INTEGRATION OF STATISTICAL SELECTION WITH SEARCH MECHANISM FOR SOLVING MULTI- OBJECTIVE SIMULATION-OPTIMIZATION PROBLEMS

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1 Proceedngs of he 006 Wner Smulaon Conference L F Perrone, F P Weland, J Lu, B G Lawson, D M Ncol, and R M Fujmoo, eds INTEGRATION OF STATISTICAL SELECTION WITH SEARCH MECHANISM FOR SOLVING MULTI- OBJECTIVE SIMULATION-OPTIMIZATION PROBLEMS Loo Hay Lee Ek Peng Chew Suyan Teng Dearmen of Indusral and Sysems Engneerng Naonal Unversy of Sngaore 10 Ken Rdge Crescen, 11960, SINGAPORE ABSTRACT In hs aer, we consder a mul-objecve smulaon omzaon roblem wh hree feaures: huge soluon sace, hgh uncerany n erformance measures, and mul-objecve roblem whch requres a se of nondomnaed soluons Our man urose s o sudy how o negrae sascal selecon wh search mechansm o address he above dffcules, and o resen a general soluon framework for solvng such roblems Here due o he mul-objecve naure, sascal selecon s done by he mul-objecve comung budge allocaon (MOCBA) rocedure For llusraon, MOCBA s negraed wh wo mea-heurscs: mul-objecve evoluonary algorhm (MOEA) and nesed arons (NP) o denfy he nondomnaed soluons for wo nvenory managemen case sudy roblems Resuls show ha, he negraed soluon framework has mroved boh search effcency and smulaon effcency Moreover, s caable of denfyng a se of non-domnaed soluons wh hgh confdence 1 INTRODUCTION Dscree even smulaon has been a commonly used ool for evaluang he erformance of sysems whch are oo comlex o be modeled analycally However, smulaon can only evaluae erformance measures for a gven se of values of he sysem decson varables, e, lacks he ably of searchng omal values for he decson varables whch would maxmze or mnmze a gven resonse or a vecor of resonses of he sysem Ths exlans he ncreasng oulary of research n negrang boh smulaon and omzaon, known as smulaon omzaon: he rocess of fndng he bes values of decson varables for a sysem where he erformance s evaluaed based on he ouu of a smulaon model of hs sysem A smulaon omzaon roblem whch mnmzes he execed value of he objecve funcon wh resec o s consran se can be exressed as: mn J ( θ ) θ Θ where J( θ) = E[ L( θ, ε)] s he erformance measure of he roblem, L( θε, ) s he samle erformance, ε reresens he sochasc effecs n he sysem, θ s a -vecor of dscree conrollable facors and Θ s he dscree consran se on θ If J ( θ ) s a scalar funcon, he roblem s sngle objecve omzaon; whereas f s a vecor, he roblem becomes mul-objecve omzaon The above smulaon omzaon roblem has been exensvely suded n he leraure When he search sace Θ s fne and relavely small, he roblem of selecng he bes sysem s usually known as Rankng and Selecon (R&S) roblem (Swsher, Jacobson, and Yücesan 003) Soluon aroaches roosed for he R&S roblem nclude: ndfference-zone rankng and selecon (Nelson e al 001), omal comung budge allocaon (Chen e al 000), and decson heorec mehods (Chck and Inoue 001) When he search sace Θ s eher nfne or fne bu wh a huge number of alernaves, search owers of omzaon rocedures need o be exlored so ha we can search effcenly hrough he gven se o fnd mrovng soluons The mos sraghforward search mehod s random search, such as he Sochasc Ruler (SR) algorhm (Yan and Muka 199), he sochasc comarson (SC) algorhm (Gong, Ho and Zha 1999), and modfed SC or SR algorhms amng a mrovng he convergence roery (Alrefae and Andradór 1997; Andradór 1999) To deal wh he randomness of he objecve funcon value, he above soluon mehods exlo he robusness of order sascs whn he random search framework To search more sysemacally raher han randomly, mos research (1) /06/$ IEEE 94

2 negraes mea-heurscs wh ceran sascal analyss echnques: Genec algorhm (GA) wh boh a mulle comarson rocedure and a R&S rocedure (Boesel, Nelson, and Ish 003); GA wh ndfference-zone R&S rocedure (Hedlund and Mollaghasem 001); modfed smulaed annealng (SA) algorhm wh confdence nerval (Alkhams and Ahmed 004); SA wh R&S rocedure (Ahmed and Alkhams 00); and nesed arons search wh wo-sage R&S rocedure (Ólafsson 1999) When he smulaon omzaon roblem has more han one erformance measure, he roblem s ofen ransformed no a sngle objecve roblem so ha exsng soluon echnques can be aled In Baesler and Seúlveda (000), goal rogrammng s ncororaed no genec algorhms o handle he mul-objecves In aers such as Buler, Morrce, and Mullarkey (001) and Swsher and Jacobson (00), he mulle arbue uly (MAU) heory s aled o form a sngle measure of effecveness of he objecves Whn such soluon frameworks, he omal soluon s a sngle comromse soluon In case a se of non-domnaed soluons are more desrable, he mulobjecve roblem needs o be solved drecly When search sace Θ s fne and relavely small, Lee e al (004, 005) develoed a mul-objecve comung budge allocaon (MOCBA) rocedure whch ncororaes he conce of Pareo omaly no he R&S scheme o fnd all non-domnaed soluons When he search sace Θ s nfne or fne bu very large, MOCBA needs o be negraed wh search rocedures for effcen exloraon of more romsng soluons, such as negraon wh mul-objecve evoluonary algorhm (MOEA) n Lee e al (006), and negraon wh nesed arons (NP) algorhm n Chew e al (006) In hs sudy, we consder a smulaon omzaon roblem formulaed n (1) wh he followng feaures: huge sze of soluon sace ( Θ ), large unceranes ( ε ) n erformance measures, and mul-objecve roblem whch requres a non-domnaed Pareo se of soluons These feaures of he roblem make boh challengng and dffcul o solve On one hand, o mrove smulaon effcency and o guaranee ha fnal soluons are nondomnaed wh hgh confdence, ceran sascal selecon rocedure s needed; on he oher hand, o search more effcenly, search mechansm of omzaon echnques s also requred Lee e al (006) and Chew e al (006) resened how o negrae MOCBA (a sascal selecon rocedure) wh MOEA and NP (mea-heurscs) In hs aer, we summarze he man deas from he above wo aers wh focus on how o ncororae he sascal selecon rocedure (such as MOCBA) no some meaheursc rocedures so ha eher of hem wll benef from he oher and ogeher hey mrove boh search effcency and smulaon effcency hrough he negraon The aer s organzed as follows In Secon, we frs dscuss why here s a need o negrae MOCBA wh a search rocedure for smulaon omzaon roblems Then based on a bref descron of he MOCBA rocedure, we resen a general soluon framework whch negraes MOCBA wh a search rocedure o effcenly allocae smulaon relcaons and denfy he non-domnaed soluons for he roblem In Secon 3, he soluon framework s llusraed by negrang MOCBA wh wo mea-heurscs (MOEA and NP) for solvng wo nvenory managemen roblems Some comuaonal resuls are also reored n hs secon Fnally some conclusons and fuure research drecons are summarzed n Secon 4 THE INTEGRATION OF STATISTICAL SELECTION WITH SEARCH IN SIMULATION OPTIMIZATION In hs secon, we dscuss why here s a need o negrae sascal selecon wh search rocedures n smulaon omzaon, and how o ncororae he selecon no he search rocedure 1 The need o negrae search and selecon n smulaon omzaon A key feaure of smulaon omzaon roblem ha makes dffcul s he need o address he search versus selecon rade-off: wh a lmed comung budge, how o allocae he budge beween searchng over he feasble sace for (oenally) beer soluons, and deermnng whch of he soluons ha have been examned are acually good Before addressng hs ssue, we frs sudy he followng wo exreme roblems n smulaon omzaon When he soluon sace Θ n (1) s fne and small, s ossble o evaluae exhausvely all members from he gven (fxed and fne) se of alernaves and comare he erformance In hs case, he focus s enrely on selecon: he comarson asec unque o he sochasc seng Anoher exreme of he roblem s o gnore he sochasc naure of he roblem To evaluae he erformance measures, a fxed number of smulaon relcaons are allocaed o each desgn alernave denfed as romsng soluon by he search rocedure In hs case, he focus s enrely on search: soluon exloraon asec moran o deermnsc omzaon roblem Obvously, for he general smulaon omzaon roblem such as he one consdered n hs sudy, s neher a ure search nor a ure selecon roblem When he soluon sace s huge, exhausve search becomes eher mraccal or mossble Hence one crcal ssue s o fnd ou whch decson scenaros are he ones desrable o be nvesgaed, and how o denfy hose good scenaros auomacally by a search rocess desgned o fnd he bes se of decsons Ths makes he search mechansm of omzaon rocedures a necessy when exlorng he soluon sace for more romsng soluons On he oher hand, when selecon 95

