Few heuristic optimization algorithms to solve the multi-period fixed charge production-distribution problem

Transcription

1 Few heursc opmzao algorhms o solve he mul-perod fxed charge produco-dsrbuo problem N. Bala a,* ad N. Jawahar b a Deparme of Mechacal Egeerg,Sr Krsha College of Egeerg ad Techology, Combaore , Ida. b Deparme of Mechacal Egeerg, Thagaraar College of Egeerg, Madura , Ida. Arcle fo: Receved: 05/09/2011 Acceped: 22/11/2011 Ole: 03/03/2012 Keywords: Mul-perod fxed charge problem, 0-1 mxed eger programmg problem, Geec algorhm, Smulaed aealg algorhm, Equvale varable cos Absrac Ths paper deals wh a mul-perod fxed charge produco-dsrbuo problem assocaed wh backorder ad veores. The obecve s o deerme he sze of he shpmes from each suppler ad backorder ad veores a each perod, so ha he oal cos curred durg he ere perod owards produco, rasporao, backorder ad veores s mmsed. A 0-1 mxed eger programmg problem s formulaed. Geec algorhm based populao search heursc, Smulaed aealg based eghbourhood search heursc ad Equvale varable cos based smple heursc are proposed o solve he formulao. The proposed mehodologes are evaluaed by comparg her soluos wh he lower boud soluos. The comparsos reveal ha Geec algorhm ad Smulaed aealg algorhm geerae beer soluos ha he Equvale varable cos soluos ad are capable of provdg soluos close o he lower boud value of he problems. 1. Iroduco I a scaered produco sysem, he produco locao also deermes he overall produco coss. I s because, he urba locao wll requre more labour ad overhead coss ha rural locao. Moreover, such scaered produco sysem wh scaered cusomers, he produco locao flueces he dsrbuo schedule ad hereby dsrbuo coss. Therefore, dusral problems where produco ad dsrbuo coss are boh of a * Correspodg auhor Emal address: rabala2001@yahoo.co. smlar magude, s ecessary o coordae he wo fucos order o lm global coss [1]. Mos compaes maage hese wo fucos depedely, wh lle or o coordao bewee produco ad dsrbuo plag. Ths decoupled approach works accepably well f here s suffce fshed goods veory o buffer he produco ad dsrbuo operaos from each oher. However, he cos of carryg veory ad he red o us--me operaos s creag pressure o reduce 77

2 JCARME N. Bala e al. Vol. 1, No. 2, March 2012 veores he dsrbuo chael. As a resul of hs pressure, may compaes are explorg closer coordao alog he maufacurg/ dsrbuo chael [2]. May compaes srve o sychroze her produco, rasporao ad repleshme plag by adopg supply cha maageme pracces such as vedor-maaged veory, effce cosumer respose, ad collaborave plag, forecasg ad repleshme. The ma obecve of hese collaborave mehods s o reduce effceces ad o elmae redudaces bewee he dffere parers [3]. The mul-perod fxed charge producodsrbuo problem (MPFCPDP) s a exeso of FCT (fxed charge rasporao) problem, where he me based sraegc decsos o sze of he shpmes from each suppler, veory, ad backorder ca make a ecoomcal dsrbuo. The MPFCPDP problem s dffcul o solve due o he presece of fxed coss, whch cause oleares he obecve fuco ad are kow o be Nodeermsc Polyomal-me NP hard [4]. The complexy of he problem s furher creased, whe suppler depede produc cos, me depede veory ad backorder are cluded he model. Ths lms he usage of he coveoal mul-perod ad fxed charge soluo procedures. There are may dffere mul-perod dsrbuo problems (MPDP) he leraure [2, 3, 5-12] volvg cosderaos of produco ad rasporao, possbly ogeher wh oher fucos. The revew o mulperod problems reveals he followg: mos of he models aemp o egrae veory ad rasporao ssues; excess avalably ay perod s held as veory a he supplers ed ad s used for he subseque perods; veory a he demad pos has o bee gve due cosderao; admsso of backorder may cosderably reduce he oal cos of logscs; o all he papers (excep [6,8,10]) have cluded he fxed charge assocaed wh rasporao; hough few papers have cluded fxed charge her models, hey do o deal exacly he mulperod fxed charge produco-dsrbuo bewee mulple sources (supplers) ad mulple desaos (cusomers) wh veory aleraves ad backorder cosderao o opmze he oal producodsrbuo cos. I he lgh of he above, hs paper cosders a MPFCPDP problem cocerg he produco, rasporao ad sorage of fshed goods from m supplers (dusral producers) o cusomers (demad ceers lke assemblg ceers, dsrbuo ceers ec.). The rasporao cos s he ma eleme he proposed produco-dsrbuo model. The oher cosderaos are sorage ad backorders. Ths paper cosders a wo-echelo veory sysem where he supplers supply capacy ad cusomers demads are deermsc. The purpose of maag wo-echelo veory s o mmze he oal dsrbuo cos whle egrag produco, rasporao, backorder ad veores. The producodsrbuo plag problem addressed here as MPFCPDP, s formulaed as a 0-1 mxed eger programmg problem. Geec algorhm (GA) based populao search heursc, Smulaed aealg algorhm (SAA) based eghbourhood search heursc ad a Equvale varable cos (EVC) based smple heursc are proposed o solve he formulao. The res of he paper s orgased as follows: Seco 2 addresses he problem evrome ad mahemacal formulao of he MPFCPDP. Seco 3 dscusses abou he proposed heurscs. Seco 4 provdes a umercal llusrao. Seco 5 dscusses he compuaoal resuls ad performace aalyss of he proposed mehodologes. A summary of he prese aalyss ad fuure research drecos are preseed he cocludg seco Problem evrome ad descrpo There are m supplers (dusral producers) o produce ad dsrbue a produc o cusomers (demad ceres lke assembly ceres, dsrbuo ceres ec.) T plag perods; each suppler =1,2, m has P us of produco each perod =1,2 T ad each cusomer =1,2, has D us of demad each perod =1,2 T. Each 78

