the aboveevaluation indexes a fitness function of genetic algorithmwas designed for assembly sequence planning as shown inthe following formula

119865 (119878) = 12059611198911+ 12059621198912+ 12059631198913 (11)

where 119878 is the assembly sequence in the population 1205961is the

weight coefficient of geometric feasibility 1205962is the weight

coefficient of assembly unit stability and 1205963is the weight

coefficient of positions of basic parts

54 Genetic Manipulation Genetic algorithm simulates theevolutionary mechanism of ldquosurvival of the fittestrdquo in theliving nature through genetic manipulation The missionof genetic manipulation is to apply certain operations onindividuals in the population according to their fitness (fit-ness evaluation) to environment thereby realizing the evo-lutionary process of survival of the fittest Common geneticmanipulation approaches include selection intersection andmutation

Selection operation refers to the selection of individualswith better fitness to produce new population For assemblysequence planning this research provided three approachesroulette selection linear ranking selection and tournamentselection Crossover operation is used to simulate geneticrecombination in the genetic evolution In the assembly

sequence planning of this paper single-point crossover two-point andmultipoint crossover as well as uniform crossoverwere applied Mutation is used to mimic the variation ofbiological heredity in the evolutionary process In this paperswitched mutation and inserted mutation were provided

55 Control Parameters and Termination Criterion of GeneticAlgorithm In order to ensure that genetic algorithm couldreach the optimal solution along the optimal research trackthe control parameter should be rationally selected whichmainly included the population size 119873 encoding length119871 crossover probability 119875

119888 mutation probability 119875

119898 and

termination conditionFor the assembly sequence planning in this research

encoding length 119871 is the number of parts in the assemblyunit The population size 119873 doubled the number of partsin assembly unit that is 119873 = 2119871 Crossover probability119875119888controls the utilization frequency of crossover operation

Generally 119875119888= 06sim10 Mutation probability 119875

119898controls the

utilization frequency of mutation Generally 119875119898

= 0005sim001 The termination condition of genetic algorithm in thispaper is when a specific algebrawas run on genetic algorithmthe calculation was terminated Besides optimal individualsin the current populationwere taken as the optimal solutions

6 Case Study

With the example of headstock of HTC2550hs CNC turningcenter this paper illustrated the technologies of concurrentassembly sequence planning

61 Assembly Modeling of Headstock of CNC Machine ToolAfter the removal of connectors in headstock there were 43function parts The assembly relationship among functionparts of the headstock was analyzed to establish ASRG ofCNC machine tool Parts were expressed by numbers asshown in Figure 2

As the headstock had a large number of function partsandASRGwas complex this paper usedGraphviz to establishASRG The function parts of headstock were expressed bynumbers The established ASRG is shown in Figure 3 Theassembly relationship information among function partsof headstock was analyzed The analytic hierarchy process(AHP) was used to generate weights of connection semanticsinformation semantic information of transmission and geo-metric constraint information The corresponding weightswere respectively 119908

1= 1199082

= 043 and 1199083

= 014 Nextedge weights in ASRG were calculated according to formula(2)

In Figure 3 as function parts were distinguished fromconnectors there were only 43 nodes in this model Inaddition the assembly relationship information was added inedges Edge information in ASRG could be revised accordingto needs so that ASRG has good maintainability

62 Assembly Unit Partition of Headstock of CNCMachine Tool

621 Selection of Basic Parts in Assembly Unit The totalnumber of function parts of this headstock was 43 According

Mathematical Problems in Engineering 7

Brak

e obj

ects6

Brak

e obj

ects29

Fric

tion

bloc

k41

Fric

tion

bloc

k43

Pisto

n ro

d20

Pisto

n ro

d35

Lid31

Back

-up

bloc

k27

Tank

cove

r28

Sign

al co

nver

ter3

6

Pisto

n ro

d30

Pisto

n ro

d7

Lid9

Lid32

Bend

ing

plat

e10

Bend

ing

plat

e34

Lid33

Fric

tion

bloc

k42

Fric

tion

bloc

k8

4 B

elt p

ulle

y37

Exp

ansio

n sle

eve

38

Exp

ansio

n sle

eve

39

Bra

ke d

isk

12

Rub

ber s

heet

40

Pro

tect

ive b

ox3 L

oop

2 H

eel c

ap1 B

ack

glan

d26

Bac

k be

arin

g ho

usin

g13

Hou

sing

14

For

mer

spac

er15

Hou

sing

21

Spa

cer

25

Spi

ndle

box

22 Locating pin

5 S

pind

le

19

Gla

nd17

Ext

erna

l gla

nd18

Lab

yrin

th ca

sing

23 Special bolts16

Hou

sing

24 Adjustingblock

11

Blin

d fr

ange

Figure 2 Explosive view of headstock

to formula (3) 119873119898

le 656 so the number of optimal unitpartition was 119873

119898= 6 According to edge weight in ASRG

formula (4) was used to calculate the importance degree ofeach part in the assembly The result was shown in Table 1

6 parts with the largest importance degrees were taken asthe basic parts of assembly unit partition In Table 1 part 2529 6 5 40 4 had the largest importance degree so that theywere selected as the basic parts of assembly unit

622 Assembly Unit Partition The assembly connectionstrength between part 119894 and other basic parts was calculated

and part 119894 was put in the unit containing basic part withthe largest connection strength to accomplish the assemblyunit partition Result of assembly unit partition was shown inTable 2 and Figure 4

623 Interference Checking and Assembly Unit CorrectionInterference checking based on the above partition resultswas carried out The interference judgment matrix of assem-bly unit 1 was shown as follows

1198621198801=

11989011198902119890311989041198905

sdot sdot sdot sdot sdot sdot sdot sdot sdot 119890511198905211989053

1198901

1198902

1198904

11989039

11989040

11989041

11989042

11989045

11989046

11989047

[[[[[[[[[[[[[[

[

00000

00000

00000

00000

00000

00000

00001

00000

00000

00000

00000

00000

00000

00000

00000

00100

00100

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

000

000

000

000

000

000

000

000

000

000

]]]]]]]]]]]]]]

]

(12)

By the analysis of interference judgment matrix in eachassembly unit it was found that part 2 17 18 and 19 wouldtrigger assembly interference among units After the disas-sembly the assembly unit without interference was obtainedIn addition part 39 shared an assembly relationship with

part 2 When part 2 was disassembled part 39 was alsodisassembled

Elements of the assembly relationship between part 19and part 5 in the interference matrix were 0 The assemblywas prioritized and it was placed in the unit where the spindle5 was Part 2 and part 39 had bolted connections with the

8 Mathematical Problems in Engineering

Table 1 The importance degree of each part in the assembly

Partnumber Importance degree Part number Importance degree Part number Importance degree Part number Importance degree

1 0796 12 088 23 0426 34 04262 0622 13 014 24 0852 35 05663 0734 14 014 25 4566 36 03984 1182 15 0112 26 1052 37 06225 1862 16 0112 27 088 38 06226 2208 17 0566 28 0482 39 04827 0566 18 0482 29 2606 40 14468 0482 19 0824 30 0566 41 04829 051 20 0566 31 051 42 048210 0426 21 0168 32 051 43 048211 088 22 0426 33 051

13 14 15 21 16 18 2 36 31 9 10

17 5 39 1 6

28 19 26 4 20 7

25 37 41 8

22 24

23

27

40 38 3

424311

12 35 30 34 33 32

29

e7 e6 e52 e10 e9 e8 e11e12 e1 e31 e30

e32

e41

e27e26e2

e13

e33

e53e43 e3

e5

e40

e45 e46

e49

e50

e51

e48

e42 e4 e15

e14

e17

e16

e18 e19

e22

e21

e24e25

e29 e28 e37 e36 e35

e20

e23

e23

e39

e34

e38

Figure 3 ASRG of headstock

Mathematical Problems in Engineering 9

Table 2 Results of assembly unit partition

Unitnumber Basic part number Results of assembly unit partition

1 25 1 17 19 22 23 24 26 27 28 362 29 30 32 33 34 35 42 433 6 7 9 8 10 20 31 414 5 2 13 14 15 16 18 21 395 40 11 126 4 3 37 38

relatively intensive connection strength Elements of assem-bly relationship in the interference matrix were 0 whichcould prioritize assembly Therefore an independent unitwas formed As the assembly connection strength between

part 17 part 18 and basic parts in other unit was smallthe partition was not conducted for independent assemblyResults of modified unit partition were shown in Table 3 andFigure 5

63 Assembly Sequence Planning and Optimization for CNCMachine Tool

631 Case of Optimized Assembly Sequence Assembly unit 3was the braking part of headstock and contained 8 functionparts The corresponding part number was 6 7 8 9 10 2031 41 as shown in Figure 6

The interference relationship between parts in an assem-bly unit at directions (plusmn119883 plusmn119884 plusmn119885) during disassembly andother parts was analyzed An integrated interference matrix119868119860was constructed as follows

119868119860

=

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

000000

111011

010000

111011

100000

011111

011111

010000

011111

000000

000100

111011

100000

000100

000100

000100

000010

100000

000000

000000

100000

000100

000100

000100

011111

011111

000000

000000

100000

001000

000100

000100

000100

010100

000100

000100

000000

000100

000100

000100

111011

100000

100000

100000

100000

000000

011111

100000

111011

100000

000000

100000

100000

111011

000000

000000

000010

100000

100000

100000

100000

000100

000100

000000

]]]]]]]]]]

]

(13)

The stability of the connectivity among parts in theassembly unit was analyzed and a stability connection matrix119862 was constructed as follows

119862 =

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

0

1

0

2

2

1

2

0

1

0

2

1

0

0

0

0

0

2

0

0

0

0

0

0

2

1

0

0

0

0

0

0

2

0

0

0

0

0

0

0

1

0

0

0

0

0

1

2

2

0

0

0

0

1

0

0

0

0

0

0

0

2

0

0

]]]]]]]]]]

]

(14)

The initial population was produced by the randomapproach and the control parameter of genetic algorithmwasset as follows encoding length 119871 = 8 population size119873 = 16crossover probability 119875

119888= 08 mutation probability 119875

119898=

001 generation gen = 100 and genetic manipulation wasrespectively selected by roulette This study conducted two-point crossover switched mutation and reverse operationPart 6 of assembly unit 3 was a basic part Weights ofgeometric feasibility assembly unit stability and positionsof basic parts in the fitness calculation formula (11) wererespectively 120596

1= 04 120596

2= 02 and 120596

3= 04

Run the program in MATLAB to solve the optimalassembly sequence of assembly unit 3 Figure 7 was thefitness variation diagram of genetic algorithm optimizationAccording to the calculation result the last generation of

maximum fitness was 152 and the corresponding optimalassembly sequence was as follows

6 997888rarr 8 997888rarr 7 997888rarr 9 997888rarr 41 997888rarr 20 997888rarr 31 997888rarr 10 (15)

During the assembly process basic parts should be firstassembled and plate cover parts should be assembled aftershaft parts According to the optimized assembly sequenceresults in assembly unit 3 braking part 6 was a basic part andwas firstly assembled Cover 9 and cover 31 were assembledafter the piston rodThus it could be found that the assemblysequence solved from genetic algorithm was rational andfeasible

For assembly units with the number of parts larger than5 the genetic algorithm was used for optimized assem-bly sequence and assembly sequences of other units wereobtained according to experiences The result was shown inTable 4

Similar to the optimized assembly sequencing of parts inthe assembly unit integrated interferencematrix and stabilityconnection matrix among assembly units were constructedAssembly unit 1 was taken as the basic unit so that theassembly sequence of each assembly unit could be obtainedas shown in the following order

Unit 1 997888rarr Unit 4 997888rarr Unit 7 997888rarr Unit 5 997888rarr Unit 6

997888rarr Unit 8 997888rarr Unit 9 997888rarr Unit 2 997888rarr Unit 3(16)

10 Mathematical Problems in Engineering

Assembly unit 1

Assembly unit 2

Assembly unit 6

Assembly unit 3

Assembly unit 4

Assembly unit 5

Figure 4 Result of assembly unit partition

Assembly unit 1

Assembly unit 2

Assembly unit 8

Assembly unit 7Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 3

Assembly unit 9

Figure 5 Result of modified assembly unit partition

Table 3 Results of modified unit partition

Unit number Result of modified assembly unit partition1 1 22 23 24 25 26 27 28 362 29 30 32 33 34 35 42 433 6 7 8 9 10 20 31 414 5 13 14 15 16 19 215 11 12 406 3 4 37 387 2 398 179 18

