A Novel Hybrid Algorithm for Multi-Period Production Scheduling of Jobs in Virtual Cellular Manufacturing Systems

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1 roceengs of he Worl Congress on Engneerng 2011 Vol I July Lonon U.K. A Novel Hybr Algorhm for Mul-ero roucon Scheulng of Jobs n Vrual Cellular Manufacurng Sysems K.L. Ma J. Ma Absrac Vrual cellular manufacurng has arace a lo of aenon n recen years because raonal cellular manufacurng s naequae uner a hghly ynamc manufacurng envronmen. In hs aer a new mahemacal moel s esablshe for generang omal roucon scheules for vrual cellular manufacurng sysems oerang uner a mul-ero manufacurng scenaro. The obecve s o mnmze he oal manufacurng cos over he enre lannng horzon. A hybr algorhm base on he echnques of scree arcle swarm omzaon an consran rogrammng s roose o solve he comlex roucon scheulng roblem. Alhough arcle swarm omzaon erforms comevely wh oher mea-heurscs for mos omzaon roblems he evoluon rocess may be sagnae as me goes on f he swarm s gong o be n equlbrum esecally for roblems wh har consrans. Consran rogrammng on he oher han s an effecve echnque for solvng roblems wh har consrans. However he echnque may be neffcen f he feasble search sace s very large. Therefore he am of he roose hybr algorhm s o combne he comlemenary avanages of arcle swarm omzaon an consran rogrammng o mrove s search erformance. The effecveness of he roose mehoology s llusrae by solvng a se of ranomly generae es roblems. Inex Terms Bacracng Consran rogrammng Dscree arcle swarm omzaon Vrual cellular manufacurng sysems I. INTRODUCTION As global mare becomes more an more comeve manufacurng nusres face relenless ressure generae from a growng enency of greaer varees of roucs wh shorer manufacurng cycles an a hghly ynamc manufacurng envronmen. Manufacurers hus shoul consanly ao effcen manufacurng sysems o reson o ynamc changes n cusomers eman n orer o ee her mare share. Cellular manufacurng (CM an vrual cellular manufacurng (VCM are wo moran manufacurng sysem esgn conces whch have arace a lo of aenon n recen ecaes. Cellular manufacurng has long been consere effcen n mrovng he roucvy of bach roucon sysems. In cellular manufacurng sysems (CMSs he ars ha K.L. Ma s rofessor a he De. of Inusral an Manufacurng Sysems Engneerng (IMSE The Unversy of Hong Kong Hong Kong (e-mal: mal@hucc.hu.h. J. Ma s a hd suen a he De. of IMSE HKU Hong Kong (e-mal: maun_nana@yahoo.cn. ISSN: (rn; ISSN: (Onlne unergo smlar manufacurng oeraons are groue ogeher o form a ar famly an he worsaons ha rouce hose ars are hyscally groue ogeher o form a manufacurng cell for manufacurng hese ars. Cellular manufacurng has he avanage n managng maeral flow easly ue o he smlary of ars an roxmy of he worsaons. However cellular manufacurng also has many rawbacs such as low machne ulzaon an unbalance worloa [1]. In orer o overcome he efcences of cellular manufacurng a new conce calle vrual cellular manufacurng was roose. The man fference beween vrual cellular manufacurng an cellular manufacurng s ha he worsaons n a vrual manufacurng cell are no groue hyscally on he roucon floor. [2] an [3] reore ha he vrual manufacurng cell aears as aa fles n a vrual cell conroller. When a ob arrves he conroller wll ae over he conrol of he relevan worsaons o form a vrual manufacurng cell. The conroller wll also oversee he manufacurng of he ob unl s fnshe. A he same me he worsaons wll no be loce u n he formaon of a vrual manufacurng cell bu are free o be assgne o rouce oher obs as long as here are remanng caaces. When he ob has been comlee he vrual manufacurng cell ermnaes an he worsaons wll be release an beome avalable for oher ncomng