A MONTE CARLO SIMULATION APPROACH TO THE CAPACITATED MULTI-LOCATION TRANSSHIPMENT PROBLEM

Transcription

1 Proceedngs of the 2003 Wnter Smulaton Conference S. Chck, P. J. Sánchez, D. Ferrn, and D. J. Morrce, eds. A MOTE CARLO SIMULATIO APPROACH TO THE CAPACITATED MULTI-LOCATIO TRASSHIPMET PROBLEM Denz Özdemr Enver Yücesan Technology Management Area ISEAD Boulevard de Constance Fontanebleau CEDEX, FRACE Yale T. Herer Faculty of Industral Engneerng and Management Technon Israel Insttute of Technology Hafa 32000, ISRAEL ABSTRACT We consder a supply chan, whch conssts of retalers and one suppler. The retalers may be coordnated through replenshment strateges and lateral transshpments, that s, movement of a product among the locatons at the same echelon level. Transshpment quanttes may be lmted, however, due to the physcal constrants of the transportaton meda or due to the reluctance of retalers to completely pool ther stock wth other retalers. We ntroduce a stochastc approxmaton algorthm to compute the order-up-to quanttes usng a sample-path-based optmzaton procedure. Gven an order-up-to S polcy, we determne an optmal transshpment polcy, usng an LP/etwork flow framework. Such a numercal approach allows us to study systems wth arbtrary complexty. ITRODUCTIO Physcal poolng of nventores has been wdely used n practce to reduce cost and mprove customer servce. On the other hand, nformaton poolng, whch entals the sharng of nventory among stockng locatons through lateral transshpments, has been less frequent. Transshpments, the montored movement of materal between locatons at the same echelon, provde an effectve mechansm for correctng dscrepances between the locatons observed demand and ther avalable nventory. As a result, transshpments lead to cost reductons and mproved servce wthout ncreasng system-wde nventores. Our research s motvated by observatons from dfferent ndustres. For example, transshpments are common n the management of spare parts. In manufacturng, factores turn to sster plants to quckly obtan a spare part before contactng the orgnal suppler. Arlnes have smlar practces. Contaner shppng lnes pool ther contaners through an exchange. Transshpments are ncreasngly common n apparel, fashon goods, and toys, partcularly by those retalers wth brck and clck outlets. All these transshpment practces, however, represent a reactve approach to unexpected stockouts. We beleve that, f we take transshpment opportuntes nto account proactvely durng the plannng phase, they can work as an effectve mechansm for correctng demand-supply dscrepances, thereby reducng cost and mprovng servce. The lterature on transshpments has generally addressed ether problems wth two retalers, e.g., Tagaras (989), Tagaras and Cohen (992), and Robnson (990), or problems wth multple, dentcal retalers, e.g., Krshnan and Rao (965) and Robnson (990). In contrast, we consder multple retalers, who may dffer both n ther cost structures and n ther demand parameters. We further consder lmts on transshpment quanttes. Such capacty constrants may reflect the physcal constrants of the transportaton meda or the reluctance of the retalers to completely share ther stock wth other retalers. Other recent work on transshpments ncludes Archbald et al. (997), Tagaras (999), Rud et al. (200), Herer and Rasht (999), and Dong and Rud (2000). In ths system, t s optmal for each retaler to follow an order-up-to S polcy. The optmalty of the order-up-to S polcy takes nto consderaton the use of transshpments among retalers, to be performed once demand s observed. We show how the values of the order-up-to quanttes can be calculated usng a a stochastc approxmaton algorthm that s based on Infntesmal Perturbaton Analyss (IPA). IPA has orgnally been ntroduced as a smulatonbased optmzaton technque (Ho et al. 979). Wth IPA, nstead of usng fnte dfferences n a gradent search method, we use the mean value of the sample path dervatve, whch s obtaned through a sngle smulaton. In other words, we conduct a sngle smulaton run, keepng track of the mpact of a change n a system parameter value on performance. We then average these changes to estmate the gradent. The mplct assumpton s that the av-

