Multi-objective balancing of assembly lines by population heuristics

Transcription

1 Mult-objectve balancng of assembly lnes by populaton heurstcs Andreas C. Nearchou To cte ths verson: Andreas C. Nearchou. Mult-objectve balancng of assembly lnes by populaton heurstcs. Internatonal Journal of Producton Research, Taylor & Francs, 0, (0), pp.-. <0.00/000>. <hal-00> HAL Id: hal-00 Submtted on Sep 0 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Internatonal Journal of Producton Research Mult-objectve balancng of assembly lnes by populaton heurstcs Journal: Internatonal Journal of Producton Research Manuscrpt ID: TPRS-0-IJPR-00.R Manuscrpt Type: Dscusson Note Date Submtted by the Author: -Aug-0 Complete Lst of Authors: Nearchou, Andreas; Unversty of Patras, Department of Busness Admnstraton Keywords: ASSEMBLY LINE BALANCING, MULTI-CRITERIA DECISION MAKING, HEURISTICS, EVOLUTIONARY ALGORITHMS Keywords (user): dfferental evoluton

3 Page of Internatonal Journal of Producton Research 0 Mult-objectve balancng of assembly lnes by populaton heurstcs Andreas C. Nearchou Department of Busness Admnstraton Unversty of Patras 0 Ro, Patras, Greece E-mal: nearchou@upatras.gr Tel.: + -0, Fax: + -

4 Internatonal Journal of Producton Research Page of 0 Abstract: Ths paper s concerned wth the soluton of the mult-objectve sngle-model determnstc assembly lne balancng problem (ALBP). Two b-crtera objectves are consdered: () mnmzng the cycle tme of the assembly lne and the balance delay tme of the workstatons, and () mnmzng the cycle tme and the smoothness ndex of the workload of the lne. A new populaton heurstc s proposed to solve the problem based on the general dfferental evoluton (DE) method. The man characterstcs of the proposed mult-objectve DE (MODE) heurstc are: (a) t formulates the cost functon of each ndvdual ALB soluton as a weghted-sum of multple objectves functons wth self-adapted weghts. (b) It mantans a separate populaton wth dverse Pareto-optmal solutons. (c) It njects the actual evolvng populaton wth some Pareto-optmal solutons. (d) It uses a new modfed scheme for the creaton of the mutant vectors. Moreover, specal representaton and encodng schemes are developed and dscussed whch adapt MODE on ALBPs. The effcency of MODE s measured over known ALB benchmarks taken from the open lterature and compared to that of two other prevously proposed populaton heurstcs, namely, a weghted-sum Pareto genetc algorthm (GA), and a Pareto-nched GA. The expermental comparsons showed a promsng hgh qualty performance for MODE approach. Key words: assembly lne balancng, dfferental evoluton, mult-objectve optmzaton, Pareto optmalty, evolutonary algorthms, manufacturng optmzaton.

5 Page of Internatonal Journal of Producton Research 0. Introducton Fndng the global optmum to a general mult-objectve optmzaton problem (MOOP) s NP-complete (Bäck ). Usually, there s no sngle optmal soluton to a MOOP but rather a set of optmal solutons known as Pareto-optmal solutons. These are solutons that are nondomnated by any other soluton n the search space when all the objectves are consdered, and they do not domnate each other n the set. Due to ther ntrnsc parallelsm, multobjectve evolutonary algorthms (EAs) have recently receved a growng research attenton for the soluton of several real-world MOOPs (Coelo, Van Veldhuzen and Lamont 00, Jones et al. 0, Twar et al. 0). Ths paper consders the smple (determnstc sngle-model) assembly lne balancng problem (ALBP) wth varous objectves. ALBP s a decson problem arsng when an assembly lne has to be (re)-confgured, and conssts of determnng the optmal parttonng of the assembly work among the workstatons n accordance wth some objectves (Baybars, Scholl ). These objectves usually take one of two forms: ether mnmzng the number of workstatons (m) gven the cycle tme (c) of the lne (SALBP-), or mnmzng c gven m (SALBP-). Any varant of the smple ALBP (SALBP) belongs to the NP-hard class of combnatoral optmzaton problems (Scholl ). Mult-objectve (MO) ALB optmzaton has attracted the research attenton n the last decade. In an nterestng work, Km et al. (), developed a MO genetc algorthm (MOGA) for SALBP wth objectve to maxmze smultaneously the workload smoothness and work relatedness. Ponnambalam et al. (00), appled a MOGA on SALBP wth the objectve to maxmze lne effcency and to mnmze workload smoothness ndex. MOGAs have also developed by (Malakoot and Kumar, Celano et al., Rekek et al. 0, Chen et al. 0, Mansour 0) for solvng the general ALBP (GALBP) wth varous cost and proft orented objectves. McMullen and Frazer () and Gambern et al. (0) addressed the stochastc MO GALBP, va smulated annealng, and a specal MO heurstc, respectvely. There s a lack n the lterature for populaton heurstcs (such as EAs) for solvng the mult-crtera SALBP-. Furthermore, the computatonal testng of most EAs has been

6 Internatonal Journal of Producton Research Page of 0 performed gnorng exstng ALB test beds (Scholl and Becker 0). Inspred n a sense, by the work of Murata et al. () who addressed the MO flow-shop schedulng problem va a MOGA; ths work presents a new populaton heurstc based on the dfferental evoluton (DE) method to deal wth MO SALBP-. Two b-crtera objectves are consdered: () mnmzng the cycle tme and the balance delay tme of the statons, and () mnmzng the cycle tme and the workload smoothness ndex of the lne. The developed mult-objectve DE (MODE) heurstc has the followng features: (a) It formulates the compound objectve functon of each possble ALB soluton as a weghted-sum of the ndvdual objectves functons wth self-adapted weghts. The proposed self-adaptable scheme for estmatng the weghts can be very easly adjusted by the decson maker to vary the emphass on the ndvdual objectves. (b) It mantans a separate populaton wth dverse Pareto-optmal solutons to the problem. Ths populaton s teratvely updated per generaton. (c) It njects the evolvng populaton wth elte solutons taken from the Pareto populaton. (d) It uses a novel scheme for the creaton of the offsprng vectors durng DE evoluton. MODE s performance s measured over publc avalable benchmarks and compared to that of two known MOGAs proposed by Km et al. () and Murata et al. (), respectvely. The rest of the paper s organzed as follows: Secton formulates SALBP. Secton analyzes the basc DE model for optmzaton over contnuous spaces. Secton presents the way DE can be appled on ALBPs, whle secton ntroduces MODE for MO SALBP-. Computatonal results concernng the performance of the algorthms are provded n secton, whle conclusons and drectons for future work are ponted out n secton.. Decson makng n ALBPs.. Formulaton of SALBP. SALBP can be stated as follows: m workstatons are arranged along an assembly lne. Manufacturng a sngle product on the lne requres the parttonng of the total work nto a set V={,,n} of n elementary operatons called tasks. Each task j s performed on exactly one

