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AdvancesinProductionEngineering&Management ISSN1854‐6250
Volume11|Number2|June2016|pp77–92 Journalhome:apem‐journal.org
http://dx.doi.org/10.14743/apem2016.2.211 Originalscientificpaper
A simulation approach to the process planning problem using a modified particle swarm optimization
Wang, J.F.a,*, Kang, W.L.a, Zhao, J.L.a, Chu, K.Y.a
aDepartment of Mechanical Engineering, North China Electric Power University, Baoding, China
A B S T R A C T A R T I C L E I N F O
Due to the complexity and variety of practical manufacturing conditions,computer‐aidedprocessplanning (CAPP)systemshavebecome increasinglyimportant in themodern production system. In CAPP, the process planning(PP) problem involves two tasks: operation determining and operation se‐quencing. To optimize theprocess plans generated from complexparts, thetraditionalparticleswarmoptimization(PSO)algorithmismodified.Efficientencoding and decoding population initialization methods have been devel‐opedtoadaptthePPproblemforthePSOapproach.Inaddition,toavoidtheproposed approach becoming trapped in local convergences and achievinglocal optimal solutions, parameters are set to control the iterations. Severalextendedoperatorsforthedifferentpartsoftheparticleshavebeenincorpo‐ratedintothetraditionalPSO.Simulationexperimentshavebeenruntoeval‐uateandverify theeffectivenessof themodifiedPSOapproach.Thesimula‐tionresultsindicatethatthePPproblemcanbemoreeffectivelysolvedbytheproposedPSOapproachthanotherapproaches.
©2016PEI,UniversityofMaribor.Allrightsreserved.
Keywords:ProcessplanningOperationdeterminingOperationsequencingParticleswarmoptimizationExtendedoperator
*Correspondingauthor:wjf266@163.com(Wang,J.F.)
Articlehistory:Received29January2016Revised27March2016Accepted10May2016
1. Introduction
In themodern computer‐integratedmanufacturing system (CIMS), the CAPP system plays animportant role [1]. Generally, process planning involves two activities: operation determiningandoperationsequencing.InCAPPsystems,thesetwoactivitiesmustbeexecutedsimultaneous‐lytoachieveagoodsolution.Thusfar,someefforthasbeenmadetoaddressthisproblem,forinstance,bydesigningamore feasiblemathematicalmodelordevelopingamoreefficient ap‐proach.Theapplicationofsomeartificial intelligenceapproaches in thePPproblemhasespe‐ciallypromotedthedevelopmentofCAPPtechnology.Theseapproachescanbecategorizedasthe genetic algorithm (GA) [1, 2], simulated annealing (SA) [2, 3], tabu search (TS) [4, 5], antcolonyoptimization(ACO)[6‐8],particleswarmoptimization(PSO)[9‐13],honeybeesmatingoptimization[14],andhybridapproaches[2,15].In1995,KennedyandEberhartproposedthePSOalgorithm[16].ThePSOisanewswarmop‐
timizationapproachthatcanoptimizeengineeringproblemsinaspectssuchasturningprocessmodelling[17],assemblysequenceplanning[18],andprocessparameteroptimization(TSP)[19].ThesearchforapplicationsofPSOinthePPproblemwasfirstintroducedbyGuoetal.[9].The
processplanparticle encoded/decoded strategy and somemodifiedoperatorsweredesigned.Kafashietal.[10]optimizedthesetupplanningusingcostindicesbasedonconstraintssuchasthe TAD (tool approach direction), the tolerance relation between features, and the featureprecedence relations.Wang et al. [11] proposed an innovative process plan representation in
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78 Advances in Production Engineering & Management 11(2) 2016
whichoperationselectionandoperationsequencingwereintroducedsimultaneouslyinaparti‐cle.TwolocalsearchoperatorsareincorporatedintothetraditionalPSOtoachievetheoptimalsolution.Lietal.[12]modifiedthetraditionalPSOfortheprocessplanningproblem.Miljkovićetal.[13]adoptedtheAND/ORnetworkrepresentationtodescribetheflexibilityofthemachine,tool,TAD,processandsequenceandaperformedmulti‐objectiveoptimizationprocedurefortheminimizationoftheproductiontimeandproductioncostusingamodifiedPSOalgorithmonthisrepresentation.Zhangetal.[1]proposedaGAforanovelCAPPmodel.Lietal.[2]incorporatedanSAintoa
GAtoimproveitssearchingefficiency.Maetal.[3]modeledthePPproblemasacombinationaloptimizationproblemwithconstraints.Anentiresolutionspace is constructed in reference toprecedenceconstraintsamongoperations.AnSAalgorithmisthenproposedtoaddressthePPproblem.Lietal.[4]appliedaTS‐basedapproachtoaddressthePPproblem.Inthisapproach,amapping relationship is established between process constraints among features and prece‐dence constraints amongoperations. Lianet al. [5]proposedamulti‐dimensional tabu search(MDTS) approach to address the PP problem. Some local search strategies for different partshavebeenintegratedintothisTSapproach.Liuetal.[6]constructedamathematicalmodelforthe PP problem by considering the process constraints and optimization objectives. The ACOapproachhasbeendeveloped to optimize thePPproblembasedon thismathematicalmodel.Wangetal.[7]representedthePPproblembyanimproveddirected/undirectedgraph.Atwo‐stage approach based on ACO was developed to optimize the process plans on the di‐rected/undirectedgraph.Wenetal.[14]proposedanewmethodbasedonthehoneybeesmat‐ingoptimization(HBMO)approachtooptimizethePPproblem.Thesolutionencoding,crosso‐veroperator,andlocalsearchstrategiesweredevelopedaccordingtothecharacteristicsofthePPproblem.Huangetal.[15]designedahybridalgorithmcombiningagraphandtheGAtoop‐timizeprocessplans.TheprecedenceconstraintsaremappedtoanoperationprecedencegraphonwhichanimprovedGAwasappliedtosolvethePPproblem.
