Fuzzy Programming Approach for a Multi-objective Single Machine Scheduling Problem with Stochastic Processing Time

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1 Proceedgs of the World Cogress o Egeerg 008 Vol II WCE 008, July - 4, 008, Lodo, U.K. Fuzzy Programmg Approach for a Mult-obectve Sgle Mache Schedulg Problem wth Stochastc Processg Tme Ira Mahdav*, Babak Javad, Al Tad Abstract - Ths paper cosders a fuzzy programmg approach for a mult-obectve sgle mache schedulg problem whe processg tmes of obs are ormal radom varables. The probablstc problem s coverted to a equvalet determstc programmg problem. The the fuzzy programmg techque has bee appled to obta a compromse soluto. A umercal example demostrates the feasblty of applyg the proposed model to sgle mache schedulg problem. Idex Terms Sgle mache schedulg, ormal radom varable, Fuzzy programmg I. ITRODUCTIO Schedulg cossts of plag ad arragg obs a orderly sequece of operatos order to meet customer s requremets []. The schedulg of obs ad the cotrol of ther flows through a producto process are the most sgfcat elemets ay moder maufacturg systems. The sgle mache evromet s bass for other types of schedulg problems. I a sgle mache schedulg, there s oly oe mache to process all obs so that optmzes system performace measures such as makespa, completo tme, tardess, umber of tardy obs, dle tmes, sum of the maxmum earless ad tardess. I sgle mache schedulg, most researches are cocered wth the mmzato of a sgle crtero. However, schedulg problems ofte volve more tha oe aspect ad therefore they requre multple crtera aalyses []. Ish ad Tada [3] cosdered a sgle mache schedulg problem mmzg the maxmum lateess of obs wth fuzzy precedece relatos. A fuzzy precedece relato relaxes the crsp precedece relato ad represets the satsfacto level wth respect to the precedece betwee two obs. Ira Mahdav, Mazadara Uversty of Scece & Techology, Departmet of Idustral Egeerg, Babol, Ira, E.mal: raarash@redffmal.com Babak Javad, Mazadara Uversty of Scece & Techology, Departmet of Idustral Egeerg, Babol, Ira, E.mal: babakavad@ustmb.ac.r Al Tad, Mazadara Uversty of Scece & Techology, Departmet of Idustral Egeerg, Babol, Ira, E.mal: al_tad@yahoo.com Adamopoulos ad Papps [4] preseted a fuzzy-lgustc approach to mult-crtera sequecg problem. They cosdered a sgle mache, whch each ob s characterzed by fuzzy processg tmes. The obectve s to determe the processg tmes of obs ad the commo due as well as to sequece the obs o the mache where pealty values are assocated wth due dates assged, earless, ad tardess. Aother approach to solve a mult-crtera sgle mache schedulg problem s preseted by Lee, et al. [5]. They proposed a approach usg lgustc values to evaluate each crtero (e.g. very poor, poor, far, good, ad very good) ad to represet ts relatve weghts (e.g. very umportat, umportat, moderately mportat, mportat, ad very mportat). Also, a tabu search method s used as a stochastc tool to fd the ear optmal soluto for a aggregated fuzzy obectve fucto. Chaas ad Kaspersk [6] cosdered two sgle mache schedulg problems wth fuzzy processg tmes ad fuzzy due dates. They defed the fuzzy tardess of a ob a gve sequece as a fuzzy maxmum of zero ad the dfferece betwee the fuzzy completo tme ad the fuzzy due date of ths ob. I the frst problem, they cosdered the mmzato of the