632 Expression of Assembly Sequence As a fishbone dia-gram can simply directly and flexibly express the assemblysequence product hierarchies and other information thispaper used the fishbone diagram to express concurrentassembly sequence Among them fish head demonstrated thefinal product of assembly Big bones connected with the fishspine represented each assembly unit Small bones connectedwith big bones represented parts Positions of big bones and

Brake object 6Bending plate 10

7 Piston rod 31 Lid41 Friction block

20 Piston rod9 Lid 8 Friction block

Figure 6 Section view of braking part

7

8

9

10

11

12

13

14

15

16

10 20 30 40 50 60 70 80 90 100

Col

ony

adap

tatio

n de

gree

Maximum colony adaptation degreeEvolutionary process

Average colony adaptation degree

Genetic algebra

Figure 7 Fitness variation diagram

Table 4 Assembly sequence planning of each unit

Assembly unitnumber Assembly sequence

1 25rarr 27rarr 26rarr 1rarr 36rarr 28rarr 24rarr 22rarr 232 29rarr 42rarr 30rarr 32rarr 43rarr 35rarr 33rarr 343 6rarr 8rarr 7rarr 9rarr 41rarr 20rarr 31rarr 104 5rarr 19rarr 21rarr 16rarr 15rarr 14rarr 135 40rarr 12rarr 116 4rarr 3rarr 38rarr 377 39rarr 28 179 18

small bones in the figure reflected the assembly sequenceThe concurrent assembly sequence of headstock was shownin Figure 8

Mathematical Problems in Engineering 11

5

19

21

16

15

14

13

40

12

11 17

29

42

30

32

43

35

33

34

25

27

26

1

36

28

24

22

23

39

2

4

3

38

37 18 10

31

20

41

9

7

8

6

Assembly unit 4

Assembly unit 5

Assembly unit 2

Assembly unit 3

Assembly unit 9

Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 8

Spindle box

Figure 8 Expression of headstock assembly sequence

7 Conclusions

In an era of global economic integration to occupy avantage ground in fierce market competition enterprisesmust enhance their product innovation capability improveproduct quality and performance also shorten the productdesign andmanufacturing cycle and reduce costs Inmodernmanufacturing industry the workload of product in theassembling stage is very heavy producing high costs andhard to realize assembly automationTherefore newmethodsand technologies are in urgent need to support the productassembly design On the basis of related research of assemblydesign this paper explored into assemblymodeling assemblyunit partition and optimized assembly sequence of CNCmachine tool In this way the assembly design efficiency ofproduct was improved This research reached the followingconclusions

(1) By distinguishing connectors from function parts asimplified ASRG was established and semantic informationof assembly was added in the edges of assembly model Thestudy on assembly modeling cases of headstock of CNCmachine tool showed that the model efficiently expressed theassembly relationship among parts

(2)According to the transmissibility of assembly connec-tion strength the assembly connection strength among partswas calculated so that the information of nonadjacent partscould be utilized The assembly unit partition based on theconnection strength changed the previous partition methodsby experiences so as to quantify the partition of assembly unitIn addition through interference checking and correction ofassembly unit the rationality and accuracy of assembly unitpartition were improved

(3) According to needs of assembly sequence planninggenerations of coding approach fitness function genetic

operator control parameters and initial population of geneticalgorithm were designedThrough case analysis of headstockof CNC machine tool the effectiveness of using geneticalgorithm for assembly sequence planning and optimizationwas verified

This research actively explored into the assembly mod-eling assembly unit partition assembly sequence planningand optimization and other aspects of CNC machine toolrealized the optimized assembly sequence of CNC machinetool and provided theoretical basis and technical supports forthe improvement of assembly quality and efficiency of CNCmachine tool

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (N120403003) the Scienceand Technology Funds of Liaoning Province (20130200522011216010) and the Research Funds of State ldquoTwelve FiverdquoSupport Program (2012BAF12B08)

References

[1] V Blanchot and A Daidie ldquoRiveted assembly modelling Studyand numerical characterisation of a riveting processrdquo Journal ofMaterials Processing Technology vol 180 no 1ndash3 pp 201ndash2092006

[2] T Gu Z Xu and Z Yang ldquoSymbolic OBDD representations formechanical assembly sequencesrdquo Computer Aided Design vol40 no 4 pp 411ndash421 2008

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

(1) 119898 assembly relationships 119864119880 = 1198901 1198902 119890

119898 in unit

119880 were extracted(2) each line of elements corresponding to 119890

1 1198902 119890

119898

was extracted from the assembly relationship interfer-ence matrix to form the intermediary matrix 119872119880

(3) each line of elements corresponding to 1198901 1198902 119890

119898

in 119872119880 was taken as zero to obtain the assembly unitinterference judgment matrix 119862119880

If the corresponding line of assembly relationship 119890119894(1 le

119894 le 119898) in assembly unit interference judgment matrix 119862119880

has 1 it indicates that assembly relationship 119890119894will interrupt

the assembly of other units that is there is interferencebetween assembly unit 119880 and other units The assemblyrelationship that produced interference was dissembled Thenewly formed assembly unit did not create interference in theprocess of assembly Disassembled parts could be adjusted toother units or partitioned into new units if needed

5 Assembly Sequence Planning Based onGenetic Algorithm

The solutions of assembly sequence planning could bedivided into two categories serial assembly and concur-rent assembly Concurrent assembly could increase the par-allelism of assembly improve assembly efficiency reduceassembly costs and could not trigger ldquocombinatorial explo-sionrdquoThus it fits the assembly sequence planning of complexproducts Genetic algorithm generates satisfactory effectsin the research of combinatorial optimization Assemblysequence planning is a typical combinatorial optimizationissue Therefore this paper used genetic algorithm in opti-mized assembly sequence of parts in assembly units andeach assembly unit and then generated concurrent assemblysequence

51 Chromosome Coding of Assembly Sequence The assemblysequence planning is under the umbrella of combinatorialoptimization and suitable for the coding approach of com-binatorial optimization Therefore this research used order-sensitive codes for the coding of parts For assembly sequenceplanning of 119873 parts the chromosome was divided into 119873

segments and each segment is the corresponding code ofparts In order to facilitate the coding part number in theassembly unit was arranged from small to large and then wascoded

52 Ways to Generate Initial Population Generally speakingunits in the initial population are randomly generated Inthis paper the automatic generation was used to generateinitial population 119910 = Randperm (119899) in MATLAB wasused for population initialization to obtain a nonrepeatedrandom structure including integers from 1 to 119899 as a validchromosome

53 Fitness Function Design of Assembly Sequence Accordingto the feasibility of assembly sequence and the incidence ofthe strengths and weaknesses indicators with the maximuminfluence and easy access to information were used to estab-lish fitness function This research mainly constructed thefitness functions from three aspects including the geometricfeasibility of assembly sequence stability of assembly unitsand the position of basic parts in assembly sequence

(1) Geometric Feasibility Geometric feasibility requires nointerference during disassembly This research used inte-grated interference matrix [15] to derive the feasible disas-sembly direction of any part in the assembly unit It wasassumed that an assembly unit 119860 = 1 2 119899 comprised119899 parts and its corresponding integrated interference matrixwas a matrix 119868

119860with 119899 rows and 6119899 columns as shown in the

following formula

119868119860

=

[[[[[[[[

[

11986811119909

11986811119910

11986811119911

119868111199091015840119868111199101015840119868111199111015840 11986812119909

11986812119910

11986812119911

119868121199091015840119868121199101015840119868121199111015840 sdot sdot sdot 119868

11198991199091198681119899119910

1198681119899119911

119868111989911990910158401198681119899119910101584011986811198991199111015840

11986821119909

11986821119910

11986821119911

119868211199091015840119868211199101015840119868211199111015840 11986822119909

11986822119910

11986822119911

119868221199091015840119868221199101015840119868221199111015840 sdot sdot sdot 119868

21198991199091198682119899119910

1198682119899119911

119868211989911990910158401198682119899119910101584011986821198991199111015840

sdot sdot sdot

1198681198991119909

1198681198991119910

1198681198991119911

119868119899111990910158401198681198991119910101584011986811989911199111015840 1198681198992119909

1198681198992119910

1198681198992119911

119868119899211990910158401198681198992119910101584011986811989921199111015840 sdot sdot sdot 119868

119898119909119868119898119910

119868119898119911

119868119898119909101584011986811989811991010158401198681198981199111015840

]]]]]]]]

]

(7)

If at one direction 119889 part 119894 is not interfered by anypart during disassembly then 119868

119894119889= 0 which indicated

that part 119894 met the geometric feasibility of disassembly andthe disassembly was activated If at one direction 119889 part119894 is interfered by any part during disassembly then 119868

119894119889=

1 which exhibited that parts did not meet the geometricfeasibility and the disassembly was not feasible When part119894 was successfully disassembled all elements at the 119894th row inintegrated interference matrix 119868

119860were 0 indicating that part

119894 did not produce interference with other parts formula (8)could evaluate the geometric feasibility of assembly sequence

1198911=

119873 (119873 minus 1)

2minus 119872 (8)

where 119873 is the total number of parts in assembly unit and119872 is the times of an assembly sequence that generatedinterferences during assembly

(2) Stability of Assembly Unit The stability connection matrix119862 was used to quantify the stability of feasible assemblysequence 119862

119894119895represented the connection relation between

part 119894 and part 119895 When a stable connection relation betweenpart 119894 and part 119895 was observed let 119862

119894119895= 2 when a contact

6 Mathematical Problems in Engineering

connection relation between part 119894 and part 119895 was observedlet 119862119894119895

= 1 when there was no connection relation betweenpart 119894 and part 119895 let 119862

119894119895= 0 The stable connection in

this research refers to connection relations with mandatoryconstraints such as screw joint interference fit and pinnedconnection Hence formula (9) was used to evaluate thestability of assembly sequence

1198912=

119873minus1

sum119894=1

119870119894 (9)

where 119873 is the total number of parts in an assembly unitand 119870

119894is a quantified connection relation between part 119894 and

part 119894 + 1 When a stable connection existed 119870119894= 2 when

a contact connection existed 119870119894

= 1 when no connectionrelation existed 119870

119894= 0

(3) Position of Basic Parts The basic part is the key functionpart in assembly unit and should be firstly assembled duringthe assembly process In initial population number 1 partin each assembly sequence is not necessarily a basic partTherefore the assembly sequence with the basic part in thevery first place should be distinguished from that with thebasic part not in the first place In this research formula (10)was used to evaluate the position of basic parts in assemblysequence

1198913= 119873 minus 119875 (10)

where 119873 is the total number of parts of assembly unit and 119875

is the position of basic parts in assembly sequence

(4) Fitness Function According to the analysis on the aboveevaluation indexes a fitness function of genetic algorithmwas designed for assembly sequence planning as shown inthe following formula

119865 (119878) = 12059611198911+ 12059621198912+ 12059631198913 (11)

where 119878 is the assembly sequence in the population 1205961is the

weight coefficient of geometric feasibility 1205962is the weight

coefficient of assembly unit stability and 1205963is the weight

coefficient of positions of basic parts

54 Genetic Manipulation Genetic algorithm simulates theevolutionary mechanism of ldquosurvival of the fittestrdquo in theliving nature through genetic manipulation The missionof genetic manipulation is to apply certain operations onindividuals in the population according to their fitness (fit-ness evaluation) to environment thereby realizing the evo-lutionary process of survival of the fittest Common geneticmanipulation approaches include selection intersection andmutation

Selection operation refers to the selection of individualswith better fitness to produce new population For assemblysequence planning this research provided three approachesroulette selection linear ranking selection and tournamentselection Crossover operation is used to simulate geneticrecombination in the genetic evolution In the assembly

sequence planning of this paper single-point crossover two-point andmultipoint crossover as well as uniform crossoverwere applied Mutation is used to mimic the variation ofbiological heredity in the evolutionary process In this paperswitched mutation and inserted mutation were provided

55 Control Parameters and Termination Criterion of GeneticAlgorithm In order to ensure that genetic algorithm couldreach the optimal solution along the optimal research trackthe control parameter should be rationally selected whichmainly included the population size 119873 encoding length119871 crossover probability 119875

119888 mutation probability 119875

119898 and

termination conditionFor the assembly sequence planning in this research

encoding length 119871 is the number of parts in the assemblyunit The population size 119873 doubled the number of partsin assembly unit that is 119873 = 2119871 Crossover probability119875119888controls the utilization frequency of crossover operation

Generally 119875119888= 06sim10 Mutation probability 119875

119898controls the

utilization frequency of mutation Generally 119875119898

= 0005sim001 The termination condition of genetic algorithm in thispaper is when a specific algebrawas run on genetic algorithmthe calculation was terminated Besides optimal individualsin the current populationwere taken as the optimal solutions