Although therehavebeen some significant improvements in solving thePPproblem, therestill remains thepotential for further improvement[20].Up tonow,someheuristicsorevolu‐tionary approaches have been applied to optimize the PP problem, but themajor difficulty isthatthesearchspaceistoolargeforpartswithcomplexfeaturestofindoptimalsolutionseffi‐ciently.Toaddressthisproblem,atraditionalPSOapproachismodifiedtosolvethePPprobleminthispaper.Themainworkincludesthefollowingtwoaspects:
Therepresentationofaprocessplanmappedtoaparticleismodifiedtofacilitatethedis‐crete PP problem. A new particle encoding/decoding strategy is adapted to make thesearchmoreefficient.
AdiversesearchingmechanismhasbeenadoptedtoimprovetheperformanceofthePSOapproach.Toavoidlocalconvergence,somemodificationstothetraditionalPSOapproachhavebeenadoptedtobetterexplorethesolutionspace.SeveraloperatorsforthedifferentpartsoftheparticleshavebeenincorporatedintothetraditionalPSO.
Section2describestheprocessplanningmodel.Section3introducestheparticleswarmop‐timizationapproach.Section4introducesanapplicationofthemodifiedPSOapproachtothePPproblem.Section5presentsthesimulationresultsoftheproposedPSOalgorithm.Finally,someconclusionsandoutlookaregiveninSection6.
2. Problem modelling
2.1 Problem description
InthePPproblem,twotaskshavetobeperformed,namely,operationdeterminingandopera‐tionsequencing.Foroperationdetermining,themethodofmappingfromfeaturestooperationsiswidelyusedinthePPproblem.Theattributesofeachfeaturedeterminesthecorrespondingmachiningmethods,whichcanbeexpressedbythealternativeoperations.Thecombinationofmachines,cuttingtools,andtoolapproachdirections(TAD)compriseallofthedifferentopera‐
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Advances in Production Engineering & Management 11(2) 2016 79
tiontypes(OPTs) fora featureofapart,whichwillbeselectedtodeterminethe finalprocessplans[1,11].
Foranoperation,thereareasetofOPTsunderwhichtheoperationcanbeexecuted.Accord‐ingly,anintegratedprocessplancanberepresentedasfollows.
, , … , , … , (1)
, , … , , … , (2)
, , (3)
isthei‐thoperation,and isthej‐thalternativeselectionoftheoperationOPi. ,and aretheIDoftheselectedmachine,toolandTADforoperation ,respectively.AnexamplepartinFig.1isusedtodemonstratetherepresentationofaprocessplan[7].The
candidateoperationsarelistedinTable1.Inadditiontooperationdetermining,operationsequencingisanothertaskinthePPproblem.
The operations should be sequenced under the conditions of satisfying the precedence con‐straintsamongoperations[4,6,12,15].TheconstraintsofPart1areshowninTable2.
According to the precedence constraints inTable 2, an available process plan for Part 1 isshowninTable3.
