A MULTI-OBJECTIVE GENETIC ALGORITHM FOR EXTEND

Transcription

1 A MULTI-OBJECTIVE GENETIC ALGORITHM FOR EXTEND Bran Kernan and John Geraghty The School of Mechancal & Manufacturng Engneerng, Dubln Cty Unversty, Dubln 9, Ireland. Correspondence: Emal: Abstract: Evolutonary-algorthms are partcularly suted for the task of searchng decson spaces, because they process a set of solutons n parallel, and explot smlartes of pareto solutons through recombnaton. The man part of ths work was to construct a pareto-optmal genetc-algorthm n Extend; a smulaton computer software package that can be used to model manufacturng systems. The algorthm developed, s a robust searchng tool that can be appled to a wde varety of systems, and can analyse decson spaces comprsed of both ntegers and real numbers. A four-stage tandem producton lne as gven n [6] was then modelled n Extend for three dfferent control polces; kanban, CONWIP, and hybrd kanban-conwip. The system had two objectves; maxmse servce level whle mnmsng the work-n-progress nventory. The decson spaces were mapped out, and the pareto-optmser then searched for the pareto-optmal solutons.e. those sets of system parameters, whch gve non-domnated objectve vectors. For the manufacturng system wth the kanban control polcy, t was found that the genetc-algorthm had a maxmum effcency ndex of 3.71 for fndng 40% of the non-domnated solutons, wth a mutaton rate of %. Ths meant that the Genetc Algorthm (GA) s 3.71 tmes more effectve than searchng the decson space randomly. The hybrd control polcy only had a maxmum effcency ndex of 1.88 for fndng 40% of the nondomnated solutons, wth a mutaton rate of %. A possble reason for the reduced effcency when compared wth the kanban control polcy could be due to the reduced confdence levels n the objectves for the decson varables. There s possbly a large potental for mprovng the genetc-algorthm. Nchng and non-nchng technques could be ntroduced, and dfferent selecton, recombnaton and mutaton technques could be used. Comparatve studes should also be carred out wth other mult-objectve genetc algorthms such as the vector evaluated genetc algorthm (VEGA) [3] and the non-domnated sortng algorthm (NSGA) [7]. Keywords: Pareto-Optmal Mult Objectve Genetc Algorthm, Extend, Producton Control Strateges, CONWIP, Kanban,.Kanban-CONWIP Hybrd. 1.0 Introducton Many real-world problems nvolve smultaneous optmsaton of multple objectves that are often competng. One way to solve mult-objectve problems s to transform the orgnal problem nto a sngle-objectve one, by weghtng the objectves wth a weght vector [1]. Ths process allows the use of any sngle-objectve optmsaton algorthm. The obtaned soluton however, depends on the weght vector used n the weghtng process. Choosng the weght vector means that some pror knowledge about the soluton or decson space s requred. Another mult-objectve optmsaton technque that does not mpose an ll-nformed weghtng process on the task of selectng a sngle optmal soluton s pareto-optmsaton. Pareto-optmsaton ams to fnd the set of non-domnated soluton ponts,.e. those vectors for whch ther components cannot be all smultaneously mproved wthout havng a detrmental effect on at least one of the remanng 83

2 components. Ths s known as the concept of pareto-optmalty. Mathematcally mult-objectve problems can be wrtten generally as [1]: Maxmse or Mnmse Subject to: x = { x,x,...,x } X f (x) where: y { f (x),f (x),...,f (x)} Y 1 N = for { 1,...,M} 1 M The vector of decson varables s x of sze N, and X s the decson space. The vector of objectves s y of sze M, and Y s called the objectve space. The soluton to a mult-objectve problem s usually no unque, but a set of equally effcent, non-nferor or non-domnated solutons; the pareto-optmal set. Mathematcally n the maxmsaton case, we can say that the soluton x 1 domnates x or s superor to x (wrtten: x 1 f x ) ff: { 1,...,M}, (x ) y(x ) y { 1,...,M} y (x ) y (x ) 1 1 > If any other x n the decson space X does not domnate x 1, then x 1 s a non-domnated pont. Two nondomnated ponts are ndfferent to each other. A pareto-optmsaton algorthm should search the decson space, fndng the entre set of pareto-optmal solutons. Gven below s a smple multobjectve problem: Maxmse: 1 ( 4 f (x) = 5 x ) Mnmse: f ( x) = 6 + ( x ) 5 Subject to: x R One way to solve ths problem s to create a sngle objectve functon. For example: Maxmse: s(x) = f1(x) 5 = f (x) 6 + ( x 4) ( x 5) Subject to: x R Fgure 1: Sngle-objectve functon Fgure : Pareto-optmal set An optmsaton algorthm can then be appled to ths sngle objectve functon whch wll result n a sngle soluton; x = If a pareto-optmsng algorthm s to be used, then the goal s to fnd the set of solutons such that: { } x, xj X f ( x) f ( xj) f ( x) f ( xj) f ( x) > f ( xj) f ( x) < f ( xj) All the x s that satsfy the above condton belong to the pareto-optmo set, P. For ths example the soluton set P would be all the real numbers n the range [4,5]. Shown n Fgures 1 and above are the graphcal solutons to the mult-objectve problem of maxmsng one objectve, whle tryng to mnmse the other. Fgure 1 shows the soluton, as would be obtaned usng a sngle-objectve optmsaton algorthm, whle Fgure shows the pareto-optmo soluton set. 84

3 In order to obtan the pareto-optmal set of solutons for a mult-objectve optmsaton problem, the decson space X needs to be explored usng a search algorthm. One such search algorthm that can be used s the genetc-algorthm (GA). GAs mmc the metaphor of natural Darwnan bologcal evoluton. They operate on a populaton of potental solutons applyng the prncple of survval of the fttest to produce successvely better approxmatons to a soluton. Evolutonary algorthms dffer substantally from more tradtonal search and optmsaton methods. The most sgnfcant dfferences are []: They search a populaton of ponts n parallel. They do not requre dervatve nformaton or other auxlary knowledge; only the objectve functon and correspondng ftness levels nfluence the drectons of search. They use probablstc transton rules, not determnstc. They are generally more straghtforward to apply, because no restrctons for the defnton of the objectve functon exst. They can provde a number of potental solutons to a gven problem. The fnal choce s left to the user. (Thus, n cases where the partcular problem does not have one ndvdual soluton, for example a famly of pareto-optmal solutons, as n the case of mult-objectve optmsaton and schedulng problems, then the evolutonary algorthm s potentally useful for dentfyng these alternatve solutons smultaneously.) GAs have been recognsed as well suted to mult-objectve optmsaton. Mult-objectve genetcalgorthms (MOGAs) are apt to searchng decson spaces whch are nosy, dscontnuous, mpractcal to search exhaustvely and contan no dervatve nformaton [1]. The GAs used for mult objectve optmsaton problems are generally smlar to a sngle objectve genetc algorthm n everyway, apart from the ftness evaluaton and selecton mechansm. The poneerng work n genetc-algorthms for mult-objectve problems was conducted by Schaffer [3]. The author developed a program named vector evaluated genetc algorthm (VEGA) to fnd paretooptmal solutons to mult-objectve problems. Snce a soluton has several objectve functons, these form a vector. For a problem wth n objectve functons, a populaton s dvded nto n subpopulatons, each of whch has the same number of ndvduals. Every subpopulaton s assocated wth one objectve functon. The solutons n a subpopulaton are evaluated usng only the assocated objectve. Selecton s performed ndependently on each of the subpopulatons, but crossover s carred out across subpopulaton boundares. Whle mplementaton s relatvely straght forward, VEGA has a strong tendency to converge on extreme ponts where only one crteron s optmsed. In other words, the algorthm has lmtatons n fndng solutons n the mddle of extremes. In order to overcome the problems assocated wth VEGA, Goldberg [4] suggested the use of a pareto optmalty rankng method called a pareto GA. The ftness of a set of ndvduals (strng) s based on the rank, whch s determned by usng a non-domnated sortng procedure. Ths procedure fnds the set of strngs that s pareto optmal n the current populaton. The hghest rank s assgned to the strngs, and then the strngs are removed from further contenton. The pareto optmal strngs from the remanng populaton are assgned the next hghest rank. The process contnues untl all the strngs are ranked. After a rank for every strng s determned, a