3 JCARME Few heursc opmzao... Vol. 1, No. 2, March 2012 suppler ca produce he produc a a produco cos of CU per u ad shp o ay cusomer a a rasporao cos of C per u for shppg from suppler o cusomer plus a fxed cos of FC cluded for operag he roue o. A ay me perod, he oal cumulave produco of he supplers may or may o be equal o he oal demad of he cusomers. The excess or shorage of produco he perod s carred over o he subseque perod +1. The excess of produco perod, addressed here as he veory, s cosdered as a addoal supply avalable for he perod +1. I s ofed as SI a a veory holdg cos of SH per u per perod a h suppler s locao ad CI a a veory holdg cos of CH per u per perod a h cusomer s locao. O he oher had, he produco shorage of he perod (excess demad), addressed here as backorder, s cosdered as a addoal demad for he perod +1. I s ofed as BL a a pealy cos of BC per u per perod a h cusomer s locao. As he proposed model cosders shor plag perods (days/weeks/mohs), he cos assocaed wh produco (CU ), rasporao (.e. C ad FC ), veory (.e. SH ad CH ) ad backorder (.e. BC ) are depede of perod. The begg perod s veory ad backorder (.e., SI 0, CI 0 ad BL 0 ) are kow quaes. Mmzao of he sum of coss of produco, rasporao, holdg veory ad pealy for he backorder supply s cosdered as he obecve crero of he problem. 4. Mahemacal model Ths model aemps o egrae produco, rasporao, backorder ad veory decsos moolhcally from a ceralzed plag po of vew. Le be a bary varable o accou fxed rasporao cos. The mahemacal model of he MPFCPDP problem s formulaed as a 0-1 Mxed Ieger Programmg (MIP) problem as gve below. T Subec o : P SI T 1 CH * CI BC * BL (1) 1 1 X 1 1 SI, =1 m &, =1 T; (2) D BL 1 CI 1 m 1 X BL CI, =1 &, =1 T; (3) 1 f X > 0;, =1 m,, =1 &, =1 T; (4) 3. Decso varables X :Number of us of shpmes from suppler o cusomer perod SI :Number of us of veory wh suppler perod CI :Number of us of veory wh cusomer perod BL :Number of us of backlog for cusomer perod :Bary varable ha specfes he produc dsrbuo from suppler o cusomer perod (.e., = 1 f X > 0 ad = 0 f X = 0) 0 f X 0;, =1 m,, =1 &, =1 T; X 0 ad eger; (5) 79