3 asec s negleced, we may encouner he followng dffcules Frsly here s no sascal guaranee on he qualy of he fnal se of soluons Secondly, he fxed number of smulaon relcaons are eher oo few o handle he sochasc nose or oo many o be of comuaonal effcency In he former case, naccurae esmaons of he erformance measures may lead he search o unroducve regons Ths s due o he fac ha search drecon s guded by he evaluaon of he objecves n any search mechansm For examle, n MOEA, arens are seleced o generae new offsrng based on fness values In case fness values are esmaed naccuraely, less f soluons may be seleced and survve no he nex generaon, whereas ruly f soluons may be negleced and lose he chance for furher consderaon Smlarly n NP, romsng ndex for each regon s calculaed based on erformance measures of he samles cked n ha regon, and fuure romsng regon s seleced accordng o he romsng ndces calculaed Wh naccurae esmaon of he erformance measures of he samles, he romsng regon ha he fuure search wll focus on may urn ou o be less romsng In he laer case, when he search needs o exlore a large number of desgn alernaves each wh a large number of smulaon relcaons, whch s usually he case for mea-heurscs, he oal smulaon cos can easly become very hgh To avod unnecessary wase n smulaon budge, a more effcen way of allocaon han unform allocaon s desrable Inuvely, for desgns wh oor erformances whch are obvously domnaed by oher desgns, s a wase f furher smulaon relcaons are o be allocaed o hem, hough he nose nvolved n he erformance measures maybe sll hgh Thus a sascal selecon rocedure s requred o sngle ou hose desgns lkely o be comeors for he bes and o omally deermne he number of smulaon relcaons needed for each of he desgns whle denfyng hose non-domnaed desgns The MOCBA rocedure resened n Lee e al (005) s secally develoed for he above uroses The MOCBA rocedure and s negraon wh search mechansms Gven a se of n desgn alernaves wh H erformance measures whch are evaluaed hrough smulaon, a mulobjecve R&S roblem s o deermne an omal allocaon of he smulaon relcaons o he desgns, so ha he non-domnaed se of desgns can be found a he leas exense n erms of smulaon relcaons In hs secon, we resen a bref descron of a soluon framework for hs roblem: he Mul-objecve Omal Comung Budge Allocaon (MOCBA) algorhm For more deals abou how MOCBA works, lease refer o Lee e al (005) 1 A Performance Index o Measure he Nondomnaed Desgns Suose we have a se of desgns ( = 1,,, n), each of whch s evaluaed n erms of H erformance measures μ k ( k = 1,,, H ) hrough smulaon Whn he Bayesan framework, μ k s a random varable whose oseror dsrbuon can be derved based on s ror dsrbuon and he smulaon ouu (Lee e al 004) We use he followng erformance ndex o measure how non-domnaed desgn s: n ψ = [1 P( μ μ )] () j= 1, j where P( μ j μ ) reresens he robably ha desgn j domnaes desgn Under he condon ha he erformance measures are ndeenden from one anoher and hey follow connuous dsrbuons, we have H k = 1 j P( μ μ ) = P( μ μ ) (3) j Performance ndex ψ measures he robably ha desgn s non-domnaed by all he oher desgns A he end of smulaon, all desgns n he Pareo se should have ψ close o 1, and hose desgns ousde of he Pareo se should have ψ close o 0, because hey are domnaed Two Tyes of Errors of he Seleced Pareo Se Durng he allocaon rocess, he Pareo se s consruced based on observed erformance Here we call he seleced Pareo se ( S ) The qualy of he seleced Pareo se deends on wheher desgns n S are all nondomnaed and desgns ousde S are all domnaed We evaluae by wo yes of errors: Tye I error ( e 1 ) and Tye II error ( e ) as follows Tye I error ( e 1 ) s defned as he robably ha a leas one desgn n he seleced non-pareo se ( S ) s nondomnaed; whle Tye II error ( e ) s defned as he robably ha a leas one desgn n he seleced Pareo se s domnaed by oher desgns When boh yes of errors aroach 0, he rue Pareo se s found The wo yes of errors can be bounded by he aroxmaed errors ae 1 and ae resecvely as gven below jk 1 1 S e ae = ψ (4) k 96

4 e ae = (1 ψ ) (5) S When he nose n he smulaon ouu s hgh, ae 1 and ae can be large However, once he seleced Pareo se aroaches he rue Pareo se, boh ae 1 and ae aroach 0, as ψ aroaches 1 for S and ψ aroaches 0 for S 3 Consrucon of he Seleced Pareo Se Rankng all he desgns n descendng order of erformance ndex ψ, hen he seleced Pareo se can be consruced accordng o he crera below C1: Assgn a maxmum number of k desgns wh he hghes ψ no S, so ha k ae = (1 ψ ) ε, where = 1 ε s a redefned error lm C: Consruc S so ha boh ae 1 and ae are mnmzed by smly ung hose desgns wh ψ 05 no S and he res no S 4 The Asymoc Allocaon rules and a Sequenal Soluon Procedure To ge he rue Pareo se wh hgh robably, we need o mnmze boh Tye I and Tye II errors In he MOCBA