Arash Motaghedi-larijani, Kamyar Sabri-laghaie & Mahdi Heydari *

Transcription

1 Internatonal Journal of Industral Engneerng & Producton Research December 2010, Volume 21, Number 4 pp ISSN: Solvng Flexble Job Shop Schedulng wth Mult Objectve Approach Arash Motaghed-larjan, Kamyar Sabr-laghae & Mahd Heydar A. Motaghed-larjan, Author s wth the Department of Industral Engneerng, Iran Unversty of scence & technology, Narmak, Tehran, Iran K.. Sabr-laghae, Author wth the same Department Tehran, Iran M. Heydar, Author s a Faculty at the same Department, Tehran, Iran KEYWORDS Flexble jobshop schedulng, Mult crtera decson makng, global programmng, Genetc algorthm, Hll clmbng ABSTRACT 1. Introducton Fndng good solutons to schedulng problems s of great mportance to ndustry, snce the producton rate and expense of a producton plant depend on the schedules used for controllng the work n the plant. Job-shop schedulng problem (JSP), whch s among the hardest combnatoral optmzaton problems [1], s a branch of producton schedulng. It s well known that ths problem s NP-hard [2]. The classcal JSP schedules a set of jobs on a set of machnes wth the objectve to mnmze a certan crteron, subjected to the constrant that each job has a specfed processng order through all machnes whch are fxed and known n advance. Flexble job-shop problem (FJSP) s an extenson of the classcal JSP that allows one operaton whch can be processed on one machne out of a set of alternatve machnes. So, ths knd of schedulng problem reduces machne constrants, and enlarges searchng scope of Correspondng author. M. Heydar Emal: a.motaghed.e@gmal.com, K_sabr@nd.ust.ac.r, Mheydar@ust.ac.r Paper frst receved May , and n revsed form Dec In ths paper flexble job-shop schedulng problem (FJSP) s studed n the case of optmzng dfferent contradctory objectves consstng of: (1) mnmzng makespan, (2) mnmzng total workload, and (3) mnmzng workload of the most loaded machne. As the problem belongs to the class of NP-Hard problems, a new hybrd genetc algorthm s proposed to obtan a large set of Paretooptmal solutons n a reasonable run tme. The algorthm utlzes from a local search heurstc for mprovng the chance of obtanng more number of global Pareto-optmal solutons. The soluton method uses from a perturbed global crteron functon for gudng the search drecton of the hybrd algorthm. Computatonal experences show that the hybrd algorthm has superor performance n contrast to prevous studes IUST Publcaton, IJIEPR, Vol. 21, No. 4, All Rghts Reserved. practcable solutons. Flexble job shop schedulng problem have many applcatons n compettve envronment. For example, t s used n flexble manufacturng systems. A flexble manufacturng system conssts of several CNC machnes. A CNC machne s a mult taskng machne. So flexble job shop schedulng s applcable for flexble manufacturng systems. Bruker and Schle (1990) were among the frst to address ths problem [3]. It s closer to the real manufacturng stuaton. Because of the addtonal needs to determne the assgnment of operatons on the machnes, FJSP s more complex than JSP, and ncorporates all the dffcultes and complextes of JSP. The problem of schedulng jobs n FJSP could be decomposed nto two sub-problems: a routng sub-problem, whch s assgnng each operaton to a machne out of a set of capable machnes and a schedulng sub-problem, whch s sequencng the assgned operatons on all selected machnes n order to obtan a feasble schedule wth optmzed objectves [4]. For solvng the realstc case wth more than two jobs, two types of approaches have been used: herarchcal approaches and ntegrated approaches. In herarchcal approaches assgnment of operatons to machnes and the sequencng of operatons on the

2 198 A. Motaghed-larjan, K. Sabr-laghae & M. Heydar Solvng Flexble Job Shop Schedulng wth resources or machnes are treated separately,.e., assgnment and sequencng are consdered ndependently, where n ntegrated approaches, assgnment and sequencng are not dfferentated. Herarchcal approaches are based on the dea of decomposng the orgnal problem n order to reduce ts complexty. Ths type of approach s natural for FJSP snce the routng and the schedulng sub-problem can be separated. Brandmarte (1993) was the frst to apply the decomposton approach nto the FJSP [5]. He solved the routng sub-problem by usng some exstng dspatchng rules and then focused on the schedulng sub-problem, whch was solved by usng a tabu search heurstc. After that Paull (1995) appled