4 JCARME N. Bala e al. Vol. 1, No. 2, March 2012, =1 m,, =1 SI 0 ad eger; &, =1 T; (6), =1 m, &, =1 T; (7) CI 0 ad eger;, =1 &, =1 T; (8) BL 0 ad eger;, =1 &, =1 T; (9) The obecve fuco gve by Eq. (1) ams o mmze he sum of he oal coss assocaed wh produco, rasporao, veory a suppler s sde ad cusomer s sde ad backorder. The frs erm of he obecve fuco provdes he oal cos of produco for he ere perod T ad he secod erm provdes he oal cos of rasporao for he ere perod T. The hrd erm addresses he oal cos of holdg veory a suppler s locaos for he ere perod T. The fourh ad he ffh erms dcae respecvely he oal cos holdg veory for he ere perod T ad he oal cos of backorder pealy for he ere perod T. Cosra se gve by Eq. (2) expresses he maeral balace a he suppler s sde bewee ay wo successve me ervals. Smlarly, cosra se (3) expresses he maeral balace a he cusomer s sde bewee ay wo successve me ervals. I he lef ad rgh sdes of he cosra se 3, eher he veory or backorder s prese. Cosra ses gve by Eqs. (4) ad (5) reur a bary value of depedg o he value of X. Cosra ses gve by Eqs. (6) o (9) esure he o-egavy aure of decso varables X, SI, CI, ad BL. 5. Soluo mehodologes I hs paper, he MPFCPDP model s solved by GA ad SAA based mea-heurscs ad EVC based smple heurscs. They are deleaed he followg secos GA ad SAA based mea-heurscs Over he las hry years, here has bee a growg eres problem solvg sysems based o he prcples of populao based ad eghborhood based search heurscs. I populao based search heurscs, he GA has bee creasgly appled o varous search ad opmzao problems ad has emerged as poeal echques o provde soluos wh accepable accuracy for NP hard problems [13-15]. I eghborhood based search heurscs, may researchers cosdered SAA for solvg may hard opmsao problems [15-20]. The proposed GA ad SAA based heurscs are srucured o solve he MPFCPDP wo sages. They are as follows. Sage I: Daa pu ad rasformao Ths sage remas commo boh GA ad SAA based heurscs. I acceps he daa of MPFCPDP uder cosderao as pu ad modulaes hem as a sgle-perod fxed charge produco-dsrbuo problem (SPFCPDP) daa se. The coverso provdes a modulaed daa suable for allocag he shpme quaes sgle-perod layou ad subsequely dervg a feasble mul-perod dsrbuo schedule he followg GA ad SAA roues, whch s deleaed Sage II. Sage II: Procedural seps of GA Sep 1: Parameers seg The parameers of GA are: pop_sze = 10; p_cross = 0.5; p_mu = 0.1; ge_o = 1; _ge 100 ( m* T)*( * T). 80

5 JCARME Few heursc opmzao... Vol. 1, No. 2, March 2012 Sep 2: Ial populao geerao I hs proposed GA, a chromosome c refers o a gee ype represeao of a dsrbuo schedule o he SPFCPDP. The chromosome c s he permuao of cell umbers of SPFCPDP marx, whch each cell s defed wh a uque The oal umber of cells he SPFCPDP, whch s also equal o he legh of he chromosome c, hus becomes (m*t)*(*t). Whe he supply ad demad are same perod (.e. c = r ), he hey form T umber of dagoal marx of sze m*. The umber of cells he dagoal marx hus becomes equal o m**t. Table 1 llusraes a example (m**t: 3*3*2) SPFCPDP marx cell umbers. A chromosome s srucured as wo pars. The frs par s framed by he cell umbers of dagoal SPFCPDP marx. The secod par s framed by he remag cell umbers of odagoal SPFCPDP marx. Table 2 shows a radomly developed chromosome wh cell umber for he above example. I he same way, a radomly developed e chromosomes form he al populao. Table 1. SPFCPDP cell umbers marx. r 1 2 c Table 2. A example chromosome of SPFCPDP. Frs par Sec. par Cell umber Cell umber Sep 3: Evaluao The chromosome, o decodg, provdes a feasble dsrbuo schedule o he MPFCPDP by allocag shpme quaes o he cells of SPFCPDP based o her prory as per he cell umber posos he chromosome. The he acual MPFCPDP dsrbuo quaes X, supplers veory SI, cusomers veory CI, ad backlog BL are derved by demodulag he allocaos of SPFCPDP. The oal produco-dsrbuo cos Z(m) correspodg o MPFCPDP dsrbuo schedule (m) s calculaed from he obecve Eq. (1). Each chromosome c he al populao s evaluaed erms of Z by he same procedure. Sep 4: Updag A he ed of frs geerao, he followgs parameers are updaed. pop _ bes global_bes ge_o ge_o 1. Sep 5: Termao checkg: The umber of geeraos s cosdered as he ermao crero. The ermao crero value of he llusrao s calculaed as follows. Termao crero = _ge =100+ (m*t)*(*t). If he ge_o value s less ha _ge go o he ex sep, else go o sep 7. Sep 6: New populao geerao The geerao of a ew populao volves hree asks: ) Seleco, ) Crossover ad ) Muao. The seleco process s repeaed as may mes as equal o pop_sze. I crossover operao, each chromosome s seleced wh probably p_cross. I muao, each gee s seleced wh probably p_mu. The, ha parcular gee s muaed. Afer geerag he ew populao he GA seps from 3 o 5 are repeaed ul reaches ermao. 81