algorhm of Lee e al (005), hs s done by eravely allocang he smulaon relcaons unl boh ae 1 and ae are whn error lm ε * accordng o some asymoc allocaon rules as follows: 1) For a desgn l S, αl Tl = Nmax α + α l S l d S αd ) For a desgn d S, Td = Nmax α + α ( + l ) wh αl = ( + m ) l S σ l σ l ρ δ l lk j j l l k j lj l l k jl m m m mk j j m mk j mj m mk jm σ σ ρ δ fxed desgn n S ; l Ωd d d S σ αd = σ d, gven m s any dkd kd where N max s he maxmum number of relcaons avalable; S and S reresen he seleced Pareo and non- Pareo se resecvely; T s he number of relcaons o α be allocaed o desgn ; σk s he samle varance of kh objecve of desgn ; δ jk s he dfference beween samle means of kh objecve of desgn and desgn j; j [desgn d d S, d, P( μ μ ) = max P( μ μ )] d jk k j S j k = 1 H s he desgn ha domnaes desgn wh he hghes robably; k j s he objecve of j ha domnaes he corresondng objecve of desgn wh he lowes robably, r {1,, H}, k objecve r of desgn j j ( ) mn ( ) P μj r μr = P μj k μk k {1,, H} wh ( )~ (, σ jk σk P μjk μk N δjk + ), and ' ' ' N j N T s he number of relcaons allocaed o desgn a he mmedae revous eraon; Ω = {desgn S, j = d} ; r Ω j r jk j r rk σ ' Tj ρ = or ρ = ' σ T j d The MOCBA algorhm s now oulned below MOCBA algorhm Se 0: Perform T 0 relcaons for each desgn Calculae he samle mean and varance for each objecve of he desgns Se eraon ndex v: = 0 v v v T1 = T = = Tn = T0 Se 1: Consruc he seleced Pareo se S accordng o creron C Calculae ae 1 and ae accordng o equaons (4) and (5) If ( ae1 < ae ), consruc S accordng o he creron C1 wh ε = ae1 Se : If (( ae1 < ε * ) and ( ae < ε * )), go o Se 5 Se 3: Increase he smulaon relcaons by a ceran amoun Δ, and calculae he new allocaon v+ 1 v+ 1 v 1 T1, T,, T + n accordng o he asymoc allocaon rules v 1 v Se 4: Perform addonal mn( δ, max(0, T + T ) ) relcaons for desgn ( =1,,n) Udae he samle mean and varance of each objecve of desgn based on cumulave smulaon ouu Se v = v+ 1 and go o Se 1 Se 5: Ouu desgns n he seleced Pareo se (S ) 97

5 5 The negraon of MOCBA wh search mechansm of omzaon rocedures In a general deermnsc search rocedure, wheher s develoed for sngle objecve or mul-objecve, wheher s oulaon based or sngle soluon based, he search mechansm works more or less n a smlarly way exce ha he mechansm of generang new soluons may dffer sgnfcanly (Fgure 1) Here erformance evaluaon and fness evaluaon are lsed searaely because, hough mos search rocedures use erformance measure of a soluon o reresen s fness, n some rocedures, such as EA and NP, fness of a soluon (regon) s defned searaely based on s erformance for deermnng he search drecon of he algorhm Meanwhle, mos rocedures kee an archve of bes soluons ever vsed and reurn as fnal soluon uon ermnaon A flow char showng how general search rocedures work s gven n Fgure 1 Inal se of Soluons Performance Evaluaon Fness Evaluaon Termnaon? No Generaon of New se of Soluons Yes Smulaon MOCBA Archve of Bes Soluons Reurn Soluons n he Archve A General Search Procedure Fgure 1: Inegraon of MOCBA wh Search Procedures s erformance; b) how o defne fness of a soluon consderng boh varably nvolved n he erformance measures and he mul-objecve naure of he roblem; c) how o selec hose bes soluons ever vsed o form he Pareo se The MOCBA rocedure hels o answer he above quesons smulaneously A general framework of negrang MOCBA (a sascal selecon rocedure) wh search rocedures s llusraed n Fgure 1 Secfcally, a each eraon, o rank he desgns n he curren oulaon, erformance ndex defned n () s used o measure he fness (non-domnance) of he desgns A desgn wh larger value of erformance ndex s non-domnaed wh hgher robably The erformance ndex acually ransforms he mul-objecve roblem no a roblem wh sngle objecve, as we can now deermne he goodness of a soluon comleely based on hs sngle erformance ndex To deermne he rgh number of smulaon relcaons needed for each desgn o accuraely esmae s fnesses, he MOCBA algorhm based on allocaon rules resened n 4 s run on he curren se of soluons A hs sage, our man urose of runnng MOCBA s no o deermne recsely he Pareo se wh very hgh confdence, bu o effcenly rank he desgns n erms of her fness values o deermne he rgh drecon of he search rocedure Therefore, o save smulaon budge and also o avod loss of ruly non-domnaed desgns durng he search rocedure, he MOCBA s ermnaed when each of he erformance ndex nsde he Pareo se s above a ceran value