herarchcal approach [6]. The schedulng problem n a flexble job shop s at least as hard as the job shop problem, whch s consdered one of the most dffcult problems n combnatoral optmzaton [7]. Zhang and Gen (2005) proposed a method called multstage based genetc algorthm to solve fjsp problem [8]. Sad Mehrabad and Fattah (2007) presented a mathematcal model and tabu search algorthm to solve the flexble job shop schedulng problem wth sequence-dependent setups [9]. They used a herarchcal approach wth two heurstcs to solve ths problem. The frst one for assgnng each operaton to a machne out of a set of two capable machnes and the second one for sequencng the assgned operatons on all machnes n order to obtan a feasble schedule for mnmzng the Makespan. Fattah, Sad Mehrabad and Jola (2007) presented a mathematcal model and heurstc approaches for flexble job shop schedulng problems [10]. They used mathematcal model to acheve optmal soluton for small sze problems. Snce FJSP s NP-hard problem, they developed two heurstc approaches nvolve of ntegrated and herarchcal approaches to solve the real sze problems and concluded that, the herarchcal algorthms have better performance than ntegrated algorthms and they used an algorthm whch use tabu search and smulated annealng for assgnment and sequencng problems consecutvely. Although an optmal soluton algorthm for the classcal JSP has not been developed yet, there s a tendency to model and solve much more complex verson of the FJSP [1]. The research on the multobjectve FJSP s much less than the mono-objectve FJSP. Hurnk, Jursch, and Thole (1994) proposed a tabu search heurstc [11]. They consdered the reassgnment and reschedulng as two dfferent types of moves. Mastrolll and Gambardella (2000) proposed some neghborhood functons for the FJSP, whch could be used n metaheurstc optmzaton technques [12]. The most mportant ssue n employng meta-heurstcs for combnatoral optmzaton problems s to develop an effectve problem mappng and soluton generaton mechansm. If these two mechansms are devsed successfully, t s possble to fnd good solutons to the gven optmzaton problem n acceptable tme. Parsopoulos and Vrahats (2002) conducted the frst research on the partcle swarm optmzaton method n mult-objectve optmzaton problems [13]. Kacem, Hammad, and Borne (2002) proposed a genetc algorthm controlled by the assgnment model whch was generated by the approach of localzaton (AL) to mono-objectve and mult-objectve FJSP [14,15]. They used the ntegrated approach consderng assgnment and schedulng at the same tme. Rgao (2004) developed two heurstcs based on tabu search: a herarchcal procedure and a multple start procedure [16]. Recently, Xa and Wu (2005) proposed a hybrd algorthm usng partcle swarm optmzaton (PSO) assgnment and smulated annealng (SA) schedulng to optmze mult-objectve FJSP, respectvely [4]. Low, Yp and WU (2006) made use of one of the multple objectve decson-makng methods, a global crteron approach, to develop a mult-objectve model for solvng FMS schedulng problems wth consderaton of three objectves [17]. Zhang, Zheng, and Wu (2007) proposed a hybrd of ant colony and partcle swarm optmzaton algorthms to solve the mult-objectve flexble job-shop schedulng problem based on the analyss of objectves and ther relatonshp [18]. Gao, Gen, Sun and Zhao (2007) developed a new genetc algorthm hybrdzed wth an nnovatve local search procedure (bottleneck shftng) for the problem [19]. Xng, Chen and Yang(2009) constructed a knowledge-based heurstc algorthm whch combned heurstc algorthm wth emprcal knowledge for the mult-objectve flexble job shop schedulng problem [20]. Also Xng, Chen and Yang (2009) presented a smulaton model to solve the multobjectve flexble job shop schedulng problem [21]. To our knowledge, research on mult-objectve FJSP s rather lmted, and most tradtonal optmal approaches used only one optmzaton algorthm for solvng mult-objectve FJSP, thereupon n ths paper, an effectve hybrd genetc algorthm s proposed to solve the mult-objectve flexble jobshop schedulng problems and accordngly comparsons wth other smlar works s added. In ths research a memetc algorthm s appled to a flexble job shop schedulng problem n whch three contradctory objectves of makespan, the total workload of machnes and maxmum workload of the machnes has been consdered. Unlke other smlar works that have ntegrated the contradctory objectves by weghtng method, a compromse programmng approach s used to deal wth these three objectves. The effect of applyng compromse programmng method s compared wth weghtng method. The remander of ths paper s organzed as follows: The notaton and problem descrpton are ntroduced n secton 2. Secton 3 presents method of the global crteron. Secton 4 descrbes genetc algorthm for global search and hll clmbng for local search. In