7 Page of Internatonal Journal of Producton Research 0 staton and requres a determnstc processng tme t j. Let S z (z=,,m) be the staton load of staton z (.e., the set of tasks assgned to z), wth a cumulated task tme ts z = j S z (z=,,m). The tasks are partally ordered by precedence relatons defnng a drected acyclc graph (DAG) G=(V,E); wth V the set of nodes denotng the tasks n G, and E the set of edges representng the precedence constrants among the tasks. The assembly lne s assocated wth a cycle tme c denotng the maxmum processng tme avalable for each staton. The objectve typcally takes one of two forms: ether mnmzng m gven c (SALBP-), or mnmzng c gven m (SALBP-). Fgure llustrates an example of a precedence graph for an -tasks ALBP. The numbers nsde the nodes of the graph correspond to the task labels, and those outsde the nodes to the processng tmes. < Insert Fgure about here > In ths work, the b-crtera SALBP- s consdered wth man objectve to mnmze the cycle tme c for a gven fxed number of statons m, and secondary objectves to mnmze: (a) The balance delay tme (BD) of the lne (see Eq. ()). BD reflects the unused capacty of the lne,.e., the summaton of the dle tmes of all the statons. (b) The smoothness ndex (SX) (Eq. ()) measurng the equalty of the dstrbuted work among the statons. The lower the value of SX the smoother the lne, resultng n reduced n-process nventory. An SX equal to zero ndcates a perfect balance of the workload among the statons. BD= SX = m ( c ts z ) z=.. The mult-objectve SALBP- m ( c ts z ) z= In MO SALBP- we deally seek for a feasble soluton that smultaneously optmzes c, as well as, BD and SX. Snce ths s almost mpossble for any MOOP (Bäck ), what we t j `() ()

8 Internatonal Journal of Producton Research Page of 0 really attempt to do s to optmze each ndvdual objectve to the greatest possble extend. MOOPs consdered n ths study can be formulated as n the followng: (A) MO SALBP- verson : Mnmze subject to: F = w c+ w BD () a partton of the set V={,,n} nto m dsjont subsets S (z =,,m) : edge (,j) E,, j V and j FL the followng holds, S A and j S B wth A B ts z t sum for all z =,,m where, t sum = t n j j= s the sum of all the tasks processng tmes and, FL = the set of mmedate successors (followers) of task. (B) MO SALBP- verson : Mnmze z (.a) (.b) F = w c+ w SX () Subject to the constrants (.a), (.b) Constrants (.a) and (.b) ensure the feasblty of an ALB soluton. In partcular, constrant (.a) guarantees the feasble assgnment of the tasks to the m statons. That s, each task s assgned to exactly one staton, and the successors of any task are not assgned to an earler staton than that of. Note that, (, j) denotes an edge between and j, wth j beng the mmedate successor of. Constrant (.b) ensure that the staton tmes of all the statons do not exceed the lne s total processng tme ( t sum ). The weghts w and w n Eqs.(),(), specfy the relatve mportance of the correspondng objectves. The determnaton of the sutable values for these weghts s n general a dffcult task and consttutes a crtcal research queston n MOO. Ths ssue wll be dscussed deeper n secton of ths study. Moreover, a new methodology wll be ntroduced for a dynamc selfadapted estmaton of these weghts.

9 Page of Internatonal Journal of Producton Research 0. Dfferental evoluton (DE) DE s a populaton heurstc ntroduced by Storn and Prce () for global optmzaton over contnuous search spaces. DE has been appled wth success on many numercal optmzaton problems outperformng other popular heurstcs ncludng GAs (Al and Törn 0, Kaelo and Al 0). Recently, the applcaton of DE has been extended wth success to combnatoral optmzaton problems wth dscrete decson varables, such as, the machne layout problem (Nearchou 0), and machne flow-shop schedulng problems (FSSPs) (Onwubolu and Davendra 0, Nearchou and Omrou 0). DE utlzes Νp, D-dmensonal parameter vectors x, k, =,,,Νp, as a populaton to search the feasble regon Ω of a gven problem. The ndex k denotes the teraton (or generaton) number of the algorthm. The ntal populaton (where, k = 0), Φ = x, x, K, }, () {,0,0 x N p, 0 s taken to be unformly dstrbuted n Ω. At each teraton, all vectors n Φ are targeted for replacement. Therefore, Νp compettons are held to determne the members of Φ for the next teratons. Ths s acheved by usng mutaton, crossover and acceptance operators. In the ) mutaton phase, for each target vector x,, =,,Νp, a mutant vector x, k s obtaned by ) k x, k = xα, k + Fs ( xβ, k xγ, k ), () where α,β,γ {,,Νp} are mutually dstnct random ndces and are also dfferent from the current target ndex. The vector x, k α s known as the base vector and F s >0 s a scalng parameter. The crossover operator (Eq.()) s then appled to obtan the tral vector ) y, k from x, k and x, k. y L, k ) x = x L, k L, k f f R R L L C R > C R or L= I, () and L I Where, I s a randomly chosen nteger n the set I,.e., I Ι ={,,,D}; the superscrpt L L represents the L-th component of respectve vectors; R (0,), drawn randomly for each L.

