Stochastic job-shop scheduling: A hybrid approach combining pseudo particle swarm optimization and the Monte Carlo method

Transcription

1 Bulletn of the JSME Journal of Advanced Mechancal Desgn, Systems, and Manufacturng Vol.10, No.3, 2016 Stochastc job-shop schedulng: A hybrd approach combnng pseudo partcle swarm optmzaton and the Monte Carlo method Kenta ARAKI* and Yasunar YOSHITOMI** *Dayo System Kahatsu Co., Ltd Edobor, Nsh-u, Osaa , Japan **Graduate School of Lfe and Envronmental Scences, Kyoto Prefectural Unversty 1-5 Naarag-cho Shmogamo, Sayo-u, Kyoto , Japan E-mal: yoshtom@pu.ac.jp Receved 15 November 2015 Abstract Many practcal problems wth uncertantes can be formulated as stochastc programmng problems, and ther optmal solutons are useful for decson-mang. However, solvng problems s generally dffcult, and feasble methods for fndng analytcal solutons are needed. The purpose of ths study s to propose a hybrd method that combnes pseudo partcle swarm optmzaton n an uncertan envronment (PPSOUCE) and the Monte Carlo (MC) method for solvng a stochastc programmng problem. As an example, we used the proposed hybrd method to solve a stochastc job-shop schedulng problem (SJSSP). We compared our proposed PPSOUCE wth the MC method to a hybrd method of a genetc algorthm n an uncertan envronment (GAUCE) wth the MC method. Numercal experments llustrate that our method provdes better solutons wth shorter CPU tmes than those of the method that combnes the GAUCE and the MC method. Key words : Stochastc programmng, Job-shop schedulng, Partcle swarm optmzaton, Monte Carlo method 1. Introducton Many practcal problems wth uncertantes can be formulated as stochastc programmng problems, and ther optmal solutons are useful for decson-mang. However, solvng these problems s generally dffcult, and feasble methods for fndng analytcal solutons are needed. To solve a stochastc programmng problem, we assume that fluctuatons n the envronment are specfed by the dstrbuton functons of stochastc varables through all generatons by a genetc algorthm (GA; Yoshtom et al., 2000). We treat the stochastc fluctuaton of the objectve functon and/or the constrant on the stochastc programmng problem as a stochastc fluctuaton of the ftness functon n the GA. Snce the ftness functon expresses the ftness of an ndvdual to that envronment, the ftness functon n the GA should fluctuate accordng to the dstrbuton functons of the stochastc varables. We then analyze all solutons through all generatons to obtan nformaton about the optmum soluton. We call ths method the GA n an uncertan envronment (GAUCE). By applyng GAUCE to several stochastc programmng problems, we confrmed that the ndvdual that recurs most frequently through all generatons gves the maxmum expected value of the objectve functon (Yoshtom et al., 2000). Frst, we successfully appled the stochastc optmal assgnment problem and the stochastc napsac problem as prelmnary examples for the verfcaton and evaluaton of GAUCE because these examples can be solved exactly by other methods (Yoshtom et al., 2000). We then successfully appled GAUCE to the stochastc mage-compresson problem, a newly formulated problem that has recently receved a great deal of attenton n the feld of computer and nformaton scence (Yoshtom et al., 2000). The job-shop schedulng problem (JSSP), whch s an NP-hard problem, s one of the most dffcult combnatoral optmzaton problems. However, snce many schedulng problems n ndustry are formulated as JSSPs, a method for solvng the JSSP has a practcal as well as an academc mportance. Thus, the JSSP has receved consderable attenton Paper No