rankng selecton scheme s appled. Another nterestng approach was proposed by Horn et al. [5]. Ths method attempts to dstrbute the populaton evenly along non-domnated fronters. Snce ths approach s based on the concept of nchng, t s named nched pareto GA. The prmal selecton s carred out usng bnary tournament adapted to control domnaton pressure. Two canddates and a comparson set are pcked at random from the populaton. If one canddate s domnated by the comparson set and the other s not, the latter s selected. If nether are or both are non-domnated by the comparson set, then a ftness sharng mechansm comes nto play. The ftness of an ndvdual s degraded by the densty of the ndvduals wthn a certan dstance from t. The sphercal regon formed by the dstance s called the nche. Snce the amount of degradaton s nversely proportonal to the number of ndvduals n a nche, a sparser nche ncreases the resultng ftness values. As a consequence, convergence nto a narrow regon of soluton space s prevented. 85

4 .0 The Pareto-Optmal Genetc Algorthm The pareto-optmal genetc algorthm developed for Extend ams to fnd the set of non-domnated solutons n a decson space comprsed of up to 8 decson varables, for an objectve vector comprsed of up to 6 components. The decson space can consst of numbers n nteger or real number format encoded as decmals. The algorthm works as follows: 1. Intalsaton An ntal populaton of 5 ndvduals s created at random from the decson space.. Domnance Check The current populaton s checked to see whch ndvduals are non-domnated. Mathematcally n the maxmsaton case, we say that ndvdual x 1 domnates x or s superor to x (wrtten: x 1 f x ) ff: { 1,...,M}, (x ) y(x ) y { 1,...,M} y (x ) y (x ) 1 1 > If any other x n the decson space X does not domnate x 1, then x 1 s a non-domnated pont. Two nondomnated ponts are ndfferent to each other. Indvduals whch are non-domnated are assgned a boolean value of true. 3. Adjustment Any ndvduals n the current populaton, whch are domnated, are replaced wth an offsprng created from two dfferent non-domnated parents. The parents are obtaned at random from the current populaton, wth equal probablty of beng chosen. If there s only one non-domnated soluton however (there wll also be at least one), then the second parent s obtaned at random from the decson space. Because the decson space can be nteger or real format, t was decded to use a hybrd recombnaton technque of dscrete and ntermedate recombnaton, whch would be referred to as trad-based recombnaton (TBR). Wth TBR, each chromosome that makes up the offsprng, can be obtaned three dfferent ways wth equal probablty: t can take on the value of the frst parent s chromosome, the value of the second parent s chromosome, or t can take on a value n between, and ncludng the two parents chromosome values. When each chromosome s created, exo-parental mutaton [] can occur, wth a probablty of occurrence defned by the user. 4. Domnance Recheck The adjusted populaton s rechecked, to see whch solutons are non-domnated. 5. Increase Populaton If all of the ndvduals of the current populaton are non-domnated, 5 new offsprng are created and added to the populaton. The offsprng are created n the same manner as descrbed n step 3 of the genetc-algorthm. 6. Repeat The algorthm s repeated by gong back to step. The algorthm can be termnated by the number of generatons or the number of deadruns specfed by the user. The algorthm wll also end f the entre decson space has been searched (vald f decson space contans nteger values only). 3.0 Verfcaton of Algorthm Once the pareto-optmser had been bult, debugged and verfed on two smple optmsaton problems, t was then appled to a case-study problem from the lterature on determnng the optmal producton control strategy for a lean manufacturng system. The manufacturng system analysed wth the GA was a four-stage tandem producton lne wth each stage consstng of a machne and an output buffer as descrbed n [6]. The machne operaton tmes were modelled usng a lognormal dstrbuton wth mean 0.98 mnutes and a standard devaton of 0.0 mnutes. The machnes faled accordng to an exponental dstrbuton wth a mean tme-between-falures of 1000 mnutes. The tme-between-falures measure spent workng tme, not clock tme. Ths s known as operaton dependent falure and mples that no machne can fal unless t s workng on a part. The repar tme of the machnes was also modelled usng an exponental dstrbuton wth a mean of 3 mnutes. The demand rate was constant at 1 part per mnute. Other assumptons across all schedulng polces are: 86

