a to the process planning problem a particle swarm...

77

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:[email protected](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

Wang, Kang, Zhao, Chu

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‐


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



Table2ConstraintsforPart1[7]Operations Precedenceconstraintdescription Constrainttypes

OP1OP1ispriortoOP2andOP3. Hard

OP1ispriortoOP4andOP5. Soft

OP2 OP2ispriortoOP3. Hard


OP4,OP5 OP4andOP5arepriortoOP6. Hard

OP6 OP6ispriortoOP2andOP3. 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)



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)



(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‐



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



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.



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.



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?





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



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



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



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


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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