2 Ara and Yoshtom, Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol.10, No.3 (2016) n the feld of operatons research. In ndustral applcatons, the stochastc job-shop schedulng problem (SJSSP) s more realstc than the JSSP. For example, n the real world, the processng tme of a machne ncludes uncertantes. Therefore, these uncertantes need to be taen nto consderaton when schedulng or when formulatng a schedulng problem such as the SJSSP. However, the SJSSP where many parameters, ncludng, for example, the processng tme of a machne, are treated as random varables s much more dffcult to solve exactly than the JSSP where every parameter has a determnstc value. For ths reason, much less research on the SJSSP than on the JSSP has been conducted. In an earler paper, we proposed a method for solvng stochastc job-shop schedulng problems usng hybrd of GAUCE and MC (Yoshtom, 2002; Yoshtom and Yamaguch, 2003). GAUCE was appled to general SJSSP where the processng tmes are treated as stochastc varables. The several ndvduals havng the hghest frequences through all generatons were selected as the good schedules. The MC method was then used to determne the approxmately optmal schedule among these good schedules. Recently, n the operatons research feld, consderable attenton has been pad to partcle swarm optmzaton (PSO; Kennedy and Eberhart, 2001) as a method for solvng optmal programmng problems, ncludng the JSSP (Lan, Jao, and Gu, 2006; Xa and Wu, 2006; Mosleh and Mahnam, 2011) and the SJSSP (Zhang et al., 2012). The purpose of ths study s to propose a hybrd method that combnes pseudo partcle swarm optmzaton n an uncertan envronment (PPSOUCE) and the MC method for solvng the stochastc programmng problem. The SJSSP s selected as an example for comparng the hybrd method of PPSOUCE wth the MC method to the hybrd method of GAUCE wth the MC method. 2. Stochastc job-shop schedulng problem The SJSSP s a stochastc programmng problem transformed from the JSSP. In actual ndustres, the processng tme of a machne almost always has uncertantes. Therefore, these uncertantes must be taen nto account when schedulng or formulatng the schedulng problem as an SJSSP. In ths paper, the processng tmes of machnes are treated as random varables, and the objectve s to fnd a schedule that mnmzes the expected value of the maespan. The other condtons of the SJSSP are the same as those of the regular JSSP. The SJSSP s formulated as follows: 1. We have n jobs, J 1,, J n, whch are to be processed on m machnes, M 1,, M m. 2. A machne can process only one job at a tme. 3. The processng of a job on a machne s called an operaton. 4. An operaton cannot be nterrupted. 5. A job conssts of at most m operatons. 6. The sequence of operatons wthn a job (the machne sequence) s gven. 7. The processng tmes for operatons are gven as random varables that have stochastc dstrbuton functons. 8. The operaton sequences on the machnes (the job sequences) are unnown. 9. On a Gantt chart, the postons of operatons wthn a job are decded n order. 10. The startng tme of an operaton wthn a job should be as early as possble. 11. A schedule conssts of the full set of job sequences. 12. The objectve s to fnd a schedule that mnmzes the expected value of the maespan. In ths study, a schedule s expressed by the job sequence on each machne. Item 10 n the above descrpton mples that no operaton can be postponed. The schedule that s unquely represented by the above condtons s called a sem-actve schedule, as descrbed n the determnstc verson (Yamada and Naano, 1992); ths means that the schedule s one n whch no operaton begns at an earler tme wthout alterng the machnng sequences. An actve schedule (Gffler and Thompson, 1969) s as follows (Yamada and Naano, 1992). Consder a O j, r sem-actve schedule and two operatons, and O, l, r n that schedule that share the same machne M r, where j and denote the respectve job numbers, and l denote the postons of operaton n the sequence of the job, 2

