A fuzzy approach to capacity constrained MRP systems *

Transcription

1 A fuzzy approach o capacy consraned MRP sysems * J. Mula CGP (Research Cenre on Producon Managemen and Engneerng) Polyechnc Unversy of Valenca Plaza Ferrándz y Carbonell, 080 Alcoy (Alcane) fmula@cgp.upv.es R. Poler CGP (Research Cenre on Producon Managemen and Engneerng) Polyechnc Unversy of Valenca Plaza Ferrándz y Carbonell, 080 Alcoy (Alcane) rpoler@cgp.upv.es J.P. García-Sabaer CGP (Research Cenre on Producon Managemen and Engneerng) Polyechnc Unversy of Valenca Campus de Vera, s/n 460 Valenca jpgarca@cgp.upv.es Absrac A model for he Maeral Requremen Plannng (MRP) problem wh unceany n a mul-produc, mul-level and mulperod manufacurng envronmen s proposed. An opmzaon model ha akes no accoun he ambguy n marke demand, ambguy n capacy daa, and ambguous coss for delayed demand s formulaed. hs work uses he approach of fuzzy programmng proposed by Gen, sujmura and da [4]. Such an approach makes possble o model he ambguy ha could be presen n he MRP sysems as rangular fuzzy numbers. he man goal s o deermne he maser producon schedule, sock levels, delayed demand, and capacy usage levels over a gven plannng horzon n such a way as o hedge agans he unceany. Fnally, he model s esed usng real daa from an auomoble sea manufacurer. Keywords: MRP, Unceany modellng, Fuzzy ses. nroducon n hs paper, for he purpose of demonsrang he usefulness and sgnfcance of he fuzzy programmng for producon plannng, a fuzzy approach s appled o a MRP problem wh ambguous daa. For a dealed descrpon of MRP, he reader s referred o Orlcky [] and Vollman, Berry and Whybark []. n fuzzy programmng, he ambguous coeffcens and vague aspraons are represened by fuzzy ses []. nuguch and chhash [6] classfy fuzzy programmng approaches no hree caegores: () mahemacal programmng wh vagueness, () mahemacal programmng wh ambguy, and () mahemacal programmng wh vagueness and ambguy. he frs ype of fuzzy programmng s called flexble programmng and he las wo ypes of fuzzy programmng are called possblsc programmng. n hs paper, wh he am of showng he usefulness and sgnfcance of MRP modellng wh fuzzy programmng, an approach of possblsc programmng s appled o a MRP problem. he man conrbuon of hs paper o he operaonal research feld s a praccal applcaon of known possblsc programmng, accompaned by expermens based on real daa. Oher applcaons of possblsc programmng n producon plannng problems can be found n [5] and [7]. hs paper s organzed as follows. Frsly, n Secon, a model for producon plannng n a capacy consraned MRP sysem wh ambguous daa s presened. n Secon, he fuzzy model s ransformed no an equvalen crsp model. Secon 4 uses a real-lfe sudy case o llusrae he poenal savngs whch can be aaned by usng fuzzy models n a fuzzy envronmen. n secon 5, hs paper offers some conclusons. * hs research has been carred ou n he framework of a projec funded by he Scence and echnology Mnsry of he Spansh Governmen, led Busness process negraon, knowledge managemen and decson suppo ools n supply chan of ndusral SMEs. Ref. DP

