PARTICLE SWARM OPTIMIZATION BASED ON BOTTLENECK MACHINE FOR JOBSHOP SCHEDULING

Transcription

1 Proceedng 7 h Inernaonal Semnar on Indusral Engneerng and Managemen PARTICLE SWARM OPTIMIZATION BASED ON BOTTLENECK MACHINE FOR JOBSHOP SCHEDULING Rahm Mauldya Indusral Engneerng Deparmen, Indusral Engneerng Faculy, Trsak Unversy rmaul@yahoo.com ABSTRACT In job shop producon sysem, boleneck machne s reaed as a cener of producon plannng so s mporan o mamze he producon capacy. Boleneck schedulng problem dvded soluon no forward and backward schedulng. The compleon me of he job unl boleneck machne s a due dae for all job and he compleon me of he job afer boleneck machne s he real compleon me for all job. In mos schedulng problem, soluon usng Parcle Swarm Opmzaon s only o mnmze makespan n forward schedulng or mnmze he number of ardy job n backward schedulng. Ths research presen he mehodology o solve job shop schedulng problem based on boleneck machne usng Parcle Swarm Opmzaon. Keywords : schedulng, job shop, boleneck, Parcle Swarm Opmzaon, due dae, makespan 1. INTRODUCTION Job shop schedulng s characerzed by he roung of each producs. n job are processed o compleon on m unrelaed machnes. The roung and processng me s known n advance. Processng me are fed. Machne s avalable from me zero and operaons are processed whou preempon. Job shop schedulng s srongly NP-hard and solve usng meaheursc algorhm. Genec algorhm, smulaed annealng algorhm, an colony algorhm and parcle swarm opmzaon s clasfed no meaheursc. Parcle swarm opmzaon (PSO) s one of he laes evoluonary opmzaon mehods nspred by naure ha ncludes evoluonary sraegy (ES), evoluonary programmng (EP), genec algorhm (GA), and genec programmng (GP). PSO s based on he meaphor of socal neracon and communcaon such as brd flockng and fsh schoolng. PSO s dsncly dfferen from oher evoluonary-ype mehods n a way ha does no use he flerng operaon (such as crossover and/or muaon), and he members of he enre populaon are mananed hrough he search procedure so ha nformaon s socally shared among ndvduals o drec he search owards he bes poson n he search space (Tasgeren, 27). Job shop schedulng problem can also be solved by usng boleneck schedulng. Boleneck machne s defned usng he longes processng me of machne. In bolleneck schedulng, here s forward schedulng and backward schedulng, backward schedulng s used unl boleneck machne and forward schedulng s used afer boleneck machne o mnmze makespan. PSO mosly use o mnmze makepan n forward schedulng. Alhough here s PSO use o mnmze due dae, bu here s no PSO use n boh sde. Ths paper presen he mehodology o solve job shop schedulng problem usng Parcle swarm opmzaon based on boleneck machne. 2. LITERATURE 2.1. Parcle Swarm Opmzaon Parcle Swarm Opmzaon (PSO) n nroduced as a mehod for nonlnear opmzaon. Ths mehod uses a model of flgh of brds (or moon of parcles) o solve he opmzaon problem n whch any poenal soluon n search space s consdered as a poenal poson for parcles. Swarm of parcles move hrough search space under a defned dynamcs of flgh and fnd he bes soluon as he opmum soluon. The poson of -h parcle PS-1 PSO based on boleneck machne (Rahm Mauldya)