10 Internatonal Journal of Producton Research Page of 0 The ultmate am of the crossover rule s to obtan the tral vector y k, wth components ) comng from the components of the target vector x, k and the mutated vector x, k. Ths s, ensured by ntroducng the crossover rate C R and the set Ι. Notce that for C R = the tral vector y, k s the replca of the mutated vector x, k untl all members of Φ are consdered. After all Νp tral vectors ). The process (mutaton and crossover) contnues y k, have been generated, acceptance s appled. In the acceptance phase, the cost of the tral vector, Cost ( ), s ( x,k compared to Cost ), the value at the target vector and the target vector s updated usng y, k f Cost( y, k ) < Cost( x, k ) x, k+ = () x, k otherwse Mutaton, crossover and acceptance phases contnue untl some stoppng condtons are met.. Applyng DE on ALBPs. Snce DE s qute smlar to an EA, wthout lose the generalty, n the followng analyss we wll use terms borrowed from the feld of Evolutonary Computaton such as, the genotype (s the real-valued vector evolved n DE), the phenotype (s the actual ALB soluton correspondng to a genotype). Every component of a vector s called a gene. Therefore, applyng DE on ALBP doman needs the specfcaton of the followng fve characterstcs: (a) a representaton mechansm,.e., a way of encodng the phenotypes to genotypes. (b) An evaluaton mechansm,.e. a way of computng the cost-functon for each genotype. (c) A way of ntalzng a populaton of genotypes. (d) The applcaton of mutaton, crossover and acceptance operators on the populaton. (e) Values to the parameters: Νp, C R and F s. Characterstcs (c)-(e) are n general same as n the standard DE, the dfferences rely on the way one mplement the frst two. y, k

11 Page of Internatonal Journal of Producton Research 0.. The representaton mechansm The natural codng for ALBPs s strngs wth ntegers. Two dfferent schemes for strng representaton applcable to ALBPs are manly dentfed n the lterature: staton-orented (sor) and task-orented representaton (tor) (Scholl and Becker 0). Both of them assume a strng length equal to n (number of the tasks to be processed). Usng sor, the components of a strng are ntegers n the range [,m] (m=number of workstatons). Thus, f the -th poston of the strng has the value z (z =,,m), then, task s assgned to staton z. Usng tor, a specfc strng s a permutaton of the ntegers,,,n. Hence, the task j n locaton of the strng wll be assgned to a staton before the task n locaton (+) of the strng. Note that, a tasks sequence s legal when t does not break the precedence relatons (constrant (.a)). For example, assume SALBP shown n Fg. wth m=, c=. A possble feasble soluton to ths problem s to assgn tasks {,} to staton, tasks {,,} to staton, and tasks {,,} to staton. Usng sor ths soluton may be represented by the strng <,,,,,,,>; whle, usng tor by <,,,,,,,>. After expermentaton wth both schemes over selected test beds (from a set of benchmarks descrbed n secton ), we found tor superor, n terms of speed of convergence and qualty of solutons, and thus, t was decded to adopt tor wth DE. DE works wth floatng-pont vectors and thus an approprate mappng s needed from the genotypc state-level (the vectors) to the phenotypc level (the actual ALB solutons). To acheve ths mappng a smple yet effectve topologcal orderng scheme has been developed based on the relatve prortes mpose by the components of a genotype. Assumng an n-tasks ALBP wth precedence relatons gven by a DAG G=(V,E), the developed encodng scheme conssts of generatng a topologcal sort of G from a specfc n-dmensonal floatng-pont vector ψ (genotype). Each vector s component ψ (=,,n) represents the relatve prorty of task ( V). The topologcal sort s therefore a rankng of all the tasks accordng to ther prortes n an approprate order to meet the precedence constrants. Ths mechansm s mplemented usng the followng procedure:

12 Internatonal Journal of Producton Research Page 0 of 0 Procedure Topologcal_orderng_encodng begn Set V = // wth V V // Repeat For all j V do f j has no predecessors then V =V {j},.e., nsert j nto the set V. Determne the gene ψ of ψ wth the maxmum value for all V Insert task nto the next avalable poston n the partal schedule (PS). V = V \ {},.e., remove task from V. Untl PS has been completed Return PS end In each step, the tasks wth no predecessors are dentfed and put n set V. Then, the task n V havng the hghest gene s value n ψ s selected, removed from V, and placed n the next avalable poston of PS. The process s repeated untl the completon of PS. Let us see how ths topologcal orderng works on genotype ψ = (0., 0., 0.0, 0., 0., 0., 0.0, 0.) concernng the -task ALBP shown n Fg.. The frst poston of array PS s taken by task (.e., PS[]=) snce ths s the only task wth no predecessors. Task s then cut from DAG and the next task wth no predecessors s task, thus PS[]=. Then, the two tasks and are canddate for the rd locaton of PS. The prortes for these tasks are 0.0 and 0., respectvely, and therefore, PS[]= snce task has the hghest prorty, consequently PS[]=. Fnally, the ALB soluton correspondng to ψ wll be (,,,,,,, ). Fg. dsplays the detaled step-by-step process for constructng the specfc feasble ALB soluton. One can see from ths fgure the partal topologcal sort, the cut (dark long dashed lnes) and the elgble nodes, as well as, the contents of the partal schedule soluton PS. < Insert Fgure about here >.. Decodng a phenotype nto an actual soluton for SALBP-. Once a specfc DEA s genotype s encoded nto a feasble ALB soluton (a phenotype) then, an approprate decodng scheme s needed to map ths phenotype to an actual soluton for SALBP-. In other words, a method s needed to assgn the tasks n the generated tasksequence nto the statons. After expermented (over representatve nstances from the test beds dscussed n secton ) wth some well known decodng schemes such as the lower and 0