6 Case Study

With the example of headstock of HTC2550hs CNC turningcenter this paper illustrated the technologies of concurrentassembly sequence planning

61 Assembly Modeling of Headstock of CNC Machine ToolAfter the removal of connectors in headstock there were 43function parts The assembly relationship among functionparts of the headstock was analyzed to establish ASRG ofCNC machine tool Parts were expressed by numbers asshown in Figure 2

As the headstock had a large number of function partsandASRGwas complex this paper usedGraphviz to establishASRG The function parts of headstock were expressed bynumbers The established ASRG is shown in Figure 3 Theassembly relationship information among function partsof headstock was analyzed The analytic hierarchy process(AHP) was used to generate weights of connection semanticsinformation semantic information of transmission and geo-metric constraint information The corresponding weightswere respectively 119908

1= 1199082

= 043 and 1199083

= 014 Nextedge weights in ASRG were calculated according to formula(2)

In Figure 3 as function parts were distinguished fromconnectors there were only 43 nodes in this model Inaddition the assembly relationship information was added inedges Edge information in ASRG could be revised accordingto needs so that ASRG has good maintainability

62 Assembly Unit Partition of Headstock of CNCMachine Tool

621 Selection of Basic Parts in Assembly Unit The totalnumber of function parts of this headstock was 43 According

Mathematical Problems in Engineering 7

Brak

e obj

ects6

Brak

e obj

ects29

Fric

tion

bloc

k41

Fric

tion

bloc

k43

Pisto

n ro

d20

Pisto

n ro

d35

Lid31

Back

-up

bloc

k27

Tank

cove

r28

Sign

al co

nver

ter3

6

Pisto

n ro

d30

Pisto

n ro

d7

Lid9

Lid32

Bend

ing

plat

e10

Bend

ing

plat

e34

Lid33

Fric

tion

bloc

k42

Fric

tion

bloc

k8

4 B

elt p

ulle

y37

Exp

ansio

n sle

eve

38

Exp

ansio

n sle

eve

39

Bra

ke d

isk

12

Rub

ber s

heet

40

Pro

tect

ive b

ox3 L

oop

2 H

eel c

ap1 B

ack

glan

d26

Bac

k be

arin

g ho

usin

g13

Hou

sing

14

For

mer

spac

er15

Hou

sing

21

Spa

cer

25

Spi

ndle

box

22 Locating pin

5 S

pind

le

19

Gla

nd17

Ext

erna

l gla

nd18

Lab

yrin

th ca

sing

23 Special bolts16

Hou

sing

24 Adjustingblock

11

Blin

d fr

ange

Figure 2 Explosive view of headstock

to formula (3) 119873119898

le 656 so the number of optimal unitpartition was 119873

119898= 6 According to edge weight in ASRG

formula (4) was used to calculate the importance degree ofeach part in the assembly The result was shown in Table 1

6 parts with the largest importance degrees were taken asthe basic parts of assembly unit partition In Table 1 part 2529 6 5 40 4 had the largest importance degree so that theywere selected as the basic parts of assembly unit

622 Assembly Unit Partition The assembly connectionstrength between part 119894 and other basic parts was calculated

and part 119894 was put in the unit containing basic part withthe largest connection strength to accomplish the assemblyunit partition Result of assembly unit partition was shown inTable 2 and Figure 4

623 Interference Checking and Assembly Unit CorrectionInterference checking based on the above partition resultswas carried out The interference judgment matrix of assem-bly unit 1 was shown as follows

1198621198801=

11989011198902119890311989041198905

sdot sdot sdot sdot sdot sdot sdot sdot sdot 119890511198905211989053

1198901

1198902

1198904

11989039

11989040

11989041

11989042

11989045

11989046

11989047

[[[[[[[[[[[[[[

[

00000

00000

00000

00000

00000

00000

00001

00000

00000

00000

00000

00000

00000

00000

00000

00100

00100

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

000

000

000

000

000

000

000

000

000

000

]]]]]]]]]]]]]]

]

(12)

By the analysis of interference judgment matrix in eachassembly unit it was found that part 2 17 18 and 19 wouldtrigger assembly interference among units After the disas-sembly the assembly unit without interference was obtainedIn addition part 39 shared an assembly relationship with

part 2 When part 2 was disassembled part 39 was alsodisassembled

Elements of the assembly relationship between part 19and part 5 in the interference matrix were 0 The assemblywas prioritized and it was placed in the unit where the spindle5 was Part 2 and part 39 had bolted connections with the

8 Mathematical Problems in Engineering

Table 1 The importance degree of each part in the assembly

Partnumber Importance degree Part number Importance degree Part number Importance degree Part number Importance degree

1 0796 12 088 23 0426 34 04262 0622 13 014 24 0852 35 05663 0734 14 014 25 4566 36 03984 1182 15 0112 26 1052 37 06225 1862 16 0112 27 088 38 06226 2208 17 0566 28 0482 39 04827 0566 18 0482 29 2606 40 14468 0482 19 0824 30 0566 41 04829 051 20 0566 31 051 42 048210 0426 21 0168 32 051 43 048211 088 22 0426 33 051

13 14 15 21 16 18 2 36 31 9 10

17 5 39 1 6

28 19 26 4 20 7

25 37 41 8

22 24

23

27

40 38 3

424311

12 35 30 34 33 32

29

e7 e6 e52 e10 e9 e8 e11e12 e1 e31 e30

e32

e41

e27e26e2

e13

e33

e53e43 e3

e5

e40

e45 e46

e49

e50

e51

e48

e42 e4 e15

e14

e17

e16

e18 e19

e22

e21

e24e25

e29 e28 e37 e36 e35

e20

e23

e23

e39

e34

e38

Figure 3 ASRG of headstock

Mathematical Problems in Engineering 9

Table 2 Results of assembly unit partition

Unitnumber Basic part number Results of assembly unit partition

1 25 1 17 19 22 23 24 26 27 28 362 29 30 32 33 34 35 42 433 6 7 9 8 10 20 31 414 5 2 13 14 15 16 18 21 395 40 11 126 4 3 37 38

relatively intensive connection strength Elements of assem-bly relationship in the interference matrix were 0 whichcould prioritize assembly Therefore an independent unitwas formed As the assembly connection strength between

part 17 part 18 and basic parts in other unit was smallthe partition was not conducted for independent assemblyResults of modified unit partition were shown in Table 3 andFigure 5

63 Assembly Sequence Planning and Optimization for CNCMachine Tool

631 Case of Optimized Assembly Sequence Assembly unit 3was the braking part of headstock and contained 8 functionparts The corresponding part number was 6 7 8 9 10 2031 41 as shown in Figure 6

The interference relationship between parts in an assem-bly unit at directions (plusmn119883 plusmn119884 plusmn119885) during disassembly andother parts was analyzed An integrated interference matrix119868119860was constructed as follows

119868119860

=

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

000000

111011

010000

111011

100000

011111

011111

010000

011111

000000

000100

111011

100000

000100

000100

000100

000010

100000

000000

000000

100000

000100

000100

000100

011111

011111

000000

000000

100000

001000

000100

000100

000100

010100

000100

000100

000000

000100

000100

000100

111011

100000

100000

100000

100000

000000

011111

100000

111011

100000

000000

100000

100000

111011

000000

000000

000010

100000

100000

100000

100000

000100

000100

000000

]]]]]]]]]]

]

(13)

The stability of the connectivity among parts in theassembly unit was analyzed and a stability connection matrix119862 was constructed as follows

119862 =

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

0

1

0

2

2

1

2

0

1

0

2

1

0

0

0

0

0

2

0

0

0

0

0

0

2

1

0

0

0

0

0

0

2

0

0

0

0

0

0

0

1

0

0

0

0

0

1

2

2

0

0

0

0

1

0

0

0

0

0

0

0

2

0

0

]]]]]]]]]]

]

(14)

The initial population was produced by the randomapproach and the control parameter of genetic algorithmwasset as follows encoding length 119871 = 8 population size119873 = 16crossover probability 119875

119888= 08 mutation probability 119875

119898=

001 generation gen = 100 and genetic manipulation wasrespectively selected by roulette This study conducted two-point crossover switched mutation and reverse operationPart 6 of assembly unit 3 was a basic part Weights ofgeometric feasibility assembly unit stability and positionsof basic parts in the fitness calculation formula (11) wererespectively 120596

1= 04 120596

2= 02 and 120596

3= 04

Run the program in MATLAB to solve the optimalassembly sequence of assembly unit 3 Figure 7 was thefitness variation diagram of genetic algorithm optimizationAccording to the calculation result the last generation of

maximum fitness was 152 and the corresponding optimalassembly sequence was as follows

6 997888rarr 8 997888rarr 7 997888rarr 9 997888rarr 41 997888rarr 20 997888rarr 31 997888rarr 10 (15)

During the assembly process basic parts should be firstassembled and plate cover parts should be assembled aftershaft parts According to the optimized assembly sequenceresults in assembly unit 3 braking part 6 was a basic part andwas firstly assembled Cover 9 and cover 31 were assembledafter the piston rodThus it could be found that the assemblysequence solved from genetic algorithm was rational andfeasible

For assembly units with the number of parts larger than5 the genetic algorithm was used for optimized assem-bly sequence and assembly sequences of other units wereobtained according to experiences The result was shown inTable 4

Similar to the optimized assembly sequencing of parts inthe assembly unit integrated interferencematrix and stabilityconnection matrix among assembly units were constructedAssembly unit 1 was taken as the basic unit so that theassembly sequence of each assembly unit could be obtainedas shown in the following order

Unit 1 997888rarr Unit 4 997888rarr Unit 7 997888rarr Unit 5 997888rarr Unit 6

997888rarr Unit 8 997888rarr Unit 9 997888rarr Unit 2 997888rarr Unit 3(16)

10 Mathematical Problems in Engineering

Assembly unit 1

Assembly unit 2

Assembly unit 6

Assembly unit 3

Assembly unit 4

Assembly unit 5

Figure 4 Result of assembly unit partition

Assembly unit 1

Assembly unit 2

Assembly unit 8

Assembly unit 7Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 3

Assembly unit 9

Figure 5 Result of modified assembly unit partition

Table 3 Results of modified unit partition

Unit number Result of modified assembly unit partition1 1 22 23 24 25 26 27 28 362 29 30 32 33 34 35 42 433 6 7 8 9 10 20 31 414 5 13 14 15 16 19 215 11 12 406 3 4 37 387 2 398 179 18

632 Expression of Assembly Sequence As a fishbone dia-gram can simply directly and flexibly express the assemblysequence product hierarchies and other information thispaper used the fishbone diagram to express concurrentassembly sequence Among them fish head demonstrated thefinal product of assembly Big bones connected with the fishspine represented each assembly unit Small bones connectedwith big bones represented parts Positions of big bones and

Brake object 6Bending plate 10

7 Piston rod 31 Lid41 Friction block

20 Piston rod9 Lid 8 Friction block

Figure 6 Section view of braking part

7

8

9

10

11

12

13

14

15

16

10 20 30 40 50 60 70 80 90 100

Col

ony

adap

tatio

n de

gree

Maximum colony adaptation degreeEvolutionary process

Average colony adaptation degree

Genetic algebra

Figure 7 Fitness variation diagram

Table 4 Assembly sequence planning of each unit

Assembly unitnumber Assembly sequence

1 25rarr 27rarr 26rarr 1rarr 36rarr 28rarr 24rarr 22rarr 232 29rarr 42rarr 30rarr 32rarr 43rarr 35rarr 33rarr 343 6rarr 8rarr 7rarr 9rarr 41rarr 20rarr 31rarr 104 5rarr 19rarr 21rarr 16rarr 15rarr 14rarr 135 40rarr 12rarr 116 4rarr 3rarr 38rarr 377 39rarr 28 179 18

small bones in the figure reflected the assembly sequenceThe concurrent assembly sequence of headstock was shownin Figure 8

Mathematical Problems in Engineering 11

5

19

21

16

15

14

13

40

12

11 17

29

42

30

32

43

35

33

34

25

27

26

1

36

28

24

22

23

39

2

4

3

38

37 18 10

31

20

41

9

7

8

6

Assembly unit 4

Assembly unit 5

Assembly unit 2

Assembly unit 3

Assembly unit 9

Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 8

Spindle box

Figure 8 Expression of headstock assembly sequence

7 Conclusions

In an era of global economic integration to occupy avantage ground in fierce market competition enterprisesmust enhance their product innovation capability improveproduct quality and performance also shorten the productdesign andmanufacturing cycle and reduce costs Inmodernmanufacturing industry the workload of product in theassembling stage is very heavy producing high costs andhard to realize assembly automationTherefore newmethodsand technologies are in urgent need to support the productassembly design On the basis of related research of assemblydesign this paper explored into assemblymodeling assemblyunit partition and optimized assembly sequence of CNCmachine tool In this way the assembly design efficiency ofproduct was improved This research reached the followingconclusions