obs. Alhough he conce of vrual cellular manufacurng has many avanages n erms of worsaon ulzaon an worloa balancng roucon scheulng for vrual cellular manufacurng sysems (VCMSs has no receve a lo of aenon from he research communy because of he comlexy of he roblem. Iran e al. [4] roose a meho base on grah heory an mahemacal rogrammng for formng vrual manufacurng cells. Bayasoglu [5] roose a smulae annealng algorhm for evelong a srbue layou for vrual cellular manufacurng cells. Ma e al. [6] eveloe a genec mehoology o generae effecve roucon scheules for vrual cellular manufacurng sysems oerang uner a sngle ero senaro. In hs aer research wll be exene no mul-ero suaon. The remaner of hs aer s organze as follows. Secon II resens he mahemacal moel an secon III he hybr algorhm. Secon IV analyses he comuaon resuls obane from solvng a se of ranomly generae

2 roceengs of he Worl Congress on Engneerng 2011 Vol I July Lonon U.K. roblems. Fnally he conclusons are gven n secon V. II. MATHEMATICAL MODEL In hs secon a mahemacal moel s esablshe o escrbe he characerscs of mul-ero VCMSs for he urose of generang omal roucon scheules. In he moel here are W worsaons ( w W. The lannng horzon conans eros ( each of whch s furher ve no a ceran number of me slces wh he same lengh ( s S. Some obs are o be rouce n each ero.the followng varables are use n he evelomen of he mahemacal moel: r = he roucon roue of ob n ero wr ( = he worsaon w use n roucon roue r O =he oeraon of ob DD = he ue ae of ob n ero V K w1 w2 =he volume of cusomer nee of ob n ero =he number of oeraons of ob =he sance beween worsaon w1 an w 2 L =he lengh of a me slce MC ws =he maxmum caacy of worsaon w n me slce s of ero wr ( =rocessng me for roucng one un of oeraon of ob on worsaon wr ( Dr ( =he oal sance of roucon roue r In aon un sance; un me; ero ; ero. Decson varables: s he cos of movng one un of ob er w s he oerang cos of worsaon w er s he nvenory holng cos of ob n s he subconracng cos of ob n R wr ( s = rocessng rae of oeraon of ob on wr ( n me slce s of ero s wr ( s = he sar me of oeraon of ob on wr ( n me slce s of ero f wr ( s = he fnsh me of oeraon of ob on wr ( n me slce s of ero X wr ( s =equal o 1 f oeraon of ob s launche by wr ( n me slce s of ero ; oherwse s 0 Y wr ( s = equal o 1 f oeraon of ob s rocesse by wr ( n me slce s of ero ; oherwse s 0 The mahemacal moel hus has he followng form: Mnmze ISSN: (rn; ISSN: (Onlne N N N V D( r IV SV W S N K Y R w wr ( s wr ( s wr ( w1 1 s1 1 1 Where DD (1 SV max{ V R IV 0} K w( r s 1 s1 w( r DD IV max{ R IV V 0} K wr ( s 1 s1 w( r S sdd 1 w( r R w1 w2 ( w1 w2 r K wr ( s Dr ( (3 Subec o X X w( r s (4 wr ( s 1 1 wr ( s R R w( r s w( r s 1 1 w( r s S( K 1 K X wr ( 1 1 s (6 wr ( s1 1 ' Ywr ( ' (1 ( ( s X wr s G w r s s (7 0 Ywr ( 1 s R wr ( ( s O w r s 0 Ywr ( 0 s S R wr ( s V O (9 s1 w( r s ( 1 SL( s1 L wr ( s O w( r s f s R wr ( s wr ( s wr ( s wr ( O w( r s (2 (5 (8 (10 (11 N K Ywr ( s Rwr ( s wr ( MCws 1 1 (12 IV 0 0 wr ( s (13 X Y {01} w s (14 wr ( s wr ( s where G s a bg number. The obecve of he mahemacal moel s o mnmze he oal manufacurng cos over he enre lannng horzon nclung maeral ransoraon cos nvenory-holng cos subconracng cos an machne oerang cos. Equaons (1 an (2 enoe he meho of calculang nvenory volume an subconrac volume of each ob n each ero resecvely. Equaons (3 show he meho of calculang he ravellng sance of a roucon roue. Consrans (4 ensure ha when oeraon of ob has been fnshe oeraon 1 of ob mus sar mmeaely. Consrans (5 mae sure ha he rocessng rae of an oeraon n a me slce mus be equal o ha of s receng oeraon n las me slce an ha of s succeeng oeraon n nex me slce n each ero. Consrans (6 ensure ha he sarng mes of all oeraons mus be whn he lannng horzon an each