3 A. Motaghed-larjan, K. Sabr-laghae & M. Heydar Solvng Flexble Job Shop Schedulng wth 199 secton 5, computatonal experments performed wth proposed approach are reported followed by the comparson wth other heurstc methods. Fnally some concludng remarks are made n secton Problem Formulaton In the flexble job shop schedulng, we have two challenges. The frst one s to assgn each operaton to a machne, and the second one s to sequence the operatons on each machne n order to mnmze makespan. The flexble job shop schedulng problem can be defned by: There are n jobs, ndexed by, and these jobs are ndependent one of each other. Each job has an operatng sequence, denoted by J (precedence constrants). J also denotes the -th job f no confuson. Each operatng sequence s an ordered set of operatons O,j for j = 1,..., l. There are m machnes, ndexed by k (the k-th machne s denoted by M k ). For each operaton O,j, there s a set of machnes capable of performng t. The set s denoted by M,j, M, {1,..., m} (routng constrants), f there s. M, j j M for at least one operaton, t s partal flexblty FJSP (P-FJSP); whle there s M,j = M for each operaton, t s total flexblty FJSP (T- FJSP) [14,15]. The processng tme of an operaton O,j on machne k s predefned and denoted by t,j,k. Tab. 1. Problem 4 5 wth 12 operatons (total flexblty) Job O,j M 1 M 2 M 3 M 4 M 5 J 1 J 2 J 3 J 4 O 1, O 1, O 1, O 2, O 2, O 2, O 3, O 3, O 3, O 3, O 4, O 4, Tab. 2. Problem 8 8 wth 27 operatons (partal flexblty) Job O,j M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 J 1 J 2 J 3 O 1, O 1, O 1, O 2, O 2, O 2, O 2, O 3, O 3, O 3, O 4, J 4 O 4, O 4, J 5 O 5, O 5, O 5, O 5, J 6 O 6, O 6, O 6, J 7 O 7, O 7, O 7, J 8 O 8, O 8, O 8, O 8, In ths paper, the followng crtera are to be mnmzed: (1) the maxmal completon tme of machnes,.e., the makespan (f 1 ); (2) the maxmal machne workload,.e., the maxmum workng tme spent on each machne (f 2 ); (3) the total workload of machnes, whch represents the total workng tme over all machnes (f 3 ). In schedulng problems makespan have a key role and t s used n objectve functon. If makespan reduce, the order can be delvered to customer earler. Second crteron balances workng tme spent on each machne. Fnally thrd crteron leads to reduce the total workng tme over all machnes. It s clear that all crtera conflct together. For example, f second crteron reduces thrd crteron maybe ncreases or mnmzng makespan lead to ncreasng second crteron. Durng the process of solvng ths problem, the followng assumptons are made: (1) Each operaton cannot be nterrupted durng ts performance (non-preemptve condton);

4 200 A. Motaghed-larjan, K. Sabr-laghae & M. Heydar Solvng Flexble Job Shop Schedulng wth (2) Each machne can perform at most one operaton at any tme (resource constrant); (3) Each machne becomes avalable to other operatons once the operatons whch are currently assgned to be completed. (4) All machnes are avalable at t = 0. (5) All jobs can be started at t = 0. (6) The precedence constrants of the operatons n a job can be defned for any par of operatons; 3. Method of the Global Crteron The method of the global crteron s also sometmes called compromse programmng In ths method, the dstance between some reference pont and the feasble objectve regon s mnmzed. The analyst has to select the reference pont and the metrc for measurng the dstances. All the objectve functons are thought to be equally mportant Dfferent Metrcs Here we examne the method where the deal objectve vector s used as a reference pont and L p -metrcs are used for measurng. In ths case, the L p -problem to be solved s: Mnmze Subject to x S Tab. 3. Problem wth 30 operatons (total flexblty) 1 k p p f ( x) z (1) = 1 From the defnton of the deal objectve vector z we know that f ( x) z for all = 1, L, k and all x S. Ths s why no absolute values are needed f we know the global deal objectve vector. If the global objectve vector s not known, the method does not necessarly work as t should. If the deal objectve vector s replaced by some other vector, t must be selected carefully. Pessmstc reference ponts must be