(1) By distinguishing connectors from function parts asimplified ASRG was established and semantic informationof assembly was added in the edges of assembly model Thestudy on assembly modeling cases of headstock of CNCmachine tool showed that the model efficiently expressed theassembly relationship among parts

(2)According to the transmissibility of assembly connec-tion strength the assembly connection strength among partswas calculated so that the information of nonadjacent partscould be utilized The assembly unit partition based on theconnection strength changed the previous partition methodsby experiences so as to quantify the partition of assembly unitIn addition through interference checking and correction ofassembly unit the rationality and accuracy of assembly unitpartition were improved

(3) According to needs of assembly sequence planninggenerations of coding approach fitness function genetic

operator control parameters and initial population of geneticalgorithm were designedThrough case analysis of headstockof CNC machine tool the effectiveness of using geneticalgorithm for assembly sequence planning and optimizationwas verified

This research actively explored into the assembly mod-eling assembly unit partition assembly sequence planningand optimization and other aspects of CNC machine toolrealized the optimized assembly sequence of CNC machinetool and provided theoretical basis and technical supports forthe improvement of assembly quality and efficiency of CNCmachine tool

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (N120403003) the Scienceand Technology Funds of Liaoning Province (20130200522011216010) and the Research Funds of State ldquoTwelve FiverdquoSupport Program (2012BAF12B08)

References

[1] V Blanchot and A Daidie ldquoRiveted assembly modelling Studyand numerical characterisation of a riveting processrdquo Journal ofMaterials Processing Technology vol 180 no 1ndash3 pp 201ndash2092006

[2] T Gu Z Xu and Z Yang ldquoSymbolic OBDD representations formechanical assembly sequencesrdquo Computer Aided Design vol40 no 4 pp 411ndash421 2008

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

connection relation between part 119894 and part 119895 was observedlet 119862119894119895

= 1 when there was no connection relation betweenpart 119894 and part 119895 let 119862

119894119895= 0 The stable connection in

this research refers to connection relations with mandatoryconstraints such as screw joint interference fit and pinnedconnection Hence formula (9) was used to evaluate thestability of assembly sequence

1198912=

119873minus1

sum119894=1

119870119894 (9)

where 119873 is the total number of parts in an assembly unitand 119870

119894is a quantified connection relation between part 119894 and

part 119894 + 1 When a stable connection existed 119870119894= 2 when

a contact connection existed 119870119894

= 1 when no connectionrelation existed 119870

119894= 0

(3) Position of Basic Parts The basic part is the key functionpart in assembly unit and should be firstly assembled duringthe assembly process In initial population number 1 partin each assembly sequence is not necessarily a basic partTherefore the assembly sequence with the basic part in thevery first place should be distinguished from that with thebasic part not in the first place In this research formula (10)was used to evaluate the position of basic parts in assemblysequence

1198913= 119873 minus 119875 (10)

where 119873 is the total number of parts of assembly unit and 119875

is the position of basic parts in assembly sequence

(4) Fitness Function According to the analysis on the aboveevaluation indexes a fitness function of genetic algorithmwas designed for assembly sequence planning as shown inthe following formula

119865 (119878) = 12059611198911+ 12059621198912+ 12059631198913 (11)

where 119878 is the assembly sequence in the population 1205961is the

weight coefficient of geometric feasibility 1205962is the weight

coefficient of assembly unit stability and 1205963is the weight

coefficient of positions of basic parts

54 Genetic Manipulation Genetic algorithm simulates theevolutionary mechanism of ldquosurvival of the fittestrdquo in theliving nature through genetic manipulation The missionof genetic manipulation is to apply certain operations onindividuals in the population according to their fitness (fit-ness evaluation) to environment thereby realizing the evo-lutionary process of survival of the fittest Common geneticmanipulation approaches include selection intersection andmutation

Selection operation refers to the selection of individualswith better fitness to produce new population For assemblysequence planning this research provided three approachesroulette selection linear ranking selection and tournamentselection Crossover operation is used to simulate geneticrecombination in the genetic evolution In the assembly

sequence planning of this paper single-point crossover two-point andmultipoint crossover as well as uniform crossoverwere applied Mutation is used to mimic the variation ofbiological heredity in the evolutionary process In this paperswitched mutation and inserted mutation were provided

55 Control Parameters and Termination Criterion of GeneticAlgorithm In order to ensure that genetic algorithm couldreach the optimal solution along the optimal research trackthe control parameter should be rationally selected whichmainly included the population size 119873 encoding length119871 crossover probability 119875

119888 mutation probability 119875

119898 and

termination conditionFor the assembly sequence planning in this research

encoding length 119871 is the number of parts in the assemblyunit The population size 119873 doubled the number of partsin assembly unit that is 119873 = 2119871 Crossover probability119875119888controls the utilization frequency of crossover operation

Generally 119875119888= 06sim10 Mutation probability 119875

119898controls the

utilization frequency of mutation Generally 119875119898

= 0005sim001 The termination condition of genetic algorithm in thispaper is when a specific algebrawas run on genetic algorithmthe calculation was terminated Besides optimal individualsin the current populationwere taken as the optimal solutions

6 Case Study

With the example of headstock of HTC2550hs CNC turningcenter this paper illustrated the technologies of concurrentassembly sequence planning

61 Assembly Modeling of Headstock of CNC Machine ToolAfter the removal of connectors in headstock there were 43function parts The assembly relationship among functionparts of the headstock was analyzed to establish ASRG ofCNC machine tool Parts were expressed by numbers asshown in Figure 2

As the headstock had a large number of function partsandASRGwas complex this paper usedGraphviz to establishASRG The function parts of headstock were expressed bynumbers The established ASRG is shown in Figure 3 Theassembly relationship information among function partsof headstock was analyzed The analytic hierarchy process(AHP) was used to generate weights of connection semanticsinformation semantic information of transmission and geo-metric constraint information The corresponding weightswere respectively 119908

1= 1199082

= 043 and 1199083

= 014 Nextedge weights in ASRG were calculated according to formula(2)

In Figure 3 as function parts were distinguished fromconnectors there were only 43 nodes in this model Inaddition the assembly relationship information was added inedges Edge information in ASRG could be revised accordingto needs so that ASRG has good maintainability

62 Assembly Unit Partition of Headstock of CNCMachine Tool

621 Selection of Basic Parts in Assembly Unit The totalnumber of function parts of this headstock was 43 According

Mathematical Problems in Engineering 7

Brak

e obj

ects6

Brak

e obj

ects29

Fric

tion

bloc

k41

Fric

tion

bloc

k43

Pisto

n ro

d20

Pisto

n ro

d35

Lid31

Back

-up

bloc

k27

Tank

cove

r28

Sign

al co

nver

ter3

6

Pisto

n ro

d30

Pisto

n ro

d7

Lid9

Lid32

Bend

ing

plat

e10

Bend

ing

plat

e34

Lid33

Fric

tion

bloc

k42

Fric

tion

bloc

k8

4 B

elt p

ulle

y37

Exp

ansio

n sle

eve

38

Exp

ansio

n sle

eve

39

Bra

ke d

isk

12

Rub

ber s

heet

40

Pro

tect

ive b

ox3 L

oop

2 H

eel c

ap1 B

ack

glan

d26

Bac

k be

arin

g ho

usin

g13

Hou

sing

14

For

mer

spac

er15

Hou

sing

21

Spa

cer

25

Spi

ndle

box

22 Locating pin

5 S

pind

le

19

Gla

nd17

Ext

erna

l gla

nd18

Lab

yrin

th ca

sing

23 Special bolts16

Hou

sing

24 Adjustingblock

11

Blin

d fr

ange

Figure 2 Explosive view of headstock

to formula (3) 119873119898

le 656 so the number of optimal unitpartition was 119873

119898= 6 According to edge weight in ASRG

formula (4) was used to calculate the importance degree ofeach part in the assembly The result was shown in Table 1

6 parts with the largest importance degrees were taken asthe basic parts of assembly unit partition In Table 1 part 2529 6 5 40 4 had the largest importance degree so that theywere selected as the basic parts of assembly unit

622 Assembly Unit Partition The assembly connectionstrength between part 119894 and other basic parts was calculated

and part 119894 was put in the unit containing basic part withthe largest connection strength to accomplish the assemblyunit partition Result of assembly unit partition was shown inTable 2 and Figure 4

623 Interference Checking and Assembly Unit CorrectionInterference checking based on the above partition resultswas carried out The interference judgment matrix of assem-bly unit 1 was shown as follows

1198621198801=

11989011198902119890311989041198905

sdot sdot sdot sdot sdot sdot sdot sdot sdot 119890511198905211989053

1198901

1198902

1198904

11989039

11989040

11989041

11989042

11989045

11989046

11989047

[[[[[[[[[[[[[[

[

00000

00000

00000

00000

00000

00000

00001

00000

00000

00000

00000

00000

00000

00000

00000

00100

00100

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

000

000

000

000

000

000

000

000

000

000

]]]]]]]]]]]]]]

]

(12)

By the analysis of interference judgment matrix in eachassembly unit it was found that part 2 17 18 and 19 wouldtrigger assembly interference among units After the disas-sembly the assembly unit without interference was obtainedIn addition part 39 shared an assembly relationship with

part 2 When part 2 was disassembled part 39 was alsodisassembled

Elements of the assembly relationship between part 19and part 5 in the interference matrix were 0 The assemblywas prioritized and it was placed in the unit where the spindle5 was Part 2 and part 39 had bolted connections with the

8 Mathematical Problems in Engineering

Table 1 The importance degree of each part in the assembly

Partnumber Importance degree Part number Importance degree Part number Importance degree Part number Importance degree

1 0796 12 088 23 0426 34 04262 0622 13 014 24 0852 35 05663 0734 14 014 25 4566 36 03984 1182 15 0112 26 1052 37 06225 1862 16 0112 27 088 38 06226 2208 17 0566 28 0482 39 04827 0566 18 0482 29 2606 40 14468 0482 19 0824 30 0566 41 04829 051 20 0566 31 051 42 048210 0426 21 0168 32 051 43 048211 088 22 0426 33 051

13 14 15 21 16 18 2 36 31 9 10

17 5 39 1 6

28 19 26 4 20 7

25 37 41 8

22 24

23

27

40 38 3

424311

12 35 30 34 33 32

29

e7 e6 e52 e10 e9 e8 e11e12 e1 e31 e30

e32

e41

e27e26e2

e13

e33

e53e43 e3

e5

e40

e45 e46

e49

e50

e51

e48

e42 e4 e15

e14

e17

e16

e18 e19

e22

e21

e24e25

e29 e28 e37 e36 e35

e20

e23

e23

e39

e34

e38

Figure 3 ASRG of headstock

Mathematical Problems in Engineering 9

Table 2 Results of assembly unit partition

Unitnumber Basic part number Results of assembly unit partition

1 25 1 17 19 22 23 24 26 27 28 362 29 30 32 33 34 35 42 433 6 7 9 8 10 20 31 414 5 2 13 14 15 16 18 21 395 40 11 126 4 3 37 38

relatively intensive connection strength Elements of assem-bly relationship in the interference matrix were 0 whichcould prioritize assembly Therefore an independent unitwas formed As the assembly connection strength between

part 17 part 18 and basic parts in other unit was smallthe partition was not conducted for independent assemblyResults of modified unit partition were shown in Table 3 andFigure 5

63 Assembly Sequence Planning and Optimization for CNCMachine Tool

631 Case of Optimized Assembly Sequence Assembly unit 3was the braking part of headstock and contained 8 functionparts The corresponding part number was 6 7 8 9 10 2031 41 as shown in Figure 6

The interference relationship between parts in an assem-bly unit at directions (plusmn119883 plusmn119884 plusmn119885) during disassembly andother parts was analyzed An integrated interference matrix119868119860was constructed as follows

119868119860

=

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

000000

111011

010000

111011

100000

011111

011111

010000

011111

000000

000100

111011

100000

000100

000100

000100

000010

100000

000000

000000

100000

000100

000100

000100

011111

011111

000000

000000

100000

001000

000100

000100

000100

010100

000100

000100

000000

000100

000100

000100

111011

100000

100000

100000

100000

000000

011111

100000

111011

100000

000000

100000

100000

111011

000000

000000

000010

100000

100000

100000

100000

000100

000100

000000

]]]]]]]]]]