5 Materal s transported n unts of one wthout delay. Informaton such as kanban s transmtted nstantaneously. Machnes operate asynchronously, so parts can be loaded whenever a part s present and the proper authorsaton has been receved. Jobs authorsed for loadng follow a frst come frst served dspatch polcy. Any demand that cannot be satsfed from the fnshed goods nventory s lost to the system. Demands occur constantly, whether or not they are beng satsfed. Ths s somewhat dfferent to the procedure used n [6], where the demand process comes to a stop untl each demand s satsfed. The objectves to be optmsed were the servce level (maxmsed) and work-n-progress (WIP) (mnmsed). Servce level s the rato of met demands to total demands and deally should be 1.0. The WIP s the total amount of nventory n the system. To reduce costs, the WIP should be kept as low as possble. These two parameters have a curve-lnear relatonshp, wth the servce level ncreasng as the amount of nventory (WIP) n the system ncreases. 3.1 Optmsaton and Decson Space for Kanban For the manufacturng system wth a kanban control polcy, cells 1, and 3 could take on kanban values n the range 1 to 5, whle cell 4 could take on kanban values n the range 1 to 10. Ths gave a total of 1,50 cases,.e. 1,50 dscrete ponts n the decson space X. The mult-objectve genetcalgorthm therefore, ams to obtan the set of pareto-optmal solutons (cell kanban) so that: { S(N ) S(N ) W(N ) W(N ) [ S(N ) > S(N ) W(N ) W(N )]} N,Nj X j j j < N = { n1,n,n3, n4 }... Cell Kanban S (N).. Servce Level for N W (N) Work-In-Progress for N j Fgure 3: Soluton space for kanban manufacturng system The entre decson space was mapped out by modelng the system usng Extend, wth 0 replcatons for 30,000 mnutes, and a warm-up perod of 5,000 mnutes, for each of the 1,50 cases. Servce levels were only calculated after the warm-up perod had fnshed, and the WIP was averaged between 5,000 and 30,000 mnutes. The decson space was placed n a data fle, so the objectve vectors for the dfferent soluton sets could be easly obtaned. The other alternatve was havng to run the smulaton to obtan servce levels and WIP levels as the genetc-algorthm demanded. By havng the decson space mapped out, the analyss and verfcaton of the genetc-algorthm on the manufacturng system was sgnfcantly qucker. Accordng to [6], the system should deally be run wth replcatons for 10,000 mnutes, wth a warm-up perod of 9,600 mnutes, to gve accurate results. Accurate results n ths case meanng, 95% confdence nterval half-wdths of for the servce level and 0.04 for the WIP would be requred. Due to computer lmtatons however, confdence level half-wdths of ths sze would have taken too long to obtan. The am of these experments however was to verfy the GA, 87

6 rather than accurately solve a manufacturng problem. The soluton space s shown n Fgure 3 for servce levels of greater than 99.9%. A hermte splne was ftted to the non-domnated solutons to show the pareto-front. In total t was found that there were 59 pareto solutons n the entre decson space. The decson space was then placed n a MATLAB data fle, and the algorthm was translated to MATLAB so a more extensve analyss of the GA could be carred out. 3. Evaluaton of the Performance of the Algorthm for Kanban In order to examne the effectveness of the genetc-algorthm on the manufacturng system, an ndex of effcency was defned. The effcency ndex E s smply the rato of the percentage of decson space searched, to the percentage of the pareto set obtaned. For example, f 50% of the decson space s searched randomly, then approxmately 50% of the pareto set wll be obtaned. Ths would gve an effcency ndex of 1.0. Before the genetc-alogrthm was appled to the problem, a stochastc paretooptmser (SPO) was frst used. The SPO works n a smlar fashon to the genetc-algorthm, except that new chldren are obtaned randomly from the decson space. The SPO was run for 150 generatons (obtans