2 Proceedng 7 h Inernaonal Semnar on Indusral Engneerng and Managemen ISSN : X s presened by an n-dmensonal vecor for an n-dmensonal search space (Rezae, 21). Algorhm of PSO mehod has seps below (Rezae, 21): 1) Problem defnon: Ths sep ncludes he defnon of cos funcon and deermnng he range of search space. The mnmum and mamum values whch he parcles could ake as coordnaes of her poson should be deermned. 2) Inalzaon: Inalze posons and veloces of parcles randomly. Also, bes poson of each parcle s se o be s nal poson and global bes poson s se o be he bes nal poson assocaed wh cos funcon. 3) Parcles moon: Fnd new poson and veloces of parcles usng equaons 3 and 4. 4) Evaluaon: Evaluae cos funcon for each parcle poson and updae bes posons of parcles and bes global poson. 5) Repea: Go o sep 3 agan and repea ll a soppng creron s sasfed. The soppng creron could be consdered as akng a predefned number of eraons Noaon Zj : random number n n parcle and job j U j : unform number n parcle and job j X : parcle n swarm a eraon, j : value poson for parcle and dmeson j where (j = 1,2,,n) and n s he number of dmenson. So, nal eraon s noed as X v j, j NP f 1, 2,..., 1n : velocy for parcle and job j a eraon : he number of job j for parcle I n permuaon a eraon : he number of parcle where he amoun s wce of number of dmenson (job) : makespan of fness value n parcle and eraon PSO based on boleneck machne (Rahm Mauldya) f pb f gb P : Fucon of he bes makespan n personal bes : Funcon of he bes makespan n global bes : Personal Bes a eraon and parcle p j : Personal Bes a parcle, job j and eraon G : Global Bes for eraon g j : Global Bes for parcle and job j a eraon w : Inera Wegh a eraon c 1, c 2 : coefsen acceleraon where c 1, c 2 = 2 r 1, r 2 : unform random number from unl 1 where r 1 s unform number for parcle poson and r 2 s unform number for parcle velocy. r 1 and r 2 are unform number from U j T : Tardy for parcle and eraon The seps for PSO are : Sep 1 Inlzaon a. Generae random number r1 usng LCG Z = (a Z 1 + b)(mod m) (1) U = Z /m (2) a. Se =, and NP = wce number of dmenson b. Defne random parcle as X, 1,2,..., NP where X usng equaon j 1, 2,..., 1 n mn ( ) * r (3) ma where mn =., ma = 4. and r 1 s unform random number beween and 1 c. Defne nal velocy for each parcle randomly v ( v v )* r (4) v j mn ma where v mn = -4 and v ma = 4, and r 2 s defne from unform random number beween 1 and. d. Generae permuaon usng Smalles mn mn Poson Value (SPV) for parcle 1 11, 12,..., 1 n (5) e. Evaluae every parcle n swarm o look for makespan of every parcle noaed as funcon f1 where = 1,2,...,NP f. For every parcle n swarm, se P X P p, p ; 1 1 2,..., n n 2 1 X 1 PS-11 (6)

3 Proceedng 7 h Inernaonal Semnar on Indusral Engneerng and Managemen g. Defne he bes fness value whch means he bes makespan from every parcle n swarm. Makespan s noaed wh: f = mn f, 1,2,..., NP (7) h. Se global bes or Gbes G = X ;G = [g,1 =,1 =,2,..,g,n =,n ](8) s parcle. G s value of Global Bes a nal eraon unll Global Bes equal o he value of parcle poson. Sep 2 Updae he eraon = +1 (9) Sep 3 Updae Inersa wegh Updae nersa wegh wh w w 1 * (1) where s decreamen facor wh value.975 and w =.9 Sep 4 Updae Velocy v w v c r ( p ) c r ( g j 1 1 j 1 1 j j 2 2 j (11) -1 Where c s coeffsen corelaon. For p j and -1 g j sue o prevous eraon. Sep 5 Updae Poson 1 v (12) j Sep 6 Defne Permuaon Usng SPV o defne new permuaon: 1 1, 2,..., n (13) Sep 7 Updae Personal Bes Each parcle s evaluaed usng permuaon. Permuaon can be renew f pb f f, 1,2,..., p (14) so personal bes wll change no P X and pb j f f, whch means f makespan n he ne eraon s smaller au equal o makespan a eraon -1, so f gb s changed no he value n parcle a he bes eraon, so ha for personal bes. Sep 8 Updae Global Bes Fnd mnmum value of Global Bes pb fl mn f (15) = 1,2,...,NP means he bes makespan of all parcle, so global bes s renewed no G X and f gb f l. Sep 9 Soppng Creron If eraon eceed mamum number of eraon, or eceed mamum me, he eraon wll be sop. j l ) 3. METHODOLOGY Parcle Swarm Opmzaon sar wh generang random number o defne he number of parcle, he poon of parcle, and parcle velocy. The number of parcle s se wo mes from he number of job. Permuaon rankng s se usng SPV. Job s schedule usng permuaon rankng. In hs research, job s scheduled based on he poson of boleneck machne. For poson before boleneck machne, defne parcle due dae and generae sequence usng backward schedulng. For poson afer boleneck machne, sequence s defned usng forward schedulng. Boh resul can defne personal bes poson wh se of P =X for each parcle. Ne sep s he same for PSO algorhm whch defne Global Bes G =X. The sequence s se based on global bes and renew a every eraon unll smales or consan global bes. In every renew eraon, s followed by he renew of wegh. 4. NUMERICAL EXAMPLE In hs paper, he daa s se n able 1. The processng me and he roung s fed and s known n advance. Table 1. Daa se JOB Processng me Roung Operaon Operaon J A B C J A C B J C B A J B C A The oal processng me of each machnes show ha machne B s he longes. Machne B s also a bolleneck machne. Machne B wll be scheduled for he frs me. Table 2. Toal processng me of each machne JOB machne Toal A B C Toal PS-12 PSO based on boleneck machne (Rahm Mauldya)