]

(13)

The stability of the connectivity among parts in theassembly unit was analyzed and a stability connection matrix119862 was constructed as follows

119862 =

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

0

1

0

2

2

1

2

0

1

0

2

1

0

0

0

0

0

2

0

0

0

0

0

0

2

1

0

0

0

0

0

0

2

0

0

0

0

0

0

0

1

0

0

0

0

0

1

2

2

0

0

0

0

1

0

0

0

0

0

0

0

2

0

0

]]]]]]]]]]

]

(14)

The initial population was produced by the randomapproach and the control parameter of genetic algorithmwasset as follows encoding length 119871 = 8 population size119873 = 16crossover probability 119875

119888= 08 mutation probability 119875

119898=

001 generation gen = 100 and genetic manipulation wasrespectively selected by roulette This study conducted two-point crossover switched mutation and reverse operationPart 6 of assembly unit 3 was a basic part Weights ofgeometric feasibility assembly unit stability and positionsof basic parts in the fitness calculation formula (11) wererespectively 120596

1= 04 120596

2= 02 and 120596

3= 04

Run the program in MATLAB to solve the optimalassembly sequence of assembly unit 3 Figure 7 was thefitness variation diagram of genetic algorithm optimizationAccording to the calculation result the last generation of

maximum fitness was 152 and the corresponding optimalassembly sequence was as follows

6 997888rarr 8 997888rarr 7 997888rarr 9 997888rarr 41 997888rarr 20 997888rarr 31 997888rarr 10 (15)

During the assembly process basic parts should be firstassembled and plate cover parts should be assembled aftershaft parts According to the optimized assembly sequenceresults in assembly unit 3 braking part 6 was a basic part andwas firstly assembled Cover 9 and cover 31 were assembledafter the piston rodThus it could be found that the assemblysequence solved from genetic algorithm was rational andfeasible

For assembly units with the number of parts larger than5 the genetic algorithm was used for optimized assem-bly sequence and assembly sequences of other units wereobtained according to experiences The result was shown inTable 4

Similar to the optimized assembly sequencing of parts inthe assembly unit integrated interferencematrix and stabilityconnection matrix among assembly units were constructedAssembly unit 1 was taken as the basic unit so that theassembly sequence of each assembly unit could be obtainedas shown in the following order

Unit 1 997888rarr Unit 4 997888rarr Unit 7 997888rarr Unit 5 997888rarr Unit 6

997888rarr Unit 8 997888rarr Unit 9 997888rarr Unit 2 997888rarr Unit 3(16)

10 Mathematical Problems in Engineering

Assembly unit 1

Assembly unit 2

Assembly unit 6

Assembly unit 3

Assembly unit 4

Assembly unit 5

Figure 4 Result of assembly unit partition

Assembly unit 1

Assembly unit 2

Assembly unit 8

Assembly unit 7Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 3

Assembly unit 9

Figure 5 Result of modified assembly unit partition

Table 3 Results of modified unit partition

Unit number Result of modified assembly unit partition1 1 22 23 24 25 26 27 28 362 29 30 32 33 34 35 42 433 6 7 8 9 10 20 31 414 5 13 14 15 16 19 215 11 12 406 3 4 37 387 2 398 179 18

632 Expression of Assembly Sequence As a fishbone dia-gram can simply directly and flexibly express the assemblysequence product hierarchies and other information thispaper used the fishbone diagram to express concurrentassembly sequence Among them fish head demonstrated thefinal product of assembly Big bones connected with the fishspine represented each assembly unit Small bones connectedwith big bones represented parts Positions of big bones and

Brake object 6Bending plate 10

7 Piston rod 31 Lid41 Friction block

20 Piston rod9 Lid 8 Friction block

Figure 6 Section view of braking part

7

8

9

10

11

12

13

14

15

16

10 20 30 40 50 60 70 80 90 100

Col

ony

adap

tatio

n de

gree

Maximum colony adaptation degreeEvolutionary process

Average colony adaptation degree

Genetic algebra

Figure 7 Fitness variation diagram

Table 4 Assembly sequence planning of each unit

Assembly unitnumber Assembly sequence

1 25rarr 27rarr 26rarr 1rarr 36rarr 28rarr 24rarr 22rarr 232 29rarr 42rarr 30rarr 32rarr 43rarr 35rarr 33rarr 343 6rarr 8rarr 7rarr 9rarr 41rarr 20rarr 31rarr 104 5rarr 19rarr 21rarr 16rarr 15rarr 14rarr 135 40rarr 12rarr 116 4rarr 3rarr 38rarr 377 39rarr 28 179 18

small bones in the figure reflected the assembly sequenceThe concurrent assembly sequence of headstock was shownin Figure 8

Mathematical Problems in Engineering 11

5

19

21

16

15

14

13

40

12

11 17

29

42

30

32

43

35

33

34

25

27

26

1

36

28

24

22

23

39

2

4

3

38

37 18 10

31

20

41

9

7

8

6

Assembly unit 4

Assembly unit 5

Assembly unit 2

Assembly unit 3

Assembly unit 9

Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 8

Spindle box

Figure 8 Expression of headstock assembly sequence

7 Conclusions

In an era of global economic integration to occupy avantage ground in fierce market competition enterprisesmust enhance their product innovation capability improveproduct quality and performance also shorten the productdesign andmanufacturing cycle and reduce costs Inmodernmanufacturing industry the workload of product in theassembling stage is very heavy producing high costs andhard to realize assembly automationTherefore newmethodsand technologies are in urgent need to support the productassembly design On the basis of related research of assemblydesign this paper explored into assemblymodeling assemblyunit partition and optimized assembly sequence of CNCmachine tool In this way the assembly design efficiency ofproduct was improved This research reached the followingconclusions

(1) By distinguishing connectors from function parts asimplified ASRG was established and semantic informationof assembly was added in the edges of assembly model Thestudy on assembly modeling cases of headstock of CNCmachine tool showed that the model efficiently expressed theassembly relationship among parts

(2)According to the transmissibility of assembly connec-tion strength the assembly connection strength among partswas calculated so that the information of nonadjacent partscould be utilized The assembly unit partition based on theconnection strength changed the previous partition methodsby experiences so as to quantify the partition of assembly unitIn addition through interference checking and correction ofassembly unit the rationality and accuracy of assembly unitpartition were improved

(3) According to needs of assembly sequence planninggenerations of coding approach fitness function genetic

operator control parameters and initial population of geneticalgorithm were designedThrough case analysis of headstockof CNC machine tool the effectiveness of using geneticalgorithm for assembly sequence planning and optimizationwas verified

This research actively explored into the assembly mod-eling assembly unit partition assembly sequence planningand optimization and other aspects of CNC machine toolrealized the optimized assembly sequence of CNC machinetool and provided theoretical basis and technical supports forthe improvement of assembly quality and efficiency of CNCmachine tool

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (N120403003) the Scienceand Technology Funds of Liaoning Province (20130200522011216010) and the Research Funds of State ldquoTwelve FiverdquoSupport Program (2012BAF12B08)

References

[1] V Blanchot and A Daidie ldquoRiveted assembly modelling Studyand numerical characterisation of a riveting processrdquo Journal ofMaterials Processing Technology vol 180 no 1ndash3 pp 201ndash2092006

[2] T Gu Z Xu and Z Yang ldquoSymbolic OBDD representations formechanical assembly sequencesrdquo Computer Aided Design vol40 no 4 pp 411ndash421 2008

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Brak

e obj

ects6

Brak

e obj

ects29

Fric

tion

bloc

k41

Fric

tion

bloc

k43

Pisto

n ro

d20

Pisto

n ro

d35

Lid31

Back

-up

bloc

k27

Tank

cove

r28

Sign

al co

nver

ter3

6

Pisto

n ro

d30

Pisto

n ro

d7

Lid9

Lid32

Bend

ing

plat

e10

Bend

ing

plat

e34

Lid33

Fric

tion

bloc

k42

Fric

tion

bloc

k8

4 B

elt p

ulle

y37

Exp

ansio

n sle

eve

38

Exp

ansio

n sle

eve

39

Bra

ke d

isk

12

Rub

ber s

heet

40

Pro

tect

ive b

ox3 L

oop

2 H

eel c

ap1 B

ack

glan

d26

Bac

k be

arin

g ho

usin

g13

Hou

sing

14

For

mer

spac

er15

Hou

sing

21

Spa

cer

25

Spi

ndle

box

22 Locating pin

5 S

pind

le

19

Gla

nd17

Ext

erna

l gla

nd18

Lab

yrin

th ca

sing

23 Special bolts16

Hou

sing

24 Adjustingblock

11

Blin

d fr

ange

Figure 2 Explosive view of headstock

to formula (3) 119873119898

le 656 so the number of optimal unitpartition was 119873

119898= 6 According to edge weight in ASRG

formula (4) was used to calculate the importance degree ofeach part in the assembly The result was shown in Table 1

6 parts with the largest importance degrees were taken asthe basic parts of assembly unit partition In Table 1 part 2529 6 5 40 4 had the largest importance degree so that theywere selected as the basic parts of assembly unit

622 Assembly Unit Partition The assembly connectionstrength between part 119894 and other basic parts was calculated

and part 119894 was put in the unit containing basic part withthe largest connection strength to accomplish the assemblyunit partition Result of assembly unit partition was shown inTable 2 and Figure 4

623 Interference Checking and Assembly Unit CorrectionInterference checking based on the above partition resultswas carried out The interference judgment matrix of assem-bly unit 1 was shown as follows

1198621198801=

11989011198902119890311989041198905

sdot sdot sdot sdot sdot sdot sdot sdot sdot 119890511198905211989053

1198901

1198902

1198904

11989039

11989040

11989041

11989042

11989045

11989046

11989047

[[[[[[[[[[[[[[

[

00000

00000

00000

00000

00000

00000

00001

00000

00000

00000

00000

00000

00000

00000

00000

00100

00100

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

000

000

000

000

000

000

000

000

000

000

]]]]]]]]]]]]]]

]

(12)

By the analysis of interference judgment matrix in eachassembly unit it was found that part 2 17 18 and 19 wouldtrigger assembly interference among units After the disas-sembly the assembly unit without interference was obtainedIn addition part 39 shared an assembly relationship with

part 2 When part 2 was disassembled part 39 was alsodisassembled

Elements of the assembly relationship between part 19and part 5 in the interference matrix were 0 The assemblywas prioritized and it was placed in the unit where the spindle5 was Part 2 and part 39 had bolted connections with the

8 Mathematical Problems in Engineering

Table 1 The importance degree of each part in the assembly

Partnumber Importance degree Part number Importance degree Part number Importance degree Part number Importance degree

1 0796 12 088 23 0426 34 04262 0622 13 014 24 0852 35 05663 0734 14 014 25 4566 36 03984 1182 15 0112 26 1052 37 06225 1862 16 0112 27 088 38 06226 2208 17 0566 28 0482 39 04827 0566 18 0482 29 2606 40 14468 0482 19 0824 30 0566 41 04829 051 20 0566 31 051 42 048210 0426 21 0168 32 051 43 048211 088 22 0426 33 051

13 14 15 21 16 18 2 36 31 9 10

17 5 39 1 6

28 19 26 4 20 7

25 37 41 8

22 24

23

27

40 38 3

424311

12 35 30 34 33 32

29

e7 e6 e52 e10 e9 e8 e11e12 e1 e31 e30

e32

e41

e27e26e2

e13

e33

e53e43 e3

e5

e40

e45 e46

e49

e50

e51

e48

e42 e4 e15

e14

e17

e16

e18 e19

e22

e21

e24e25

e29 e28 e37 e36 e35

e20

e23

e23

e39

e34

e38

Figure 3 ASRG of headstock

Mathematical Problems in Engineering 9

Table 2 Results of assembly unit partition

Unitnumber Basic part number Results of assembly unit partition

1 25 1 17 19 22 23 24 26 27 28 362 29 30 32 33 34 35 42 433 6 7 9 8 10 20 31 414 5 2 13 14 15 16 18 21 395 40 11 126 4 3 37 38

relatively intensive connection strength Elements of assem-bly relationship in the interference matrix were 0 whichcould prioritize assembly Therefore an independent unitwas formed As the assembly connection strength between

part 17 part 18 and basic parts in other unit was smallthe partition was not conducted for independent assemblyResults of modified unit partition were shown in Table 3 andFigure 5

