Ganesh Subramaniam. American Solutions Inc., 100 Commerce Dr Suite # 103, Newark, DE 19713, USA

Transcription

1 238 Int. J. Sulaton and Process Modellng, Vol. 3, No. 4, 2007 Sulaton-based optsaton for ateral dspatchng n Vendor-Managed Inventory systes Ganesh Subraana Aercan Solutons Inc., 100 Coerce Dr Sute # 103, Newark, DE 19713, USA Abhjt Gosav* Sulaton-Optzaton Laboratory, Departent of Industral Engneerng, State Unversty of New York at Buffalo, 317 Bell Hall, Buffalo, NY , USA E-al: agosav@buffalo.edu *Correspondng author Abstract: Ths paper exanes a proble related to replenshng nventores at retalers n dstrbuton networks operated under the paradg of Vendor Managed Inventory (VMI). In the real world, coplexty n analysng such systes arses out of nuerous factors, ncludng several rando varables and a ulttude of costs n the syste. Much of the lterature akes splfyng assuptons about these factors to develop tractable atheatcal odels. In ths paper, we develop a sulaton-optsaton approach that can accoodate ost of these factors. In our experents, our approach outperfors a heurstc used n a local ndustry and a heurstc developed fro the newsvendor odel. Keywords: retaler networks; supply chan; sulaton-based optsaton; Vendor-Managed Inventory; VMI. Reference to ths paper should be ade as follows: Subraana, G. and Gosav, A. (2007) Sulaton-based optsaton for ateral dspatchng n Vendor-Managed Inventory systes, Int. J. Sulaton and Process Modellng, Vol. 3, No. 4, pp Bographcal notes: Ganesh Subraana receved hs MS fro the Unversty at Buffalo, State Unversty of New York n Septeber He s now eployed as a Statstcal Analyst wth Aercan Solutons, Newark, Delaware. Hs research nterests nclude sulaton and supply chan anageent. Abhjt Gosav s an Assstant Professor n the Departent of Industral Engneerng at the Unversty at Buffalo, State Unversty of New York. Hs research areas nclude sulaton-based optsaton achne learnng preventve antenance and revenue anageent. 1 Introducton The scence of supply chan anageent has provded several scentfc procedures for reducng nventory and optsng servce levels n dstrbuton networks of anufactured goods. For any copanes, the applcaton of these technques has resulted n achevng sgnfcant cost reductons For exaple, Geon (see Haer, 2001) obtaned cost proveents by ntegratng ts operatons wth those of ts supplers and custoers for nventory anageent, t pleented advanced replenshent strateges lke VMI. There are nuerous exaples n the ndustry of organsatons that have utlsed effcent nventory-anageent systes to reduce the total expenses ncurred n transportaton, storage, and provdng servce. Exaples of such copanes are producers of food products and petroleu. An portant proble n any dstrbuton networks operated under the VMI paradg s to deterne the optal quanttes of ateral to be dspatched to each retaler fro the local warehouse Usually, the goal s to optse the trade-off between antanng excessve nventory at the retalers, whch ncreases the operatng costs, and not antanng enough nventory, whch can cause stock-outs and lost sales. In addton, one can not gnore the ssue of transportaton costs whch can be reduced by shppng large batches that unfortunately tend to ncrease nventory-holdng costs. In ths paper, we consder a retaler network of the type found n any real-world Copyrght 2007 Inderscence Enterprses Ltd.