maxmal expected value of a fuzzy tardess. I the secod oe, they cosdered the mmzato of the expected value of a maxmal fuzzy tardess. Chaas ad Kaspersk [7] cosdered the sgle mache schedulg problem wth parameters gve the form of fuzzy umbers. It s assumed that the optmal schedule such a problem caot be determed precsely. I ther paper, t s show how to calculate the degrees of possble ad ecessary optmalty of a gve schedule oe of the specal cases of the sgle mache schedulg problems. Azzoglu, et al. [8] studed the b-crtera schedulg problem of mmzg the maxmum earless ad the umber of tardy obs o a sgle mache. They assumed that dle tme serto s ot allowed. Frst, they examed the problem of mmzg maxmum earless whle keepg the umber of tardy obs to ts mmum value. They also developed a geeral procedure to fd the effcet schedule mmzg a composte fucto of the two crtera by evaluatg oly a small fracto of the effcet solutos. They adopted the geeral procedures for the b-crtera ISB: WCE 008

2 Proceedgs of the World Cogress o Egeerg 008 Vol II WCE 008, July - 4, 008, Lodo, U.K. problem of mmzg the maxmum earless ad umber of tardy obs. Ere ad Guer [9] cosdered a b-crtera schedulg problem wth sequece depedet setup tmes o a sgle mache. The obectve fucto s to mmze the weghted sum of total completo tme ad total tardess. A teger programmg model s developed for the problem whch belogs to a P-Hard class. For solvg problems cotag a large umber of obs, a specal heurstc s also used for large obs problems. To mprove the performace of the tabu search (TS) method, the result of the proposed heurstc algorthm s take as a tal soluto of the TS method. Tavakkol-Moghaddam, et al. [0] preseted a fuzzy goal programmg based approach for solvg a mxed-teger model of a sgle mache schedulg problem mmzg the total weghted flow tme ad total weghted tardess. Because of the exstg coflct of these two obectves, they proposed a fuzzy goal programmg model based approach to solve the exteded mathematcal model of a sgle mache schedulg problem. Ths approach s costructed based o desrablty of the decso maker (DM) ad toleraces cosdered o goal values. Huo et al. [] cosdered b-crtera sgle mache schedulg problems volvg the maxmum weghted tardess ad umber of tardy obs. They gave P-hardess proofs for the schedulg problems whe oe of these two crtera s the prmary crtero ad the other oe s the secodary crtero. They cosdered complexty relatoshps betwee the varous problems ad proposed polyomal algorthms for some specal cases as well as fast heurstcs for the geeral case. It s well kow that the optmal soluto of sgle obectve models ca be qute dfferet f the obectve s dfferet (e.g., for the smplest model of oe mache wthout ay addtoal costrat, the shortest processg tme (SPT) rule s optmal to mmze F - (.e., mea flow tme) but the earlest due date (EDD) rule s optmal to mmze the maxmal tardess (T max )). I fact, each partcular decso maker ofte wats to mmze the gve crtero. For example a compay, the commercal maager s terested by satsfyg customers ad the mmzg the tardess. O the other had, the producto maager wshes to optmze the use of the mache by mmzg the makespa or the work process by mmzg the maxmum flow tme. I addto, each of these obectves s vald from a geeral pot of vew. Sce these obectves are coflctg, a soluto may perform well for oe obectve, but