13 Page of Internatonal Journal of Producton Research 0 upper bound search methods (Scholl ), we fnally decded to adopt a scheme proposed by Km et al. (). The use of ths scheme (see below) n DE was found to be superor n terms of qualty of solutons. The dea s to face SALBP- through an terated procedure that solves the correspondng SALBP- wth a cycle tme value beng progressvely decreased untl reachng a near-optmum value wthn a specfc permtted range. Procedure decode_salbp- begn Step. Set c ntally equal to the theoretcal mnmum cycle tme,.e., c = t sum /m. Step. Assgn as many as possble tasks nto the frst m- workstatons. Assgn all the remanng tasks to the last workstaton, m. Step. Calculate the work load W z for each workstaton z (z=,,,m), and the potental workload PW z (z=,,,m-) as follows: W z =the staton tme ts z (z=,,,m). PW z = ts z + the processng tme of the frst task assgned to (z+)st staton (z=,,,m-). Step. Set c W = max { W, W,,W m } and c = mn { PW, PW,,PW m- } Step. f (c W > c) then goto step else Return c W end.. The evaluaton mechansm Ths mechansm corresponds to the computaton of the cost (objectve) functon for each phenotype soluton. As analyzed n sub-secton., the objectve s to mnmze the functons gven by Eqs. (),(). Hence, for a phenotypeρ (=,,Np) the cost functon s gven by, Cost = ( Ρ ) Fj () Where, F j (j=,) s the j th composte objectve functon gven by Eqs. (), (), respectvely.. A mult objectve DE (MODE) for SALBP-. When solvng a MOOP often the attempt s to fnd a Pareto set of optmal solutons. Pareto set contans all those non-domnated solutons to the problem under nvestgaton such that no other solutons are superor to them n respect to all the dscrete objectves. To that purpose, MODE tres to fnd a set of non-domnated ALB solutons rather a sngle ALB soluton. MODE (see Fg. for a schematc overvew) has the followng two man features: a) It mantans a separate populaton of dverse Pareto-optmal ALB solutons teratvely updated generaton by generaton.

14 Internatonal Journal of Producton Research Page of 0 b) It uses an eltst preservng strategy wth whch a porton of the evolvng populaton s randomly replaced by a number of elte Pareto solutons. < Insert Fgure about here > In a general MOOP a soluton wth the best values for each objectve can be regarded as an elte soluton. Hence, for the b-crtera ALBPs examned n ths work, there are two elte (extreme) solutons n the evolvng populaton each of whch optmzes one objectve. These solutons are always coped nto Pareto populaton. Pareto set s further completed by addtonal elte solutons usng a mechansm explaned below. The Pareto populaton of the fnal generaton contans the near-optmal solutons to the MOO ALBP. The decson maker can then select that soluton accomplshng more her or hs preferences. Moreover, two addtonal features are ncluded wthn MODE resultng to modfcatons to the standard mutaton rule gven by Eq. (). These features are as follows: () In each teraton k, mutant vectors ) x = r x ) x, k (=,,Νp) are created usng the relaton, + ( r) x, + Fs ( x, x,, k α, k β k γ k δ k where r s a random number unformly taken wthn (0,) and α β γ δ {,,Νp} are dstnct random ndces. Ths scheme was found more robust n prelmnary experments than other standard mutant schemes such as the one gven by Eq. (). Partcularly, the proposed mutaton scheme enhances the ablty of the algorthm for convergng faster to a nearoptmum soluton. () Tradtonally, the mutaton-factor F s takes a value wthn the range (0,] and ths value remans constant through the lfe cycle of the DE algorthm. In ths study, we propose the followng dynamc scheme for estmatng F s. ) (0) f Cost 0. Cost then Fs = F0 else Fs = Θ Fs () MIN AVG where, CostMIN and Cost AVG are the populaton mnmum and average costs values, respectvely. Ths scheme gves to Fs a hgh value at the begnnng of the run and decreases ths rate slowly by the dversty of the populaton. Fs s ntally defned to be equal to F 0 =0., and decreased n each new teraton by a factor Θ = 0. usng the lnear relaton F s =Θ F s. If the mnmum

15 Page of Internatonal Journal of Producton Research 0 populaton cost becomes almost the same to the average populaton cost, then a very small dversty s encountered n the populaton and thus F s s reset to the ntal value F 0. MODE s gven below n pseudo-code format: Algorthm MODE for SALBP- Pre-processng step: Read nput data concernng the DAG G=(V,E) of a specfc n-tasks ALBP:.e., the set of tasks V, the set of edges E, the processng tmes t j (j=,,n), the number of statons m. Intalzaton step: Set values for the control parameters (Νp, C R ); Set mutaton scale factor Fs = F 0 ; Set the sze of the Pareto populaton, Γ_sze; Intalze generaton counter k = 0; Generate a populaton Φ = x, x, K, x } of n-dmensonal floatng-pont vectors; The components of k {, k, k Np, k x, ( =,,Νp) are randomly chosen wthn the range [0,]; Γ = { }; // Create an ntal empty Pareto populaton // Repeat for = to Νp do // create populaton Φ of the new generaton // Mutaton step: ) Generate a mutant vector x usng Eq. (0), k Crossover step: ) Generate a tral vector y, k by crossng x, k and x, k usng Eq. (). Soluton Interpretaton Step: // Buld the phenotypes correspondng to the genotypes x, k and y, k // x, k P = Topologcal_orderng_encodng ( x, k ); y, k P = Topologcal_orderng _encodng ( y, k ); // Decode phenotypes to actual solutons for SALBP- usng the scheme presented n sub-secton.. Then evaluate the cost of each soluton usng Eq.() // f MO SALBP- verson then x,k ( x k Cost P ) = F (decode_salbp-( P, )) y,k k Cost( P ) = F (decode_salbp-( P )) else f MO SALBP- verson then x,k ( x k Cost P ) = F (decode_salbp-( P, )) y,k ( k Cost P ) = F (decode_salbp-( P )) endf Acceptance step: y,k f ( x,k Cost P ) < Cost( P ) then x, k+ = y, k else x, k+ = x, k endfor Update Pareto Populaton Step: // Check each one of the ndvdual solutons n Φ whether consttutes a Pareto soluton // cφ = cγ = 0 ; // ntalze counters for the members n Φ and Γ, respectvely // Whle (cγ Γ_sze) and (cφ Νp) do cφ = cφ + ; Compare Φ(cΦ) (.e., the cφ member of Φ) wth all Pareto solutons n Γ; y, y,

