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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
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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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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
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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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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Decision SciencesAdvances in
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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
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
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
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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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
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
Recommended