gvg bad results for others. For ths reaso, schedulg problems ofte have a mult-obectve ature (Loukl, et al. []). The chace costraed programmg was frst developed by Chares ad Cooper []. Subsequetly, some researchers lke Segupta [3], Cot [4], Leclercq [5], Teghem et al. [6] ad may others have establshed some theoretcal results the feld of stochastc programmg. Stacu-Masa ad Wets [7] have preseted a revew paper o stochastc programmg wth a sgle obectve fucto. The fuzzess occurs may of the real lfe decso makg problems. Decso makg a fuzzy evromet was frst developed by Bellma ad Zadeh [8]. Zmmerma [9] preseted a applcato of fuzzy lear programmg to the lear vector-maxmum problem ad showed that the soluto obtaed by fuzzy lear programmg s always effcet. Haa [0], arasmha [], Leberlg [] ad may others have made cotrbutos fuzzy goal programmg ad fuzzy mult-obectve programmg. Thus, the am of ths paper s to develop a fuzzy programmg approach for solvg a mult-obectve sgle mache schedulg problem whe processg tmes of obs are ormal radom varables ad the costrats follow a ot probablty dstrbuto. Ths probablstc model s frst coverted to a equvalet determstc model, to whch fuzzy programmg techque s appled to solve a multobectve sgle mache schedulg problem to obta a compromse soluto. II. PROBLEM FORMULATIO The followg otato ad deftos are used to descrbe the mult-obectve sgle mache schedulg problem. A. otato Idexes: = umber of obs, p = processg tme of ob (ormal radom varable) ( =,, ), R = release tme of ob ( =,, ), d = due date of ob ( =,, ), W = mportace factor (or weght) related to ob ( =,, ), M = a large postve teger value. Decso Varables: f ob s scheduled after ob, X 0 otherwse. B. Mathematcal Model I ths model, the obectve s to fd the best (or optmal) schedule mmzg the weghted completo tme (.e., Z ) ad total weghted tardess (.e., Z ) of a maufacturg system. M Z = WC () = M Z = WT () = subect to C R + P (3) X + X =, ; (4) C C + M X P (5) ISB: WCE 008

3 Proceedgs of the World Cogress o Egeerg 008 Vol II WCE 008, July - 4, 008, Lodo, U.K. T = max{ 0, C D} (6) X { 0,} (7) Costrat (3) esures that the completo tme of a ob s greater tha ts release tme plus processg tme. Costrat (4) specfes the order relato betwee two obs scheduled. Costrat (5) stpulates relatve completo tmes of ay two obs. M should be large eough for costrat (5). Costrat (6) specfes the tardess of each ob. III. MULTI-OBJECTIVE CHACE COSTRAIED PROGRAMMIG PROBLEM A mult-obectve chace costraed programmg problem wth a ot probablty costrat ca be stated as M Z k ( k ) (x) = C x ; k=,,k (8) = Pr ax b, ax b,..., ax m bm α = = = (9) x 0 ; =,, (0) Where b 's are depedet ormal radom varables wth kow meas ad varaces. Eq. (9) s a ot probablstc costrat ad 0 α s a specfed probablty. We assume that the decso varables x 's are determstc. Let the mea ad stadard devato of the ormal depedet radom varable b be gve by E (b ) ad σ (b ), respectvely. Hece the equvalet determstc model of probablstc problem ca be preseted as (Sha et al. [3]) M Z k ( k ) (x) = C x ; k=,, K () 3β e 3 β β = π + ( φ( β ) ) ax Eb ( ) = where β = σ ( b ) m ; =,,m () φ ( β ) α ; =,,m (3) = ax βσ ( b ) = Eb ( ) ; =,,m (4) = ( ) ; =,,m (5) 0 φ β x 0 ; =,, (6) We ow preset the methodology to solve a multobectve stochastc programmg problem usg fuzzy programmg approach. The algorthm