2 Formulaon of he problem A Lnear Programmng (LP) model for he capacy consraned MRP problem orgnally proposed n [9] and called MRPDe s adoped as he bass for our work. MRPDe s a model for he opmzaon of he medum erm producon plannng problem n a capacy consraned MRP, mul produc, mul level and mul perod manufacurng envronmen. Equaon () shows he oal coss o be mnmzed: coss of he nvenores c, coss of he exra me used by resources, cex, and coss of he lazy me of resources, coc. he MRPDe ncludes a plan o sasfy he delayed demands penalzed wh a cos, crd. s assumed ha hs cos s lnear o he number of backlogs n every perod. he balance equaons for he nvenory are gven by he group of consrans (). hese equaons ake no accoun he backlogs of he demand whch behave as a negave nvenory. s mpoan o hghlgh he consderaon of he parameer RP ha guaranees he connuy of he MRP along he successve explosons carred ou durng a gven plannng horzon. he producon n every perod s lmed by he avalably of a group of shared resources. he equaon () consders he lms of capacy of hese resources. hs equaon has been hough n a smlar way ha n he model proposed by Bllngon, Mcclan and homas [] alhough he seup mes have no been ncluded. he decson varables oc and ex are no lmed by any esablshed parameer bu are penalzed wh he correspondng coss n he objecve funcon. hs s o provde he mos possble generaly o he model. he lmaon of hese varables for specfc applcaons could be easly consdered akng no accoun ha f hose lms are exceeded he soluon of he model could be no feasble. A consran has also been added (4) o fnsh wh he delays n he las perod () of he plannng horzon. he model also conemplaes he non negavy consrans (5) for he decson varables. Fnally, he decson varables P, NV and Rd wll be defned as connuous or neger varables dependng on he manufacurng envronmen where he model s appled. Le us consder he followng fuzzy formulaon of he MRPDe model. Decson varables and parameers for he model are defned n able. Mnmze z = cp P + c NV + crd Rd + ( coc oc + cex ex ) Subjec o NV = AR = = R r = =, + P, S + RP NV Rd aj ( Pj RPj ) Rd d,, + + = =, = () j= r P + oc ex = CAP r = R, = () Rd =0 = (4) P, NV, Rd, oc, ex 0 =, r = R, = (5) () where crd = (crd, crd, crd ), d ), AR r = (AR r, AR r, AR r ) and d = (d, d, CAP = (CAP, CAP, CAP ) are posve rangular fuzzy numbers (FNs). 548

3 able : Decson varables and model parameers. Ses Se of perods n he plannng horzon ( = ) Se of producs ( = ) J Se he paren producs n he bll of maerals (j = J) R Se of resources (r = R) Decson Varables Daa P Quany o produce of he produc on perod d Marke demand of he produc on perod NV nvenory of he produc a he end of perod a j Requred quany of o produce an un of he Rd Delayed demand of he produc a he end of produc j perod oc Undeme hours of he resource r on perod S Lead me of he produc Decson Varables Daa ex Oveme hours of he resource r on perod NV0 nvenory of he produc on perod 0 Objecve Funcon Cos Coeffcens Rd0 Delayed demand of he produc on perod 0 cp Varable cos of producon of an un of he produc RP Programmed recepons of he produc on perod c nvenory cos of a un of he produc echnologcal Coeffcens crd Delayed demand cos of a un of he produc ARr Requred me of he resource r for un of producon of he produc coc Undeme hour cos of he resource r on perod CAP Avalable capacy of he resource r n he perod cex Oveme hour cos of he resource r on perod A fuzzy programmng model he approach of Gen, sujmura and da. [4] s a generalzed ransformaon mehod ha s applcable o any ype of fuzzy consran and objecve funcon (maxmzaon or mnmzaon), where he fuzzy parameers are represened by FNs whch membershp funcon s defned n [4] as: he operaor used o aggregae he fuzzy objecve funcon and consrans s he mn-operaor []. he soluon of he fuzzy problem may be acheved solvng he followng mahemacal programmng problem: r µ r ( x) = r r r ( x r ) + ( x r ) + 0 f f f ( r ( r x r ) x r ) ( x r r, x) (6) Mnmze z = cpp + c NV + (( ) crd + αcrd) Rd + ( cococ + cexex ) Subjec o NV NV = = = = R α (7) r = =, + P, S + RP NV Rd aj Pj + RPj + Rd d +,, ) ( ) j= ( α αd =, = (8), + P, S + RP NV Rd aj Pj + RPj + Rd d +,, ) ( ) j= ( α αd =, = (9) (( α ) AR + αar ) P + oc ex ( α CAP + αcap r = R, = (0) r r ) (( α ) AR + αar ) P + oc ex ( α CAP + αcap r = R, = () r r ) 549