Fig.1Anexamplepart–Part1
Table1CandidateoperationsforPart1
Feathers Operations Operationtypes Machines Tools TADs Description
F1 Milling(OP1)OPT11,OPT12 M2 T1
+X,+Z
M1:DrillingpressM2:VerticalmillingmachineT1:MillingcutterT2:Drill1T3:TappingtoolT4:Drill2T5:Reamer1T6:SlotcutterT7:ChamfercutterT8:Drill3T9:Reamer2
F2Drilling(OP2) OPT21,OPT22
M1,M2T2
−ZTapping(OP3) OPT31,OPT32 T3
F3Drilling(OP4) OPT41,OPT42
M1,M2T4
−XReaming(OP5) OPT51,OPT52 T5
F4 Milling(OP6) OPT61 M2 T6 +Z
F5 Milling(OP7) OPT71,OPT72 M2 T7 +Y,−Z
F6 Drilling(OP8) OPT81,OPT82 M1,M2 T8 +X
Reaming(OP9) OPT91,OPT92 T9
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Table2ConstraintsforPart1[7]Operations Precedenceconstraintdescription Constrainttypes
OP1OP1ispriortoOP2andOP3. Hard
OP1ispriortoOP4andOP5. Soft
OP2 OP2ispriortoOP3. Hard
OP4 OP4ispriortoOP5. Hard
OP4,OP5 OP4andOP5arepriortoOP6. Hard
OP6 OP6ispriortoOP2andOP3. Hard
OP8 OP8ispriortoOP9. Hard
OP8,OP9 OP8andOP9arepriortoOP7. Hard
Table3AvailableprocessplanforPart1
Operation Machine Tool TADOP1 M2 T1 +XOP8 M1 T8 +XOP9 M1 T9 +XOP4 M1 T4 −XOP5 M1 T5 −XOP6 M2 T6 +ZOP7 M2 T7 −ZOP2 M1 T2 −ZOP3 M1 T3 −Z
2.2 Mathematical model
Thecriterionofminimizingproductioncosts(CP)isusuallyusedtoevaluatetheprocessplan.The CP includes themachine cost (CM), cutting tool cost (CT),machine‐changing cost (CMC),cuttingtoolchangingcost(CTC),andset‐upcost(CSC)[2,3,6,8,11,14,15].
Themachinecostis
, , … , , … , (4)
where isthenumberofmachines.Thecuttingtoolcostis
, , … , , … , (5)
where isthenumberofcuttingtools.AsshowninEq.2andEq.3,anoperationisselectedfromseveralalternativeOPTs.Thema‐
chinecostforanoperationvariesaccordingtothealternativeOPTs,sothemachinecost foranOPT canbegivenas
(6)
where isexplainedinEq.3.Thecuttingtoolcost foranOPT canbegivenas
(7)
where isexplainedinEq.3.Themachinechangingcost betweenOPT andOPT canbegivenas
Φ , (8)
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Advances in Production Engineering & Management 11(2) 2016 81
where isthecostofmachinechanging,whichisconsideredtobethesameforeachmachinechange.Φ , canbecalculatedasfollows:
Φ , 10
(9)
Thecuttingtoolchangingcost betweenOPT andOPT canbegivenas
Ω Φ , , Φ , (10)
where isthecuttingtoolchangingcost,whichisconsideredtobethesameforeachcuttingtoolchange.Ω , canbecalculatedasfollows.
Ω , 0 01
(11)
Theset‐upcost betweenOPT andOPT canbegivenas
Ω Φ , , Φ , (12)
where isexplainedinEq.3,and isthecostforaset‐up,whichisconsideredtobethesameforeachset‐up.
Thedefinitionsofmachinechanging,toolchanging,andset‐upchanginghavebeenexplainedinreference[2].Basedontheaboveanalysis,themathematicalmodelofthePPproblemisfor‐mulatedasfollows[6]:
Objectives:
(i)AcombinationofCM,CT,CMC,CTC,andCSwillbeconsideredasCP,andminimizingCPistheobjectiveofPP.
Min . . . ′ ′′
′ ′
′
′
. . ′ ′ . ′ ′ . ′ ′ (13)
Subjectto:
(ii)noperationshavetobeselectedwhilemachiningapart.
(14)
(iii)Foranoperation,oneandonlyoneOPTcanbeselectedfromitsmalternativeOPTs.
1 1, 2, … , (15)
(iv)Foraprocessplanconsistingofnoperations,thenumbersofoperationchangesisn–1,andthechangesofcombinationsofmachines,cuttingtools,andset‐upsareaccompaniedbyopera‐tionchanges.
′ ′′
′ ′
′
′
1 (16)
(v)ForeachOPT,onlyoneadjacentoperationislinedupbeforeit.
′ ′ 1 (17)
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(vi)ForeachOPT,onlyoneadjacentoperationislinedupafterit.
′ ′
′′
1 (18)
(vii) isanintegerenumerationvariable.
1ifOPT is selected for operation0 otherwise
(19)
(viii) ′ ′isanintegerenumerationvariable.