3 Ara and Yoshtom, Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol.10, No.3 (2016) and r denotes the respectve machne number. If machne operaton M r has an dle perod longer than O j, r O, p j, r O l, r O, O,,,, s processed pror to j, r, and pror to processng l r, (ts processng tme), then t s possble to reassgn tass so that, s processed pror to, l r. Such reassgnng s called a permssble left shft. A schedule havng the property that no operaton can be processed earler by a permssble left shft s called an actve schedule. 3. PPSOUCE-MC approach 3.1 PSO and PPSOUCE In the PSO procedure, many partcles move n multdmensonal vector space. The poston and the velocty of partcle 1,, N at moment t are gven by the vector x (t) and the vector (t), respectvely, where N denotes the number of partcles. The best poston found thus far by the partcle s expressed by a vector pbest (t) ; ths gves the best value thus far for the evaluaton functon: pbest (t) v f. The best poston found thus far by the group havng n partcles s expressed by the vector gbest (t), whch gves the best group value thus far for the evaluaton functon: gbest (t) f. To obtan the approxmate optmal soluton for the SJSSP, we modfy the algorthm (Lan et al., 2006) ntroduced for the GA and used for the movement of a partcle n the PSO. We also use the statstcal method n GAUCE. We refer to the proposed method as the pseudo partcle swarm optmzaton n an uncertan envronment (PPSOUCE). 3.2 Partcle movement usng the GA functons In the proposed method, the equaton of moton of a partcle s as follows: v 1 pbest j 1 chld ( m, 0) x v, m 1, 2 (2) chld ( m, n) Mn chld m 1, 2, n 1, 2,, N m, 0 max 1 x, f x f chld a, b chld a, b, f x f chld a, b chld ( a, b) mn f chld ( m, n) x (4) f where 1,, N (5) 1m2, 0nN max s the respectve partcle number, j s the respectve number of the partcle selected randomly, s the crossover operator, chld (m,0) s the chld partcle generated by the crossover, M n s the mutaton operator, and chld ( m, n) ( n 0) s the chld partcle generated by the mutaton, m s the number of respectve chld partcle, n s the mutaton ndex, and s the step ndex correspondng to a dscrete tme. In ths study, the respectve job number s used for expressng the element of a vector for a partcle, the crossover s that proposed by Hrano (1995), and the mutaton s the shft change proposed by Ono (1997) PPSOUCE-MC approach For each step, the processng tme s gven as the random number generated accordng to the stochastc dstrbuton functon for the processng tme. The basc procedure of the proposed method s as follows. (1) (3) 0 0 Step 1: Intal nformaton x, v 1,, N for each partcle s randomly gven. Step 2: The followng procedure of substeps [1] and [2] s repeated n numercal order by the preset loop frequency. [1] A random number generated accordng to the stochastc dstrbuton functon s gven for each random valuable. 23

4 Ara and Yoshtom, Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol.10, No.3 (2016) [2] The nformaton x v, pbest 3.2., for each partcle s updated accordng to Eqs. (1) through (5), stated n Secton Step 3: Usng the MC method to calculate the approxmate expected maespan, an approxmately optmal soluton s found; t s the soluton wth hgh frequency n the Step 2 loop that gves the mnmum approxmate expected maespan. Here, 1,, N s the respectve partcle number, and s the step ndex correspondng to a dscrete tme scale. The respectve job number s used for expressng the element of the vector x for a partcle. The vector v s 0 0 gven by the ntal nformaton x, v and Eqs. (1) through (5) n Secton 3.2. We consder only actve schedules at each step. When a sem-actve schedule s generated n the PPSOUCE process, that schedule s changed to ts actve verson by a permssble left shft. Because the number of actve schedules for the problem may be huge, MC s appled for selectng the best soluton among several good solutons wth hgh frequency n the loop n Step Numercal experments and dscusson 4.1 Development envronment The development of ths system and the experments for evaluaton of the proposed method were performed n the followng envronment: personal computer: DELL OPTIPLEX 780 (CPU: Intel Core 2 Duo E GHz, RAM: 4.00 GB); OS: Mcrosoft Wndows 7 Professonal; Development language: Mcrosoft Vsual C Express Edton. 