and each of he erformance ndex n he non-pareo se s below a ceran value Then desgns n he Pareo se are u no he archve A he fnal eraon, MOCBA s run agan on desgns n he archve wh very low lms on Tye I and Tye II errors, so ha ruly non-domnaed desgns are denfed wh leas ossble smulaon relcaons uon ermnaon of he search rocedure Though wh slgh modfcaons, sngle soluon based search rocedure can work wh MOCBA for solvng mul-objecve smulaon omzaon roblems, he deal search rocedures should be able o move wh a se of soluons from one eraon o he nex, because he se of soluons can rovde a bass for generang (aroxmang) he Pareo se of soluons a each eraon by runnng MOCBA Moreover, MOCBA can omally deermne he number of smulaon relcaons needed for each of he se of soluons and herefore can effecvely avod wase of smulaon budge Mea-heurscs such as mulobjecve evoluonary algorhm (MOEA) and nesed arons (NP) are nce canddaes for he above uroses However, when we aly such a search rocedure o solve mul-objecve smulaon omzaon roblems, we need o consder: a) he rgh number of smulaon relcaons needed for each desgn o accuraely evaluae 98

6 3 INTEGRATION OF MOCBA WITH META- HEURISTIC PROCEDURES -- TWO CASE STUDY PROBLEMS In hs secon, we use wo examle mea-heursc rocedures (MOEA and NP) o llusrae how o negrae a sascal selecon rocedure wh a search rocedure The wo negraed frameworks are hen aled o solve wo nvenory managemen case sudy roblems Ths secon brefly summarzes he soluon rocedures and some comuaonal resuls For more deals regardng roblem descron and resuls resenaon, refer o Lee e al (006) and Chew e al (006) 31 The negraon of MOCBA wh MOEA 311 The negraed MOEA framework MOEA s an adave heursc search algorhm whch smulaes he survval of he fes among ndvduals over consecuve generaons for solvng a mul-objecve roblem Based on an nal oulaon randomly generaed, a each generaon, MOEA evaluaes he chromosomes and ranks hem n erms of her fness; he fer soluons wll be seleced o generae new offsrng by recombnaon and muaon oeraors Ths rocess of evoluon s reeaed unl he algorhm converges o a oulaon whch covers he non-domnaed soluons MOEA has been successfully aled n solvng deermnsc mul-objecve roblems (Fonseca and Flemng 1995, Hanne and Nckel 005) To make MOEA work well for smulaon omzaon roblems where varably s a man concern, MOCBA s needed manly n he followng wo asecs: fness evaluaon and formaon of ele oulaon A dealed descron of he negraed soluon framework s gven below Oulne of he Inegraed MOEA Se 0: Inalzaon: Randomly generae an nal feasble oulaon POP of sze N ; se ele oulaon E = ; se generaon ndex = 0 Se 1: Run MOCBA (Secon 4) o deermne he number of relcaons for each desgn n oulaon POP, and selec desgns no he Pareo se Se : Formaon of ele oulaon: Form he ele oulaon E wh desgns n he Pareo se Se 3: Check he ermnaon condon If s no sasfed, go o Se 5 Se 4: Termnaon: Run MOCBA on he Ele Poulaon E wh boh yes of errors whn error lm ε * Ouu he Pareo se as he fnal se of nondomnaed desgns Se 5: Evaluaon and fness assgnmen: Use erformance ndex ψ a he ermnaon of MOCBA as he fness value of chromosome Se 6: New oulaon: Se = + 1 ; le POP = POP 1 Creae a new oulaon by reeang he followng ses unl M ars of arens are seleced 1) Selecon: selec one ar of chromosomes by ournamen selecon from oulaon POP 1 ) Crossover: Perform arhmec crossover and one-on crossover wh robably 05 each, resulng n a ar of offsrng Check he feasbly of he offsrng 3) Add he new offsrng no oulaon POP Se 7: Muaon: Run MOCBA on oulaon POP o deermne fness value for he new offsrng If N > Nmax, delee ( N Nmax ) desgns wh leas fness value For each chromosome n POP, erform muaon wh robably Pm = 01ψ Check he feasbly of he muaed chromosome Go o Se 1 31 The arcraf sare ars nvenory allocaon roblem case sudy 1 When a rearable em on an arcraf becomes defecve, s removed and relaced by anoher em from he sare sock The defecve ar hen goes no some rear cycle a he Cenral Rear Deo (CRD) If he aror does no have he sare ar n sock, he arcraf wll be grounded unl an ncomng flgh brngs a relacemen ar from he CRD or from a neghborng aror To reduce dearure delays due o unancaed falures, arlnes need o kee nvenory of sare ars a he assocaed arors, as well as a he CRD Suose we have an aror nework whch consss of S arors and 1 CRD We assume ha N rearable sares of he same ye are avalable n he nework A every manenance check, a falure occurs