′ ′1ifOPT ′ ′is selected after OPT is executed0 otherwise
(20)
Eq.13means that theobjective functionof thePP is tominimize the totalproductioncost.TheconstraintsareinEq.14toEq.20.ConstraintsinEq.14andEq.15ensurethatalltheopera‐tionsare carriedout.The constraint inEq. 16 ensures thatn−1 changing costsofmachines,cuttingtools,andset‐upsareaddedintothePCamongnoperations.ConstraintsinEq.17andEq.18ensurethateveryoperationexceptthefirstandthelasthaveonlyoneadjacentoperationoneachside.
3. Introduction of the particle swarm optimization approach
PSO is a swarm optimization approach [16]. Every particle in the population representsanN‐dimensional solution that constructs a search space for every particle. The particles flyfreelytosearchfortheoptimalpositionatagivenvelocity.Hence,fortheparticlei,thevectors and at the t‐th iteration can be denoted as , , … , , … , and, , … , , … , ,respectively.Withtheemergenceoftheoptimalpositionatiterationt+
1,thevectors and canbeupdatedasfollows:
∗ ∗ ∗ ∗ ∗ (21)
(22)
InEq.21andEq.22, isthevelocityondimensionk,and isthepositionondimensionk. isthelocaloptimalpositionondimensionk,and istheglobaloptimalpositionondi‐mensionk.Theweightwisusedtocontroltheiteration.Theconstantsc1andc2controlthebal‐ancebetweenalocaloptimalpositionandtheglobaloptimalposition.Rand()islimitedin[0,1].ThetraditionalPSOapproachiscarriedoutinfoursteps:
Step1:Populationinitialization.Step2:Updatethevectors and accordingtoEq.21andEq.22.Step3:Update and accordingtotheperformanceofthepopulationStep4:LooptoStep2untilaterminationconditionismet.
4. The proposed PSO approach
4.1 Solution representation
ItisnecessarytomodifythetraditionalPSOforthePPproblemtoinclude,forexample,particlerepresentation and the particle movement strategy. In applying the PSO approach to the PPproblem, three tasks have to be performed, namely, particle encoding, particle validation andparticledecoding.Thefirsttaskistoencodeaprocessplantoanappropriateparticle.Becauseoperationdeterminingandoperationsequencinghavetobeperformedsimultaneously inpro‐
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Advances in Production Engineering & Management 11(2) 2016 83
cessplanning,theparticlestructureshouldbeconsideredtocomprisetheinformationofdeter‐miningandsequencingoperations.ThedetailsofaparticlearelistedinTable4.
Accordingtothedefinitionofaparticle,wemodifiedaparticleiofa2×nmatrixtorepresentaprocessplanatiterationt[11],i.e.
, , … ,, , … ,
(23)
Thefirstrow istheOperationDetermining(OD)partandrepresentstheoperationselec‐tionforeachfeature.Thesecondrow istheOperationSequencing(OS)partandrepresentsthepriorityamongoperations. isinitializedrandomlyintherangeof0and1accordingtothepriorityamongoperations,and canbecalculatedbyEq.24.
∗ _ ∗ _ _ / (24)
InEq.24,ix_m,ix_c,andix_taregeneratedrandomlyfromthecandidatemachineset,cuttingtoolset,andTADsetforexecutingtheoperation.acanbegivenbytheequation
, , 1 (25)
whereNSisthenumberofTADs.Toillustratetheparticleencoding,theprocessplaninTable3istakenasanexampleandthe
correspondingencodingof isshowninFig.2.Forinstance,thevaluesofix_m,ix_c,andix_taresetto1iftheOPTsofM2,T1and+X,respectively,areselectedtoprocessoperationOP1.IftheOPTsofM2,T1and+ZareselectedtoprocessoperationOP1,thevaluesofix_m,ix_c,andix_twillbe1,1,and3,respectively.
FortheprocessplaninTable3,afeasibleparticleislistedinTable5.InTable5,thefirstrowmeanstheoperations.ThesecondrowrepresentsthecalculatedvalueofitsassignedOPT,andthethirdrowrepresents thepriorityamongtheoperations.Thesecondtask is tovalidate theparticlestoaccordwiththeprecedenceconstraints.Aparticlerepresentsaprocessplan.Never‐theless,processplansgeneratedbyparticles flying freely in solution spaceareusually invalidagainsttheprecedenceconstraints.Tovalidateeachparticle,ann×nconstraintmatrixPispro‐posedtoincorporatetheprecedenceconstraintsamongtheoperationsintotheparticles.
Table4Detailofaparticle
Datatype Variable DescriptionInteger ix_o Idofanoperation,whichcorrespondstotheindexoftheoperationset.Integer ix_m Idofamachine,whichcorrespondstotheindexofthemachineset.Integer ix_c Idofacuttingtool,whichcorrespondstotheindexofthecuttingtoolset.Integer ix_t IdofaTAD,whichcorrespondstotheindexoftheTADset.