4.2 Condtons The data for the 6 6, 10 10, and 20 5 JSSPs, taen from Muth and Thompson (1963), are shown n Tables 1 3, respectvely. The dataset ncludes the routng of each job on each machne and the processng tme for each operaton (n parentheses). Tables 1 3 gve the data, whch are well-nown benchmars. Table job-shop schedulng problem (Muth and Thompson, 1963) Job Operaton routng (processng tme) 1 3(1) 1(3) 2(6) 4(7) 6(3) 5(6) 2 2(8) 3(5) 5(10) 6(10) 1(10) 4(4) 3 3(5) 4(4) 6(8) 1(9) 2(1) 5(7) 4 2(5) 1(5) 3(5) 4(3) 5(8) 6(9) 5 3(9) 2(3) 5(5) 6(4) 1(3) 4(1) 6 2(3) 4(3) 6(9) 1(10) 5(4) 3(1) Table job-shop schedulng problem (Muth and Thompson, 1963) Job Operaton routng (processng tme) 1 1(29) 2(78) 3(9) 4(36) 5(49) 6(11) 7(62) 8(56) 9(44) 10(21) 2 1(43) 3(90) 5(75) 10(11) 4(69) 2(28) 7(46) 6(46) 8(72) 9(30) 3 2(91) 1(85) 4(39) 3(74) 9(90) 6(10) 8(12) 7(89) 10(45) 5(33) 4 2(81) 3(95) 1(71) 5(99) 7(9) 9(52) 8(85) 4(98) 10(22) 6(43) 5 3(14) 1(6) 2(22) 6(61) 4(26) 5(69) 9(21) 8(49) 10(72) 7(53) 6 3(84) 2(2) 6(52) 4(95) 9(48) 10(72) 1(47) 7(65) 5(6) 8(25) 7 2(46) 1(37) 4(61) 3(13) 7(32) 6(21) 10(32) 9(89) 8(30) 5(55) 8 3(31) 1(86) 2(46) 6(74) 5(32) 7(88) 9(19) 10(48) 8(36) 4(79) 9 1(76) 2(69) 4(76) 6(51) 3(85) 10(11) 7(40) 8(89) 5(26) 9(74) 10 2(85) 1(13) 3(61) 7(7) 9(64) 10(76) 6(47) 4(52) 5(90) 8(45) 24

5 Ara and Yoshtom, Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol.10, No.3 (2016) Table job-shop schedulng problem (Muth and Thompson, 1963) Job Operaton routng (processng tme) 1 1(29) 2(9) 3(49) 4(62) 5(44) 2 1(43) 2(75) 4(69) 3(46) 5(72) 3 2(91) 1(39) 3(90) 5(12) 4(45) 4 2(81) 1(71) 5(9) 3(85) 4(22) 5 3(14) 2(22) 1(26) 4(21) 5(72) 6 3(84) 2(52) 5(48) 1(47) 4(6) 7 2(46) 1(61) 3(32) 4(32) 5(30) 8 3(31) 2(46) 1(32) 4(19) 5(36) 9 1(76) 4(76) 3(85) 2(40) 5(26) 10 2(85) 3(61) 1(64) 4(47) 5(90) 11 2(78) 4(36) 1(11) 5(56) 3(21) 12 3(90) 1(11) 2(28) 4(46) 5(30) 13 1(85) 3(74) 2(10) 4(89) 5(33) 14 3(95) 1(99) 2(52) 4(98) 5(43) 15 1(6) 2(61) 5(69) 3(49) 4(53) 16 2(2) 1(95) 4(72) 5(65) 3(25) 17 1(37) 3(13) 2(21) 4(89) 5(55) 18 1(86) 2(74) 5(88) 3(48) 4(79) 19 2(69) 3(51) 1(11) 4(89) 5(74) 20 1(13) 2(7) 3(76) 4(52) 5(45) To transform a JSSP to an SJSSP, the processng tme s treated as a random varable. The stochastc dstrbuton functon of the processng tme s a normal dstrbuton havng as the mean values the values n parentheses n Tables 1 3. The standard devaton of the normal dstrbuton s gven as the rato of the devatons to the mean. In general, n ndustral applcatons, the devaton of the processng tme from ts mean value s not very large. Therefore, as condtons for the stochastc processng tme, the ratos of the standard devaton to the mean value were selected as () 0 (determnstc), () 0.1, and () 0.2. The case where the rato s 0 was selected for comparson between the JSSP and the SJSSP. The normal dstrbuton was used as the stochastc dstrbuton functon because t s one of the smplest dstrbutons for computer smulatons. When PPSOUCE-MC s appled n an ndustral applcaton, the most approprate dstrbuton should be used. However, snce ths study s the frst step n applyng PPSOUCE-MC to an SJSSP, t s approprate to use a smple dstrbuton for the stochastc dstrbuton functon. In ths experment, as the condtons of PPSOUCE, the number of partcles was set to 4000, and the loop frequency was set to 700; these were chosen after tryng varous values n