4 Rd =0 = () P, NV, Rd, oc, ex 0 =, r = R, = () where 0 α s a cu value. n order o solve he problem α s sele down paramercally o oban he value of he objecve funcon for each one of hose α [0, ]. he resul s, however, a fuzzy se and he planner have o decde ha par (α, z) consders opmal f he wans o oban a crsp soluon. 4 mplemenaon and resoluon he models have been mplemened wh a hgh level language for mahemacal programmng models, he modellng language MPL [8]. Resoluon has been carred ou wh he opmzaon solver CPLEX []. Lasly, he npu daa and oupus of he models are managed hrough a Mcrosof Access 000 daabase. 5 Applcaon n an auomoble sea company hs secon uses a real-lfe example o llusrae he poenal savngs whch can be aaned by usng fuzzy models n a fuzzy envronmen. he proposed models are appled n a frm ha manufacurng and assemblng seas for auomobles. he hypoheses o carry ou he compuaonal expermen are summarzed as follows: he sudy consders a sngle pa (wh s bll of maerals). he decson varables, P, NV and Rd are neger. Exernal demand only exss for he fnal produc. Backorders of he delayed demand for he fnal produc PNR are consdered. Only he producve resource resrcs he producon: he assembly lne. For he fuzzy model, he parameer α s a cu value (0 α ) wh a sep of 0.. Also, he fuzzy cos coeffcen, crd = (crd, crd, crd ), he fuzzy rgh-hand-sde number, d = (d, d, d ), and he fuzzy echnologcal coeffcens, AR r, AR r ) and AR r = (AR r, CAP = (CAP, CAP, CAP ), requred by he fuzzy model have been defned followng company crera. For nsance, n he case of he demand nformaon: o d s obaned by decreasng 0% he value of d, o d s consdered he demand nformaon receved by he company, o d s obaned by ncreasng 0% he value of d. hey are no consdered producon varaons by qualy or machne falures. has been consdered a sx monhs plannng horzon wh a weekly perod plannng. he planned orders are recalculaed n every plannng perod, whle he programmed recepons of he componens are consdered frm. he execuon performance ndcaors for each MRP exploson are: producon, nvenory, delayed demand and oveme coss. he company receves (daly) from he OEM he demand nformaon wh a plannng horzon for sx monhs. However, hese demand forecass are rarely precse (see [0]). herefore, hs secon wll valdae f he fuzzy model for producon plannng, proposed n hs paper, can be a useful ool for he decson makng process of he producon planners. he expermen has been carred ou on a PC, wh AMD Ahlon processor a 600 MHz and wh 56 MB of RAM memory, n he followng way: consders he echncal and economc nformaon of he pa. Moreover, he demand nformaon for a plannng horzon of 0 weeks. Each MRP s execued for each one of he weeks updang he demand values, he nvenory, he delayed demand and he programmed recepons of componens. he dealed daa of hs compuaonal expermen and he MPL models can be found n Mula (004). 550