avoded snce the method cannot fnd solutons better than reference pont. We should know that, the soluton obtaned by the model depends greatly on the value chosen for p. Wdely used choces are. p = 1, 2 or. Instead of the terms f ( x) z, denomnators may be added to problem (1.1) to normalze the components, that s, to use the terms f ( x) z / z nstead. Some other denomnators, lke z nad z, can also be used. The reason for employng denomnators s that (7) Settng up tme of machnes and move tme between operatons are neglgble; (8) Machnes are ndependent from each other; (9) There are no precedence constrants among operatons of dfferent jobs. (10) Nether release tmes nor due dates are specfed. Job O,j M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 10 J 1 O 1, O 1, O 1, sometmes t s worthwhle to use relatve dstances n the calculatons. For example, usng the components of z forms the contour of the metrc to reflect better the locaton of the deal objectve vector. Naturally, the denomnators z cannot be used f some of them equals zero [22]. Brefly, the method of the global crteron s a smple method to use f the am s smply to obtan a soluton where no specal hopes are set. Ths s a method that combnes all objectve functons to form a sngle functon. Advantages of the method are: (1) t focuses on a sngle objectve wth lmts on others, (2) t always provdes a weakly Pareto optmal pont, assumng the formulaton gves a soluton, (3) t s not necessary to normalze the objectve functons, (4) t gves Pareto optmal soluton f one exsts and s unque. The only dsadvantage s that the optmzaton problem may be nfeasble f the bounds on the objectve functons are not approprate. 4. Hybrd Algorthm Wth Hll Clmbng Approach 4.1 Genetc Algorthms Genetc algorthm (GA) s a stochastc search method for optmzaton problems based on the mechancs of natural selecton and natural genetcs (.e., survval of the fttest). GA has demonstrated consderable success n provdng good solutons to many complex optmzaton problems and receved more and more attentons durng the past three decades. When the objectve functons to be optmzed n the optmzaton problems are multmodal or the search spaces are partcularly rregular, algorthms need to be hghly robust n order to avod gettng stuck at a local optmal soluton. The advantage of GA s just able to obtan the global optmal soluton farly. In addton, GA does not requre the specfc mathematcal analyss of optmzaton problems, whch makes GA easly coded by users who are not necessarly good at mathematcs and algorthms. One of the mportant techncal terms n

5 A. Motaghed-larjan, K. Sabr-laghae & M. Heydar Solvng Flexble Job Shop Schedulng wth 201 GA s chromosome, whch s usually a strng of symbols or numbers. A chromosome s a codng of a soluton of an optmzaton problem, not necessarly the soluton tself. GA starts wth an ntal set of randomly generated chromosomes called a populaton. The number of ndvduals n the populaton s a predetermned nteger and s called populaton sze. All chromosomes are evaluated by the so-called evaluaton functon, whch s some measure of ftness. A new populaton wll be formed by a selecton process usng some samplng mechansm based on the ftness values. The cycle from one populaton to the next one s called a generaton. In each new generaton, all chromosomes wll be updated by the crossover and mutaton operatons. The revsed chromosomes are also called offsprng. The selecton process selects chromosomes to form a new populaton and the genetc system enters a new generaton. After performng the genetc system a gven number of cycles, we decode the best chromosome nto a soluton whch s regarded as the optmal soluton of the optmzaton problem [23]. The am of ths secton s to ntroduce an effectve GA for solvng complex optmzaton problems. The overall structure of our GA can be descrbed as follows: Intal Populaton: The ntal chromosomes are obtaned by a random procedure Ftness Functon: Chromosomes n the populaton wll be evaluated by the objectve functon whch s a combnaton of f1, f 2 and f 3 accordng to the descrpton n Secton problem formulaton. In ths paper the global crteron method s used to transform the dfferent objectves nto one objectve. Accordngly, our combned objectve functon can be expressed as, Mnmze Tab. 4. Problem wth 56 operatons (total flexblty) ( ) ( ) ( ) F= f x z f x z f x + + z z z z Where z = ( z 1, z2, z3 ), as explaned n secton Method of global crteron, s the deal vector. In order to have a good estmaton of deal vector, n ths paper, we have run a genetc algorthm for each ndvdual objectve functon to obtan z vector. Job O,j M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 10 J 1 J 2 O 1, O 1, O 1, O 1, O 2, O 2, O 2, O 2, J 3 O 3, O 3, O 3, O O 4, J 4 O 4, O 4, O 4, J 5 O 5, O 5, O 5, O 5, J 6 O 6, O 6, J 7 O 7, O 7,