2 Özdemr, Yücesan, and Herer erage of these changes represents the change n expectatons hence, t yelds an unbased estmator. Glasserman (99) establshed the general condtons for the unbasedness of the IPA estmator. Applcatons of perturbaton analyss have been reported n smulatons of Markov chans (Glasserman 992), nventory models (Fu 994), manufacturng systems (Glasserman 994), fnance (Fu and Hu 997), and control charts for statstcal process control (Fu and Hu 999). IPA-based methods have also been ntroduced to analyze supply chan problems (Glasserman and Tayur 995). Smulaton-based dervatve estmates help the search for an mproved polcy whle allowng for complex features that are typcally outsde of the scope of analytcal models. Whle the optmal order-up-to quanttes have to be found once for the entre system, an optmal transshpment strategy has to be found on a perod-by-perod bass, gven the perod s demand realzaton. We show how these transshpment quanttes can be found usng an LP / etwork flow framework. The contrbuton of ths paper s twofold. Frst s the development of an ntegrated IPA/LP algorthm for a system that allows capactated transshpments. The system we consder s more general than prevously studed systems wth transshpments n that we consder multple retalers, whch dffer both n ther cost structure and n ther demand parameters. Second s a methodologcal contrbuton, obtaned by formulatng and valdatng IPA dervatve estmators for the transshpment problem. Formulatng these estmators means ntroducng approprate algorthms; valdatng them calls for showng that they converge to the correct values, where convergence s over the number of ndependent smulaton replcatons used to estmate the dervatve nformaton. 2 PROBLEM DESCRIPTIO We consder a system wth one suppler and nondentcal retalers, assocated wth dstnct stockng locatons. The system nventory s revewed perodcally. The demand dstrbuton of each retaler n a perod s assumed to be known and statonary over tme. The frst event n each perod s the arrval of orders placed n the prevous perod. These orders are used to satsfy any outstandng backlog and to ncrease the nventory level. ext n the perod s the occurrence of demand. Snce the realzaton of demand represents the only uncertan event of the perod, once t s observed all the decsons of the perod,.e., transshpment and replenshment quanttes, are made. Lateral transshpments are then executed, and subsequently the demand s satsfed. Unsatsfed demand s backlogged. At ths pont, backlogs and nventores are observed, and penalty and holdng costs, respectvely, are ncurred. The nventory s carred, as usual, to the next perod. ote that tems n stock elsewhere n the system are suppled mmedately through transshpments whle backlogged tems have to wat untl the begnnng of the next perod. Thus, the advantage of usng transshpments s n ganng a source of supply whose reacton tme s shorter than that of the regular supply. We now ntroduce the notaton used. We wll represent the vector of quanttes descrbed below, as well as the ones that we wll ntroduce later n the paper, by droppng the subscrpts, thus, d = ( d,..., d ). = number of retalers; D = random varable assocated wth the perodc demand at retaler wth E[D ] < ; f (D) = ont probablty densty functon for the demand vector D ; d = actual demand at retaler n an arbtrary perod; h = holdng cost ncurred at retaler per unt held per perod; p = penalty cost ncurred at retaler per unt backlogged per perod; c = replenshment cost per unt at retaler ; t ˆ = drect transshpment cost per unt trans- shpped from retaler to retaler ; t = effectve transshpment cost, or smply the transshpment cost, per unt transshpped from retaler to retaler, t = tˆ + c c. 3 OPTIMAL POLICIES In each perod, the replenshment and transshpment quanttes must be determned. For the uncapactated verson of the problem, Herer et al. (999) have already proven that, f transshpments are only made to satsfy the actual demand and not to buld up nventory, there exsts an optmal orderup-to S = ( S, S2,..., S ) replenshment polcy for all possble transshpment decsons. In the capactated system, ths property s preserved. We therefore adopt a base stock replenshment polcy. Gven an order-up-to S polcy for the replenshment quanttes, the optmal transshpment quanttes need to be determned each perod between every two retalers. We defne the decson varables n Table. In partcular, let represent the quantty transshpped from retaler to F B M retaler. Recall that S s the order-up-to level at retaler. We also use the followng auxlary varable: I = nventory level at retaler mmedately after transshpments and demand satsfacton = S F + F BM B M = = d