prelmnary experments. The averaged maespans for the solutons that had the 100 hghest frequences through all updates of nformaton of the partcles n PPSOUCE were obtaned from 100,000 MC samples for each schedule. The number 100,000 was also determned by prelmnary experments. The MC method was used for selectng the best among several good solutons, all of whch showed hgh frequency n all nformaton updates n PPSOUCE. In PPSOUCE, the problem transforms from a stochastc problem to a determnstc one at each update of nformaton, as outlned n the followng. Frst, a random number s generated for each processng tme accordng to ts stochastc dstrbuton functon. In ths study, n the conventonal method, the unform random number s transformed to a normally dstrbuted random number wth the mean and standard devaton taen from Tables 1 3 and the stochastc condton, respectvely. The random number s then used to transform the stochastc problem nto a determnstc problem each tme the partcle nformaton s updated. For each processng tme, the random number s generated ndependently. When the random number generated for a processng tme s negatve, the processng tme s treated as 0. In ths case, the correspondng operaton s not performed. Negatve or zero processng tmes come from usng the normal dstrbuton as the stochastc dstrbuton functon. The above treatment for negatve or zero processng tmes may not be the best way to reflect the stochastc nature of the 25

6 Ara and Yoshtom, Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol.10, No.3 (2016) processng tme when usng PPSOUCE-MC n an ndustral applcaton. However, n ths numercal example, because the ratos of the standard devaton to the mean value are from 0 to 0.2, the probablty of havng a negatve or zero processng tme for each operaton s from 0 to Therefore, ths treatment of negatve or zero processng tmes has lttle nfluence on the overall numercal results. In the framewor of PPSOUCE, a good soluton s one wth hgh frequency through all updates of the partcle nformaton. The expected value of the maespan for the soluton s approxmated by the averaged maespan gven by the MC process, n whch the maespan s calculated 100,000 tmes for each schedule. We compare the results by our method for the SJSSP wth those obtaned by conventonal methods (GAUCE-MC), and we also supply the results for the JSSP as a reference. 4.3 Results and dscusson Fgs. 1 3 show the convergence of PPSOUCE for JSSPs and SJSSPs when usng the best value of N max for the approxmately optmal soluton on each condton of the stochastc processng tme. Fgs. 1 3 show the cases of 6 6, 10 10, and 20 5 for JSSP and SJSSPs, respectvely. In all cases of 6 6, 10 10, and 20 5 SJSSPs and durng all teratons, the fluctuatons of both the average and the mnmum maespans are ncreased by the ncrease of the stochastc fluctuaton on the processng tme. Table 4 shows a comparson of the proposed method (PPSOUCE-MC) and the conventonal methods (method I (GA) (Yamad and Naano, 1992), method II (GAUCE-MC) (Yoshtom and Yamaguch, 2003), and method III (GAUCE-MC) (Furutan, 2009)). As shown n Table 4, the proposed method s superor to the conventonal methods (II and III), n terms of both the average value of the maespan and the calculaton tme. On the approxmately optmal soluton, the superorty of the proposed method to the conventonal method III s more remarable n the case of SJSSPs than that n the case of 6 6 SJSSPs. Therefore, the proposed method s more effectve for fndng out the approxmately optmal soluton n larger soluton space, compared wth the conventonal method III. In addton, on the optmal soluton, the superorty of the proposed method to the conventonal method I s only shown n the case of 20 5 JSSP. Therefore, the proposed method s more effectve for fndng out the optmal soluton n larger soluton space, compared