2 Sulaton-based optsaton for ateral dspatchng n Vendor-Managed Inventory systes 239 systes. Generally, trucks are dspatched fro the warehouse to the retalers carryng the ateral needed. We assue that the trucks follow a fxed route n whch each retaler s vsted once (see Fgure 1). When the truck (s) leaves the warehouse t carres the ateral needed for replenshng each retaler. Fgure 1 A scheatc of the route followed by a truck Contrbutons of ths paper: We develop a sulaton-optsaton ethodology to solve the proble of deternng the optal quanttes to be delvered to each retaler. We show that the ethodology presented here can accoodate a large nuber of features of real-world systes and can outperfor two heurstcs. One of the heurstcs outperfored s used n a local ndustry. The other heurstc s derved fro the newsvendor odel. An nterestng fndng s that our newsvendor heurstc produces reasonably good solutons although, of course, t s outperfored by sulaton-based optsaton. Ths ples that anagers not nterested n pursung an elaborate sulaton-optsaton approach can resort to the spler newsvendor odel for soluton purposes. In addton, we also prove that under the general assuptons ade here, the cost functon to be nsed s non-convex. A nuber of factors contrbute to randoness n the syste. Soe exaples of such factors are: randoness n custoer arrval and sze of custoer deand and the randoness n the transportaton. As entoned prevously, there are three sources of costs n these systes: transportaton costs, nventory-holdng costs, and stock-out costs. Modellng such a syste atheatcally s often dffcult and challengng. Although any atheatcal odels have been developed n the lterature, they usually dsregard soe coplexty ether a cost or a governng rando varable to keep the odel tractable. As a result, we focus on developng a sulaton-based odel that accoodates any features of a real-world syste. The sulaton odel s cobned wth optsaton technques to generate pleentable solutons. The proble consdered n ths paper s of treendous relevance to anagers of warehouses who have to dspatch trucks wth the rght aounts of ateral. VMI-based systes face ths proble on a daly or a weekly bass. In fact, ths study was otvated by a proble faced n a local ndustry. Solutons usng the ethods proposed n ths paper can be obtaned easly wth coputer progras, whch can be run on personal coputers, and used drectly n the decson-akng process. (Coputer codes are avalable fro the second author upon request). Snce the sulaton odel s very general, the anager can easly change the syste paraeters for retalers, travel tes, and the deand rates as and when needed. Senal work on the statc nventory-allocaton proble s fro Clark and Scarf (1960) and Eppen and Schrage (1981), whch fors the foundaton of the underlyng scence n ths feld. Much of the exstng lterature s devoted to the developent of atheatcal odels, and as such, t gnores ether one or ore of the three costs (entoned above) nvolved or else akes splfyng assuptons about governng rando varables. For nstance, McGavn et al. (1993) gnore nventory-holdng costs, Federgruen and Zpkn (1984) develop a so called yopc odel whch optses n the current te perod but dsregards costs n future te perods, and Nahas and Sth (1994) develop odels for the negatve-bnoal dstrbuton. Soe other works that look at dynac allocaton and especally routng an aspect that we do not consder are Kuar et al. (1995), Mnkoff (1993), Beran and Larson (2001) and Kleywegt et al. (2002). Many of these use stochastc dynac prograng. The rest of the paper s organsed as follows: Secton 2 provdes a proble stateent. Secton 3 descrbes the soluton ethodology adopted and the coputatonal results obtaned. Conclusons drawn fro ths work are presented n Secton 4. 2 A proble descrpton The dstrbuton network, generally, has a herarchcal structure n whch a warehouse serves a set of retalers. The proble consdered n ths paper s to deterne the optal quanttes to be delvered fro the warehouse (also called the transhpent pont) to each of the retalers, so as to nse the expected long-run cost of operatng the syste. Our odel takes the followng costs nto consderaton. nventory-holdng costs stock-out costs, whch nclude the cost of lost sales and the loss of goodwll transportaton costs, whch nclude the operatng cost of the truck and the cost of transportng goods, whch, n turn, depends on the quanttes transported. The rando varables governng our syste are: the nter-arrval te of custoers at each retaler, the quantty deanded by the custoers, the servce te for each truck, and the travel te between the warehouse and the retalers