4 Proceedng 7 h Inernaonal Semnar on Indusral Engneerng and Managemen ISSN : X Now, focus s on machne B as boleneck machne, Boleneck schedulng Akve schedule Makespan Flowme Idle Tme 32 19,5 A = 5 B = 1 C = ,5 A = 11 B = 7 C = 9 5. CONCLUSION Fgure 1. roung produc Random number s generae usng hs daa: a : mulpler = 23 b : ncremen facor = m : modulus = 11 Z : Random number wll resul r 1 dan r 2, hen s used o defne permuaon n able 3. Table 3. Sequence Parkel X 1 X 8 Parcle Sequence X 1 J2 J3 J4 J1 X 2 J1 J3 J2 J4 X 3 J4 J1 J2 J3 X 4 J4 J2 J3 J1 X 5 J3 J2 J4 J1 X 6 J1 J2 J4 J3 X 7 J1 J3 J4 J2 X 8 J3 J1 J2 J4 Each permuaon n = s observed o ge makespan for nal. Due dae s se unl machne boleneck. Due dae s he longes me unll machne boleneck. Parcle whch gve he bes resul s J4-J1-J3-J2. Gan char for he sequence s n fgure 2. PSO proposed Fgure 2. Gan char Table 4. Resul Makespan Flowme Idle Tme 32 19,5 A = 5 B = 1 C = 3 The PSO based on boleneck machne s effecve for large number of job. Tesng he PSO no flowhop schedulng s he ne research and defne he boleneck machne usng heory of consran. 6. REFERENCES (a) Baker, Kenneh R. (1974). Inroducon o Sequencng and Schedulng. John Wley & Sons, Inc.Canada (b) Brucker, Peer. (26). Schedulng Algorhms: Sprnger 5 h edon. German (c) Clerc, M. (25). Parcle Swarm Opmzaon. Lavoser Ise. London (d) Dvan, Vana. (212). Usulan Penjadwalan Job Dengan Meode Parcle Swarm Opmzaon Unuk Memnmas Number Of Tardy Jobs D PT. Faco Global Engneerng. Unversas Trsak. Jakara (e) Law, A.M dan Kelon, A.W. (2). Smulaon Modelng and Analyss: MGH 3r d Edon. New Jersey (f) Moron, T. and Penco, D. (1993). Heursc schedulng sysem. John Wley Inerscence (g) Pnedo, Mchael. (22). Schedulng: Theory, Algorhms, and Sysems. New Jersey: Prence Hall Inc. 2 nd Edon (h) Rezae, K., Nazar, S. S., Nazar- Shrkouh Salman, dan Ghods Reza. (21). Theory of Consrans and Parcle Swarm Opmzaon Approaches for Produc M. Ausralan Journal of Basc and Appled Scences, ISSN () Tasgeren, M. F., Lang, Y.-C., Sevkl, M., dan Gencylmaz, G. (27). A Parcle Swarm Opmzaon Algorhm for Makespan and Toal Flowme Mnmzaon n Permuaon Flowshop Sequencng Problem. European Journal PSO based on boleneck machne (Rahm Mauldya) PS-13

5 Proceedng 7 h Inernaonal Semnar on Indusral Engneerng and Managemen of Operaonal Research, Vol 177, AUTHOR BIOGRAPHIES Rahm Mauldya s a lecurer n Deparmen of Indusral Engneerng, Faculy of Indusral Technology, Trsak Unversy, Jakara. She receved her Maser of Indusral Engneerng from Insu Technology Bandung n 25. Her research neress are n he area of Producon Plannng & Conrol, Roboc and Desgn Produc. She s a member of he Producon Sysem Laboraory. Her emal address s <rmaul@yahoo.com> PS-14 PSO based on boleneck machne (Rahm Mauldya)