6 202 A. Motaghed-larjan, K. Sabr-laghae & M. Heydar Solvng Flexble Job Shop Schedulng wth Contnuance Tab. 4. Problem wth 56 operatons (total flexblty) Job O,j M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 10 J 8 O 8, O 8, O 8, O 8, J 9 O 9, O 9, O 9, O 9, J 10 O 10, O 10, O 10, O 10, J 11 O 11, O 11, O 11, O 11, J 12 O 12, O 12, O 12, O 12, J 13 O 13, O 13, O 13, O 13, J 14 O 14, O 14, O 14, O 14, J 15 O 15, O 15, O 15, O 15, Representaton of A-strng O 1,1 O 1,2 O 1,3 O 2,1 O 2,2 O 2,3 O 3,1 O 3,2 O 3,3 O 3,4 O 4,1 O 4, M 1 M 2 M 3 M 4 M 5 M 1 M 2 M 3 M 4 M 5 Fg. 1. Representaton of A-strng

7 A. Motaghed-larjan, K. Sabr-laghae & M. Heydar Solvng Flexble Job Shop Schedulng wth 203 Representaton of B-strng O 1,1 O 2,1 O 1,2 O 3,1 O 4,1 O 2,2 O 1,3 O 3,2 O 2,3 O 4,2 O 3,3 O 3,4 In a generaton, after the evaluaton of each chromosome, the best ftness wll be the lowest one and should be transferred to the next generaton durng the search process Encodng Scheme: In order to apply GA successfully to a mult-objectve FJSP problem, approprate representaton of chromosomes s of a great mportance. In ths matter, Zhang G. et al.(2008) dvded the chromosomes nto two parts, A-strng and B-strng. In ths mproved representaton, A-strng represents machnes assgned to operatons and B-strng defnes the sequence of operatons. On the other hand n ths representaton the feasblty s consdered automatcally [24]. So there s no need of any repar mechansm to mantan feasblty. For nstance, consder the problem n the table 1. The problem s to execute 4 jobs on 5 machnes. An nstance of a possble encodng of A and B-strng can be seen n Fg. 1 and Fg. 2 respectvely. The elements of A-strng represent the machnes assgned to operatons. The frst element s representatve of operaton O 1,1, the second s representatve of operaton O 1,2 and n lke manner the last element s representatve of operaton O 4,2. The number located n each element s the number of machne assgned to each operaton. B-strng s as long as A-strng. Numbers n B-strng show a sequence of job numbers n whch job number occurs n tmes. The schedule s always feasble f each operaton s replaced by the correspondng job ndex. As llustrated n Fg 2, one possble B-strng may be The lst of ordered operatons related to ths B-strng could be translated as: O 1,1 - O 2,1 - O 1,2 - O 3,1 - O 4,1 - O 2,2 - O 1,3- O 3,2 - O 2,3 - O 4,2 - O 3,3 - O 3,4. When an ndvdual s decoded, at frst the Fg. 2. Representaton of B-strng Chld sequence of operatons s extracted from B-strng. Then Parent 1 Parent Chld 2 Fg. 3. Cross over of A-strng operatons are assgned to machnes accordng to A- strng Crossover Operatons: Crossover s the process of takng two parent solutons and producng from them a chld. Crossover operator s appled to the matng pool wth the hope that t creates a better offsprng. Therefore, Crossover s a recombnaton operator that proceeds n three steps:. The reproducton operator selects at random a par of two ndvdual strngs for the matng.. A cross ste s selected at random along the strng length.. Fnally, the poston values are swapped between the two strngs followng the cross ste [23]. In ths hybrd algorthm the crossover operatons of both A- strng and B-strng perform separately, and ther performance s dfferent. Accordng to the method that [24] used, the crossover operaton for A and B-strng s as follows. For A-strng two postons are randomly chosen from the parents and the whole substrngs between these two postons are swapped and then two new strngs are created. The procedure could be llustrated as Fg 3. In order to perform crossover operaton on B-strng, we dvde all jobs nto two groups by a random procedure, J1 and J2. At the next stage, we retan the jobs from parent 1 that belong to J1 and we do so for parent 2 and J2. Then we remove the other elements of the parents. The remanng elements of parent 1 and parent 2 are respectvely transferred to chld 1 and chld 2 wth the same postons. The empty postons of chld 1 and chld 2 wll be orderly flled wth the elements of parent 2 and parent 1 that belong to ther prevous sequence. Ths procedure could be llustrated as Fg. 5.