63 Assembly Sequence Planning and Optimization for CNCMachine Tool

631 Case of Optimized Assembly Sequence Assembly unit 3was the braking part of headstock and contained 8 functionparts The corresponding part number was 6 7 8 9 10 2031 41 as shown in Figure 6

The interference relationship between parts in an assem-bly unit at directions (plusmn119883 plusmn119884 plusmn119885) during disassembly andother parts was analyzed An integrated interference matrix119868119860was constructed as follows

119868119860

=

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

000000

111011

010000

111011

100000

011111

011111

010000

011111

000000

000100

111011

100000

000100

000100

000100

000010

100000

000000

000000

100000

000100

000100

000100

011111

011111

000000

000000

100000

001000

000100

000100

000100

010100

000100

000100

000000

000100

000100

000100

111011

100000

100000

100000

100000

000000

011111

100000

111011

100000

000000

100000

100000

111011

000000

000000

000010

100000

100000

100000

100000

000100

000100

000000

]]]]]]]]]]

]

(13)

The stability of the connectivity among parts in theassembly unit was analyzed and a stability connection matrix119862 was constructed as follows

119862 =

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

0

1

0

2

2

1

2

0

1

0

2

1

0

0

0

0

0

2

0

0

0

0

0

0

2

1

0

0

0

0

0

0

2

0

0

0

0

0

0

0

1

0

0

0

0

0

1

2

2

0

0

0

0

1

0

0

0

0

0

0

0

2

0

0

]]]]]]]]]]

]

(14)

The initial population was produced by the randomapproach and the control parameter of genetic algorithmwasset as follows encoding length 119871 = 8 population size119873 = 16crossover probability 119875

119888= 08 mutation probability 119875

119898=

001 generation gen = 100 and genetic manipulation wasrespectively selected by roulette This study conducted two-point crossover switched mutation and reverse operationPart 6 of assembly unit 3 was a basic part Weights ofgeometric feasibility assembly unit stability and positionsof basic parts in the fitness calculation formula (11) wererespectively 120596

1= 04 120596

2= 02 and 120596

3= 04

Run the program in MATLAB to solve the optimalassembly sequence of assembly unit 3 Figure 7 was thefitness variation diagram of genetic algorithm optimizationAccording to the calculation result the last generation of

maximum fitness was 152 and the corresponding optimalassembly sequence was as follows

6 997888rarr 8 997888rarr 7 997888rarr 9 997888rarr 41 997888rarr 20 997888rarr 31 997888rarr 10 (15)

During the assembly process basic parts should be firstassembled and plate cover parts should be assembled aftershaft parts According to the optimized assembly sequenceresults in assembly unit 3 braking part 6 was a basic part andwas firstly assembled Cover 9 and cover 31 were assembledafter the piston rodThus it could be found that the assemblysequence solved from genetic algorithm was rational andfeasible

For assembly units with the number of parts larger than5 the genetic algorithm was used for optimized assem-bly sequence and assembly sequences of other units wereobtained according to experiences The result was shown inTable 4

Similar to the optimized assembly sequencing of parts inthe assembly unit integrated interferencematrix and stabilityconnection matrix among assembly units were constructedAssembly unit 1 was taken as the basic unit so that theassembly sequence of each assembly unit could be obtainedas shown in the following order

Unit 1 997888rarr Unit 4 997888rarr Unit 7 997888rarr Unit 5 997888rarr Unit 6

997888rarr Unit 8 997888rarr Unit 9 997888rarr Unit 2 997888rarr Unit 3(16)

10 Mathematical Problems in Engineering

Assembly unit 1

Assembly unit 2

Assembly unit 6

Assembly unit 3

Assembly unit 4

Assembly unit 5

Figure 4 Result of assembly unit partition

Assembly unit 1

Assembly unit 2

Assembly unit 8

Assembly unit 7Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 3

Assembly unit 9

Figure 5 Result of modified assembly unit partition

Table 3 Results of modified unit partition

Unit number Result of modified assembly unit partition1 1 22 23 24 25 26 27 28 362 29 30 32 33 34 35 42 433 6 7 8 9 10 20 31 414 5 13 14 15 16 19 215 11 12 406 3 4 37 387 2 398 179 18

632 Expression of Assembly Sequence As a fishbone dia-gram can simply directly and flexibly express the assemblysequence product hierarchies and other information thispaper used the fishbone diagram to express concurrentassembly sequence Among them fish head demonstrated thefinal product of assembly Big bones connected with the fishspine represented each assembly unit Small bones connectedwith big bones represented parts Positions of big bones and

Brake object 6Bending plate 10

7 Piston rod 31 Lid41 Friction block

20 Piston rod9 Lid 8 Friction block

Figure 6 Section view of braking part

7

8

9

10

11

12

13

14

15

16

10 20 30 40 50 60 70 80 90 100

Col

ony

adap

tatio

n de

gree

Maximum colony adaptation degreeEvolutionary process

Average colony adaptation degree

Genetic algebra

Figure 7 Fitness variation diagram

Table 4 Assembly sequence planning of each unit

Assembly unitnumber Assembly sequence

1 25rarr 27rarr 26rarr 1rarr 36rarr 28rarr 24rarr 22rarr 232 29rarr 42rarr 30rarr 32rarr 43rarr 35rarr 33rarr 343 6rarr 8rarr 7rarr 9rarr 41rarr 20rarr 31rarr 104 5rarr 19rarr 21rarr 16rarr 15rarr 14rarr 135 40rarr 12rarr 116 4rarr 3rarr 38rarr 377 39rarr 28 179 18

small bones in the figure reflected the assembly sequenceThe concurrent assembly sequence of headstock was shownin Figure 8

Mathematical Problems in Engineering 11

5

19

21

16

15

14

13

40

12

11 17

29

42

30

32

43

35

33

34

25

27

26

1

36

28

24

22

23

39

2

4

3

38

37 18 10

31

20

41

9

7

8

6

Assembly unit 4

Assembly unit 5

Assembly unit 2

Assembly unit 3

Assembly unit 9

Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 8

Spindle box

Figure 8 Expression of headstock assembly sequence

7 Conclusions

In an era of global economic integration to occupy avantage ground in fierce market competition enterprisesmust enhance their product innovation capability improveproduct quality and performance also shorten the productdesign andmanufacturing cycle and reduce costs Inmodernmanufacturing industry the workload of product in theassembling stage is very heavy producing high costs andhard to realize assembly automationTherefore newmethodsand technologies are in urgent need to support the productassembly design On the basis of related research of assemblydesign this paper explored into assemblymodeling assemblyunit partition and optimized assembly sequence of CNCmachine tool In this way the assembly design efficiency ofproduct was improved This research reached the followingconclusions

(1) By distinguishing connectors from function parts asimplified ASRG was established and semantic informationof assembly was added in the edges of assembly model Thestudy on assembly modeling cases of headstock of CNCmachine tool showed that the model efficiently expressed theassembly relationship among parts

(2)According to the transmissibility of assembly connec-tion strength the assembly connection strength among partswas calculated so that the information of nonadjacent partscould be utilized The assembly unit partition based on theconnection strength changed the previous partition methodsby experiences so as to quantify the partition of assembly unitIn addition through interference checking and correction ofassembly unit the rationality and accuracy of assembly unitpartition were improved

(3) According to needs of assembly sequence planninggenerations of coding approach fitness function genetic

operator control parameters and initial population of geneticalgorithm were designedThrough case analysis of headstockof CNC machine tool the effectiveness of using geneticalgorithm for assembly sequence planning and optimizationwas verified

This research actively explored into the assembly mod-eling assembly unit partition assembly sequence planningand optimization and other aspects of CNC machine toolrealized the optimized assembly sequence of CNC machinetool and provided theoretical basis and technical supports forthe improvement of assembly quality and efficiency of CNCmachine tool

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (N120403003) the Scienceand Technology Funds of Liaoning Province (20130200522011216010) and the Research Funds of State ldquoTwelve FiverdquoSupport Program (2012BAF12B08)

References

[1] V Blanchot and A Daidie ldquoRiveted assembly modelling Studyand numerical characterisation of a riveting processrdquo Journal ofMaterials Processing Technology vol 180 no 1ndash3 pp 201ndash2092006

[2] T Gu Z Xu and Z Yang ldquoSymbolic OBDD representations formechanical assembly sequencesrdquo Computer Aided Design vol40 no 4 pp 411ndash421 2008

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 1 The importance degree of each part in the assembly

Partnumber Importance degree Part number Importance degree Part number Importance degree Part number Importance degree

1 0796 12 088 23 0426 34 04262 0622 13 014 24 0852 35 05663 0734 14 014 25 4566 36 03984 1182 15 0112 26 1052 37 06225 1862 16 0112 27 088 38 06226 2208 17 0566 28 0482 39 04827 0566 18 0482 29 2606 40 14468 0482 19 0824 30 0566 41 04829 051 20 0566 31 051 42 048210 0426 21 0168 32 051 43 048211 088 22 0426 33 051

13 14 15 21 16 18 2 36 31 9 10

17 5 39 1 6

28 19 26 4 20 7

25 37 41 8

22 24

23

27

40 38 3

424311

12 35 30 34 33 32

29

e7 e6 e52 e10 e9 e8 e11e12 e1 e31 e30

e32

e41

e27e26e2

e13

e33

e53e43 e3

e5

e40

e45 e46

e49

e50

e51

e48

e42 e4 e15

e14

e17

e16

e18 e19

e22

e21

e24e25

e29 e28 e37 e36 e35

e20

e23

e23

e39

e34

e38

Figure 3 ASRG of headstock

Mathematical Problems in Engineering 9

Table 2 Results of assembly unit partition

Unitnumber Basic part number Results of assembly unit partition

1 25 1 17 19 22 23 24 26 27 28 362 29 30 32 33 34 35 42 433 6 7 9 8 10 20 31 414 5 2 13 14 15 16 18 21 395 40 11 126 4 3 37 38

relatively intensive connection strength Elements of assem-bly relationship in the interference matrix were 0 whichcould prioritize assembly Therefore an independent unitwas formed As the assembly connection strength between

part 17 part 18 and basic parts in other unit was smallthe partition was not conducted for independent assemblyResults of modified unit partition were shown in Table 3 andFigure 5

63 Assembly Sequence Planning and Optimization for CNCMachine Tool

631 Case of Optimized Assembly Sequence Assembly unit 3was the braking part of headstock and contained 8 functionparts The corresponding part number was 6 7 8 9 10 2031 41 as shown in Figure 6

The interference relationship between parts in an assem-bly unit at directions (plusmn119883 plusmn119884 plusmn119885) during disassembly andother parts was analyzed An integrated interference matrix119868119860was constructed as follows

119868119860

=

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

000000

111011

010000

111011

100000

011111

011111

010000

011111

000000

000100

111011

100000

000100

000100

000100

000010

100000

000000

000000

100000

000100

000100

000100

011111

011111

000000

000000

100000

001000

000100

000100

000100

010100

000100

000100

000000

000100

000100

000100

111011

100000

100000

100000

100000

000000

011111

100000

111011

100000

000000

100000

100000

111011

000000

000000

000010

100000

100000

100000

100000

000100

000100

000000

]]]]]]]]]]

]

(13)

The stability of the connectivity among parts in theassembly unit was analyzed and a stability connection matrix119862 was constructed as follows

119862 =

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

0

1

0

2

2

1

2

0

1

0

2

1

0

0

0

0

0

2

0

0

0

0

0

0

2

1

0

0

0

0

0

0

2

0

0

0

0

0

0

0

1

0

0

0

0

0

1

2

2

0

0

0

0

1

0

0

0

0

0

0

0

2

0

0

]]]]]]]]]]

]

(14)

The initial population was produced by the randomapproach and the control parameter of genetic algorithmwasset as follows encoding length 119871 = 8 population size119873 = 16crossover probability 119875

119888= 08 mutation probability 119875

119898=

001 generation gen = 100 and genetic manipulation wasrespectively selected by roulette This study conducted two-point crossover switched mutation and reverse operationPart 6 of assembly unit 3 was a basic part Weights ofgeometric feasibility assembly unit stability and positionsof basic parts in the fitness calculation formula (11) wererespectively 120596

1= 04 120596

2= 02 and 120596

3= 04

Run the program in MATLAB to solve the optimalassembly sequence of assembly unit 3 Figure 7 was thefitness variation diagram of genetic algorithm optimizationAccording to the calculation result the last generation of

maximum fitness was 152 and the corresponding optimalassembly sequence was as follows

6 997888rarr 8 997888rarr 7 997888rarr 9 997888rarr 41 997888rarr 20 997888rarr 31 997888rarr 10 (15)