3 roceengs of he Worl Congress on Engneerng 2011 Vol I July Lonon U.K. ob can only have a unque roucon roue n each ero. Consrans (7 mean ha no oeraon can sar before he me slce from whch roucon s launche n each ero. Consrans (8 resrc he rocessng rae o be greaer han or equal o zero. Consrans (9 enoe he relaonsh beween he rocessng rae an roucon volume of each ob n each ero. Consrans (10 an (11 escrbe he consrans of sarng me an fnshng me of each oeraon. Consrans (12 mae sure ha all obs assgne o a machne can be fnshe n each me slce of each ero. Consrans (13 mean ha here s no nvenory of any ob a he begnnng of he lannng horzon. Consrans (14 ncae ha hese varables are bnary. To llusrae consrans (4 an (5 of he mahemacal moel Table 1 roves an examle of he roucon ouus of a ob n a ero. In hs able he value n he brace s he maxmum rocessng rae of an oeraon n a me slce. For examle he maxmum rocessng rae of oeraon 1 n me slce 1 s 5. The maxmum rocessng rae of oeraon 2 n me slce 2 s 8 an ha of oeraon 3 n me slce 3 s 6. Thus he feasble rocessng rae of oeraon 1 n me slce 1 (oeraon 2 n me slce 2 an oeraon 3 n me slce 3 s 5. Ths ensures ha no wor-n-rocess exss beween oeraons of a ob. Afer scheulng of hs ob worsaon 2 sll has a ceran amoun of remanng caacy n me slce 2 whch allows o be assgne o rouce oher ncomng obs. TABLE 1 A SAMLE OF RODUCTION OUTUTS OF A JOB Seral number of oeraons Worsaon No Tme slce No. 1 5(5 3( (2 5(8 3(9 3 3(4 2(5 5( (3 2( (6 III. HYBRID OTIMIZATION ALGORITHM In orer o fn an effcen an effecve roucon scheule hs aer evelos a new hybr algorhm base on consran rogrammng (C an scree arcle swarm omzaon (DSO. A. Consran rogrammng Consran rogrammng [7] s an effecve mehoology for solvng ffcul combnaoral roblems by reresenng hem as consran sasfacon roblems (CSs. A consran sasfacon roblem usually consss of a se of varables a oman for each varable an a se of consrans resrcng he values ha he varables can smulaneously ae. Bacracng aragm s a basc consran roagaon echnque use o solve consran sasfacon roblems. The basc oeraon s o c one varable a a me an conser one value n s oman a a me mang sure ha he newly ce label s comable wh he nsanae aral soluon so far. If he newly ce label volaes ceran consrans hen an alernave value f exss s ce. If no value can be assgne o a varable whou volang any consran wll bacrac o he mos recenly nsanae varable. Ths rocess connues unl a feasble soluon s foun or all ossble combnaons of labels have been re an fale [7]. The roceure of consran rogrammng wh bacracng roagaon s resene n Fgure 1. Fgure 1 The roceure of consran rogrammng B. Dscree arcle swarm omzaon arcle swarm omzaon (SO a oulaon-base omzaon aroach nsre by he observaons of br flocng an fsh schoolng was roose by Kenney an Eberhar n 1995 [8]. The basc ea of hs aroach s o locae he omal or near omal soluon hrough cooeraon an sharng of nformaon among nvuals n he swarm. The swarm s comose of a grou of arcles n a search sace wh wo moran characerscs namely oson an velocy. Each arcle reresens a oenal soluon whch fles hrough he hyersace an has wo essenal reasonng caacbls: he memory of s own bes oson an he nowlege of he global or s neghborhoo s bes oson. arcles whn he swarm communcae nformaon wh each oher an aus her own oson an velocy base on he nformaon accorng o followng equaons: V V cr 11( X c2r2( G X (15 X X V (16 where X reresens he oson of arcle V reresens he velocy of arcle s he bes revously vse oson of arcle G s he global bes oson ω s nera wegh ha conrols he mac of revous velocy on s curren one whch s usually reuce ynamcally o max ecrease he search area : ( max mn mn n max whch max an mn enoe he maxmum value an he mnmum value of neral wegh resecvely an max s he maxmum number of eraons. In racce many omzaon roblems such as roucon scheulng are se n a sace feaurng scree or ISSN: (rn; ISSN: (Onlne