8 204 A. Motaghed-larjan, K. Sabr-laghae & M. Heydar Solvng Flexble Job Shop Schedulng wth Mutaton: Another mportant genetc operator s mutaton, Mutaton prevents the algorthm to be trapped n a local mnmum. Mutaton plays the role of recoverng the lost genetc materals as well as for randomly dsturbng genetc nformaton. It s an nsurance polcy aganst the rreversble loss of genetc materal Mutaton s vewed as a background operator to mantan genetc dversty n the populaton. Chld 1 Parent 1 Parent 2 Chld 2 Objectve functon H Iteraton Fg. 5. Comparson of GA and HGA performance on problem 4 5 Objectve functon J 1={1,4} & J 2={2,3} Fg. 4. Cross over of B-strng GA 0 50 Iteraton Fg. 6. Comparson of GA and HGA performance on problem 8 8 It ntroduces new genetc structures n the populaton by randomly modfyng some of ts buldng blocks. Mutaton helps escape from local mnma s trap and mantans dversty n the populaton. The mportant parameter n the mutaton technque s the mutaton probablty. The mutaton probablty decdes how often parts of chromosome wll be mutated. Mutaton s appled wth small probablty [23]. Here s presented just one mutaton operator whch s performed by the B-strng of each chromosome. Zhang G. et al(2008) used the followng procedure; by generatng two random numbers, two postons of B- strng wll be selected, then the substrng between ths two postons wll be nverted to get a new B-strng. The new B-strng wll be replaced wth the old one [24] Hll Clmbng Method In computer scence, hll clmbng s a mathematcal optmzaton technque whch belongs to the famly of local search. It s relatvely smple to mplement, makng t a popular frst choce. Although more advanced algorthms may gve better results, n some stuatons hll clmbng works just as well. Hll clmbng can be used to solve problems that have many solutons, some of whch are better than others. It starts wth a random (potentally poor) soluton, and teratvely makes small changes to the soluton, each tme mprovng t a lttle. When the algorthm cannot see any mprovement anymore, t termnates. Ideally, at that pont the current soluton s close to optmal, but t s not guaranteed that hll clmbng wll ever come close to the optmal soluton. In smple hll clmbng, the frst closer node s chosen, whereas n steepest ascent hll clmbng all successors are compared and the closest to the soluton s chosen. Both forms fal f there s no closer node, whch may happen f there are local maxma n the search space whch are not solutons. Steepest ascent hll clmbng s smlar to best-frst search, whch tres all possble extensons of the current path nstead of only one [25]. In ths paper, hll clmbng approach s used to mprove derved solutons of the GA. In fact, hll clmbng method s appled to change the order of assgned jobs. Wth regard to neghborhood functon hll clmbng approach looks for the best soluton. In ths algorthm, neghborhood functon s just defned to change the sequence of assgned jobs to each machne. Hence, algorthm s just able to mprove C max and doesn t change Max WL and WL Hybrd Algorthm: Now the framework of hybrd algorthm s explaned. As llustrated n Fg. 8 t conssts of fve major steps.

9 A. Motaghed-larjan, K. Sabr-laghae & M. Heydar Solvng Flexble Job Shop Schedulng wth 205 Inputs {GA parameters} Step 1. Generate ntal populaton randomly accordng to the encodng scheme. Step 2. Compute C max, Max WL, WL and then evaluate each chromosome by ftness functon. Step 3. Perform crossover and mutaton operatons on the populaton at hand and then generate new populaton. Step 4. Perform hll clmbng procedure on each chromosome. (a) Search the neghborhood subset to fnd the best possble soluton. (b) If the soluton s of better C max then repeat (a), else go to step 5. Step 5. Extract the best found ftness functon. Step 6. If the termnaton crteron s not met go to step 4; otherwse, go to step 3. The termnaton crteron s a predefned number of generatons. 5. Expermental Results To llustrate the effectveness and performance of the hybrd algorthm proposed n ths paper, the algorthm procedure was mplemented n Java eclpse on a vostro 1500 wth 2.2 GHz CPU and 2 GB Ram. Four representatve nstances based on practcal data have been selected. Tables 1 to 4 represent the mentoned nstances. Each nstance can be characterzed by the followng parameters: number of jobs (n), number of machnes (m), and each operaton O,j of job. Four problem nstances (problem 4 5, problem 8 8, problem 10 10, and problem 15 10) are all taken from [14,15]. In tables 5,6,7 and 8 the column labeled addtve objectve refers to the objectve functon of weghted method that objectve functons are weghted equally. The Hybrd Algorthm s compared wth followng publshed algorthms n the comparatve