approxmately 3% pareto-set) wth 50 replcatons. The effcency ndex was found to be ± wth 90% confdence ntervals. Ths value of effcency ndex s consstent wth theory. Table 1: Non-domnated soluton set for manufacturng system wth kanban control Kanban Kanban Kanban Kanban Kanban Kanban Kanban Kanban Fgure 4: Effcency Vs. Mutaton Fgure 5: Effcency Vs. Pareto % The genetc-algorthm should have an effcency ndex greater than one, because t uses smple artfcal ntellgence to search the space. If there s a pattern to the pareto-optmal soluton set, then the algorthm wll fnd t. In ths case, t would be expected that the non-domnated solutons would have a hgher number of kanban n the fourth cell, then n the other three cells (see Table 1). Fgure 4 shows the effcency ndces for fndng 70% of the pareto-set wth dfferent mutaton rates. Fgure 5 shows the effcency ndces of the genetc-algorthm for fndng dfferent percentages of the pareto-set, wth a % mutaton rate. A % mutaton rate was chosen because accordng to the data n Fgure 4 t gves the largest effcency ndex for fndng 70% of the pareto-set. The effcency ndces were calculated by runnng the genetc-algorthm for 100 generatons. Confdence ntervals (90%) for the effcency ndces are ndcated by the arrows on the fgures. In both graphs, the two sets of data have been ftted wth a four-pont hermte splne, to show the general trend of the effcency ndex as a functon of mutaton rate and pareto-set percentage. 3.3 Dscusson of Performance of the Algorthm on Kanban System Accordng to the data n Fgure 4, the effcency of the algorthm peaks at, approxmately, 3.38 wth a mutaton rate of %, and then steadly decreases. Ths s consstent wth theory; a mutaton rate that s 88

7 too low, means that the genetc-algorthm can get stuck n a local set of solutons, and a mutaton rate that s too hgh can adversely affect the breedng of new solutons. For most pareto-optmal genetcalgorthms, mutaton rates are generally between 1% and 30%. It should be noted however, that the best mutaton rate s dependent on the actual problem. Also, for ths problem a mutaton rate of % gves the hghest effcency for fndng 70% of the pareto set. It may be the case, that dfferent mutaton rates gve hgher effcences for fndng dfferent percentages of the pareto set. Accordng to the data n Fgure 5, the effcency steadly ncreases from 10% of the pareto-set and reaches a peak at 40% of the pareto-set, and then steadly decreases. Agan ths s consstent wth theory. The effcency s low for fndng 10% of the pareto-set, because the genetc-algorthm does not have enough ft ndvduals yet to determne the complete pattern to the pareto-set (f there s any). The effcency peaks at 40%, ndcatng that 40% of the pareto-set follows a pattern that the genetcalgorthm can fnd. The remnng 60% of the set starts to follow a more random pattern, and so the effcency of the genetc-algorthm decreases. The effcency for fndng the entre non-domnated soluton set s 1.6,.e. 6% more effcent then searchng the decson space randomly. The reason ths value s so low, s because the pareto soluton set s obtaned from a system wth stochastc nputs, and therefore some solutons wll probably be completely anomalous and random; consequently t s very dffcult for the genetc-algorthm to fnd these solutons. It s mportant to note that the system was only run for 30,000 mnutes wth 0 replcatons. Ths may not gve suffcent confdence for the pareto-soluton set. Orgnally the system was run for 30,000 mnutes wth 10 replcatons and gave only 47 non-domnated solutons (compared wth 59 non-domnated solutons for 30,000 mnutes wth 0 replcatons, see Table 1). In theory, runnng the system for a longer tme and wth addtonal replcatons, wll gve more confdence for the pareto soluton set, and consequently the effcency of the genetc-algorthm should ncrease. Another way to check the valdty of the genetc-algorthm s to run the system wth dfferent control polces such as CONWIP, and hybrd kanban-conwip, as mentoned n [6], and to use the MOGA to obtan the pareto-soluton set for these control polces. 