During the assembly process basic parts should be firstassembled and plate cover parts should be assembled aftershaft parts According to the optimized assembly sequenceresults in assembly unit 3 braking part 6 was a basic part andwas firstly assembled Cover 9 and cover 31 were assembledafter the piston rodThus it could be found that the assemblysequence solved from genetic algorithm was rational andfeasible

For assembly units with the number of parts larger than5 the genetic algorithm was used for optimized assem-bly sequence and assembly sequences of other units wereobtained according to experiences The result was shown inTable 4

Similar to the optimized assembly sequencing of parts inthe assembly unit integrated interferencematrix and stabilityconnection matrix among assembly units were constructedAssembly unit 1 was taken as the basic unit so that theassembly sequence of each assembly unit could be obtainedas shown in the following order

Unit 1 997888rarr Unit 4 997888rarr Unit 7 997888rarr Unit 5 997888rarr Unit 6

997888rarr Unit 8 997888rarr Unit 9 997888rarr Unit 2 997888rarr Unit 3(16)

10 Mathematical Problems in Engineering

Assembly unit 1

Assembly unit 2

Assembly unit 6

Assembly unit 3

Assembly unit 4

Assembly unit 5

Figure 4 Result of assembly unit partition

Assembly unit 1

Assembly unit 2

Assembly unit 8

Assembly unit 7Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 3

Assembly unit 9

Figure 5 Result of modified assembly unit partition

Table 3 Results of modified unit partition

Unit number Result of modified assembly unit partition1 1 22 23 24 25 26 27 28 362 29 30 32 33 34 35 42 433 6 7 8 9 10 20 31 414 5 13 14 15 16 19 215 11 12 406 3 4 37 387 2 398 179 18

632 Expression of Assembly Sequence As a fishbone dia-gram can simply directly and flexibly express the assemblysequence product hierarchies and other information thispaper used the fishbone diagram to express concurrentassembly sequence Among them fish head demonstrated thefinal product of assembly Big bones connected with the fishspine represented each assembly unit Small bones connectedwith big bones represented parts Positions of big bones and

Brake object 6Bending plate 10

7 Piston rod 31 Lid41 Friction block

20 Piston rod9 Lid 8 Friction block

Figure 6 Section view of braking part

7

8

9

10

11

12

13

14

15

16

10 20 30 40 50 60 70 80 90 100

Col

ony

adap

tatio

n de

gree

Maximum colony adaptation degreeEvolutionary process

Average colony adaptation degree

Genetic algebra

Figure 7 Fitness variation diagram

Table 4 Assembly sequence planning of each unit

Assembly unitnumber Assembly sequence

1 25rarr 27rarr 26rarr 1rarr 36rarr 28rarr 24rarr 22rarr 232 29rarr 42rarr 30rarr 32rarr 43rarr 35rarr 33rarr 343 6rarr 8rarr 7rarr 9rarr 41rarr 20rarr 31rarr 104 5rarr 19rarr 21rarr 16rarr 15rarr 14rarr 135 40rarr 12rarr 116 4rarr 3rarr 38rarr 377 39rarr 28 179 18

small bones in the figure reflected the assembly sequenceThe concurrent assembly sequence of headstock was shownin Figure 8

Mathematical Problems in Engineering 11

5

19

21

16

15

14

13

40

12

11 17

29

42

30

32

43

35

33

34

25

27

26

1

36

28

24

22

23

39

2

4

3

38

37 18 10

31

20

41

9

7

8

6

Assembly unit 4

Assembly unit 5

Assembly unit 2

Assembly unit 3

Assembly unit 9

Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 8

Spindle box

Figure 8 Expression of headstock assembly sequence

7 Conclusions

In an era of global economic integration to occupy avantage ground in fierce market competition enterprisesmust enhance their product innovation capability improveproduct quality and performance also shorten the productdesign andmanufacturing cycle and reduce costs Inmodernmanufacturing industry the workload of product in theassembling stage is very heavy producing high costs andhard to realize assembly automationTherefore newmethodsand technologies are in urgent need to support the productassembly design On the basis of related research of assemblydesign this paper explored into assemblymodeling assemblyunit partition and optimized assembly sequence of CNCmachine tool In this way the assembly design efficiency ofproduct was improved This research reached the followingconclusions

(1) By distinguishing connectors from function parts asimplified ASRG was established and semantic informationof assembly was added in the edges of assembly model Thestudy on assembly modeling cases of headstock of CNCmachine tool showed that the model efficiently expressed theassembly relationship among parts

(2)According to the transmissibility of assembly connec-tion strength the assembly connection strength among partswas calculated so that the information of nonadjacent partscould be utilized The assembly unit partition based on theconnection strength changed the previous partition methodsby experiences so as to quantify the partition of assembly unitIn addition through interference checking and correction ofassembly unit the rationality and accuracy of assembly unitpartition were improved

(3) According to needs of assembly sequence planninggenerations of coding approach fitness function genetic

operator control parameters and initial population of geneticalgorithm were designedThrough case analysis of headstockof CNC machine tool the effectiveness of using geneticalgorithm for assembly sequence planning and optimizationwas verified

This research actively explored into the assembly mod-eling assembly unit partition assembly sequence planningand optimization and other aspects of CNC machine toolrealized the optimized assembly sequence of CNC machinetool and provided theoretical basis and technical supports forthe improvement of assembly quality and efficiency of CNCmachine tool

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (N120403003) the Scienceand Technology Funds of Liaoning Province (20130200522011216010) and the Research Funds of State ldquoTwelve FiverdquoSupport Program (2012BAF12B08)

References

[1] V Blanchot and A Daidie ldquoRiveted assembly modelling Studyand numerical characterisation of a riveting processrdquo Journal ofMaterials Processing Technology vol 180 no 1ndash3 pp 201ndash2092006

[2] T Gu Z Xu and Z Yang ldquoSymbolic OBDD representations formechanical assembly sequencesrdquo Computer Aided Design vol40 no 4 pp 411ndash421 2008

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 2 Results of assembly unit partition

Unitnumber Basic part number Results of assembly unit partition

1 25 1 17 19 22 23 24 26 27 28 362 29 30 32 33 34 35 42 433 6 7 9 8 10 20 31 414 5 2 13 14 15 16 18 21 395 40 11 126 4 3 37 38

relatively intensive connection strength Elements of assem-bly relationship in the interference matrix were 0 whichcould prioritize assembly Therefore an independent unitwas formed As the assembly connection strength between

part 17 part 18 and basic parts in other unit was smallthe partition was not conducted for independent assemblyResults of modified unit partition were shown in Table 3 andFigure 5

63 Assembly Sequence Planning and Optimization for CNCMachine Tool

631 Case of Optimized Assembly Sequence Assembly unit 3was the braking part of headstock and contained 8 functionparts The corresponding part number was 6 7 8 9 10 2031 41 as shown in Figure 6

The interference relationship between parts in an assem-bly unit at directions (plusmn119883 plusmn119884 plusmn119885) during disassembly andother parts was analyzed An integrated interference matrix119868119860was constructed as follows

119868119860

=

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

000000

111011

010000

111011

100000

011111

011111

010000

011111

000000

000100

111011

100000

000100

000100

000100

000010

100000

000000

000000

100000

000100

000100

000100

011111

011111

000000

000000

100000

001000

000100

000100

000100

010100

000100

000100

000000

000100

000100

000100

111011

100000

100000

100000

100000

000000

011111

100000

111011

100000

000000

100000

100000

111011

000000

000000

000010

100000

100000

100000

100000

000100

000100

000000

]]]]]]]]]]

]

(13)

The stability of the connectivity among parts in theassembly unit was analyzed and a stability connection matrix119862 was constructed as follows

119862 =

6 7 8 9 10 20 31 41

6

7

8

9

10

20

31

41

[[[[[[[[[[

[

0

1

0

2

2

1

2

0

1

0

2

1

0

0

0

0

0

2

0

0

0

0

0

0

2

1

0

0

0

0

0

0

2

0

0

0

0

0

0

0

1

0

0

0

0

0

1

2

2

0

0

0

0

1

0

0

0

0

0

0

0

2

0

0

]]]]]]]]]]

]

(14)

The initial population was produced by the randomapproach and the control parameter of genetic algorithmwasset as follows encoding length 119871 = 8 population size119873 = 16crossover probability 119875

119888= 08 mutation probability 119875

119898=

001 generation gen = 100 and genetic manipulation wasrespectively selected by roulette This study conducted two-point crossover switched mutation and reverse operationPart 6 of assembly unit 3 was a basic part Weights ofgeometric feasibility assembly unit stability and positionsof basic parts in the fitness calculation formula (11) wererespectively 120596

1= 04 120596

2= 02 and 120596

3= 04

Run the program in MATLAB to solve the optimalassembly sequence of assembly unit 3 Figure 7 was thefitness variation diagram of genetic algorithm optimizationAccording to the calculation result the last generation of

maximum fitness was 152 and the corresponding optimalassembly sequence was as follows

6 997888rarr 8 997888rarr 7 997888rarr 9 997888rarr 41 997888rarr 20 997888rarr 31 997888rarr 10 (15)

During the assembly process basic parts should be firstassembled and plate cover parts should be assembled aftershaft parts According to the optimized assembly sequenceresults in assembly unit 3 braking part 6 was a basic part andwas firstly assembled Cover 9 and cover 31 were assembledafter the piston rodThus it could be found that the assemblysequence solved from genetic algorithm was rational andfeasible

For assembly units with the number of parts larger than5 the genetic algorithm was used for optimized assem-bly sequence and assembly sequences of other units wereobtained according to experiences The result was shown inTable 4

Similar to the optimized assembly sequencing of parts inthe assembly unit integrated interferencematrix and stabilityconnection matrix among assembly units were constructedAssembly unit 1 was taken as the basic unit so that theassembly sequence of each assembly unit could be obtainedas shown in the following order

Unit 1 997888rarr Unit 4 997888rarr Unit 7 997888rarr Unit 5 997888rarr Unit 6

997888rarr Unit 8 997888rarr Unit 9 997888rarr Unit 2 997888rarr Unit 3(16)

10 Mathematical Problems in Engineering

Assembly unit 1

Assembly unit 2

Assembly unit 6

Assembly unit 3

Assembly unit 4

Assembly unit 5

Figure 4 Result of assembly unit partition

Assembly unit 1

Assembly unit 2

Assembly unit 8

Assembly unit 7Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 3

Assembly unit 9

Figure 5 Result of modified assembly unit partition

Table 3 Results of modified unit partition

Unit number Result of modified assembly unit partition1 1 22 23 24 25 26 27 28 362 29 30 32 33 34 35 42 433 6 7 8 9 10 20 31 414 5 13 14 15 16 19 215 11 12 406 3 4 37 387 2 398 179 18

632 Expression of Assembly Sequence As a fishbone dia-gram can simply directly and flexibly express the assemblysequence product hierarchies and other information thispaper used the fishbone diagram to express concurrentassembly sequence Among them fish head demonstrated thefinal product of assembly Big bones connected with the fishspine represented each assembly unit Small bones connectedwith big bones represented parts Positions of big bones and

Brake object 6Bending plate 10

7 Piston rod 31 Lid41 Friction block

20 Piston rod9 Lid 8 Friction block

Figure 6 Section view of braking part

7

8

9

10

11

12

13

14

15

16

10 20 30 40 50 60 70 80 90 100

Col

ony

adap

tatio

n de

gree

Maximum colony adaptation degreeEvolutionary process

Average colony adaptation degree

Genetic algebra

Figure 7 Fitness variation diagram

Table 4 Assembly sequence planning of each unit

Assembly unitnumber Assembly sequence

1 25rarr 27rarr 26rarr 1rarr 36rarr 28rarr 24rarr 22rarr 232 29rarr 42rarr 30rarr 32rarr 43rarr 35rarr 33rarr 343 6rarr 8rarr 7rarr 9rarr 41rarr 20rarr 31rarr 104 5rarr 19rarr 21rarr 16rarr 15rarr 14rarr 135 40rarr 12rarr 116 4rarr 3rarr 38rarr 377 39rarr 28 179 18

small bones in the figure reflected the assembly sequenceThe concurrent assembly sequence of headstock was shownin Figure 8