Fig.2Encodingof fortheprocessplaninTable3
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Table5AfeasibleparticlefortheprocessplaninTable3
Operation OP1 OP2 OP3 OP4 OP5 OP6 OP7 OP8 OP9
0.111 0.126 0.136 0.144 0.154 0.164 0.176 0.181 0.191
1 0.50 0.40 0.85 0.75 0.70 0.60 0.95 0.90
Thesecondtaskistovalidatetheparticlestoaccordwiththeprecedenceconstraints.Aparti‐
clerepresentsaprocessplan.Nevertheless,processplansgeneratedbyparticlesflyingfreelyinsolutionspaceareusuallyinvalidagainsttheprecedenceconstraints.Tovalidateeachparticle,ann×nconstraintmatrixP isproposedto incorporatetheprecedenceconstraintsamongtheoperationsintotheparticles.
(26)
1 is prior to or0 otherwise
(27)
TheprecedenceconstraintmatrixfortheprecedenceconstraintsinTable2isshownasfollows:
1 0 0 0 0 0 0 0 01 1 0 0 0 1 0 0 01110010001001000001001100000001110000 0 0 0 0 0 1 1 10 0 0 0 0 0 0 1 00 0 0 0 0 0 0 1 1
(28)
Thethirdtaskistodecodeaparticleintoasolution.Accordingtotheinputofencodedposi‐
tionmatrix,aparticlecanbedecodedtoobtainaprocessplan.Theparticledecoding includesthefollowingtwosteps.
First,determinethemachine,cuttingtoolandTADaccordingtothevalue .Thedecodingapproachfor is
∗ (29)
_ / (30)
_ _ ∗ / (31)
_ _ ∗ _ ∗ (32)
where isaninteger.Second,sequencethesedeterminedoperationsaccordingtotheprecedencevalue .Ifthe
sequenceofoperationsviolatestheprecedenceconstraints,theoperationsshouldbesequencedagainaccordingtotheprecedencevalue withthehelpoftheprecedencematrixPm.
(33)
Similarly,forparticlei,avelocitymatrixatiterationtcanberepresentedas
, , … , , … , (34)
where isinitializedrandomlyintherangeof–1to1.
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4.2 Population initialization
Theparticleswarmisinitializedinthreesteps:
(i) SetthepopulationsizePmaxandthemaximumiterationnumberNmax.(ii) Initializeeachparticle.Theinitialpositionandvelocityofeachoftheparticlesinthepopu‐
lationaregenerated.(iii)DecodeeveryparticletotheprocessplanaccordingtoEq.29toEq.33,andthencalculate
CP.Getthelocaloptimalposition andtheglobaloptimalposition .
4.3 Iteration and control
Foreveryselectedparticle,thepositionandvelocitycanbeupdatedaccordingtoEq.21andEq.22.Toensuretheprocessplanvalidity,decodingeverynewlygeneratedparticletotheprocessplanisnecessary.Ifthenewprocessplansviolatetheprecedenceconstraints,thecorrespondingparticleswillbenormalizedbyusingtheconstraintmatrixP.Foreachvalidparticle, theCPofthe correspondingprocessplanwill be calculated. If a lowerCP is achieved, the local optimalposition andtheglobaloptimalposition willbeupdated.
WhenthetraditionalPSOisappliedinthePPproblem,aquicklocalconvergenceatanearlystageofthePSOusuallyhastobefaced.Thequicklocalconvergencewillmakefurtherexplora‐tiondifficult and cangenerate anundesirable solution.To solve thisproblem, somemodifica‐tionsaresuggestedtoenhancetheperformanceofthetraditionalPSOalgorithm[9,10,13].FouroperatorsareincorporatedintothetraditionalPSOapproach.Becausethepositionofaparticleisexpressedasa2×nmatrix,inwhichthefirstrowistheODpartandthesecondrowistheOSpart,theoperatorsforthedifferentrowsvaryindependently.FortheODpart,twotypesofmutationoperatoraredesignedtogenerateanewsolution.
Mutationoperator1
Oneparticleintheswarmischosenforamutationoperationwithapredefinedprobability(pms).First,fortheODpartofthisparticle,onepositionpointL israndomlyselected.Second,decodetheparticletoaprocessplan,andobtainthemachine,cuttingtool,andTAD(M,T,TAD)oftheLthoperationinthisprocessplan.Third,fromthemachineset,cuttingtoolset,andTADsetoftheLth operation, an alternative selection ofmachine, cutting tool, andTAD is determined toreplacethecurrentmachine,cuttingtool,andTAD.