wth the conventonal method I. Accordngly, the hgher ablty for fndng out the (approxmately) optmal soluton n larger soluton space s the characterstc of the proposed method, compared wth conventonal methods (I and III). In other words, the proposed method has the hgher ablty of global search, compared wth conventonal methods (I and III). In addton, the proposed method s superor to the conventonal method II, n terms of the average value of the maespan n the case of 6 6 SJSSPs. As a functon of global search, Eqs. (1) and (3) n Secton 3.2 have the mportant role n the proposed method. Tables 5-7 show the approxmately optmal solutons obtaned by PPSOUCE-MC. In Tables 5-7, the solutons are expressed n the order of jobs processed by each machne. For example, n Table 5(a) means that J 1, J 4, J 3, J 6, J 2, J 5 are processed by M 1 n that order. In Tables 6 and 7, the name of each machne s representatvely expressed n (a). In Table 6, job 10 s ndcated by 0. In Table 7, the respectve job number s expressed by the vgesmal numeral system. In Tables 5 7, Frequency means the number of partcles that correspond to the soluton. In Tables 5 7, ran means the order of frequency. For example, (1) n Table 4(a) means that the correspondng partcle has the hghest frequency of any partcle appearng n the loop of Step 2, descrbed n the Secton 3.3. In all cases of 6 6, 10 10, and 20 5 SJSSPs, the frequences of the best soluton are decreased by the ncrease of the stochastc fluctuaton on the processng tme. Ths tendency was also shown n the references usng conventonal methods (GAUCE-MC method II (Yoshtom and Yamaguch, 2003) for the case of 6 6 SJSSPs and GAUCE-MC method III (Furutan, 2009) for both cases of 6 6 and SJSSPs). Ths tendency mght suggest that t s more dffcult to fnd out the approxmately optmal soluton havng very low value of the approxmate expected maespan n the case of bgger stochastc fluctuaton on the processng tme. 5. Conclusons Ths paper proposed a method for solvng the stochastc job-shop programmng problem usng a hybrd of PPSOUCE wth the MC method. PPSOUCE was appled to general SJSSPs, where the processng tmes were treated as stochastc varables and the objectve functon was the expected value of the maespan. Solutons wth very hgh 26

7 Ara and Yoshtom, Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol.10, No.3 (2016) frequency n all teratons were selected as good schedules. The MC method was then used to determne the approxmate optmal schedule from these good schedules. We performed numercal experments that proved that the proposed method s superor to the hybrd of GAUCE wth the MC method n terms of both the approxmately optmal soluton and the requred calculaton tme. Fg. 1 Convergence of PPSOUCE for 6 6 JSSP and SJSSPs. Fg.2 Convergence of PPSOUCE for JSSP and SJSSPs. 27

8 Ara and Yoshtom, Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol.10, No.3 (2016) Fg.3 Convergence of PPSOUCE for 20 5 JSSP and SJSSPs. Table 4 Comparson of PPSOUCE-MC and two versons of GAUCE-MC (a) Average value of maespan (ran) Standard devaton /Average jobs, 6 machnes 10 jobs, 10 machnes Mnmum GA GAUCE-MC a 55(2) 56.82(1) 59.80(7) GAUCE-MC b 55(1) 56.11(1) 58.04(20) PPSOUCE-MC 55(1) 55.94(3) 58.01(2) Mnmum GA GAUCE-MC b 977(1) (11) (10) PPSOUCE-MC 930(1) (27) (31) Mnmum jobs, GA machnes PPSOUCE-MC 1165(1) (44) (27) (b) Calculaton tme (sec.) Standard devaton /Average jobs, 6 machnes 10 jobs, 10 machnes 10 jobs, 10 machnes GAUCE-MC b PPSOUCE-MC GAUCE-MC b PPSOUCE-MC PPSOUCE-MC Notes: GA: conventonal method I (Yamada and Naano, 1992) GAUCE-MC a : conventonal method II (Yoshtom and Yamaguch, 2003) GAUCE-MC b : conventonal method III (Furutan, 2009) 28