cludes the followg steps: C. Algorthm Step : Frst, covert the gve stochastc programmg problem to a equvalet determstc programmg problem by chace costraed programmg techque as dscussed. Step : Solve the mult-obectve determstc problem obtaed from Step, usg oly oe obectve at a tme ad gorg the others. Repeat the process K tmes for the K dfferet obectve fuctos. Let X () ; X () ; ; X (K) be the respectve deal solutos of the K obectve fuctos. Step 3: Usg the solutos obtaed Step, fd the correspodg value of all the obectve fuctos at each of the solutos. Step 4: From Step 3, obta the upper ad lower bouds (U k ad L k, k =,, K) for each of the obectve fuctos. Step 5: Usg a lear membershp fucto, formulate a crsp model. By troducg a augmeted varable formulate sgle obectve o-lear programmg problem. Hece, the model ca be formulated as Max λ (7) ( k ) ( ) + ( ) λ ; k=,,k (8) Z x U L U β k k k 3β π e ( y + ) ; =,,m where 3 β y = φ β (9) m = ( ) y α (0) ax βσ ( b ) = Eb ( ) ; =,,m () = 0 y ; =,,m () x, x,..., x 0 (3) λ 0 & β, β,..., βm are urestrcted sg V. UMERICAL EXAMPLE Table summarzes the data that form the umercal example. We cosder the followg assumptos:. The processg tmes (P ) s tegers ad s geerated from a ormal dstrbuto.. The due dates (d ) are computed by d = μ P (- M) as gve []. s the umber of obs ad M the uformly radom umber betwee 0 ad. 3. The ready tmes (R ) are geerated from a uform dstrbuto o [, 0], 4. The obs weghts (w ) are uformly geerated from dscrete uform dstrbuto o [, 0]. ISB: WCE 008

4 Proceedgs of the World Cogress o Egeerg 008 Vol II WCE 008, July - 4, 008, Lodo, U.K. Table. Geerated data Job μ P s σ P d R w A mult-obectve sgle mache schedulg problem wth stochastc processg tme s preseted as follows: M Z M Z s.t. = w C (4) = = = w T (5) C R P, Pr 0.85, ; C C + MX P (6) X + X =, ; (7) = max{ 0, } (8) T C D X {,} 0, ; (9) We obta the equvalet determstc programmg problem for the above mult-obectve stochastc programmg problem by usg Eqs. ()- (6). MZ = WC (30) = MZ = WT (3) = s.t. C - R - σ Pβ = μp, ; (3) C C + MX σ β = μ, ; (33) P P β.5334( + y)( 3 β ) 3β exp 6 = (34) y 0.85 (35) X + X =, ; (36) max{ 0, } { } T = C D (37) X 0,, y 0, ; (38) All the computatoal expermets are carred out wth a brach-ad-boud (B&B) method the Lgo 8.0 software by a Itel.6 GHz processor wth 5 Mb RAM. Solvg the problem for obectve Z ad Z, The deal solutos are as follows: Z = 4.063, Z = β =, β =, β3 =, β4 =, β5 =, y =, y = y4 =, y5 =. 0.85, y3 =, Usg the lear membershp fucto, we formulate the followg fuzzy programmg problem: Z 4 -Z (39) μ ( Z) 4 Z 4 0 Z Z Z μ ( Z) 58 Z Z 958 (40) Max λ (4) - WC λ ( 4) (4) = 958- WT λ ( ) (43) = C - R - σ Pβ, ; = μp (44) C C + MX σ β = μ, ; (45) P P β.5334( + y)( 3 β ) 3β exp 6 = (46) y 0.85 (47) X + X =, ; (48) = max{ 0, } (49) T C D X {,} 0, ; (50) Solvg the above fuzzy programmg problem, we get the compromse soluto as λ = 0.59 Z = , Z = β =.695, β =.695, β 3 =.695, β 4 =.66, β 5 =.695, y =, y =, y 3 =, y 4 = 0.85, y =. 5 ISB: WCE 008