3 240 G. Subraana and A. Gosav and the sae between the retalers. We have assued that each retaler s dstnct and has unque values for syste paraeters. The rate of arrval of custoers, the nventory-holdng costs and the stock-out costs are dfferent for each retaler. Our objectve s to nse the average cost per unt te of operatng the entre syste. We now present detals of the sulaton odel developed. 2.1 A sulaton odel The sulator s wrtten usng the standard procedure for dscrete-event systes (see Chapter 2 of Law and Kelton, 2000). Our progras were wrtten n C language, but could be just as easly wrtten n coercal sulaton packages. Fgure 2 shows the event clock for the sulator. Let q denote the quantty to be delvered to the th retaler and n be the nuber of retalers. Then q = ( q1, q2,, q n ) wll represent the vector of delvery quanttes. We wll now defne the followng quanttes: L ( q, t): the total nuber of lost sales by te t n the sulaton (te starts at 0) at the th retaler, and I ( qt, ): the postve nventory at te t at the th retaler C tr : the operatng cost per unt te of a truck carryng a unt quantty of ateral (transportaton cost) C l : the stock-out (lost-sales) cost per unt quantty of sales lost at the th retaler, and C s : the nventory-holdng cost per unt quantty at the th retaler. Fgure 2 The event clock of the sulator showng the dfferent types of events Then, atheatcally, the proble s to deterne the soluton vector q n order to nse the average cost per unt te.e., to nse n n g ( qt, ) Gq ( ) = Ctr q + l, (1) t = 1 = 1 t where t g( q, t) = ClL( q, t) + Cs I( q, τ )d τ, (2) 0 such that q 0 for = 1, 2,, n. The frst ter n equaton (1) denotes the transportaton cost per unt te, and the second ter denotes the expected cost of stock-outs and holdng nventory on a unt te bass. The frst ter on the rght-hand sde of equaton (2) denotes the costs due to stock-outs and the second ter denotes the costs of holdng nventory. In equaton (2) L ( q, t) s evaluated wth a separate counter for the th retaler that s ncreented whenever a lost sale occurs at the th retaler, whle the second quantty s estated as follows: t 0 I ( q, τ )d τ = j (, t) j= 0 j where j (, t) denotes the total duraton of te nterval, startng fro te 0 untl the sulaton clock strkes t, durng whch the th retaler has j unts of postve nventory. j (, t) can be easly valuated n the sulaton progra. In the next sub-secton, we provde a bref descrpton of the an optsaton technque used n ths paper. 2.2 Sultaneous Perturbaton (SP) Gradent-descent ethods are defned as those ethods n whch values of partal dervatves of the objectve functon are used n the optsaton (search) process. Dervatves can be calculated nuercally wth a fnte dfference technque when the closed for s unavalable. The age-old gradent-descent rule (Cauchy, 1847), whch s also called the steepest descent rule, can be expressed as follows: x x µ f( x), f( x) f( x) f( x) f( x) =,,,. x(1) x(2) x( n) In the above equaton, f( x)/ x( ) denotes a partal dervatve of f(.) wth respect to x() where f (.) : R n + R, and µ denotes the step sze. In the fnte dfferences ethod, the gradent s calculated nuercally. Ths s useful when we do not have the closed for of the objectve functon. In odel-free sulaton-based optsaton (Carson and Mara, 1997; Andradóttr, 2002; Fu and Hu, 1997; Gosav, 2003), we do not have access to the closed for, but snce the perforance easure of the sys te ay always be evaluated wth the sulator, the fnte dfferences approach becoes especally useful. To calculate the gradent nuercally, we can use the central dfferences forula whch s gven by f( w) f( w+ h) f( w h) =. x 2h The dffculty wth the above s that the sulator has to be run twce for each decson varable n every teraton of the search algorth once to calculate f (w + h) and once to calculate f (w h). Thus the sulator has to be run 2n tes f n denotes the nuber of decson varables. Then, as n