16 Internatonal Journal of Producton Research Page of 0 If Φ(cΦ) s not contaned n Γ then If t domnates some Pareto solutons then Add Φ(cΦ) nto Γ and delete the solutons domnated by t; Increment accordngly counter cγ; else f there s empty space n Γ then Add Φ(cΦ) nto Γ. cγ = cγ + ; endf endf endwhle Eltst Preservng Strategy Step: Determne the two elte Pareto solutons n Γ; Randomly select two members n Φ and replace them wth the two elte Pareto solutons from Γ; Adaptaton of parameter Fs Step: Determne worst and average cost functons n Φ; then, adapt Fs usng Eq. (); k = k + // ncrement teraton counter // Untl k > MAXI; // MAXI stands for Maxmum Iteratons // Return Γ ; As t was referred n sub-secton., weghted-sum method s used to construct the composte objectve functons F and F gven by Eqs.() and (), respectvely. In the lterature, there are two general methods to compute the weghts w (=,,Q) for a weghtedsum objectve functon wth Q objectves: the fxed-weght method and the random-weght method. The former uses constant weghts satsfyng the relaton, Q w = = w > 0 for all = K,,Q However, as Murata et al. () shown, usng constant weghts wthn an EA the search drecton s fxed, and for ths reason t s dffcult for the search process to obtan a varety of non-domnated solutons. To overcome ths drawback, Murata et al. (), proposed the use of random weghts accordng to the followng formula, where w = random =,, K,Q random + random + L+ random Q random (=,,Q) are non-negatve random numbers. () ()

17 Page of Internatonal Journal of Producton Research 0 Furthermore, Gen and Cheng (00) proposed an adaptve weght approach wthn a MOGA, whch readjusts the weghts by utlzng some useful nformaton from the current populaton. Ths method computes the weghts by, where max z w =, for all =, K,Q () max mn z z and mn z are the maxmal and mnmal values of the th objectve n the populaton. In ths work, a new, self-adapted method for the estmaton of the weghts w and w (used n Eqs.(),()) has been developed. The proposed method s gven by the followng relaton: d w = λ e + µ w = w where, λ and µ are user-defned coeffcents used to normalze the upper and lower bounds of w, respectvely. In ths study, we set λ = µ = 0.. The exponent d, s determned by, () d = c * c () c * s the theoretcal mnmal cycle tme of the assembly lne. Note that the proposed scheme gves to the frst objectve hgher prorty assumng that ths s the most sgnfcant crteron n the composte objectve functon. In SALBP- the man crteron s to mnmze c, hence, w s ncreased whle d approaches zero,.e., when a generated ALB soluton approaches the optmal soluton regardng the cycle tme crteron. Fgure shows the rate of change of the w n respect to the changes of parameter d. It s worth pontng that, the applcaton of the proposed scheme on any b-crtera optmzaton problem s straghtforward provded by the decson maker the man and the secondary optmzaton crtera. < Insert Fgure about here >

18 Internatonal Journal of Producton Research Page of 0. Numercal Results and dscusson. Expermental Setup MODE was compared aganst two representatve MOGAs dentfed n the lterature: the frst s a Pareto-nched GA proposed by Km et al. () for solvng MO SALBP, whle the second s a Pareto weght-sum GA developed by Murata et al. () to address MO FSSP. We wll refer to these algorthms wth the abbrevatons MOGA and MOGA, respectvely. Although MOGA was ntroduced for FSSP, as wll be explaned below, very easly t can be extended and appled on SALBP, as well. Three versons of MODE were mplemented each one dffer on the way the weghts n the objectve functon are estmated. The frst verson uses fxed and equal value weghts ( w = w = 0.), the second verson uses random weghts estmated by Eq. (), and the thrd verson uses the proposed adaptve scheme gven by Eq. (). In the followng analyss we wll refer to the three MODEs as, MODE (wth fxed weghts), MODE (random weghts), and MODE (adaptve weghts), respectvely. All the heurstcs were mplemented n Delph Pascal and run on a Pentum (. GHz) PC. The experments were carred out on known ALBP benchmarks taken from the open lterature (Scholl ). Note that the upper bounds on the optmal objectve functon values for these benchmarks concern the mnmal cycle tme. In the experments we nclude the avalable test nstances concernng the followng fve ALBPs: Buxey (n=, n#=), Sawyer (n=, n#=), Gunther (n=, n#=0), Klbrdge (n=, n#=), and Tonge (0, n#=). n denotes the number of the tasks n the correspondng precedence graphs and n# the number of the test nstances ncluded n the specfc ALBP. To be far wth the stochastc behavor of the fve heurstcs, we run each one of them ten tmes over every test nstance and the solutons qualty were averaged. Ths means that, each heurstc was run over (++0++) 0=0 test experments n total. All the examned heurstcs were defned to evolve a fxed sze populaton of n ndvdual solutons, and run for a maxmum of 00n teratons.