3 Özdemr, Yücesan, and Herer ote that denote: I may be ether postve or negatve, and we + I = max{ I,0}, I = max{ I,0}. ow, the total cost of the system n a gven perod s gven by: TC = t F = = BM + h I = + + p I = In the above equaton we have not fully accounted for the replenshment costs. Snce we are usng an Order-up-to S replenshment polcy at each retaler, the total amount replenshed system-wde wll be exactly equal to the system-wde demand. Recall, however, that the replenshment cost dfferentals were ncluded n the defnton of t. Thus, to fully account for the replenshment costs, we = would need to add the term c d to Equaton (). () However, snce ths term s ndependent of our decson varables, t s omtted. Snce the optmal polcy s to order up to S unts at retaler, the pont n tme after an order arrves s a regeneraton pont. That s, the system returns to the same state ( S unts at each retaler and no backorders) ust after the start of each and every perod. Thus, we can vew the mult-perod problem as a seres of sngle-perod problems. Consder the movement of materal n an arbtrary perod. At the begnnng of the perod there are S unts n stock at each retaler. Ths stock can be used n one of three dfferent ways: satsfy demand at retaler, satsfy demand at retaler (.e., a transshpment from retaler to ), and hold n nventory at retaler. Whle t s true that t s physcally possble to move stock from retaler to another retaler, e.g.,, for storage, ths s precluded. At the end of the perod replenshment arrves from the suppler. Ths materal can be used n two dfferent ways: to satsfy a backorder at a retaler or to buldup nventory at a retaler so that the perod wll end wth the rght amount of stock. The stock at the begnnng of the perod and the replenshment that arrves from the suppler are the only two sources of materal durng a perod. On the other hand, the demand at retaler, d, can be satsfed n one of three dfferent ways: from the ntal nventory at retaler, from the ntal nventory at another retaler (.e., a transshpment from retaler to retaler ), or from replenshment at the end of the perod. Another snk for materal s the requrement that each retaler ends the perod wth S unts n nventory. These unts can come from one of two sources: the startng nventory at retaler or replenshment at the end of the perod. As dscussed above, nventory from another retaler wll not be used to buldup nventory levels at retaler. Usng the observatons above we model the movement of stock durng a perod as a network flow problem. In partcular we have a source node, B, to represent the begnnng nventory at retaler, and a source node, R, to represent the replenshment that occurs at the end of the perod. The snk node assocated wth the demand at retaler wll be denoted M and the snk node assocated wth the endng nventory at retaler wll be denoted E. The arcs n the network flow problem are exactly those actvtes descrbed above and are summarzed (wth ther assocated cost per unt flow) n Table. The complete network flow representaton of the problem can be found n Fgure for three retalers. ote that the graph s bpartte, though our representaton of the graph, whch was chosen to show the connecton to the underlyng nventory problem, does not emphasze ths pont. The LP formulaton assocated wth ths network flow problem s as follows: Problem (P) s.t. Z ( S, d) = mn h F + t F + p F B E B M = = = = S = FB M + FB M + F B E = FB M + FB M + FRM = d = d = FRM + F RE = = = FB E M FRE = S =,..., =,..., RM (2) (3) (4) + =,..., (5) Tr F B C,, =,2,...,, (6) FB E, FB M, FB M, FRM, FRE 0, =,...,, =,...,. Equatons (2), (3), (4) and (5), respectvely, represent the nventory balance constrant at the B, M, R and E nodes. Equaton (6) reflects a physcal constrant, Tr C, on the quantty that can be transshpped from locaton to locaton. Alternatvely, each locaton may wsh to allocate only a porton, say β, of ts on-hand nventory to trans-

4 Özdemr, Yücesan, and Herer Table : The Defnton of the Arcs n the etwork Flow Problem Arc Varable Cost per Meanng unt flow ( B, E ) h nventory s held at retaler ( B, M ) ( B, M ) ( R, M ) ( R, E ) F B E F B M 0 stock at retaler s used to satsfy demand at retaler F B M t ( t = 0 ) p stock at retaler s used to satsfy demand at retaler,.e., transshpment from retaler to retaler F RM shortage at retaler s satsfed through replenshment F RE 0 nventory at retaler s ncreased through replenshment d S B C Tr 3 C Tr 2 M E S S 2 B 2 C Tr 2 M 2 d 2 C Tr 23 E 2 S 2 C Tr 3 C Tr 32 d 3 S 3 B 3 M 3 E 3 S 3 Fgure : etwork Flow Representaton of a Sngle Perod R d shpments. Ths practce, typcally referred to as partal poolng, can be represented through the followng constrants: F B M β S, =,...,. (7) = 