6 JCARME N. Bala e al. Vol. 1, No. 2, March 2012 Sep 7: Oupu The dsrbuo schedule ( (bes) ) ad dsrbuo cos (Z (ear op) ) he global_bes are he soluos o he problem ad are gve as oupu. Sage II: Procedural seps of SAA Sep 1: Ialzao of SAA parameers ad couers The parameers of SAA are: TE = 475, ACCEPT = 0, TOTAL = 0, FREEZE = 0, α =0.90 ad Termao codo = (FREEZE = 5 or TE = 20). Sep 2: Geerao of al seed srg I hs proposed SAA, a srg S refers o a gee ype represeao of a dsrbuo schedule o he SPFCPDP. The srg S s he permuao of cell umbers of SPFCPDP marx, whch each cell s defed wh a uque The oal umber of cells he SPFCPDP, whch s also equal o he legh of he srg S, hus becomes (m*t)*(*t). Whe he supply ad demad are same perod (.e. c = r ), he hey form T umber of dagoal marx of sze m*. The umber of cells he dagoal marx hus becomes equal o m**t. Table 3 llusraes a example (m**t: 3*3*2) of SPFCPDP marx cell umbers. r Table 3. SPFCPDP cell umbers marx. 1 2 c SPFCPDP marx. The secod par s framed by he remag cell umbers of o-dagoal SPFCPDP marx. Table 4 shows a example of a radomly developed seed srg wh cell umber for he above example. Frs par Sec. par Table 4. A example seed srg of SPFCPDP. Cell umber Cell umber Sep 3: Evaluao The srg, o decodg, provdes a feasble dsrbuo schedule o he MPFCPDP by frs allocag shpme quaes o he cells of SPFCPDP based o her prory as per he posos he srg. The he acual MPFCPDP dsrbuo quaes X, supplers veory SI, cusomers veory CI, ad backlog BL are derved by demodulag he allocaos of SPFCPDP. The oal produco-dsrbuo cos Z(m) correspodg o MPFCPDP dsrbuo schedule (m) s calculaed from he obecve Eq. (1). Sep 4: Geerao of eghborhood seed srg ad s evaluao A eghborhood seed srg S o he curre seed srg S s geeraed va muao operaor [21]. Muao for hs research s a uary radom muao. A radom umber U bewee 0 ad 1 s geeraed correspodg o every eleme of he seed ad f he radom umber U s less ha mu (0.5), he hose wo parcular elemes are erchaged (muaed). The muao exchages he sequece elemes wh he seed for maag he feasbly. Ths process s carred ou separaely he frs ad secod par of he srg. A srg s srucured as wo pars. The frs par s framed by he cell umbers of dagoal 82