19 Page of Internatonal Journal of Producton Research 0 Remark : MOGA combnes a Pareto GA and a nched GA. Rankng s performed usng the Goldberg s rankng method (Goldberg ). Ths method assgns to Pareto solutons the same rank and to all the others solutons a some less desrable rank. The nche radus n the shared ftness functon was defned to be equal to popsze, wth popsze denotng the populaton sze. Chromosomes are nteger strngs and are mapped nto ALB solutons through sequence-orented representaton. If a task sequence breaks the precedence constrants then a sutable reparng method s appled to correct t. Crossover and mutaton are performed through the partally mapped (PMX) crossover and recprocal exchange, respectvely. The rates of crossover and mutaton operators were set (same as n Km et al. ()) equal to 0. and 0., respectvely. The reparng procedure s also appled after the applcaton of these operators on the chromosomes. Selecton s done by a tournament strategy (Goldberg ). Remark : In MOGA, a chromosome s a permutaton of the nteger numbers,,,n, wth each gene n the chromosome denotng a dfferent task label. A scalar ftness functon s used formulated as a weghted sum functon wth the weghts estmated usng Eq. (). For each chromosome x, =,,,popsze the probablty of beng selected for reproducton s gven by popsze the rato ( f ( x ) f ) ( f ( x ) f ). Where, ( ) and mn = mn f x s the ftness functon of x f mn the mnmum populaton ftness functon. Once these selecton probabltes are estmated for all the vectors n the entre populaton, selecton s performed usng the roulette wheel strategy (Goldberg ). Crossover s performed by a two-pont crossover procedure wth a rate equal to.0, whle mutaton s performed va the shft mutaton operator wth a rate /n. To map a chromosome to a feasble ALB soluton we used the same representaton mechansm, as well as, the same reparng method as n MOGA. The tasks n an ALB soluton are assgned to the statons accordng to the scheme descrbed n sub-secton.. MOGA mantans a separate set wth Pareto solutons, and apples an eltst strategy wth these solutons on the entre populaton.

20 Internatonal Journal of Producton Research Page of 0. Choce of the control parameters settngs for MODE Much nvestgaton on the selecton of the approprate settngs of the control parameters (populaton sze Νp, crossover rate C R [0,], and mutaton-scale factor Fs (0,]) was undertaken n prelmnary tests. Here, we descrbe the expermental desgn methodology used to determne these settngs. In partcular, two levels of Νp (n, n) were examned. C R was defned to take values wthn the dscrete range {0., 0., 0., 0., 0.}, whle for F s two dfferent control schemes were used: (a) a statc scheme based on whch Fs takes values n the range [0., 0., 0.,.], and (b) the proposed dynamc scheme gven by Eq. (). < Insert Table about here > < Insert Table about here > After runs of MODE algorthm over representatve ALBP nstances wth the above control schemes we determned that the best results obtaned were due to the combnaton Νp=n, C R =0. and Fs beng estmated by Eq.(). Table presents the effect of C R, and F s (when Νp=n) on MODE s performance, over Buxey s nstances. For each dfferent par ( F s,c R ) two numbers are reported n ths table. The frst number corresponds to the mean percentage devaton of the generated cycle tme from the exstng optmum soluton, and the second number enclosed n brackets corresponds to the mean % effort (see Eq ()) spent by the algorthm untl attaned ths soluton. Kopt % effort= 00 MaxK where K opt s the number of teratons attaned the best soluton and MaxK=00n s the maxmum permtted number of teratons. As one can see from Table, the best results are due to ( F s,c R )=(adaptve, 0.). Wth these settngs MODE acheved ALB solutons wth a mean devaton from the global optmum equal to 0.% after spendng approx. the % of the total permtted teratons. Hence, n the followng dscusson, all the expermental results obtaned by MODE heurstcs are due to the above optmal combnaton of settngs. ()

21 Page of Internatonal Journal of Producton Research 0. Comparson of MODE and MOGA heurstcs Tables and dsplay the comparatve results obtaned by the fve heurstcs over the benchmarks nstances descrbed n sub-secton.. The results n Table concern the experments wth the objectve gven by Eq.(), whle the results presented n Table concern the experments wth the objectve gven by Eq.(). The frst and second columns of Tables and ndcate the problems tested and ther sze, respectvely, whle the thrd column of the tables ndcates the method used. The rest columns of the tables provde the followng nformaton: c%dev = the average relatve devaton from optmum n percentage; estmated by ((c c*)/c*) 00, wth c* the exstng optmal cycle tme, and c the cycle tme of the best soluton generated by a specfc heurstc. BD = the average balance delay tme correspondng to the optmal soluton attaned by a heurstc (ths nformaton s ncluded n Table ). SX = the average smoothness ndex of the optmal soluton attaned by a heurstc (ncluded n Table ). Compound cost = ncludes the best, worst, and mean populaton costs (Eq. ()) generated by each heurstc. %effort = denotes the convergence rato of a heurstc n % gven by Eq. (). cpu-tme = the average actual processng tme n seconds spent by each heurstc untl the convergence to the best possble soluton. < Insert Table about here > As one can observe from Table the best results concernng c%dev have been obtaned by MODE wth a mean offset from optmum approx. equal to 0.% (BD ) for Buxey s problems,.% (BD ) for Sawyer s problems, 0.% (BD ) for Gunther s problems, etc. The second best performance s reported by MODE wth (c%dev, BD) approx. equal to (0.%, ) for Buxey s problems, (.%, ) for Sawyer s problems, and so on. Smlar hgh performance for MODE and MODE heurstcs s reported n Table. Agan, the

22 Internatonal Journal of Producton Research Page of 0 proposed MODE outperformed all the other heurstcs generatng solutons of hgher qualty n respect to both objectves, ether c%dev, or SX. Near to ths performance stays that attaned by MODE. Furthermore, a sgnfcant observaton from these tables s that the use of SX as the second optmzaton crteron n the objectve functon results n much hgher qualty solutons (ndependently of the heurstc used) than those obtaned when usng BD as second optmzaton crteron. Another observaton from Tables and s that usng random weghts n the compound objectve functon of the MODE heurstc (case of MODE) results to a lower qualty performance n comparson to the other heurstcs. In regard to the cpu-tme spent by the heurstcs, the two MOGAs are n general faster than MODEs heurstcs (see last column n Tables and, respectvely) wth MOGA beng the fastest. < Insert Fgure about here > For better llustraton of the generated results we bult Fgures and. In partcular, Fg. (a) shows graphcally the fluctuaton of c%dev over the fve ALBPs when BD s used as second optmzaton crteron. The assocated fluctuaton of the average BD values s shown n Fg. (b). It s clear from Fg. that MODE s superor from the other heurstcs. Fg. shows the fluctuaton of the mean c%dev (Fg. (a)) and the assocated average SX values (Fg. (b)) n regard to the compound objectve gven by Eq. (). < Insert Fgure about here > Fnally, Fg. shows the %effort spent by the fve heurstcs over the varous benchmarks n regard to the objectves of Eq. () (Fg. (a)) and Eq. () (Fg. (b)). As one can see from these fgures, MODE (the lowest curve) spends n average less teratons untl the convergence to the near-optmum soluton than the other heurstcs. < Insert Fgure about here >. Conclusons Any varant of the smple assembly lne balancng problem (SALBP) s NP-hard and thus, t s justfed to address large-sze nstances of the problem through the use of heurstcs. Ths paper ntroduced a mult-objectve (MO) dfferental evoluton (DE) based approach for