Mathematical Problems in Engineering 11

5

19

21

16

15

14

13

40

12

11 17

29

42

30

32

43

35

33

34

25

27

26

1

36

28

24

22

23

39

2

4

3

38

37 18 10

31

20

41

9

7

8

6

Assembly unit 4

Assembly unit 5

Assembly unit 2

Assembly unit 3

Assembly unit 9

Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 8

Spindle box

Figure 8 Expression of headstock assembly sequence

7 Conclusions

In an era of global economic integration to occupy avantage ground in fierce market competition enterprisesmust enhance their product innovation capability improveproduct quality and performance also shorten the productdesign andmanufacturing cycle and reduce costs Inmodernmanufacturing industry the workload of product in theassembling stage is very heavy producing high costs andhard to realize assembly automationTherefore newmethodsand technologies are in urgent need to support the productassembly design On the basis of related research of assemblydesign this paper explored into assemblymodeling assemblyunit partition and optimized assembly sequence of CNCmachine tool In this way the assembly design efficiency ofproduct was improved This research reached the followingconclusions

(1) By distinguishing connectors from function parts asimplified ASRG was established and semantic informationof assembly was added in the edges of assembly model Thestudy on assembly modeling cases of headstock of CNCmachine tool showed that the model efficiently expressed theassembly relationship among parts

(2)According to the transmissibility of assembly connec-tion strength the assembly connection strength among partswas calculated so that the information of nonadjacent partscould be utilized The assembly unit partition based on theconnection strength changed the previous partition methodsby experiences so as to quantify the partition of assembly unitIn addition through interference checking and correction ofassembly unit the rationality and accuracy of assembly unitpartition were improved

(3) According to needs of assembly sequence planninggenerations of coding approach fitness function genetic

operator control parameters and initial population of geneticalgorithm were designedThrough case analysis of headstockof CNC machine tool the effectiveness of using geneticalgorithm for assembly sequence planning and optimizationwas verified

This research actively explored into the assembly mod-eling assembly unit partition assembly sequence planningand optimization and other aspects of CNC machine toolrealized the optimized assembly sequence of CNC machinetool and provided theoretical basis and technical supports forthe improvement of assembly quality and efficiency of CNCmachine tool

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (N120403003) the Scienceand Technology Funds of Liaoning Province (20130200522011216010) and the Research Funds of State ldquoTwelve FiverdquoSupport Program (2012BAF12B08)

References

[1] V Blanchot and A Daidie ldquoRiveted assembly modelling Studyand numerical characterisation of a riveting processrdquo Journal ofMaterials Processing Technology vol 180 no 1ndash3 pp 201ndash2092006

[2] T Gu Z Xu and Z Yang ldquoSymbolic OBDD representations formechanical assembly sequencesrdquo Computer Aided Design vol40 no 4 pp 411ndash421 2008

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Assembly unit 1

Assembly unit 2

Assembly unit 6

Assembly unit 3

Assembly unit 4

Assembly unit 5

Figure 4 Result of assembly unit partition

Assembly unit 1

Assembly unit 2

Assembly unit 8

Assembly unit 7Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 3

Assembly unit 9

Figure 5 Result of modified assembly unit partition

Table 3 Results of modified unit partition

Unit number Result of modified assembly unit partition1 1 22 23 24 25 26 27 28 362 29 30 32 33 34 35 42 433 6 7 8 9 10 20 31 414 5 13 14 15 16 19 215 11 12 406 3 4 37 387 2 398 179 18

632 Expression of Assembly Sequence As a fishbone dia-gram can simply directly and flexibly express the assemblysequence product hierarchies and other information thispaper used the fishbone diagram to express concurrentassembly sequence Among them fish head demonstrated thefinal product of assembly Big bones connected with the fishspine represented each assembly unit Small bones connectedwith big bones represented parts Positions of big bones and

Brake object 6Bending plate 10

7 Piston rod 31 Lid41 Friction block

20 Piston rod9 Lid 8 Friction block

Figure 6 Section view of braking part

7

8

9

10

11

12

13

14

15

16

10 20 30 40 50 60 70 80 90 100

Col

ony

adap

tatio

n de

gree

Maximum colony adaptation degreeEvolutionary process

Average colony adaptation degree

Genetic algebra

Figure 7 Fitness variation diagram

Table 4 Assembly sequence planning of each unit

Assembly unitnumber Assembly sequence

1 25rarr 27rarr 26rarr 1rarr 36rarr 28rarr 24rarr 22rarr 232 29rarr 42rarr 30rarr 32rarr 43rarr 35rarr 33rarr 343 6rarr 8rarr 7rarr 9rarr 41rarr 20rarr 31rarr 104 5rarr 19rarr 21rarr 16rarr 15rarr 14rarr 135 40rarr 12rarr 116 4rarr 3rarr 38rarr 377 39rarr 28 179 18

small bones in the figure reflected the assembly sequenceThe concurrent assembly sequence of headstock was shownin Figure 8

Mathematical Problems in Engineering 11

5

19

21

16

15

14

13

40

12

11 17

29

42

30

32

43

35

33

34

25

27

26

1

36

28

24

22

23

39

2

4

3

38

37 18 10

31

20

41

9

7

8

6

Assembly unit 4

Assembly unit 5

Assembly unit 2

Assembly unit 3

Assembly unit 9

Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 8

Spindle box

Figure 8 Expression of headstock assembly sequence

7 Conclusions

In an era of global economic integration to occupy avantage ground in fierce market competition enterprisesmust enhance their product innovation capability improveproduct quality and performance also shorten the productdesign andmanufacturing cycle and reduce costs Inmodernmanufacturing industry the workload of product in theassembling stage is very heavy producing high costs andhard to realize assembly automationTherefore newmethodsand technologies are in urgent need to support the productassembly design On the basis of related research of assemblydesign this paper explored into assemblymodeling assemblyunit partition and optimized assembly sequence of CNCmachine tool In this way the assembly design efficiency ofproduct was improved This research reached the followingconclusions

(1) By distinguishing connectors from function parts asimplified ASRG was established and semantic informationof assembly was added in the edges of assembly model Thestudy on assembly modeling cases of headstock of CNCmachine tool showed that the model efficiently expressed theassembly relationship among parts

(2)According to the transmissibility of assembly connec-tion strength the assembly connection strength among partswas calculated so that the information of nonadjacent partscould be utilized The assembly unit partition based on theconnection strength changed the previous partition methodsby experiences so as to quantify the partition of assembly unitIn addition through interference checking and correction ofassembly unit the rationality and accuracy of assembly unitpartition were improved

(3) According to needs of assembly sequence planninggenerations of coding approach fitness function genetic

operator control parameters and initial population of geneticalgorithm were designedThrough case analysis of headstockof CNC machine tool the effectiveness of using geneticalgorithm for assembly sequence planning and optimizationwas verified

This research actively explored into the assembly mod-eling assembly unit partition assembly sequence planningand optimization and other aspects of CNC machine toolrealized the optimized assembly sequence of CNC machinetool and provided theoretical basis and technical supports forthe improvement of assembly quality and efficiency of CNCmachine tool

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (N120403003) the Scienceand Technology Funds of Liaoning Province (20130200522011216010) and the Research Funds of State ldquoTwelve FiverdquoSupport Program (2012BAF12B08)

References

[1] V Blanchot and A Daidie ldquoRiveted assembly modelling Studyand numerical characterisation of a riveting processrdquo Journal ofMaterials Processing Technology vol 180 no 1ndash3 pp 201ndash2092006

[2] T Gu Z Xu and Z Yang ldquoSymbolic OBDD representations formechanical assembly sequencesrdquo Computer Aided Design vol40 no 4 pp 411ndash421 2008

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

5

19

21

16

15

14

13

40

12

11 17

29

42

30

32

43

35

33

34

25

27

26

1

36

28

24

22

23

39

2

4

3

38

37 18 10

31

20

41

9

7

8

6

Assembly unit 4

Assembly unit 5

Assembly unit 2

Assembly unit 3

Assembly unit 9

Assembly unit 6

Assembly unit 7

Assembly unit 1

Assembly unit 8

Spindle box

Figure 8 Expression of headstock assembly sequence

7 Conclusions

In an era of global economic integration to occupy avantage ground in fierce market competition enterprisesmust enhance their product innovation capability improveproduct quality and performance also shorten the productdesign andmanufacturing cycle and reduce costs Inmodernmanufacturing industry the workload of product in theassembling stage is very heavy producing high costs andhard to realize assembly automationTherefore newmethodsand technologies are in urgent need to support the productassembly design On the basis of related research of assemblydesign this paper explored into assemblymodeling assemblyunit partition and optimized assembly sequence of CNCmachine tool In this way the assembly design efficiency ofproduct was improved This research reached the followingconclusions

(1) By distinguishing connectors from function parts asimplified ASRG was established and semantic informationof assembly was added in the edges of assembly model Thestudy on assembly modeling cases of headstock of CNCmachine tool showed that the model efficiently expressed theassembly relationship among parts

(2)According to the transmissibility of assembly connec-tion strength the assembly connection strength among partswas calculated so that the information of nonadjacent partscould be utilized The assembly unit partition based on theconnection strength changed the previous partition methodsby experiences so as to quantify the partition of assembly unitIn addition through interference checking and correction ofassembly unit the rationality and accuracy of assembly unitpartition were improved

(3) According to needs of assembly sequence planninggenerations of coding approach fitness function genetic

operator control parameters and initial population of geneticalgorithm were designedThrough case analysis of headstockof CNC machine tool the effectiveness of using geneticalgorithm for assembly sequence planning and optimizationwas verified

This research actively explored into the assembly mod-eling assembly unit partition assembly sequence planningand optimization and other aspects of CNC machine toolrealized the optimized assembly sequence of CNC machinetool and provided theoretical basis and technical supports forthe improvement of assembly quality and efficiency of CNCmachine tool

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (N120403003) the Scienceand Technology Funds of Liaoning Province (20130200522011216010) and the Research Funds of State ldquoTwelve FiverdquoSupport Program (2012BAF12B08)

References

[1] V Blanchot and A Daidie ldquoRiveted assembly modelling Studyand numerical characterisation of a riveting processrdquo Journal ofMaterials Processing Technology vol 180 no 1ndash3 pp 201ndash2092006

[2] T Gu Z Xu and Z Yang ldquoSymbolic OBDD representations formechanical assembly sequencesrdquo Computer Aided Design vol40 no 4 pp 411ndash421 2008

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[3] Z Xu Y Li J Zhang H Cheng S Jiang and W Tang ldquoAdynamic assembly model for assembly sequence planning ofcomplex product based on polychromatic sets theoryrdquoAssemblyAutomation vol 32 no 2 pp 152ndash162 2012

[4] X Guo L Ma and X Xu ldquoHierarchical assembly model undercomplex constraint conditionrdquo Computer Engineering vol 37no 18 pp 260ndash265 2011

[5] R B Gottipolu and K Ghosh ldquoA simplified and efficient repre-sentation for evaluation and selection of assembly sequencesrdquoComputers in Industry vol 50 no 3 pp 251ndash264 2003

[6] J Ko E Nazarian H Wang and J Abell ldquoAn assemblydecomposition model for subassembly planning consideringimperfect inspection to reduce assembly defect ratesrdquo Journalof Manufacturing Systems vol 32 no 3 pp 412ndash416 2013

[7] R Wu Research on Assembly Key Technologies Oriented toProcess Reuse National University of Defense TechnologyChangsha China 2008

[8] YWang and J Liu ldquoAssembly unit partitioning for collaborativeassembly planningrdquo Journal of Mechanical Engineering vol 45no 10 pp 172ndash179 2009

[9] H Wang Y Rong and D Xiang ldquoMechanical assembly plan-ning using ant colony optimizationrdquo Computer Aided Designvol 47 pp 59ndash71 2014

[10] C Sinanoglu and H R Borklu ldquoAn approach to determinegeometric feasibility to assembly states by intersection matricesin assembly sequence planningrdquo Journal of Intelligent Manufac-turing vol 15 no 4 pp 543ndash559 2004

[11] Y Zhong K Xue and D Shi ldquoAssembly unit partitioning forhull structure in shipbuildingrdquo Computer-Aided Design vol 45no 12 pp 1630ndash1638 2013

[12] Y Wang J H Liu and L S Li ldquoAssembly sequences mergingbased on assembly unit partitioningrdquoThe International Journalof AdvancedManufacturing Technology vol 45 no 7-8 pp 808ndash820 2009

[13] Y-M Li C-T Wu and C-Y Lai ldquoA social recommendermechanism for e-commerce combining similarity trust andrelationshiprdquo Decision Support Systems vol 55 no 3 pp 740ndash752 2013

[14] F Demoly A Matsokis and D Kiritsis ldquoA mereotopologicalproduct relationship description approach for assembly ori-ented designrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 6 pp 681ndash693 2012

[15] S Li Y Liu J Wang and H Zeng ldquoAn intelligent interactiveapproach for assembly process planning based on hierarchicalclassification of partsrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9-12 pp 1903ndash1914 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of