5 he evaluaon mehod consss on he comparave analyss of he performance of he wo models,.e. he fuzzy model and he deermnsc model, accordng o a group of parameers defned n [9]: () he oal coss; () he servce level; () he levels of nvenory; (v) he plannng nervousness respec o he planned perod and he planned quany (see able ); and (v) he compuaonal effcency (able ). n he case of he fuzzy model ha provdes a fuzzy soluon, he fuzzy se of he decson has been obaned. Nex, has been chosen as crsp soluon whch obans he bes resuls n he hghes number of he evaluaed ndcaors. Along hs secon wll only be consdered for hs model he seleced crsp soluon, where he parameer α was esablshed a 0. able : Evaluaon of he resuls. Number of mnmum nvenory levels Plannng nervousness (perod ) Plannng nervousness (quany) oal coss ( ) Servce Model level (%) MRPDe ,07 Fuzzy MRP ,69 able : Effcency of he compuaonal expermens for a MRP execuon (frs week). Model eraons Varables neger Consrans Elemens non zero Array Densy (%) CPU me (seconds) MRPDe Fuzzy MRP Boh models presen an average servce level above 99.5% he fuzzy model provdes a lghly beer value n hs performance ndcaor. Conrarly, he fuzzy model generaes a hgher number of mnmum nvenory levels,.e. 9 from he 46 evaluaed ems presen lower nvenores wh he producon plans obaned by he fuzzy model n conras wh he 8 ems wh lower nvenores provded by he MRPDe model. Boh models have presened a smlar nervousness wh respec o he planned me perod. On he oher hand, he fuzzy model presens he bes value of nervousness wh respec o he planned quany. Also, he fuzzy model generaes lower oal coss han MRPDe. hese dfferences n he oal coss are due, manly, o wo aspecs: () he consderaon of possble fuure varaons of he demand ha orgnaes larger producon and nvenores wh he objecve of avodng he srongly penalzed demand backlogs, and () he src consrans of he deermnsc model, where he requred capacy and he avalable capacy of he assembly lne are fxed rgdly. Fnally, he fuzzy model has provded freedom of acon wh regard o problems where ambguous values appear wh a mnmum ncremen of he requremens of nformaon sorage and a moderae ncremen of he requred CPU me. 6. Conclusons. n many manufacurng envronmens, such as he auomoble ndusry, he producon plannng decsons have o be made under condons of unceany n parameers as mpoan as coss, marke demand or capacy daa. A model based on fuzzy mahemacal programmng for producon plannng under condons of unceany has been proposed. n a general way, he srucure of he fuzzy model has been able o ncrease he group sasfacon (level of servce, nvenory levels, plannng nervousness and oal coss) whou causng an explosve growh of he compuaonal effo. Presened he research conclusons o he company, he saff n charge of plannng have shown her neres n he new model ha would allow hem o consder n a beer way he demand varably and o nroduce her percepons. References [] Bellman, R. and Zadeh, L.A., Decson-makng n a fuzzy envronmen, Managemen Scence, vol. 7, pp. 4-64,

6 [] Bllngon, P.J., Mcclan, J.O. and homas, L.J. Mahemacal programmng approaches o capacy consraned MRP sysems: Revew, formulaon and problem reducon, Managemen Scence, vol. 9, pp. 6-4, 98. [] CPLEX Opmzaon nc., Usng he CPLEX callable lbrary, 994. [4] Gen, M. sujmura, Y. and da, K., Mehod for solvng mulobjecve aggregae producon plannng problem wh fuzzy parameers, Compuers and ndusral Engneerng, vol., (-4), pp. 7-0, 99. [5] Hsu, H. and Wang, W., Possblsc programmng n producon plannng of assemble-o-order envronmens, Fuzzy Ses and Sysems, vol. 9, pp , 00. [6] nuguch, M. and chhash, H., Relave modales and her use n possblsc lnear programmng, Fuzzy Ses and Sysems, vol. 5, pp. 0-, 990. [7] nuguch, M., Sakawa, M. and Kume, Y., he usefulness of possblsc programmng n producon plannng problems, nernaonal Journal of Producon Economcs, vol., pp. 45-5, 994. [8] Maxmal Sofware ncorporaon, MPL modellng sysem. Release 4., USA, 000. [9] Mula, J., Models for producon plannng under unceany. Applcaon n a company of he auomoble secor, PhD (n Spansh), Polyechnc Unversy of Valenca, Span. Ed. SP-UPV, SBN: , 004. [0] Mula, J., Poler, R., García, J.P. and Oz, A., Supply plannng and demand behavour n an auomoble ndusry supply chan, nernaonal Conference on ndusral Engneerng and Producon Managemen (EPM 0), Poo (Pougal), May, 6-8, 00. [] Orlcky, J., Maeral Requremens Plannng, McGraw Hll, London, 975. [] Vollmann,.E., Berry, W.L. and Whybark, D.C., Manufacurng plannng and conrol sysems. hrd Edon, rwn, Homewood, llnos, 99. [] Zadeh, L.A., Fuzzy ses, nformaon Conrol, vol. 8, pp. 8-5,