23 Page of Internatonal Journal of Producton Research 0 solvng the b-crtera SALBP. The man objectve was to mnmze the cycle tme of the lne and secondary objectves to mnmze balance delay tme and workload smoothness ndex. MODE dffers from exstng MO populaton heurstcs n at least four features: the encodng scheme used to represent the feasble ALB solutons, the evaluaton mechansm to compute the multple objectves, the procedure for generatng the new ndvdual solutons, and n the way t seeks and mantans the set of the non-domnated solutons. Partcularly, MODE has the followng man characterstc: (a) utlzng a robust encodng scheme maps real-valued vectors (genotypes) to nteger strngs correspondng to feasble ALB solutons (phenotypes). (b) Every objectve functon assgnng a cost value to a genotype s formulated as a weghted-sum of the ndvdual objectves wth the weght coeffcents beng dynamcally adjusted by a new effcent method. The applcaton of ths method on any b-crtera optmzaton problem s straghtforward provded by the decson maker (DM) the man and the secondary optmzaton crtera. (c) Mutant vectors are generated usng a modfed effcent strategy. (d) It mantans and updates teratvely a set of non-domnated solutons separately of the actual evolvng populaton, as an attempt to obtan qualty and dverse Pareto-optmal solutons. (e) It uses an eltst strategy to preserve non-domnated solutons found over generatons from gettng lost. MODE s smple, and very easly mplemented. Extensve expermental comparsons over publc avalable ALB benchmarks between MODE and two exstng MO evolutonary algorthms showed a superor performance for the former n terms of qualty of solutons. In practce, many MO optmzaton problems (MOOPs) have multple conflctng objectve functons expressed n dfferng unts, and wth an nverse, nonlnear relatonshp among themselves. These objectves may be even mprecse (or fuzzy) n nature to be defned. In ts present form MODE cannot address such problems. Hybrdzng MODE wth mechansms borrowed from the feld of nonlnear goal programmng (GP) (Lee ) may result to a promsng optmzaton tool for these problems. GP needs DM to provde a numerc goal (together wth a prorty level) for each objectve, and then seeks for a soluton that mnmzes the weghted-sum of the devatons of the objectve functons from ther respectve

24 Internatonal Journal of Producton Research Page of 0 goals. Ths dea wll consttute a central research drecton n the near future. Moreover, there are MOOPs wth a huge set of Pareto-optmal solutons for whch evaluatng ths set to select the best one becomes unpractcal for DM. Perhaps, a soluton for these problems can be obtaned by tryng to get compromses, based on the DM s nformaton. Compromse solutonbased ftness assgnments (Gen and Chen 00), s an nterestng approach to be nvestgated wthn MODE. Moreover, ths work s lmted n the determnstc sngle-model ALBP, however, t represents a good start pont for further studes focused on more dffcult ALBPs such as the stochastc or dynamc ALBP. In realty, tasks processng tmes are rarely determnstc and may vary more or less. When these varatons are consderable then we have the stochastc ALBP. Dynamc ALBP consders operaton tmes beng varyng over tme, e.g. due to learnng effects, or successve mprovements of the producton process (Scholl ). Our ntuton s that, MODE can be rather easly extended to address the stochastc SALBP provded that a sutable statstcal model wll be developed that transforms the stochastc task tmes to determnstc ones, and realzed dfferent cycle tmes so that to avod blockng and starvng of the workstatons. Another avenue for further research s to consder the mxed-model ALBP (MALBP). Ths problem s much more complex than SALBP snce, the attempt s to manufacture dfferent versons (models) of the same basc product n the same lne (e.g., PCs wth or wthout DVD drve, wth varous CPU types, etc.) n arbtrarly ntermxed sequence. A frst dea s to address the feasblty MALBP;.e., gven the cycle tme c and the number m of the statons determne whether or not, a feasble mxed-model assgnment wth m statons exst. Acknowledgments The author thanks the anonymous referees for ther valuable comments and suggestons on ths paper. Ths work s ntegrated n the Innovatve Producton Machnes and Systems (I*PROMS) Network of Excellence.

25 Page of Internatonal Journal of Producton Research 0 References Al M.M. and Törn A. 0, Populaton set-based global optmzaton algorthms: some modfcatons and numercal studes, Computers & Operatons Research,, 0-. Bäck T. (), Evolutonary algorthms n theory and practce. Oxford Unversty Press, New York. Baybars I.,, A survey of exact algorthms for the smple assembly lne balancng problem, Management Scence,, 0-,. Celano G., Fchera S., Grasso V., Commare U. Perrone G., An evolutonary approach to mult-objectve schedulng of mxed model assembly lnes, Computers & Industral Engneerng,, -. Chen R.-S., Lu K.-Y., and Yu S.-C., 0, A hybrd genetc algorthm on mult-objectve assembly plannng problem, Engneerng Applcatons of Artfcal Intellgence,, -. Coelo C.A.,, A comprehensve survey of evolutonary-based multobjectve optmzaton, Knowledge and Informaton Systems, /, -. Gambern R., Grass A., and Rmn B., 0, A new mult-objectve heurstc algorthm for solvng the stochastc assembly lne re-balancng problem, Int. Journal of Producton Economcs, 0/, -. Gen M. and Cheng R. 00, Genetc algorthms and engneerng optmzaton, A Wley- Interscence publcaton, New York. Goldberg D.E., Genetc algorthms n search, optmzaton, and machne learnng. Addson-Wesley, Readng, MA. Jones D.F., Mrrazav S.K, and Tamz M. 0, Mult-objectve meta-heurstcs: An overvew of the current-state-of-the-art, European Journal of Operatonal Research,, -. Kaelo P. and Al M.M., 0, A numercal study of some modfed dfferental evoluton algorthms, European Journal of Operatonal Research, (), -. Km Y.K., Km Y.-J., and Km Y.,, Genetc algorthms for assembly lne balancng wth varous objectves, Computers Industral Engneerng, /, -.