studes. The comparatve results are shown n Tables 5, 6, 7, 8. For these tables, rows have been labeled as: (1) Star refers to the best soluton found for each sngle objectve functon. (2) A1: Temporal Decomposton refers to [14]; (3) A2: Classc GA refers to [14]; (4) A3: Approach by Localzaton refers to [14]; (5) A4: AL+CGA refers to [14]; (6) A6: PSO+SA refers to [4]; (7) moga refers to [8]; (8) hga refers to [19]; (9) PSO+TS refers to [24]; (10) smulaton modelng refers to [21]; (11) GA+HC refers to our Hybrd algorthm In order to attan the star objectves that that are the best solutons found for each sngle objectve functon, every nstance s run for 100 tmes. Then each nstance accordng to ts scale s run between tmes and all among solutons wth the best objectve functon the most repeated soluton s returned Problem 4 5 At frst, we use a small scale nstance to dsplay the optmzng ablty and evaluate the effectveness of ths Fg. 7. The structure of hybrd algorthm hybrd algorthm. Ths nstance s taken from [15]. We have appled t to the nstance n Table 1. The GA parameters are as follow: populaton sze=200, teraton number=100, mutaton rate=0.1. The best values by the hybrd algorthm are shown as follows: Soluton1: Makespan = 13, Max(Wtw) = 7, Wtw = 33 Soluton2: Makespan = 12, Max(Wtw) = 8, Wtw = 32 Where Makespan s maxmum complete tme, Max(Wtw) s used to represent the crtcal machne workload, and Wtw represents the total workload of all machnes. From Table 5 we could obvously see that the proposed hybrd algorthm has reduced two objectve under the same value of Max(Wtw): the makespan (11 nstead of 13) and the Wtw (32 nstead of 34). By comparng between the data n these two columns, t s clearly shown that the proposed novel hybrd algorthm s effectve. Fg. 8. Optmzaton soluton of problem Problem 8 8 Table 2 dsplays the nstance of the mddle scale problem 8 8. The GA parameters are as follow: populaton sze=100, teraton number=500, mutaton

10 206 A. Motaghed-larjan, K. Sabr-laghae & M. Heydar Solvng Flexble Job Shop Schedulng wth rate=0.1. The computatonal results by the proposed hybrd algorthm are shown n the followng: Soluton 1: Makespan =14, Max(Wtw) = 11, Wtw = 78 Soluton 2: Makespan =14, Max(Wtw) = 12, Wtw = 77 Tab. 5. Comparson of results on problem 4 5 Objectves Makespan Max(Wtw) Wtw Addtve Objectve Fg. 9. Optmzaton soluton of problem 8 8 Tab. 6. Comparson of results on problem 8 8 Global Objectve Stars AL + GA PSO + S GA HC objectves Makespan Max(Wtw) Wtw Addtve Objectve Global crteron star Temporal decomposton Classc GA Approach by localzaton AL + CGA PSO + SA moga PSO + TS hga smulaton modelng GA+HC The soluton 1 has been graphcally represented n Fg. 9. The comparson of our hybrd algorthm wth other algorthm s shown n Table Problem A mddle scale nstance whch s taken from [15] s shown n Table 3. And t was taken to evaluate the effcency of proposed hybrd algorthm. The GA parameters are as follow: populaton sze=500, teraton number=500, mutaton rate=0.1 The computatonal results obtaned by the proposed hybrd algorthm are gven n the followng: Soluton1: Makespan = 7, Max(Wtw) = 5, Wtw = 43 Soluton2: Makespan = 8, Max(Wtw) = 5, Wtw = 42 The schedule s Gantt chart representaton correspondng to the soluton s shown n Fg. 10. The comparson of the proposed hybrd algorthm wth other algorthms s shown n Table 7. Tab. 7. Comparson of results on problem Objectves Makespan Max(Wtw) Wtw Addtve Objectve Global crteron Stars Temporal decomposton Classc GA Approach by localzaton AL + CGA PSO + SA moga hga PSO + TS Smulaton modelng GA+HC Tab. 8. Comparson of results on problem Objectves Makespan Max(Wtw) Wtw Addtve Objectve Global crteron Star AL + CGA PSO + SA hga PSO + TS Smulaton modelng GA+HC

11 A. Motaghed-larjan, K. Sabr-laghae & M. Heydar Solvng Flexble Job Shop Schedulng wth Problem To evaluate the effcency and valdty of our algorthm on larger scale problems, we have chosen the nstance shown n Table 4 whch s taken from [15]. The GA parameters are as follow: populaton sze=600, teraton number=500, mutaton rate=0.1. The best computatonal results by ths hybrd algorthm are shown n the followng: Soluton1: Makespan = 11, Max(Wtw) = 11, Wtw = 91 Soluton2: Makespan = 12, Max(Wtw) = 10, Wtw = 93 The computatonal result s schedule s represented by Gantt chart n Fg. 11.The comparson of proposed hybrd algorthm wth other algorthm s shown n Table 8. other authors algorthms for four representatve nstances. The obtaned results show the effectveness of the proposed algorthm. Also Effect of neghborhood searchng by tables 5, 6, 7 and 8 has been shown. In ths paper, a Pareto approach s proposed to solve the mult objectve flexble job shop schedulng problems. The objectves