a he robably of α Uon a ar falure, re-suly of he sare ar comes from eher he nvenory a he aror, he CRD or from he neghborng arors The rear me for a defecve ar s assumed o follow unform dsrbuon beween [ R1, R ] The roblem s o deermne he allocaon of he sare ars among he arors, and he relacemen olcy (where o ge a relacemen ar: aror s own nvenory, neghborng aror or CRD) uon he occurrence of a ar falure, so ha he average cos nvolved n he sysem s mnmzed and boh he average fll-rae and he mnmum fll-rae of he enre nework are maxmzed Here he cos s defned n erms of nvenory cos and he ransoraon cos of shng he defecve ar The fll-rae of an aror s defned as he ercenage of falures servced by s own sare ar nvenory Wh hree objecves evaluaed hrough smulaon, he roblem has he followng dffcul feaures: huge search sace, mul-objecve, and hgh varably To ad- 99

7 dress hese dffcules, we aly he negraed MOEA o solve hs roblem In he negraed MOEA, he codng scheme defnes each chromosome as S + 1 genes L ( = 1,,, S + 1), each of whch reresens he nvenory level of sares a aror (or CRD) A each MOEA generaon, crossover and muaon are erformed o generae new offsrng, and a se of bes chromosomes are ke n he ele oulaon When ermnaon condon (no new soluons are added no he ele oulaon or a maxmum number of generaons has been execued) s me, MOCBA s run on he fnal ele oulaon, and he resuled Pareo se s ouu as he fnal non-domnaed se of soluons Fgures and 3 llusrae he mrovemen of Pareo fron as he number of MOEA generaons ncreases from 1 o 00 Normalsed Cos Average Servce Level vs Cos s Run 100h run 00h run Normalzed Servce Level Fgure Imrovemen of Pareo Se n erms of Average Servce Level and Cos Normalsed Mn Servce Level Cos vs Mn Servce Level Normalsed Cos 1s Run 100h Run 00h Run Fgure 3 Imrovemen of Pareo Se n erms of Mnmum Servce Level and Cos Fgures and 3 show ha, all hree erformance measures of he desgns n he Pareo se are mroved sgnfcanly by he negraed MOEA from he 1 s o he 100 h generaon The negraed MOEA s caable of denfyng hose non-domnaed desgns wh a lower cos, and a hgher average as well as mnmum servce level Meanwhle, from he 100 h o 00 h generaon of he negraed MOEA, he mrovemen of he Pareo se can be observed bu no as aaren as n he frs 100 genec generaons The new desgns generaed and added no he Pareo se are mosly n he same regon as hose generaed durng he frs 100 generaons, less han 0 new desgns have been found are sueror o he desgns n he Pareo se found a he 100 h generaon Ths mles ha, he negraed MOEA has sared o converge o a local omum 3 The negraon of MOCBA wh NP 31 The negraed NP framework The NP mehod, roosed by Sh and Ólafsson (000), s a randomzed mehod based on he conce of adave samlng for solvng sngle objecve global omzaon roblems I sysemacally arons he feasble regon and concenraes he search n regons ha are he mos romsng I combnes aronng, random samlng, a selecon of a romsng ndex, and backrackng o creae a Markov chan ha converges o a global omum For he sngle objecve deermnsc NP o be alcable for solvng our roblem, he erformance ndex ψ nroduced n MOCBA s used o calculae he romsng ndex o deermne he nex romsng regon Moreover, MOCBA s also used for effecve allocaon of smulaon relcaons as well as formng he global Pareo se The negraed NP framework (descrbed below) s develoed for roblems wh wo-dmensonal search sace: s and Q Oulne of he Mul-objecve Inegraed NP Se 0: Inalzaon: Deermne he feasble regon of [ s, Q ] and denoe as Ω ; le be he curren mos romsng regon Ω b ; se he surroundng regon as Ω s = ; se he global Pareo se S = Se 1: Paron and samlng: Paron Ω b no M subregons; ake ω samles from each sub-regon and he surroundng regon Ω s Se : Promsng ndex calculaon: Run r relcaons of he smulaon model for each samle aken; calculae erformance ndex ψ for each samle; calculae he romsng ndex for each regon Denoe he regon wh he hghes romsng ndex as Ω h Add hose samles wh ψ > 05 no he global Pareo se S Se 3: Udae of global Pareo se S : Run MOCBA on desgns n S Fnd he Pareo se S ' wh boh yes of errors whn error lm ε * Relace S ' wh S Se 4: Check ermnaon condon If no sasfed, go o Se 6 300