3. Fndng the Optmal Order-up-To Values In the most general settng, exact computaton of optmal order-up-to levels by analytcal methods s dffcult. Ths s n fact the problem of optmzng an expected value functon. Snce the correspondng expectaton functon cannot be computed exactly, t s approxmated through Monte Carlo samplng. Usng the notaton of Shapro (200), ths represents a class of optmzaton problems of the form: mn { g( x) : = E[ G( x, ω)] }, x X (8) where the expectaton g(x) s well defned for every x X. The functon G(x,ω) s n tself an optmzaton problem. In our case, G s the optmal value of a network flow problem, where retaler demand s the random data of the problem. We solve the optmzaton problem (8) by Monte Carlo smulaton, that s, by generatng an IID random sample and calculatng the correspondng sample average: U gˆ ( x) : = U G( x, ω ). U = The optmzaton problem (8) s then approxmated by: mn ˆ x X g U ( x). We propose an IPA-based approach to solve (9). The dea s to use the expected value of the sample path dervatve obtaned va smulaton nstead of usng the dervatve of the expected cost n a gradent search method. In other (9)

5 words, the gradent of nterest s de[tc]/ds whereas our numercal procedure computes E[dTC/dS]. To valdate ths approach, that s, to ustfy the nterchange of the dervatve and the ntegral, we need to show that the obectve functon s ontly convex and smooth n the S varables (Glasserman and Tayur 995). For the capactated transshpment problem, ths s done n Özdemr et al. (2003). We now turn to the descrpton of the IPA algorthm. 3.2 Descrpton of the IPA Algorthm The algorthm starts wth an arbtrary value for the orderup-to levels, S. An nstance of the demand s generated at each retaler. Once the demand s observed, problem (P) s solved n a determnstc fashon to compute the mnmumcost soluton. The gradent of the total cost (dervatves wth respect to the order-up-to levels) s estmated and accumulated through ndependent replcatons; the average gradent value s then used to update the values of S. The procedure s summarzed n a pseudo-code format, where K denotes the number of steps taken n a path search, U represents the number of ndependent replcatons, ak represents the step sze at teraton k, and represents the k S order-up-to level for retaler at the k th teraton. Intalze K Intalze U Set k For each retaler, set ntal order-up-to Özdemr, Yücesan, and Herer k = 0 levels, S k= Repeat Set dtc 0 Set u 0 Repeat. Generate an nstance of the demand at each retaler, d, from f(d). Solve problem (P) to determne optmal transshpment quanttes. Accumulate the desred gradents (dervatves) of the total cost, dtc v. u u + Untl u = U v. Calculate the desred gradent(s), dtc / U v.update the order-up-to-levels, S : 4. Expermental Desgn k k S S a (dtc /U) + v. k k + Untl k = K k In step () of the algorthm, we use IPA to compute the gradent. To llustrate the sample-path dervatve dea, suppose that we end a perod wth nventory at retaler. In ths case, rasng S by unt would result n ncreasng total cost by h. In the computer mplementaton, for each retaler, we could partally code Step () as: dtc = dtc + h, f nventory at retaler s postve, at the end of Step (). Startng wth dtc = 0 for all at the begnnng of the smulaton and dvdng dtc by U n Step (v) yeld the dervatve estmates. Our network flow formulaton greatly smplfes computatons. Increasng S corresponds to ncreasng the supply at source node B and the demand at snk node E. From a network flow perspectve, dtc/ds = h, f the arc ( B, E ) s basc or, equvalently, the flow F B s postve. If the arc s non-basc, then snce any basc soluton E corresponds to a spannng tree n the network, there exsts a unque augmentng path from B to E whose total cost yelds the gradent value. For example, the augmentng path may go from B to M to R to E, wth an assocated cost of t - p. Such a path represents a transshpment from retaler to retaler (wth a cost of t ), a reducton n backorders at retaler (wth a savngs of p ) and a purchase of another unt at retaler (cost of zero). Furthermore, our mplementaton of the dervatve computaton n Step () s very effcent. Snce the value of the gradent s equal to the total cost along the unque path from B to E for each retaler, ths quantty can be calculated drectly as the dfference between the holdng cost at retaler and the reduced cost of the arc (B,E ), whch s readly avalable from the lnear programmng soluton n Step (). In Step (v) of the algorthm, one typcally mposes condtons on the step sze, a k such that 2 k k= a and a <. The frst condton facltates convergence by ensurng that the steps do not become too small too fast. However, f the algorthm s to converge, the step szes must eventually become small, as ensured by the second condton. ote that when the gradent estmator s unbased (as s the case here), step (v) represents a Robbns-Monro algorthm (95) for stochastc search. 