7 JCARME Few heursc opmzao... Vol. 1, No. 2, March 2012 Sep 5: Calculao of uphll accepace parameer dela The ew seed srg S s seleced by calculag he value of he dela. Dela s he cos dfferece bewee he eghborhood seed srg dsrbuo schedule ad he al seed srg dsrbuo schedule..e., Dela = Z(m ) Z(m); If Dela 0 proceed o sep 6 (dowhll move), else (Dela > 0) go o sep 7 (uphll move). Sep 6: Dowhll move Assg m =m Z(m) = Z(m ) ad ACCEPT ACCEPT 1 If Z (m) < Z (ear op) he se (bes) =m ad Z (ear op) = Z (m), else go o sep 9. Sep 7: Uphll move (-Dela /TE ) P e ad sample Compuao of R (Radom o. geeraed (0, 1)).If proceed o sep 8 else proceed o sep 9. Sep 8: R P Assg m=m, Z(m) = Z(m ) ad ACCEPT ACCEPT 1. Sep 9: Se TOTAL= TOTAL + 1. Sep 10: Check for ermao The algorhm s froze. The ermao of he SAA s acheved whe FREEZE couer reaches he specfed value (FREEZE=5) or he emperaure TE falls o a pre-specfed value (TE=20). Now (bes) coas he bes MPFCPDP dsrbuo schedule ad Z (ear op) has he mmum Z(m). If (TOTAL > (m**)) or (ACCEPT > (m**)/2), he proceed o sep 11 else go back o sep 4 ul sasfes he codo sep 10. Sep 11: Oupu The dsrbuo schedule ( (bes) ) ad dsrbuo cos (Z (ear op) ) he global_bes are he soluos o he problem ad are gve as oupu Equvale varable cos heursc A lear dsrbuo model ca be obaed by relaxg he egral resrcos [22] of he olear problem wh equvale varable rasporao cos EVC, whch s defed as: EVC C FC / M( P, D, =1 m,, =1 ad, =1 T ; (10) M Z T m T T Subec o : m 1 1 ( EVC ( CU * * X T ) X ) ) T 1 m 1 1 SH * SI CH * CI BC * BL (11) Cora ses (2), (3), (6), (7), (8) ad (9). The above lear programmg problem ca be solved opmally usg LINGO Solver. The subsuo of he opmal soluo Eqs. (1) ad (12) respecvely provdes Equvale varable cos soluo Z (EVC) ad lower boud value Z (L) Numercal llusrao A llusrao for he above hree mehodologes s gve below wh a example problem. The daa used for he llusrao are as follows: m 3, 3, 3. Tables 5a ad 5c show he daa relaed o rasporao, suppler ad cusomer respecvely. 83

8 JCARME N. Bala e al. Vol. 1, No. 2, March 2012 Table 5a. Trasporao cos daa (C & FC ). Table 5b. Supplers daa (P, CU, SH & SI 0 ). P Table 5c. Cusomers daa (D, CH, BC, BL 0 & CI 0 ). D D CH BC BL CI GA ad SAA based mea-heurscs soluos The dsrbuo schedule (Tables 6 ad 7) ad oal produco-dsrbuo cos (Z) of he above example problem are gve as follows (solved usg GA ad SAA separaely). Z (ear op) = 27, (GAs soluo) FC 900 C P SH SI CU Z (ear op) = 27, (SAAs soluo) 5.5. Equvale varable cos soluo The equvale varable cos marx ad he opmal dsrbuo schedule (solved usg LINGO solver) of he relaxed problem are gve Tables 8 ad 9 respecvely. The subsuo of he opmal soluo Eqs. (1) ad (11) respecvely provdes equvale varable cos soluo Z (EVC) ad lower boud value Z (L). Z (L) = 27, (Lower boud value) Z (EVC) = 28, (Equvale varable cos soluo) 6. Compuaoal resuls ad performace aalyss To evaluae he performace of he proposed heurscs, compuaoal expermes were doe o 40 es problems. Fory es problems alog wh her oupus are cosdered for hs performace comparso. The comparsos reveal he followgs: Equvale varable cos mehod provdes oly approxmae soluos o all he es problems bu very few of hem are close or equal o GA ad SAA soluos. EVC heursc ca also provde he lower boud value of he problem; GA ad SAA based heurscs geerae beer soluos ha he EVC heursc ad are capable of provdg soluos close or equal o lower boud values. The average perceage devao of GA based heursc wh lower boud value s 2.12%. The average perceage devao of SAA based heursc wh lower boud value s 2.21%. The average perceage devao of EVC heursc wh lower boud value s 8.89%. They are depced Fg

9 JCARME Few heursc opmzao... Vol. 1, No. 2, March 2012 Table 6. Dsrbuo schedule usg GA. SI 1 SI BL CI SI 3 Table 7. Dsrbuo schedule usg SAA. SI 1 SI 2 SI BL CI Table 8. Equvale varable cos (EVC ) marx Table 9. Opmal dsrbuo schedule for he relaxed problem. SI 1 SI 2 SI BL CI