26 Internatonal Journal of Producton Research Page of 0 Lee S.,, Goal programmng for decson analyss, Auerbach Publshers, Phladelpha. Malakoot B. and Kumar A.,, A knowledge-based system for solvng mult-objectve assembly lne balancng problems, Int. Journal of Producton Research, (), -. Mansour S.A., 0, A mult-objectve genetc algorthm for mxed-model sequencng on JIT assembly lnes, European Journal of Operatonal Research,, -. McMullen P.R. and Frazer G.V.,, Usng smulated annealng to solve a multobjectve assembly lne balancng problem wth parallel workstatons, Int. Journal of Producton Research, /0, -. Murata T., Ishbuch H., and Tanaka H.,, Mult-objectve genetc algorthms and ts applcaton to flowshop schedulng, Computers and Industral Engneerng, /, -. Nearchou A.C., 0, Meta-heurstcs from nature for the loop layout desgn problem, Int. Journal of Producton Economcs, 0/, -. Nearchou A.C and Omrou S.L., 0, Dfferental evoluton for sequencng and schedulng optmzaton, Journal of Heurstcs, (to appear). Onwubolu G.O. and Davendra D., 0, Schedulng flow shops usng dfferental evoluton, European Journal of Operatonal Research,, -. Ponnambalam S. G., Aravndan P, and Nadu M.G., 00, A mult-objectve genetc algorthm for solvng assembly lne balancng problem, Int. Journal of Advanced Manufacturng Technology,, -. Rekek B., De Lt P., Pellchero F., L Englse T., Fouda P., Falkenauer E., and Delchambre A., 0, A multple objectve groupng genetc algorthm for assembly lne desgn, Journal of Intellgent Manufacturng,, -. Scholl A.,, Balancng and sequencng of assembly lnes, Physca-Verlag publ., Hedelberg, Germany. Scholl A. and Becker C., 0, State of the art exact and heurstc soluton procedures for smple assembly lne balancng, European Journal of Operatonal Research, (), -.

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28 Internatonal Journal of Producton Research Page of 0 Lst of Fgures Fgure. A precedence graph for an -tasks ALBP. Fgure. The applcaton of topologcal orderng encodng method on genotype ψ = (0., 0., 0.0, 0., 0., 0., 0.0, 0.). Fgure. The general structure of the proposed MODE. Fgure. Rate of change of the weghtng coeffcent w n respect to the parameter d. Fgure. Comparsons of the three MODEs and the two MOGAs n regard to: (a) c%dev and (b) the mnmum balance delay tme on the selected ALB benchmarks. Fgure. Expermental comparsons of the heurstcs n regard to: (a) c%dev and (b) the mnmum smoothness ndex on the selected ALB benchmarks. Fgure. Comparsons of the fve heurstcs n regard to the % effort and the number of the tasks to be assembled. (a) Mnmzng c and BD. (b) Mnmzng c and SX. Lst of Tables Table. Choosng the correct settngs for the parameters C R and F s. Average costs over characterstcs runs concernng Buxey s ALB benchmarks. Table. Mnmzng cycle tme and balance delay tme. Table. Mnmzng cycle tme and smoothness ndex.

29 Page of Internatonal Journal of Producton Research 0 Fgure

30 Internatonal Journal of Producton Research Page of 0 Fgure

31 Page of Internatonal Journal of Producton Research 0 Fgure

32 Internatonal Journal of Producton Research Page of 0 Fgure,0 w 0, d

33 Page of Internatonal Journal of Producton Research 0,0 c%dev,0 BD,0,0,0 0, Fgure MODE MODE MODE MOGA MOGA Buxey Gunther Klbrdge Sawyer Tonge (a) ALBPs MODE MODE MODE MOGA MOGA Buxey Gunther Klbrdge Sawyer Tonge (b) ALBPs

34 Internatonal Journal of Producton Research Page of 0 Fgure MODE MODE MODE MOGA MOGA,0,0,0 c%dev,0,0 0,0 Buxey Gunther Klbrdge Sawyer Tonge ALBPs (a) MODE MODE MODE MOGA MOGA SX 0 0 Buxey Gunther Klbrdge Sawyer Tonge ALBPs (b)

35 Page of Internatonal Journal of Producton Research ` Fgure MODE MODE MODE MOGA MOGA tasks numbers (a) MODE MODE MODE MOGA MOGA 0 0 tasks numbers (b)

36 Internatonal Journal of Producton Research Page of 0 Table Fs C R =Θ C R 0.. (.). (.). (.). (.) 0. (.). (.) 0.. (.). (.). (.) 0. (.). (.). (.) 0.. (.0). (.). (.0). (.). (.). (.0).. (.). (.). (.). (.) 0. (.). (.) F s =Θ F s.0 (.). (.). (.) 0. (.0). (.). (.) C R

37 Page of Internatonal Journal of Producton Research 0 Table : Compound cost functon ALBP n Method c%dev BD best worst mean %effort cpu-tme (sec) Buxey MODE MODE MODE MOGA MOGA Sawyer MODE MODE MODE MOGA MOGA Gunther MODE MODE MODE MOGA MOGA Klbrdge MODE MODE MODE MOGA MOGA Tonge 0 MODE MODE MODE MOGA MOGA

38 Internatonal Journal of Producton Research Page of 0 Table Compound cost functon ALBP n Method c%dev SX best worst mean %effort cpu-tme (sec) Buxey MODE MODE MODE MOGA MOGA Sawyer MODE MODE MODE MOGA MOGA Gunther MODE MODE MODE MOGA MOGA Klbrdge MODE MODE MODE MOGA MOGA Tonge 0 MODE MODE MODE MOGA MOGA