4 roceengs of he Worl Congress on Engneerng 2011 Vol I July Lonon U.K. qualave sncons beween varables. To mee hs eman a scree verson of arcle swarm omzaon was roose [9]. DSO has wo man fferences from he orgnal one. Frs he arcle s comose of bnary varables. Secon he velocy mus be ransforme no he change of robably whch s he chance of he varable ang he value 1. Usually he ransformaon s acheve hrough followng sgmo funcon. 1 sv ( (17 1 ex( V Where s( V enoes he robably of corresonng b ang value 1. In hs research arcle a eraon can be resene 1 as X ( X... X... X where X ( S M S enoes he ob roucon sequence n ero an M enoes worsaon assgnmen of obs n ero. The bes soluon foun by arcle unl eraon s 1 enoe as (... an he bes soluon foun by 1 he swarm unl eraon s enoe as g ( g... g. The velocy of arcle a eraon can be resene as 1 V ( V... V... V where V ( VS VM VS enoes he velocy of ob roucon sequence n ero an VM enoes he velocy of worersaon assgnmen of obs n ero. To faclae unersanng of he roose mehoology a ob roucon sequence n a ero s use as an examle o llusrae he consrucon of a arcle. In S S ( S1... S N an S ( s 1 s 2... s N. s s bnary where s equal o 1 f ob s n he h osonof he roucon sequence; oherwse s 0. For examle suose he ob roucon sequence n ero 1 s ( n a arcle. Then s21 s32 s43 s14 1 an all oher bs are equal o zero. In VS VS ( VS1... VS N an VS ( vs 1 vs 2... vs N. Hgher value of vs means ha ob s more lely o be lace n he h oson n ero whle lower value ncaes ha s beer o lace he ob n anoher oson. In each eraon he velocy s uae accorng o equaon (15 an hen convere o he change of robably va he followng sgmo funcon. 1 s( vs N (18 1ex( vs where N s he number of obs n ero svs ( enoes he robably of lacng ob n he h oson of he roucon sequence n ero. In he eraon rocess of scree arcle swarm omzaon each arcle shoul be ecoe no a comlee roucon scheule. In he ob roucon sequence examle he consrucon of a ob roucon sequence n ero sars from a null sequence an hen laces an unscheule h ob n he oson from 1 o N accorng o followng robably [10]: q svs ( (19 ( ( svs ' ' U where U s he se of of unscheule obs n ero an h q ( s he robably of lacng ob n he oson. A comlee ob roucon sequence of a ero has been consruce when each of he obs n hs ero has been assgne o a oson. C. The roose hybr algorhm Se 1. Inalzaon Se 1.1 Inalze arameers arcle sze K max Se 1.2 Inalze arcles osons X an veloces V ranomly. Se 1.3 Evaluae obecve funcon value for each arcle. Inalze an g Se 2. erform eraon rocess whle ( max Se 2.1 for 1 o K Uae velocy of arcle for 1 o Uae ob roucon sequence n ero of arcle whle ob roucon sequence of ero s no emy en-whle en-for en-for Se 2.2 Uae Uae g c he frs ob n he roucon sequence Uae wor saon assgnmen of ob Chec conssency whle (no conssen Deec crcal machne an a o volaon se f here s alernaves of he crcal machne change a new assgnmen chec conssency else ranomly assgn a suable machne se conssen en-whle calculae roucon ouu of ob n ero elee hs ob from he ob roucon sequence Se 2.3 Incremen of eraon coun 1 en-whle Se 3. Reor he bes soluon of he swarm an corresonng obecve funcon value Fgure 2. The roceure of he hybr algorhm ISSN: (rn; ISSN: (Onlne