8 Se 5: Termnaon: Ouu he global Pareo se S as he fnal se of non-domnaed soluons Se 6: If Ω h Ω b, se Ω b =Ωh, Ω s =Ω Ω h Go o Se 1 Se 7: Backrackng: Backrack o a suer regon of Ω b Regard as he curren mos romsng regon Ω b, and aggregae he surroundng regon as Ω s Go o Se 1 3 The dfferenaed servce nvenory roblem case sudy In a dfferenaed servce nvenory roblem, cusomers are classfed no dfferen grous accordng o her morance o he decson makers The roblem s o deermne how o relensh nvenores and how o allocae hese nvenores o dfferen demand classes accordng o some erformance measures such ha each demand class s offered wh dfferen servce level In hs sudy, we assume ha here are m dfferen demand classes where he demand for each class s sochasc, and he nvenory s relenshed accordng o a connuous ( s, Q ) nvenory model Under he dynamc hreshold olcy develoed n Chew, Lee, and Lu (005) o dfferenae demand classes and o offer dfferen servces, he roblem s o oban a se of non-domnaed reorder on s and order quany Q wh he bes combnaon of cos and servce level Here he cos s defned n erms of average annual cos, whch consss of seu cos, nvenory holdng cos and backorder cos The servce level s defned n erms of average backorder of each cusomer class, whch s calculaed as he average number of backorders er year Ths roblem nvolves wo objecves whch are also evaluaed by smulaon models To search for he nondomnaed (, s Q ) olces whn he feasble regon whch s an area formed by uer bounds of boh s and Q, we emloy he negraed NP o aron he search sace eravely and focus on he more romsng regon n searchng for beer soluons The uer bound of Q s esmaed by he EOQ model The uer bound of reorder on s s deermned by frs mnmzng he average annual cos and hen mnmzng he average backorder Wh smlar ermnaon condon as n he negraed MOEA, desgns n he fnal global Pareo se are llusraed n Fgures 4 and 5 Fgures 4 and 5 ndcae ha, desgns n he fnal run global Pareo se of he negraed NP form a curve whch reresens he effcen Pareo froner All desgns from he nal run are above hs curve and herefore are domnaed Moreover, we can observe ha, n he fnal run, unlke he nal run, desgns are more evenly dsrbued along he Pareo froner Ths ndcaes ha, he negraed NP s caable of fndng a full secrum of non-domnaed desgns for he decson makers, whch s hghly desrable Class 1 cusomer backorder Class 1 cusomer average backorder vs cos Cos Inal 160 runs Fgure 4 Pareo Fron Imrovemen of Class 1 Cusomers Class cusomer backorder Class cusomer average backorder vs Cos Cos Inal 160 runs Fgure 5 Pareo Fron Imrovemen of Class Cusomers n he racce Moreover, he algorhm s caable of searchng dfferen regons of he search sace raher han concenrang on a ceran regon, herefore has a hgher chance of fndng he global omal se of non-domnaed soluons 4 CONCLUSIONS For he smulaon omzaon roblem consdered n hs sudy, neher a sascal selecon rocedure nor a search rocedure s enough o address he dffcul ssues nvolved n he roblem: huge soluon sace, hgh varably and mul-objecve We herefore roose a soluon framework whch negraes he wo rocedures for effcen allocaon of smulaon relcaons as well as search and denfcaon of non-domnaed soluons Due o he fac ha he erformance ndex nroduced n MOCBA ransforms he mul-objecves no a sngle measure of effecveness whch can be used o evaluae he soluons n erms of non-domnance, search rocedures develoed for boh sngle objecve and mul-objecve omzaon roblems are alcable here We llusrae he soluon framework by negrang MOCBA wh wo meaheursc rocedures whch move wh a se of soluons a each eraon: MOEA and NP The wo negraed frameworks are hen aled o solve wo nvenory managemen case sudy roblems Resuls show ha, hroughou he negraed search rocedure, he Pareo se s mroved 301

9 grealy n erms of boh ndvdual soluon qualy and he dsrbuon of he soluons along he Pareo froner The negraed soluon framework s caable of denfyng a se of non-domnaed soluons wh hgh confdence In hs sudy, he convergence and he qualy of he Pareo se are analyzed by observng how erformance measures of he non-domnaed desgns n he Pareo se change as he search rogresses; and he ermnaon condon of he algorhms s based on hs observaon In fuure research, may be worhwhle o sudy he convergence roery more sysemacally Meanwhle, s also moran o nvesgae how o evaluae he qualy of he fnal Pareo se n comarson wh rue Pareo se REFERENCES Ahmed, M A, T M Alkhams 00 Smulaon-based omzaon usng smulaed annealng wh rankng and selecon Comuers & Oeraons Research 9, Alkhams, T M, M A Ahmed 004 Smulaon-based omzaon usng smulaed annealng wh confdence nerval Proceedngs of he 004 Wner Smulaon Conference, Alrefae, M H, S Andradór 1997 Accelerang he convergence