5 Proceedgs of the World Cogress o Egeerg 008 Vol II WCE 008, July - 4, 008, Lodo, U.K. The optmal sequece of obs s show as follows: J -J 4 -J 5 -J -J 3 V. COCLUSIO We have cosdered a mult-obectve probablstc sgle mache schedulg problem to mmze the total weghted completo tme ad total weghted tardess wth ot costrats, where oly processg tme of obs are cosdered as depedet ormal radom varables. Usg the stated procedures a probablstc mult-obectve sgle mache schedulg problem wth ot costrats ca be easly trasformed to a determstc mult-obectve o-lear programmg problem ad the solved by the fuzzy programmg techque to obta the compromse soluto. [9] H.J. Zmmerma, Fuzzy programmg ad lear programmg wth several obectve fuctos, Fuzzy Sets ad Systems, Vol., pp.45-56, 978. [0] E.L. Haa, O fuzzy goal programmg, Decso Scece, Vol., pp.5-53, 98. [] R. arsmha, Goal programmg a fuzzy evromet, Decso Sceces, Vol., pp , 980. [] H. Leberlg, O fdg compromse solutos mult-crtera problems usg the fuzzy m-operator, Fuzzy Sets ad Systems, Vol.6, pp.05-8, 98. [3] S.B. Sha, S. Hulsurkar, M.P. Bswal, Fuzzy programmg approach to mult-obectve stochastc programmg problems whe b's follow ot ormal dstrbuto, Fuzzy Sets ad Systems, Vol.09, pp.9-96, 000. REFERECES [] D.R. Sule, Idustral Schedulg, PWC Publshg Compay, 997. [] T. Loukl, J. Teghem ad D. Tuyttes, Solvg mult-obectve producto schedulg problems usg metaheurstcs, Europea Joural of Operatoal Research,Vol.6,pp. 4-6, 005. [3] H. Ish ad M. Tada, Sgle mache schedulg problem wth fuzzy precedece relato, Europea Joural of Operatoal Research, Vol.87, pp.84-88, 995. [4] G.I. Adamopoulos ad C.P. Papps, a fuzzy-lgustc approach to a mult-crtera sequecg problem, Europea Joural of Operato Research,Vol.9,pp , 996. [5] H.T. Lee, S.H. Che ad H.Y. Kag, Mult-crtera schedulg usg fuzzy theory ad tabu search, Iteratoal Joural of Producto Research,Vol.40,o.5,pp.-34, 00. [6] S.Chaas, A. Kaspersk, o two sgle mache schedulg problems wth fuzzy processg tmes ad fuzzy due dates, Europea Joural of Operatoal Research, Vol.47, pp. 8-96, 003. [7] S. Chaas, A.Kaspersk, Possble ad ecessary optmalty of solutos the sgle mache schedulg problem wth fuzzy parameters, Fuzzy Sets ad Systems, Vol.4, pp , 004. [8] M. Azzoglu, S. Kodakc, M. Köksala, Sgle mache schedulg wth maxmum earless ad umber tardy, Computers ad Idustral Egeerg,Vol.45, o.,pp.57-68,003. [9] T. Ere, E. Güer, A b-crtera schedulg wth sequece-depedet setup tmes, Appled Mathematcs ad Computato,Vol.79,pp , 006. [0] R. Tavakkol-Moghaddam, B. Javad,. Safae, Solvg a mxed-teger model of a sgle mache schedulg problem by a fuzzy goal programmg approach, Wseas Trasactos o Busess ad Ecoomcs,Vol.3, o.,pp. 45-5, 006. [] Y. Huo, J.Y.-T. Leug, H. Zhao, B-crtera schedulg problems: umber of tardy obs ad maxmum weghted tardess, Europea Joural of Operatoal Research,Vol.77,o.,pp. 6-34, 007. [] Chares, W.W. Cooper, Chace costraed programmg, Maagemet Scece,Vol.6,pp ,959. [3] [3] J.K. Segupta, Stochastc Programmg: Methods ad Applcatos, orth-hollad, Amsterdam, 97. [4] B. Cot, A stochastc approach to goal programmg, Operato Research, Vol.6, pp , 978. [5] J.P. Leclercq, Stochastc programmg: A teractve mult-crtera approach, Europea Joural of Operato Research,Vol.0 pp.33-4, 98. [6] J. Jr. Teghem, D. Dufrace, M. Thauvoye, P. Kuch, Strage: a teractve method for mult-obectve lear programmg uder ucertaty, Europea Joural of Operatoal Research, Vol.6, pp.65-8, 986. [7] I.M. Stacu-Masa, M.J. Wets, A research bblography stochastc programmg, Operato Research, Vol.4, pp.078-9, 976. [8] RE. Bellma, L.A. Zadeh, Decso-makg a fuzzy evromet, Maagemet Scece, Vol.7, pp. 4 64, 970. ISB: WCE 008