consdered are to mnmze the maxmal completon tme of machnes, the maxmal machne workload and the total workload of machnes. Ths approach gets dea from compromse programmng. Frst the problem have been solved by consderng only one objectve functon and obtaned soluton s named star soluton. Then global objectve functon s made. The contrbuton of ths paper can be summarzed as follows: It presented a hybrd algorthm to solve the flexble job shop schedulng problems wth multobjectves of mnmzng makespan, the total workload of machnes and maxmum workload of the machnes In ths paper mult objectve FJSP s solved by method that get dea from compromse programmng. Ths method s more reasonable wth comparson of weghtng method. The effect of local search has been compared wth genetc algorthm only. Fgures 6,7, 12 and 13 have llustrated that hybrd algorthm can be reached to acceptable soluton n less teraton. Fg. 10. Optmzaton soluton of problem Future Research and Concluson In ths paper, an effectve hybrd genetc algorthm s proposed to solve the mult-objectve flexble jobshop schedulng problems. In ths hybrd algorthm, genetc algorthm perform global search and hll clmbng s used to local search. When the ntal answer was obtaned by the GA, local search starts wth Hll clmbng. the assgnment s consdered as a neghborhood structure for the problem. The performance of the presented algorthm s evaluated n comparson wth the results obtaned from Fg. 11. Optmzaton soluton of problem 15 10

12 208 A. Motaghed-larjan, K. Sabr-laghae & M. Heydar Solvng Flexble Job Shop Schedulng wth The future researches of ths paper are as follows: Improvng the local search by defnng new neghborhood structures. Elmnatng one of the assumpton n ths paper. For example, elmnatng non preemptve condton. It can be enlarge soluton space. Proposng an approach that can avod local optmum solutons. Fg. 12. Comparson of GA and HGA performance on problem Fg. 13. Comparson of GA and HGA performance on problem References [1] Baykasoglu, A., Ozbakr, L., Sonmez, A.I., "Usng Multple Objectve Tabu Search and Grammars to Model and Solve Mult-Objectve Flexble Job-Shop Schedulng Problems." Journal of Intellgent Manufacturng, Vol. 15(6), 2004, pp [2] Garey, M.R., Johnson, D.S., Seth, R., "The Xomplexty of Flow Shop and Jobshop Schedulng," Mathematcs of Operatons Research, Vol. 1, 1976, pp [3] Bruker, P., Schle, R., "Job Shop Schedulng wth Mult- Purpose Machne", Computng, Vol. 45, 1990, pp [4] Xa, W.J., Wu, Z.M., "An Effectve Hybrd Optmzaton Approach for Multobjectve Flexble Job-Shop Schedulng problems," Computers and Industral Engneerng, Vol. 48(2), 2005, pp [5] Brandmarte, P., "Routng and Schedulng n a Flexble Job Shop by Taboo Search,". Annals of Operatons Research, Vol. 41(3), 1993, pp [6] Paull, J., "A Herarchcal Approach for the FMS Schedulng Problem," European Journal of Operatonal Research Vol. 86, 1995, pp [7] Lawler, E.l., Lenstra, J.K., Rnnooy Kan, A.H.G., Shmoys, D.B., Sequencng and Schedulng: Algorthms and Complexty, In S. C. Graves et al. (Eds.), Logstcs of producton and nventory Amsterdam: North Holland, 1993, pp [8] Zhang, H., Gen, M., "Multstage-Based Genetc Algorthm for Flexble Job-Shop Schedulng Problem," Journal of Complexty Internatonal, Vol. 11, 2005, pp [9] Sad Mehrabad, M., Fattah, P., "Flexble Job Shop Schedulng wth Tabu Search Algorthm." Internatonal Journal of Advanced Manufacturng Technology, Vol. 32, 2007, pp [10] Fattah, P., Sad Mehrabad, M., Jola, F., "Mathematcal Modelng and Heurstc Approaches to Flexble job Shop Schedulng Problem," Journal of Intellgent Manufacturng, Vol. 18, 2007, pp [11] Hurnk, E., Jursch, B., Thole, M., " Tabu Search for the Job Shop Schedulng Problem wth Mult-Purpose Machne," Operatons Research Spektrum, Vol. 15, 1994, pp [12] Mastrolll, M., Gambardella, L.M., "Effectve Neghborhood Functons for the Flexble Job Shop Problem," Journal of Schedulng, Vol. 3(1), 2000, pp [13] Parsopoulos, K.E., Vrahats, M.N., "Recent Approaches to Global Optmzaton Problems Through Partcle Swarm Optmzaton. Natural Computng," Vol. 1(2-3), 2002, pp [14] Kacem, I., Hammad, S., Borne, P., "Approach by Localzaton and Multobjectve Evolutonary Optmzaton for Flexble Job-Shop Schedulng Problem," IEEE transactons on systems, man, and cybernetcs, Part C, Vol. 32(1), 2002, pp [15] Kacem, I., Hammad, S., Borne, P., "Pareto-Optmalty Approach for Flexble Job-Shop Schedulng Problems: Hybrdzaton of Evolutonary Algorthms and Fuzzy Logc," Mathematcs and Computers n Smulaton, Vol. 60, 2002, pp [16] Rgao, C., "Tardness Mnmzaton n a Flexble Job Shop: A Tabu Search Approach," Journal of Intellgent Manufacturng, Vol. 15(1), 2004, pp

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