5 roceengs of he Worl Congress on Engneerng 2011 Vol I July Lonon U.K. arcle swarm omzaon s an effecve algorhm for solvng many yes of omzaon roblems. However f he swarm s gong o be n equbrum he evoluon rocess wll be saganee as me goes on [11]. Consran rogrammng s secalze for solvng roblems wh har consrans bu may be neffcen when he feasble search sace s very large. Hence a hybr algorhm whch combnes her comlemenary avanages o mrove he search rocess s roose n hs research. The roose hybr algorhm s summarze brefly n Fgure 2. In hs research he lannng horzon consss of mul eros. Due o caacy lmaons of he varous worsaons some obs may no be fnshe before her ue aes n some eros whle some worsaons may have remanng caaces n some eros. Hence s necessary o mae a rae-off among nvenory holng cos subconracng cos an worsaon ulzaon. A conce calle exra caacy s frsly nrouce. The exra caacy of a ob n a ero s he number of uns of he ob ha can be rouce n excess n hs ero whou affecng oher scheule obs. When a ob canno be fnshe before s ue ae n a ero an has exra caacy n revous eros he exra caacy wll be ulze o rouce an he nvenory s carre o hs ero. If here s sll baclog of hs ob afer ulzng exra caacy he amoun wll be subconrace n orer o mee cusomers eman. In racce subconracng cos of a ob s usually much hgher han s nvenory holng cos an manufacurng cos. Hence n he roose algorhm f a ob canno be fnshe before s ue ae even afer ulzng he exra caacy of revous eros wll be reae as nconssency an he crcal worsaon wll be eece o ry a new assgnmen. In aon he roceure ncaes ha consran roagaon n hs research aes he form of sngle-level bacracng. When nconssency occurs he algorhm wll eec he crcal resource an chec wheher hs crcal resource has any alernaves. If yes anoher worsaon selece from he alernaves wll be assgne o erform he oeraon; oherwse he algorhm wll no bacrac o he mos recenly scheule ob an us ranomly assgn a suable worsaon for accorng o DSO mechansm regarless of conssency an hen connue o scheule he nex ob unl all obs have been scheule. IV. COMUTATION RESULTS In hs secon he erformance of he roose hybr algorhm s analyse by comarng he resuls obane from solvng a se of ranomly generae es roblems wh ha of DSO. A. Tes roblem Se an arameers The values of arameers use n he algorhm are as follows. arcle sze s 100 maxmum number of eraons s 100 maxmum neral wegh s 0.8 mnmum neral wegh s 0.2. c 1 an c2 are boh equal o 2 he velocy of arcles s resrce n he range [-4 4]. Each ero conans 30 me slces he lengh of whch s 300 secons. There are 3 or 4 oeraons requre for roucng each un of a ob. Each oeraon has a rocessng me ranomly generae from [20 40]. The number of uns ha shoul be rouce (a ob n a ero s ranomly generae from [50 80]. Table 2 shows he scheme use o generae he es roblem. In he able ( nsm s use o enoe he arameer combnaon where s he number of eros n s he number of obs n each ero enoes he ue ae s enoes ob subconracng cos an m enos he number of worsaons. For examle scheme 1 means ha here are 5 eros he number of obs n a ero s ranomly generae from [5 10] he ue ae of each ob s ranomly generaes from [ S S] where α 0.4 β 0.7 he subconracng cos of each ob er un s generae from [ ] he number of worsaons s 12. TABLE 2 ARAMETER SCHEME No. arameer value 1 (5 [5 10] [ ] [ ] 12 2 (5 [5 10] [ ] [ ] 12 3 (5 [5 10] [ ] [ ] 12 4 (5 [5 10] [ ] [ ] 12 5 (5 [5 10] [ ] [ ] 20 6 (5 [5 10] [ ] [ ] 20 7 (5 [5 10] [ ] [ ] 20 8 (5 [5 10] [ ] [ ] 20 9 (10 [5 10] [ ] [ ] (10 [5 10] [ ] [ ] (10 [5 10] [ ] [ ] (10 [5 10] [ ] [ ] (10 [10 15] [ ] [ ] (10 [10 15] [ ] [ ] (10 [10 15] [ ] [ ] (10 [10 15] [ ] [ ] 12 B. Comarson wh DSO Uner each scheme fve es roblems are ranomly generae. The erformances of he hybr algorhm an DSO are obane by averagng he resuls of runnng he algorhms fve mes for each roblem. Table 3 shows he erformance comarson of he wo algorhms afer execung boh algorhms 100 eraons. In he able cos ff an me ff are calculae accorng o equaons (20 an (21. cos of DSO-cos of hybr algorhm cos ff= cos of DSO CU me of hybr algorhm-cu me of DSO me ff= CU me of DSO (20 (21 From Table 2 s clear ha he roose hybr algorhm can oban much beer scheule soluons n he sacrfce of longer comuaonal me. Snce he hybr algorhm nees longer comuaonal me for he same number of eraons. I s necessary o comare he erformance of hese wo algorhms wh he same comuaonal me. Table 4 lss he comarson resuls. In hs able Tso reresens he comuaonal me of DSO runnng 100 eraons. ISSN: (rn; ISSN: (Onlne