of he sochasc ruler mehod for dscree sochasc omzaon Proceedngs of he 1997 Wner Smulaon Conference, Andradór, S 1999 Accelerang he convergence of random search mehods for dscree sochasc omzaon ACM Transacons on Modelng and Comuer Smulaon 9 (4), Baesler, F F, J A Seúlveda 000 Mul-resonse smulaon omzaon usng sochasc genec search whn a goal rogrammng framework Proceedngs of he 000 Wner Smulaon Conference, Boesel, J, B L Nelson, and N Ish 003 A framework for smulaon-omzaon sofware IIE Transacons 35, 1 9 Buler, J, D J Morrce, and P W Mullarkey 001 A mulle arbue uly heory aroach o rankng and selecon Managemen Scence 47(6): Chen, C H, J W Ln, E Yücesan, and S E Chck 000 Smulaon budge allocaon for furher enhancng he effcency of ordnal omzaon Dscree Even Dynamc Sysems: Theory and Alcaons 10: Chew, E P, L H Lee, and S D Lu 005 Dynamc raonng olces for nvenory sysems wh mulle demand classes Workng aer, Dearmen of Indusral & Sysems Engneerng, Naonal Unversy of Sngaore Chew E P, L H Lee, S Y Teng, and C H Koh 006 Mul-objecve smulaon omzaon for dfferenaed servce nvenory roblems Submed o Journal of he Oeraonal Research Socey Chck, S E and K Inoue 001 New wo-sage and sequenal rocedures for selecng he bes smulaed sysem Oeraons Research 49 (5): Fonseca, C M and P J Flemng 1995 An overvew of evoluonary algorhms n mulobjecve omzaon Evoluonary Comuaon 3(1), 1 16 Gong, W B, Y C Ho, and W G Zha 1999 Sochasc comarson algorhm for dscree omzaon wh esmaon SIAM Journal on conrol and omzaon 10, Hanne, T and S Nckel 005 A mulobjecve evoluonary algorhm for schedulng and nsecon lannng n sofware develomen rojecs Euroean Journal of Oeraonal Research 167, Hedlund, H E, and M Mollaghasem 001 A genec algorhm and an ndfference-zone rankng and selecon framework for smulaon omzaon Proceedngs of he 001 Wner Smulaon Conference, Lee, L H, E P Chew, S Y Teng, and D Goldsman 004 Omal comung budge allocaon for mulobjecve smulaon models In Proceedngs of he 004 Wner Smulaon Conference (ed R G Ingalls, M D Rosse, J S Smh, and B A Peers), Lee, L H, E P Chew, S Y Teng, and D Goldsman 005 Fndng he non-domnaed Pareo se for mulobjecve smulaon models Submed o IIE Transacons Lee, L H, E P Chew, S Y Teng, and Y K Chen 006 Mul-objecve smulaon-based evoluonary algorhm for an arcraf sare ars allocaon roblem Submed o Euroean Journal of Oeraonal Research Nelson, B L, J Swann, D Goldsman, and W M Song 001 Smle rocedures for selecng he bes smulaed sysem when he number of alernaves s large Oeraons Research 49: Ólafsson, S 1999 Ierave rankng-and-selecon for large-scale omzaon Proceedngs of he 1999 Wner Smulaon Conference, Sh, L and S Ólafsson 000 Nesed arons mehod for global omzaon Oeraons Research 48 (3), Swsher, J R and S H Jacobson 00 Evaluang he desgn of a famly racce healhcare clnc usng dscree-even smulaon Healh Care Managemen Scence 5 (): Swsher, J R, S H Jacobson, and E Yücesan 003 Dscree-even smulaon omzaon usng rankng, selecon, and mulle comarson rocedures: A survey ACM Transacons on Modelng and Comuer Smulaon 13 ():

10 Yan D, and H Muka 199 Sochasc dscree omzaon SIAM Journal on conrol and omzaon 30, AUTHOR BIOGRAPHIES LOO HAY LEE s an Assocae Professor n he Dearmen of Indusral and Sysems Engneerng, Naonal Unversy of Sngaore He receved hs BS (Elecrcal Engneerng) degree from he Naonal Tawan Unversy n 199 and hs SM and PhD degrees n 1994 and 1997 from Harvard Unversy He s currenly a senor member of IEEE, commee member of ORSS, and a member of INFORMS Hs research neress nclude smulaonbased omzaon, roducon schedulng and sequencng, logscs and suly chan lannng, and vehcle roung Hs emal address s <seleelh@nusedusg> and hs webse s <wwwsenusedusg/saff/ leelh/> EK PENG CHEW s an Assocae Professor and Deuy Head (Academc) n he Dearmen of Indusral and Sysems Engneerng, Naonal Unversy of Sngaore He receved hs PhD degree from he Georga Insue of Technology Hs research neress nclude logscs and nvenory managemen, sysem modelng and smulaon, and sysem omzaon Hs emal address s <sece@nusedusg> and hs web address s <wwwsenusedusg/saff/chewe/> SUYAN TENG s currenly a Research Fellow n he Dearmen of Indusral and Sysems Engneerng, Naonal Unversy of Sngaore She receved her PhD degree from Naonal Unversy of Sngaore n 004 Her research neress nclude smulaon-based omzaon and heursc algorhms develomen for solvng vehcle roung and schedulng roblems Her e-mal address s <sesy@nusedusg> 303