4 COMPUTATIOAL STUDY An llustratve example of the system wth four retalers s shown n Fgure 2. Let us call retaler 0 the central retaler and all other retalers the remote retalers. We begn by consderng the case of dentcal retalers, the cost parameters are as follows: h h = $ per unt, p p = $4 per unt, and the basc drect transshpment cost, c t = $0.5 per unt, when transshpments are allowed. Each retaler faces an ndependent demand stream dstrbuted unformly over (0, 200). ote that t 0, =,2,,, represents the transshpment cost from the central retaler to remote retalers, t 0, =,2,,, represents the transshpment cost from the re-

6 Özdemr, Yücesan, and Herer t t t t t t t t k k t k t k t k 200 k t k Fgure 2: Confguraton Used n umercal Testng mote retalers to the central retaler, and t,,=,2,,, denotes the transshpment cost from remote retaler to remote retaler. As summarzed n Table 2, we consder fve alternatve system confguratons. ote that t = mples that transshpments are not allowed between retalers and. Table 2: System Confguratons System t 0 t 0 t 2 c t 3 c t c t 4 c t c t 2 c t 5 c t c t c t System, where no materal movement s allowed among retalers, represents + ndependent newsvendor problems. It thus serves as a benchmark. In system 2, transshpments are allowed only from the central retaler to the remote retalers. System 3 extends the scenaro n system 2 by allowng transshpments from the remote retalers to the central retaler as well. In system 4, all materal movement s possble. However, transshpments between any two remote retalers are twce as expensve as the transshpments from/to the central retaler. Fnally, all transshpment costs are dentcal n system Results and Analyss In all systems, we observe an ncrease n the total cost and n the total nventory levels when capacty consderatons are ncorporated. However, as the number of unts allocated for transshpment ncreases, both the total cost and the total nventory levels decrease. Fgure 3 depcts the total cost as a functon of transshpment capacty for the fve systems wth 0 retalers. In systems 2 and 3, where transshpment opportuntes are restrcted, decreasng capacty leads to ncreased costs. In systems 4 and 5, however, a tghtenng transshpment capacty s fully compensated for through the avalablty of transshpment opportuntes between any par of retalers. Total Cost X=25 X=50 X=75 X=00 IF Transshpment Capacty Sys Sys 2 Sys 3 Sys 4 Sys 5 Fgure 3: Total Cost n the Presence of Transport Constrants Fgure 4 llustrates an nterestng redstrbuton of nventory n system 3 n the presence of transshpment capactes. Recall that, n system 3, stockng locaton behaves as a clearnghouse for all other stockng locatons. In other words, when there s no constrant on the quantty that can be transshpped, most of the materal s stocked n locaton and shpped to the other locatons, as needed. In fact, up to 25% of the system-wde nventory s kept at locaton. However, when transshpment quanttes are tghtly constraned, locaton fnds tself unable to support any of the other locatons through transshpments. It therefore does not have any reason to carry extra nventory. In fact, n a tghtly capactated envronment, the other locatons carry addtonal stock for locaton. Even for very small transshpment capactes, there s enough stock (9 tmes the transshpment capacty) carred by the other locatons that can be sent to locaton n case of a shortfall there. 5 SUMMARY In ths paper, we consdered the mult-locaton dynamc transshpment problem, where transshpment quanttes may be restrcted. Our approach ncludes several nnovatons. Frst, an arbtrary number of non-dentcal retalers s consdered wth possbly dependent stochastc demand. Second, we model the dynamc behavor of the system n an arbtrary perod as a network flow problem. Fnally, we employ a smulaton-based method usng nfntesmal perturbaton analyss for optmzaton. Our smulatonbased optmzaton approach therefore provdes a flexble platform to analyze transshpment problems of arbtrary complexty. ACKOWLEDGMETS Ths work has been partally supported by PrceWaterhouseCooper s grant to ISEAD s ntatve on Hgh- Performance Organzatons.