Mutationoperator2
Oneparticle ischosenforamutationoperationwithapredefinedprobability(pss).FortheODpartof thisparticle, twoadjacentpositionpointsL1andL2arerandomlyselected.Decode theparticletoaprocessplan,andobtainthemachines,cuttingtools,andTADs(Ms,Ts,TADs)oftheL1‐thandL2‐thoperationsinthisprocessplan.If ,let or ,orotherwisethepositionpointsL1,L2willbereselected.WithrespecttotheOSpart,twooperatorsareemployed,crossoverandshift.
Crossoveroperator
Twoparticles,AandB,areselectedtoexecuteacrossoveroperationwithapredefinedprobabil‐ity(pcq).FortheOSpartofthosetwoparticles,onepositionpointLisrandomlydetermined.TheOSpartisdividedintotwoparts.Subsequently,thevalueofthefrontpartofOSAistakenoutandinsertedbeforethecuttingpointofOSB,andthevalueoftheleftpartofOSBistakenoutandin‐sertedbeforethecuttingpointofOSA.
Shiftoperator
Oneparticleischosenforashiftoperationwithapredefinedprobability(psq).FortheOSpartofthisparticle, twopositionpointsL1,L2 are randomly selected, and their values , ex‐changed.
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4.4 Termination
IfthemaxnumberofiterationsNmaxisreached,theiterationwillbeterminated.Decodetheob‐tainedparticleposition toachievethefinalprocessplan.TheflowchartisdescribedinFig.3.
Fig.3FlowchartofthemodifiedPSOapproach
5. Experiments and results
Twocharacteristicpartsareusedforthesimulationexperiments.ThefirstpartisshowninFig.4[1](Part2),andthesecondpart isshowninFig.5[2](Part3).Thedetailed informationonPart2andPart3isintroducedintheresearchofLietal.[4].AllofthesimulationexperimentswillbeperformedonaPCwith2.8GHzandtheWindows7operatingsystem.
Iterate
YES
YES
Initialize parameters Pmax, Nmax, w, c1, c2, pms, pss, pcq, psq, Rmax
Select particles to do two types of mutation operati‐
ons for OS part with probabilities of pms and pss
If TPCi<TPC of Pl?
Update Pl with the position of Pi
N ++
N< Nmax
NO
YES
Output the optimal solution and TPC
Update position and velocity of partic‐
le i.
Update the Pg with the Pl
Each particle i
Starting
Initialize every particle i, decode particle i
and calculate CPi. Initialize Pl, Pg,
Select particles to do cross and shift operations for
OQ part with probabilities of pcq and psq
If TPC of Pl<TPC of Pg?
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Advances in Production Engineering & Management 11(2) 2016 87
Fig.4Anexamplepart–Part2
Fig.5Anexamplepart–Part3
5.1 Simulation experiments
WhileapplyingPSOtosolve theprocessplanningproblem, thekeyparametershavetobede‐terminedtofacilitatetheperformanceofthemodifiedPSO.Accordingly,manypreliminarysimu‐lationexperimentshavetobecarriedouttodeterminethoseparameters.TheprocessplanningproblemforPart2 isusedto illustratehowthekeyparametersaredetermined. It isassumedthat to inEq.13aresetas1.
Thekeyparametersmaybeanalysedfromtwoaspects,namely,theswarmcharacteristicpa‐rametersofPSO(Pmax,Nmax,w,c1,c2)andtheproblemdata(pms,pss,pcq,psq).TheswarmsizePmaxanditerationnumberNmaxwillbeincreasedwiththeincreasedcomplexityofthepart.ForPart2,aftermanytrials,PmaxandNmaxarefixedat2000and300,respectively.
Theconstantsc1andc2areusedtobalancethevelocitytendencytothelocaloptimalpositionandtheglobaloptimalposition .Ifc1andc2aretoolarge,thesearchspaceoftheparticles
willbeexpanded,whichmayevenleadtonoconvergenceofthePSO.Ifc1andc2aretoosmall,slowconvergencemaycausethecomputationtimetobeverylong.InthecaseofproblemswithPmax=2000,Nmax=300,w=0.75,pms=0.6,pss=0.6,pcq=0.2,psq=0.2,50trialswereseparatelyconductedbyvaryingthevaluesofc1=c2∈1,1.5,2.Theaverageresultsaresummarized inFig.6.