6 roceengs of he Worl Congress on Engneerng 2011 Vol I July Lonon U.K. TABLE 3 COMARISON WITHIN THE SAME ITERATIONS No. DSO HYBRID Cos ff Tme ff cos me cos me % 27.14% % 27.59% % 32.41% % 29.55% % 17.72% % % 19.66% % 15.66% % 21.08% % 23.16% % 20.33% % 18.39% % 39.61% % 39.02% % 39.84% % 41.49% TABLE 4 COMARISON WITHIN THE SAME COMUTATION TIME Cos ff No. 1/3 T so 1/2 T so 2/3 T so T so % 6.29% 5.89% 5.58% % 9.23% 9.26% 8.54% % 19.68% 18.06% 18.63% % 13.36% 13.53% 14.08% % 6.26% 6.67% 7.26% % 2.56% 3.10% 2.74% % 14.46% 14.27% 13.94% % 4.44% 3.79% 3.96% % 7.44% 7.82% 7.83% % 6.53% 6.70% 6.46% % 13.56% 13.14% 12.47% % 11.82% 11.95% 11.32% % 9.64% 9.51% 9.70% % 10.10% 10.32% 10.11% % 16.99% 17.40% 17.62% % 22.00% % Comuaonal exermens usng a se of ranomly generae es roblems emonsraes ha he hybr algorhm can generae beer roucon scheules wh he same number of eraons or he same amoun of comuaonal mes esecally when he sze of he roblem s large. ACKNOWLEDGEMENT The wor escrbe n hs aer was suore by a gran from he research Grans Councl of he Hong Kong Secal Amnsrave Regon Chna (roec No. HKU E. REFERENCE [1] U. Wemmerlov an N. Hyer Cellular manufacurng n he US nusry: a survey of users In. J. ro. Res vol. 27 no [2] J.R. Drole Scheulng vrual cellular manufacurng sysems hd Dsseraon urue Unversy Wes Lafayee IN [3] V.R. Kannan an S. Ghosh Cellular manufacurng usng vrual cells In. J. O. ro. Manage vol. 16 no [4] S.A. Iran T.M. Cavaler an.h. Cohen. Vrual manufacurng cells: Exlong layou esgn an nercell flows for he machne-sharng roblem. In. J. ro. Res ( [5] A. Bayasoglu. Caably-base srbue layou aroach for vrual manufacurng cells. In. J. ro. Res ( [6] K.L. Ma J.S.K. Lau an X.X. Wang A genec scheulng mehoology for vrual cellular manufacurng sysems: an nusral alcaon In. J. ro. Res vol. 43 no June [7] E..K. Tsang Founaon of Consran Sasfacon. Acaemc ress [8] J. Kenney an R.C. Eberhar arcle swarm omzaon IEEE nernaonal conference on neural newor [9] J. Kenney an R.C. Eberhar A scree bnary verson of he arcle swarm omzaon The worl mulconference on sysems cybernecs an nformacs [10] C.J. Lao C.T. Tseng an. Luarn A scree verson of arcle swarm omzaon for flowsho scheulng roblems Comuers an Oeraons Research vol [11].S. Sheloar. Sarry V.K. Jayaraman an B.D. Kularn. arcle swarm an an colony algorhms hybrze for mrove connuous omzaon Ale Mahemacs an Comuaon I s clear ha he roose hybr algorhm has beer erformance n locang goo scheules whn he same comuaonal me. Furhermore he mrovemen s more obvous when roblem sze becomes larger ue ae becomes gher or subconrac cos becomes hgher. Hence s suable for solvng real nusral roblems whch usually have large roblem sze. V. CONCLUSIONS Ths aer focuses on solvng he roucon scheulng roblem for vrual cellular manufacurng sysems oerang uner a mul-ero manufacurng senaro. The obecve s o mnmze he oal manufacurng cos whn he enre lannng horzon. A new mahemacal moel has been esablshe o escrbe he characerscs of a vrual cellular manufacurng sysem an a hybr algorhm whch combnes he avanages of he echnques of consran rogrammng an scree arcle swarm omzaon has been eveloe o generae effecvely he omal roucon scheule for he manufacurng sysem. ISSN: (rn; ISSN: (Onlne