10 JCARME N. Bala e al. Vol. 1, No. 2, March 2012 Fg. 1. Perceage devao of proposed heurscs soluos. 7. Coclusos Ths paper proposes GA ad SAA based meaheurscs ad EVC based smple heurscs o solve he MPFCPDP problem. The proposed mehodologes are evaluaed by comparg her soluos wh lower boud values. The comparso of resuls reveal ha he GA ad SAA geerae beer soluos ha he EVC soluos ad are capable of provdg soluos close or equal o he lower boud values. Ths paper coceraes o sgle-sage mul-perod fxed charge model. As a fuure research, he sgle perod formulao o he proposed mulperod fxed charge problem faclaes s scope for exedg hs o mul-sage supply cha problems. Refereces [1] C. D. Flppo ad G. Fke, A egraed model for a dusral produco- dsrbuo problem, IIE Trasacos, Vol. 33, pp , (2001). [2] P. Chadra, ad M. L. Fsher., Coordao of produco ad rasporao plag, Europea Joural of Operaoal Research, Vol. 72, pp ,(1994). [3] N. Rzk, A. Marel, S. D. Amours, Sychrozed produco rasporao plag a Sgle-pla muldesao ework, Joural of he Operaoal Research Socey, Vol.59, pp , (2008). [4] D. Km, P. M. Pardalos, A soluo approach o he fxed charge ework flow problem usg a dyamc slope scalg procedure, Operaos Research Leer, Vol. 24, pp ,(1999). [5] N. Haq, Prem Vra, ad Aru Kada, A egraed produco-dsrbuo model for maufacure of urea: a case, Ieraoal Joural of Produco Ecoomcs, Vol. 39, pp , (1991). [6] G. Barbarosogluad ad D. Ozgur, Herarchcal desg of a egraed produco ad 2-echelo rasporao sysem, Europea Joural of Operaoal Research, Vol. 118, pp , (1999). [7] J. U. Km, ad Y. D. Km, A Lagraga relaxao approach o mulperod veory/dsrbuo plag, Joural of he Operaoal Research Socey, Vol.51, pp , (2000). [8] P. Kamsky ad D. Smch-Lev, Produco ad rasporao lo szg a wo sage supply cha, IIE 86

11 JCARME Few heursc opmzao... Vol. 1, No. 2, March 2012 Trasacos, Vol.35, pp , (2003). [9] Y. B. Park, A egraed approach for produco ad dsrbuo plag supply cha maageme, Ieraoal Joural of Produco Research, Vol. 43(6), pp , (2005). [10] T. F. Abdelmagud ad M. M. Dessouky, A Geec algorhm approach o he egraed veory-dsrbuo problem, Ieraoal Joural of Produco Research, Vol. 44 (21), pp , (2006). [11] F. Z. Sargu ad H. E. Rome, Capacaed produco ad subcoracg a seral supply cha, IIE Trasacos, Vol. 39, pp , (2007). [12] A. S. Safae, S. M. Moaar Husse, R. Z. Faraha, F. Jola ad S. H. Ghodsypour, Iegraed mul-se produco-dsrbuo plag supply cha by hybrd modelg, Ieraoal Joural of Produco Research, Vol.48 (14), pp , (2010). [13] D. E. Goldberg, Geec algorhms search, Opmzao ad Mache Leag, Addso Wesley Lmed: U.S.A, (2000). [14] M. Ge, A. Kumar, J. R. Km, Rece ework desg echques usg evoluoary algorhms, Ieraoal Joural of Produco Ecoomcs, Vol. 98, pp [15] W. Dullaer, M. Sevaux, K. Sorese, Applcaos of Mea-heurscs, Europea Joural of Operaoal Research, Vol. 179, pp , (2007). [16] S. Parhasarahy ad C. Raedra,, A expermeal evaluao of heurscs for schedulg a real lfe flow shop wh sequece-depede se up me of obs, Ieraoal Joural of Produco Ecoomcs, Vol. 49, pp ,(1997). [17] S.G. Poambalam, N. Jawahar, ad P. Aravda, A Smulaed Aealg Algorhm for Job Shop Schedulg, Produco Plag ad Corol, Vol.10, pp , (1999). [18] V. Jayarama ad A. Ross, A smulaed aealg mehodology o rasporao ework desg ad maageme, Europea Joural of Operaoal Research, Vol.144, pp , (2003). [19] B. Suma ad P. Kumar,A survey of smulaed aealg as a ool for sgle ad mul-obecve opmzao, Joural of Operaoal Research Socey, Vol. 57, pp , (2006). [20] A. N. Bala ad N. Jawahar, A Smulaed Aealg Algorhm for a wo-sage fxed charge dsrbuo problem of a Supply Cha, Ieraoal Joural of Operaoal Research, Vol. 7(2), pp , (2010). [21] S. A. Masour, A smulaed aealg approach o a b-crera sequecg problem a wo-sage supply cha, Compuers & Idusral Egeerg, Vol.50, pp , (2006). [22] U.S. Palekar, M. H. Karwa, ad S. Zos, A brach-ad-boud mehod for he fxed charge rasporao problem, Maageme Scece, 36 (9), pp , (1990). 87