Z F6
F2 F10F11
F12
F13
F14
F1 F3
F4
F5
X Y
F7
F9
F8
a
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Fig.6CPoftheproposedPSOwithdifferentconstantsc1andc2
FromFig.6,whenc1andc2arebothsettobe1,thePSOalgorithmshowsfastconvergence
andgoodcomputationalefficiency.Theinertiaweightwissettocoordinatethelocalexplorationandtheglobalexploration.Ifw
istoolarge,theremaybeaquicklocalconvergenceatanearlystageofthePSO.Ifwistoosmall,thecomputationtimeforeachiterationwillbelong,andtheoptimizationratewillbecomeveryslow.InthecaseofproblemswithPmax=2000,Nmax=300,c1=c2=1,pms=0.6,pss=0.6,pcq=0.2,psq=0.2,50trialswereseparatelyconductedbyvaryingthevaluesofw∈0.75,1,1.25.TheaverageresultsaresummarizedinFig.7.
Fig.7CPoftheproposedPSOwithdifferentinertiaweightsw
FromFig.7,theoptimalefficiencyandstabilityareachievedundertheconditionofw=1.The
problemdata (pms,pss,pcq,psq) aredetermined tohelp theapproachescape from local conver‐gences.Fiftytrialsareconductedin8groupcombinationsofthefourparameterspms,pss,pcq,psqtoillustratetheselectionoftheseparameters.TheaverageCPsforthe50trialsarelistedinTa‐ble6.Itisshownthatthecombinationofpms=0.6,pss=0.6,pcq=0.2andpsq=0.2canyieldthebestperformance. In conclusion, theperformanceof themodifiedPSO forPart2 is good, andPmax=2000,Nmax=300,w=1,c1=c2=1,pms=0.6,pss=0.6,pcq=0.2,psq=0.2.ThebestsimulationresultandaveragesimulationresultareshowninTable7andTable8,respectively.
Table6DeterminationoffourprobabilitiesofthemodifiedPSO (Mutation1:pms,Mutation2:pss,Crossover:pcq,Shift:psq)
(0.6,0.4,0.2,0.4)
(0.4,0.6,0.4,0.2)
(0.6,0.6,0.4,0.4)
(0.6, 0.6,0.2,0.2)
(0.4, 0.4,0.2,0.2)
(0.4, 0.4,0.4,0.4)
(0.4,0.6,0.2,0.4)
(0.6, 0.4,0.4,0.2)
Mean 1198.5 1198.8 1308.4 1131.8 1136.6 1541.3 1206.1 1186.7Maximum 1263 1318 1713 1163 1188 1998 1318 1263Minimum 1143 1143 1148 1128 1128 1158 1143 1143
1100
1150
1200
1250
1300
1350
1 30 59 88 117 146 175 204 233 262 291
CP
Iterations
c1=c2=1 c1=c2=1.5 c1=c2=2
1100
1150
1200
1250
1300
1350
1 30 59 88 117 146 175 204 233 262 291
CP
Iterations
w=0.75 w=1 w=1.25
A simulation approach to the process planning problem using a modified particle swarm optimization
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Table7BestsimulationresultforPart2Operation Machine Tool TAD Result
6 2 2 −Z CM=490CT=98CTC=60CS=480CP=1128
1 2 1 −Z7 2 1 −Z9 2 1 −Z12 2 1 −Z5 2 5 −Z3 2 5 +Y4 2 5 +Y8 2 5 +X10 2 5 −Y11 2 5 −Y13 2 5 −Y14 2 1 −Y2 2 8 −Y
Table8Averagesimulationresultof50trialsforPart2
Type Mean Maximum Minimum StandarddeviationCM 490 490 490 0CT 98.2 103 98 0.98CMC 0 0 0 0CTC 60.9 75 60 3.56CS 480 480 480 0CP 1128.9 1143.0 1128 3.56
The above method of determining the key parameters is based on Part 2. Themethod of
choosingparametersforPart3issameasforPart2.InthecaseofproblemswithPmax=2000,Nmax=500,w=1.25,c1=c2=1,pms=0.6,pss=0.6,pcq=0.3,psq=0.3,50trialswereseparatelycon‐ducted.ThebestprocessplansarelistedinTable9,andtheaverageresultsarelistedinTable10.
Table9BestsimulationresultforPart3Operation Machine Tool TAD Result
1 2 6 +Z CM=770CT=235CMC=320CTC=200CS=1000CP=2525
3 2 6 +X5 2 6 +X6 2 6 −Z2 2 6 −Z18 2 6 −Z11 2 7 −Z12 2 2 −Z13 2 9 −Z17 2 7 −X7 2 7 −a8 2 2 −a9 2 9 −a19 2 9 +Z14 4 10 −Z20 4 10 +Z10 4 10 −a4 1 2 −Z15 1 1 −Z16 1 5 −Z
Table10Averagesimulationresultof50trialsforPart3
Type Mean Maximum Minimum StandarddeviationCM 770 770 770 0CT 240 267 235 10.13CMC 320 320 320 0CTC 197.2 180 200 6.94CS 1000 1000 1000 0CP 2527.2 2535.0 2525.0 3.28
Wang, Kang, Zhao, Chu
90 Advances in Production Engineering & Management 11(2) 2016
5.2 Extensive comparative experiments
The following three conditions areused to verify themodifiedPSOapproach for the exampleparts[2,4,6]:
(a) Allmachinesandcuttingtoolsareavailable,and to inEq.13aresetas1.(b) Allmachinesandcuttingtoolsareavailable,and = =0, = = =1.(c) M2andT7aredown, = =0, = = =1.