3.4 Applcaton of the Algorthm for Hybrd Polcy A smlar analyss was conducted for the kanban-conwip hybrd producton control strategy. Ths producton control strategy dstrbutes Kanbans to ndvdual workcentres and places a lmt, known as a WIP Cap, on the total amount of nventory that may be n the system at any gven pont n tme. Once ths lmt has been reached nventory may not enter the system untl a demand event removes a corespondng amount of nventory from the system. Overall, the genetc-algorthm was neffcent at searchng the hybrd decson space. The hghest effcency obtaned was only 1.88 for obtanng 40% of the pareto set. In order to obtan 100% of the pareto set, the effcency was only 1.1,.e. 89% of the decson space needed to be searched. The effcency ndces are so low because the genetc-algorthm was unable to fnd a pattern among the pareto optmal solutons. The reason that there appears to be no pattern to the pareto soluton set, s probably because there was nsuffcent confdence n the values for servce level and WIP n the decson space. An mportant observaton, s that some of the ndvduals n the non-domanted soluton set, do not make sense from a practcal vewpont. For nstance the frst non-domnated soluton found had a cap of 1, and 3 kanbans crculatng n the frst cell; there s no need to have 3 kanband crculatng n a cell, snce only one part s allowed to be n the system at any one tme. In fact, there were a total of 4 ndvduals out of the 97 non-domnated ndvduals.e. 5%, that have kanban values n ndvdual cells that are hgher than the cap for the system. If these ndvduals are removed from the pareto soluton set, and f the genetc-algorthm s told to gnore ndvduals whch have cell kanban hgher than the cap, then the effcency ndex for fndng 40% of the pareto-set s now.1 ± Although ths s only a margnal mprovement, t s an mportant consderaton when analysng manufacturng systems wth hybrd kanban-conwip control polces; the cap should always be greater than or equal to the ndvdual kanban values for each cell. In theory, runnng the system for more replcatons and/or a longer smulaton tme, wll gve more confdence for the pareto soluton set, and consequently the effcency of the genetc-algorthm should ncrease. In order to reduce processng tme, the ndvduals n the decson space that have cell kanbans hgher than the cap can be gnored. Ths means that the decson space of 3,15 ndvduals s reduced to,75 ndvduals (a 13% decrease). However, t may be the case that the ndvduals that make up the pareto soluton set wll not have a dscernable pattern that the genetc-algorthm can easly fnd. 89

8 Regardless, the genetc-algorthm stll operates up to twce as effectvely as a SPO would, for ths system wth a hybrd control polcy. 3.5 Comparson of Control Strateges Shown below n Fgure 6 are the pareto-fronts for the 4-stage tandem producton wth the three dfferent control polces; kanban, CONWIP and hybrd. The fronts were obtaned by fttng hermte splnes to the non-domnated solutons. The hybrd control polcy appears to be the best schedulng polcy (best n ths case, apples to both servce level and WIP). If a servce level of 99.9% s desred, then the kanban control polcy would result n a WIP of approxmately 13, CONWIP 1 and hybrd only 10. (These values are sgnfcantly dfferent to those obtaned n [6] whch are kanban 15.8, CONWIP, 14.6 and hybrd 13.93). The dfference s probably due to the dfferent demand processes). Essentally the hybrd control polcy s superor to kanban and CONWIP, because t combnes the ndvdual advantages of both: It places a lmt on the total amount of nventory n the system (CONWIP), and uses kanban to ensure that ndvdual buffer levels do not get too hgh when bottleneckng occurs. Kanban CONWIP Hybrd Fgure 6: Pareto-fronts for manufacturng system wth dfferent control polces It s mportant to note that for the kanban and hybrd control polces, the hghest effcency ndces are for fndng 40% of the pareto soluton set (kanban 3.71, hybrd 1.88). If a manufacturng system s beng analysed to see whch s the optmal control polcy, then 40% of the pareto-soluton set, should be suffcent to determne the pareto front. Tryng to fnd the complete pareto soluton set for stochastc systems, wll probably result n reduced effectveness of the genetc-algorthm, due to anomalous ndvduals n the pareto-set. 4.0 Dscusson and Future Work The genetc-algorthm performs reasonably well on the gven problem of optmsng a manufacturng system. Unfortunately, no