7 Özdemr, Yücesan, and Herer IF %S 45%S 40%S 35%S 30%S 25%S 20%S 5%S 0%S 5%S Locaton Locaton Fgure 4: Optmal Base Stock Levels n the Presence of Transport Constrants REFERECES Archbald, T.W., A.A.E. Sassen, and L.C. Thomas An Optmal Polcy for a Two-Depot Inventory problem wth Stock Transfer. Management Scence 43: Dong, L., and. Rud Supply Chan Interacton Under Transshpments. Workng paper. The Smon School, Unversty of Rochester, Rochester, Y, U.S.A. Fu, M.C Sample Path Dervatves for (s,s) Inventory Systems. Operatons Research 42: Fu, M.C., and Hu, J.Q Condtonal Monte Carlo: Gradent Estmaton and Optmzaton Applcatons. Kluwer Academc Publshers. Fu, M.C., and Hu, J.Q Effcent Desgn and Senstvty Analyss of Control Charts usng Monte Carlo Smulaton. Management Scence 45: Glasserman, P. 99. Gradent Estmaton va Perturbaton Analyss. Kluwer Academc Publshers. Glasserman, P Dervatve Estmates from Smulaton of Contnuous-Tme Markov Chans. Operatons Research 40: Glasserman, P Perturbaton Analyss of Producton etworks. In Stochastc Modelng and Analyss of Manufacturng Systems (Yao, Ed.). Sprnger-Verlag. Glasserman, P. and S. Tayur Senstvty Analyss for Base Stock Levels n Mult-Echelon Producton- Inventory Systems. Management Scence 4: Herer, Y.T. and A. Rasht Lateral Stock Transshpments n a Two-Locaton Inventory System wth Fxed and Jont Replenshment Costs. aval Research Logstcs 46: Herer Y. T., M. Tzur, E. Yücesan The Mult- Locaton Transshpment Problem. Workng paper. Department of Industral Engneerng, Tel Avv Unversty. Ho, Y.C., M.A. Eyler, and T.T. Chen A Gradent Technque for General Buffer Storage Desgn n a Seral Producton Lne. Internatonal Journal of Producton Research 7: Krshnan, K.S. and V. R. K. Rao Inventory Control n Warehouses. Journal of Industral Engneerng 6: Özdemr, D., E. Yücesan, and Y.T. Herer Mult- Locaton Transshpment Problem wth Capactated Transportaton. Workng paper. Technology Management Area, ISEAD. Robbns H. and S. Monro. 95. A Stochastc Approxmaton Method. Annals of Mathematcal Statstcs 22: Robnson, L.W Optmal and Approxmate Polces n Multperod, Multlocaton Inventory Models wth Transshpments. Operatons Research 38: Rud,., S. Kapur, and D. Pyke A Two-locaton Inventory Model wth Transshpment and Local Decson Makng. Management Scence 47: Shapro, A Monte Carlo Smulaton Approach to Stochastc Programmng. Proceeedngs of the 200

8 Wnter Smulaton Conference (Peters, Smth, Mederos, and Rohrer, eds.) Tagaras, G Effects of Poolng on the Optmzaton and Servce Levels of Two-Locaton Inventory Systems. IIE Transactons 2: Tagaras, G Poolng n Mult-Locaton Perodc Inventory Dstrbuton Systems. Omega Tagaras, G. and M. Cohen Poolng n Two- Locaton Inventory Systems wth on-eglgble Replenshment Lead Tmes. Management Scence 38: AUTHOR BIOGRAPHIES DEIZ ÖZDEMIR s a doctoral student n the Technology Management Area at the European Insttute of Busness Admnstraton, ISEAD, n Fontanebleau, France. She can be contacted at <denz.ozdemr@nsead.edu>. EVER YÜCESA s a Professor of Operatons Management at the European Insttute of Busness Admnstraton, ISEAD, n Fontanebleau, France. He has an undergraduate degree n Industral Engneerng from Purdue Unversty. Hs MS and PhD, both n Operatons Research, are from Cornell Unversty. He served as the Proceedngs Co-Edtor for WSC 02. He can be contacted by e-mal at <enver.yucesan@nsead.edu>. YALE T. HERER oned the Faculty of IndustralEngneerng and Management at the Technon --- Israel Insttute of Technology n 990 mmedately after the completon of hs graduate studes at Cornell Unversty and, except for a few year hatus at the Department of Industral Engneerng at Tel-Avv Unversty, he has been there ever snce. Presently Yale s servng as the Head of the Industral Engneerng Area. He has worked for several ndustral concerns, both as a consultant and as an advsor to proect groups. In 996 Yale receved the IIE Transactons best paper award. Yale s a member of the Insttute for Operatons Research and Management Scences (IFORMS), Insttute of Industral Engneers (IIE), and the Operatons Research Socety of Israel (ORSIS). He serves on the Edtoral Board of IIE Transactons. Hs research nterests nclude supply chan management, especally when ntegrated wth transshpments. He s also nterested n producton control and producton system desgn. He can be contacted by e-mal at <yale@technon.ac.l>. Özdemr, Yücesan, and Herer