ForPart2,50trialswereperformedunderconditions(a)and(b).Apenaltycostisincludedin theCP to facilitate the comparisonwithotherapproaches,which is200 forPart2 [4,7].Acomparisonwith the resultsobtainedusing theGAandSAapproaches [2],TS [4], andHBMO[14],aswellasthetwo‐stageACO[7],isprovidedinTable11.
Undercondition(a),thisapproachissameasSA,TS,HBMO,andtwo‐stageACOandisbetterthanGAusingtheminimummachinecosts.Usingthemaximummachinecosts,theCP(1328.0)isthesameasthatofHBMO,anditisbetterthantheotherfourapproaches.Thisapproachhasthebestperformanceinthemeanmachinecost(1328).ItisobviousthatthesameCP(1328.0)isachieved50timesandissuperiortoalloftheotherapproaches.
Undercondition(b),theperformanceofthisapproachisthesameasthoseofHBMOandthetwo‐stageACO.Usingtheminimummachinecosts,thisapproachisbetterthanGA,anditisthesameastheotherfourapproaches.Usingthemaximumandmeanmachinecosts,thisapproachisbetterthanGA,SA,andTS.
ForPart3,50trialswerecarriedoutunderconditions(a),(b),and(c).AcomparisonoftheresultswiththoseofTS[4],PSO[9],andHBMO[14],aswellasthetwo‐stageACO[7]isprovid‐edinTable12.
Table11Resultscomparedtootherapproaches
Condition Proposedapproach GA SA TS HBMO Two‐stageACO(a) Mean 1328 1611.0 1373.5 1342.6 1328 1329
Maximum 1328 1778 1518 1378 1328 1348Minimum 1328 1478 1328 1328 1328 1328
(b) Mean 1170 1482 1217 1194 1170 1170
Maximum 1170 1650 1345 1290 1170 1170Minimum 1170 1410 1170 1170 1170 1170
Table12Resultscomparedtootherapproaches
ConditionProposedapproach
TS PSO HBMO Two‐stageACO
(a) Mean 2527.2 2609.6 2680.5 2543.5 2552.4
Maximum 2535 2690 ‐ 2557 2557Minimum 2525 2527 2535 2525 2525
(b) Mean 2093.0 2208.0 ‐ 2098.0 2120.5
Maximum 2120 2390 ‐ 2120 2380Minimum 2090 2120 ‐ 2090 2090
(c) Mean 2593.2 2630.0 ‐ 2592.4 2600.8
Maximum 2600 2740 ‐ 2600 2740Minimum 2590 2580 ‐ 2590 2590
A simulation approach to the process planning problem using a modified particle swarm optimization
Under condition (a), the minimum machine cost is the same as that of HBMO and the two-stage ACO, and it is better than TS and PSO. Among the 50 trial results, CP (2525) occurs 23 times, CP (2527) occurs 20 times, and CP (2535) occurs 7 times. Accordingly, this approach is superior to all of the other approaches on the mean machine cost. Under condition (b), the min-imum machine cost (2090) is the same as that of HBMO and the two-stage ACO. CP (2090) oc-curs 45 times, and CP (2120) occurs 5 times in 50 trials; the mean machine cost is the best among all of the approaches. Under condition (c), CP (2590) occurs 34 times, and CP (2600) oc-curs 16 times in 50 trials. Generally, the performance of this approach is similar to that of HBMO under condition (c) and is superior to those of the two-stage ACO and TS.
6. Conclusions A traditional PSO approach is modified to solve the PP problem. Efficient encoding and decoding, population initialization, and iteration and control methods have been designed. Meanwhile, to avoid local convergence, several new operators for the different parts of the particles have been designed and incorporated into the traditional PSO to improve the particles’ movements. Simu-lation experiments show that the modified PSO algorithm can perform the process plan optimi-zation competently and consistently and generate better solutions compared with other ap-proaches.
In the simulation experiment, a small change in the parameters induces computational result saltation. Hence, a deep discussion of selecting the modified PSO approach parameters will be conducted. Additionally, integrated process planning and scheduling that considers the better performance of manufacturing systems may be a direction for further study [21, 22].
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