research papers on pareto-optmal genetc-algorthms appled to ths knd of problem could be found, so no comparatve studes could be carred out. Consequently there s no way to comparatvely quantfy the effectveness of the genetc-algorthm. Further work would nvolve buldng other evolutonary-algorthms such as the vector evaluated genetc algorthm (VEGA) presented by Schaffer [3], and the non-domnated sortng algorthm (NSGA) developed by Srnvas and Deb [7], and dong a comparatve study on the manufacturng system. VEGA conducts selecton for each objectve separately. In detal, the matng pool s dvded nto parts of equal sze and ndvduals are chosen at random from the current populaton accordng to each objectve. Afterwards the matng pool s shuffled and crossover and mutaton are performed as usual. Schaffer mplemented ths method n combnaton wth ftness proportonate selecton. Although some serous drawbacks are known, ths algorthm has been a strong pont of reference n the feld of evolutonary-algorthms. 90

9 Wth NSGA, the ftness assgnment s carred out n several steps. In each, the non-domnated solutons consttutng a non-domnated front are assgned the same dummy ftness value. These solutons are shared wth ther dummy ftness values and gnored n the further classfcaton process. Fnally, the dummy ftness s set to a value less than the smallest shared ftness value n the current non-domnated front. Then the next front s extracted. Ths procedure s repeated untl all ndvduals n the populaton are classfed. In order to get more accurate results from the evolutonary algorthms, mproved confdence n the decson space for the manufacturng system wth kanban and hybrd control polces would also need to be obtaned. By comparng and analysng the dfferent MOGAs t wll probably be possble to buld a more effectve and robust genetc-algorthm. It s mportant to remember that evolutonary algorthms use a wde and vared set of technques and methods and that there s potentally a large scope for mprovng the operaton of the genetc-algorthm. Some of the possble changes to the genetc algorthm nclude: Encodng chromosomes n bnary format Currently, the chromosomes that make up the decson space are encoded n decmal format. The chromosomes could be encoded as bnary numbers usng Gray code or Bnary Coded Decmals (BCDs). Each bnary bt would correspond to the actvaton or deactvaton of a gene. Nchng and non-nchng technques The am of both of these technques s to preserve dversty n the populaton and therefore try to prevent premature convergence. Nchng technques are charactersed by ther capablty to promote the formulaton and mantenance of stable subpopulatons (nches). Ftness sharng s used most frequently, whch s a nchng technque based on the dea that ndvduals n a partcular nche have to share the avalable resources. The more ndvduals are located n the neghborhood of a certan ndvdual, the more ts ftness value s degraded. The neghborhood s defned n terms of a dstance measure and specfed by the so-called nche radus σ share. Currently, most MOGAs mplement ftness sharng. Among the non-nchng technques, restrcted matng s the most common n mult-crtera functon optmsaton. Bascally two ndvduals are allowed to mate only f they are wthn a certan dstance (gven by the parameter σ mate ) to each other. Ths mechansm may avod the formaton of unft ndvduals and therefore mprove the onlne performance. Nevertheless, as mentoned n [8], t does not appear to be wdespread n the feld of mult-objectve evolutonary algorthms. Dfferent selecton processes The selecton process currently used n the genetc-algorthm s farly smplstc, and there s potentally large room for mprovement. One selecton process that could be mplemented s that used n the nched pareto genetc algorthm (NPGA) [5]. The NPGA combnes tournament selecton wth the concept of pareto domnance. Two competng ndvduals and a comparson set of other ndvduals are pcked at random from the populaton; the sze of the comparson set s determned by a comparson parameter. If one of the competng ndvduals s domnated by any member of the set, and the other s not, then the latter s chosen as the wnner of the tournament. If both ndvduals are domnated (or nondomnated), the result of the tournament s decded by sharng: The ndvdual that has