9 Ara and Yoshtom, Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol.10, No.3 (2016) Table 5 Approxmately optmal solutons obtaned by PPSOUCE-MC (6 6 JSSP and SJSSP) () Standard devaton /Average = 0 (determnstc), wthout mutaton Frequency Soluton Maespan () Standard devaton /Average = 0.1, N max =4 Frequency Soluton Average value of maespan () Standard devaton /Average = 0.2, N max =5 Frequency Soluton Average value of maespan Table 6 Approxmately optmal solutons obtaned by PPSOUCE-MC (10 10 JSSP and SJSSP) () Standard devaton /Average = 0 (determnstc), N max =6 Frequency Soluton Maespan M 1 : M 2 : M 3 : M 4 : M 5 : M 6 : M 7 : M 8 : M 9 : M 10 : () Standard devaton /Average = 0.1, N max =4 Frequency Soluton Average value of maespan () Standard devaton /Average = 0.2, N max =8 Frequency Soluton Average value of maespan Table 7 Approxmately optmal solutons obtaned by PPSOUCE-MC (20 5 JSSP and SJSSP) () Standard devaton /Average = 0 (determnstc), N max =10 Frequency Soluton Maespan M 1 :H5KGF2B1CJ9ID7A8E346 M 2 :G5HBKJF1A27CI6D843E M 3 :5H8CKJ61AFDE2I973G4B M 4 :5HBGKJ912CDAF8I7E M 5 :5HGFBJKIC16D2A4938E7 () Standard devaton /Average = 0.1, N max =4 Frequency Soluton Average value of maespan 1F5HKIJD9GB2CA7E4368 G51JKHBIFA6D7243CE HJK6CDAEI7F293B4G 51HJBK9GDA2I7CE86F HIFKJBG6AD247C389E () Standard devaton /Average = 0.2, N max =1 Frequency Soluton Average value of maespan FH9JKG215CBDIAE63748 JGHFB5K61A2CI3D7E HJ6CK1FAED29I7B34G HJB95GK12AFCDE67I FHJ6B5GKIA1C2D473E89 29

10 Ara and Yoshtom, Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol.10, No.3 (2016) References Furutan, T., A method for approxmately solvng large-scale stochastc job-shop schedulng problem usng genetc algorthm n uncertan envronment and Monte Carlo method, Master Thess, Kyoto Prefectural Unversty, Kyoto, Japan (2009) (n Japanese). Gffler, B. and Thompson, G. L., Algorthm for solvng producton schedulng problem, Operatons Research, Vol. 8 (1969), pp Hrano, H., Genetc algorthms wth cluster averagng method for solvng job-shop schedulng problems, Transactons of the Japanese Socety for Artfcal Intellgence, Vol. 10 (1995), pp Kennedy, J. and Eberhart, R. C., Swarm Intellgence, Morgan Kaufmann Publshers, San Francsco, Calforna (2001). Lan, Z., Jao, B., and Gu, X., A smlar partcle swarm optmzaton algorthm for job-shop schedulng to mnmze maespan, Appled Mathematcs and Computaton, Vol. 183 (2006), pp Mosleh, G. and Mahnam, M., A pareto approach to mult-objectve flexble job-shop schedulng problem usng partcle swarm optmzaton and local search, Internatonal Journal of Producton Economcs, Vol. 129 (2011), pp Muth, J. F. and Thompson, G. L., Industral Schedulng, Prentce-Hall, Englewood Clffs, N.J. (1963). Ono, I., Genetc algorthms for optmzaton tang account of characterstcs preservaton, Doctor Thess, Toyo Insttute of Technology, Toyo, Japan (1997) (n Japanese). Xa, W. and Wu, Z., A hybrd partcle swarm optmzaton approach for the job-shop schedulng problem, The Internatonal Journal of Advanced Manufacturng Technology, Vol. 29 (2006), pp Yamada, T. and Naano, R., A genetc algorthm applcable to large scale job-shop problems, Proceedngs of the Second Internatonal Conference on Parallel Problem Solvng from Nature, (1992) pp Yoshtom, Y., Genetc algorthm approach to solvng stochastc job-shop schedulng problems, Internatonal Transactons n Operatonal Research, Vol. 9 (2002), pp Yoshtom, Y., Ienoue, H., Taeba, T., and Tomta, S., Genetc algorthm n uncertan envronments for solvng stochastc programmng problem, Journal of the Operatons Research Socety of Japan, Vol. 43 (2000), pp Yoshtom, Y. and Yamaguch, R., A genetc algorthm and the Monte Carlo method for stochastc job-shop schedulng, Internatonal Transactons n Operatonal Research, Vol. 10 (2003), pp Zhang, R., Song, S., and Wu, C., A two-stage hybrd partcle swarm optmzaton algorthm for the stochastc job shop schedulng problem, Knowledge-Based System, Vol. 27 (2012), pp