the least ndvduals n ts nche s selected for reproducton Dfferent recombnaton processes Dfferent methods of creatng offsprng from parents such as lne recombnaton and mult-pont crossover could be used. The latter recombnaton technque can only be used f the decson space s encoded n bnary format. Dfferent mutaton processes There are a number of dfferent ways the mutaton process could be altered. For example, f the decson space s encoded n bnary format, then mutaton could be allowed to occur on one chromosome per new offsprng created. Also t may be possble to ntroduce a varable mutaton rate that changes throughout an evolutonary run. When dealng wth genetc-algorthms t s mportant that the algorthm not be calbrated to the problem used for valdatng the algorthm. It may be the case that the GA wll work excellently on the 91

10 valdatng problem, but not on any other optmsaton problems. To overcome ths, a wde and vared array of problems should be used to verfy the algorthm. In ths project, the GA needs to be able to search decson spaces wth real numbers as well as ntegers, so t should be tested on large nosy, dscontnuous decson spaces comprsed of real numbers. Genetc-algorthms are currently a fertle ground for research and applcaton development. There are a plethora of dfferent ways a genetcalgorthm can be bult; some are robust and desgned to optmse a wde varety of problem, whle others are desgned for specfc problems. Whle a rch set of technques and models are avalable coverng a range of domans, there are stll many areas n the feld of evolutonary algorthms remanng to be understood and exploted. 5. Conclusons One of the man conclusons to be drawn from ths work s that pareto-optmal MOGAs can be an effcent method of searchng nosy dscontnuous decson spaces, although ther effectveness and robustness s dependent on the ablty of the algorthm to fnd smlartes between non-domnated solutons. The MOGA bult n Extend was tested on a four-stage producton lne for three dfferent control polces; kanban, CONWIP, and hybrd kanban-conwip. For the manufacturng system wth kanban control polcy, t was found that the genetc-algorthm had a maxmum effcency ndex of 3.71 for fndng 40% of the non-domnated solutons, wth a mutaton rate of %. The CONWIP control polcy only had a decson space of 5 varables and 4 of those were non-domnated; consequently t was determned that t s neffcent for a GA to search such a small decson space. The system wth the hybrd control polcy only had a maxmum effcency ndex of 1.88 for fndng 40% of the non-domnated solutons, wth a mutaton rate of %. Because no comparatve studes could be carred out, there s currently way to comparatvely quantfy the effectveness of the genetc-algorthm. However t s expected that the effcency ndces of the MOGA wll ncrease f there s ncreased confdence n the objectve vectors for the decson space. There s large potental for mprovng the genetc-algorthm. Nchng and non-nchng technques could be ntroduced, and dfferent selecton, recombnaton and mutaton technques could be used. Comparatve analyses should also be carred out wth other mult-objectve genetc algorthms such as the vector evaluated genetc algorthm (VEGA) and the non-domnated sortng algorthm (NSGA). References [1] N. Srnvas and K. Deb. Multobjectve Optmzaton usng Nondomnated Sortng n Genetc Algorthms. Tech. Rep., Dept. Mechancal Engneerng, Kanput, Inda, [] Evolutonary Algorthms (accessed 1 st June 04) [3] J. D. Schaffer. Multple Objectve Optmzaton wth Vector Evaluated Genetc Algorthms. Ph.D. dssertaton, Vanderblt Unv., Nashvlle, TN, [4] D. E. Goldberg. Genetc Algorthms n Search, Optmzaton and Machne Learnng. Addson-Wesley, Readng MA, [5] J. Horn, N. Nafplots and D. E. Goldberg. A Nched Pareto Genetc-Algorthm for Mult- Objectve Optmzaton. Proc. of 1 st IEEE-ICEC Conference. 1994, pp [6] A. M. Bonvk, C. E. Couch, and S. B. Gershwn. A Comparson of Producton-Lne Control Mechansms. Internatonal Journal of Producton Research, 35(3):pp , [7] N. Srnvas and K. Deb. Mult-Objectve Optmzaton usng Non-Domnated Sortng n Genetc Algorthms. Evoluton Computng, vol., no. 3, pp. 1 48, [8] C. M. Fonseca and P. J. Flemng. An Overvew of Evolutonary Algorthms n Mult- Objectve Optmzaton. Evoluton Computng, vol. 3, no.1, pp. 1 16,