4 Sulaton-based optsaton for ateral dspatchng n Vendor-Managed Inventory systes 241 ncreases, the nuber of runs of the sulator also ncreases and consequently, the coputatonal burden ncreases consderably. A way of workng around the above-entoned dffculty s va the Sultaneous Perturbaton (SP) ethod (Spall, 1992). Ths s a recent but rearkable developent. SP requres only two functon evaluatons per teraton of the algorth regardless of the nuber of decson varables. Hence, n the context of sulaton-based optsaton, the ethod s of specal nterest. The algorth s a stochastc search algorth. What we present below s a slghtly odfed verson of the orgnal algorth; the odfcaton s n Steps 6 and 7. Step 1: Set = 1. Intalse x, the soluton vector n the th teraton, to a feasble soluton obtaned fro the proble specfc heurstc. The algorth wll be ternated when the step sze µ becoes saller than a pre-deterned value µ n. Defne a sequence, whose th ter s c = l/ ζ. We wll fx ζ to a value n the open nterval (0, 1). Defne A and B such that they satsfy: 0 < A < 1, 0 < B < 1 and B < A. Step 2: Generate a rando nuber H(), for 1, 2,, n, fro a Bernoull dstrbuton, whose two perssble and equally lkely values are 1 and 1. Copute the values of h() usng the followng forula. h () = H() c. Step 3: Copute f ( x + h ) and f ( x h ) as follows, where f ( x + h) = f( x (1) + h(1), x (2) + h(2),..., x ( n) + h( n)), f ( x h) = f( x (1) h(1), x (2) h(2),..., x ( n) h( n)). Step 4: For varables = 1, 2,, n we obtan the partal dervatves usng f( x ) f( x + h) f( x h). x () 2 h ( ) Step 5: Copute y usng the followng rule: f( x ) y x () µ for = 1,2,, n. x () 1 Step 6: If f( y) < f( x ) then set x + = y and µ µ B. Otherwse, go to Step 7. 1 Step 7: Increase by 1, set x + = x, and update µ usng: µ µa. Step 8: If µ µ n, stop. Otherwse, return to Step 2. SP takes one to the vcnty of a good soluton very rapdly. However, one can fne tune the perforance of SP by usng the well-known eta-heurstc, Sulated Annealng (SA) (Krkpatrck et al., 1983), after usng SP. In other words, the soluton generated by SP can be used as a startng soluton for SA. In the next subsecton, we dscuss the SA algorth. 2.3 Sulated Annealng (SA) SA starts at a user-provded soluton and searches n the neghbourng regon for a better soluton. The power of SA depends on the neghbourhood search strategy used. It has a so-called exploratory property whch allows the algorth to worsen the objectve functon value at tes. It s claed that ths can help n fndng the global optu n a syste havng a lot of local opta. The algorth accepts the worse soluton wth a probablty whch decays wth the nuber of teratons of the algorth. The algorth s ternated when the probablty of ovng to a worse neghbour approaches 0. At each teraton, the best soluton obtaned so far s stored. As a result, the best of all the solutons tested by the algorth s returned as the near-optal soluton. An portant paraeter, called teperature s used to control the exploratory property. It has been establshed that the algorth does provde the optal soluton f the teperature s decreased properly (Lundy and Mees, 1986). We wll skp the steps n the algorth referrng the reader to any standard text on eta-heurstcs. However, we do defne our neghbour-generatng strategy next. If x current denotes the current soluton n the SA algorth, a neghbour (.e., the new soluton) s selected as follows. Let π denote the thckness of the neghbourhood. When a neghbour s to be selected, generate a rando nuber u for = 1, 2,, n between 0 and 1 fro the unfor dstrbuton (0, 1). If u 0.5 then for = 1, 2,, n, set: x () = x () + π, neghbour current and otherwse set: x () = ax(0, x () π ). neghbour current In our experents, π was set to Proble-specfc heurstcs We now descrbe soe proble-specfc heurstcs that can be used on ths proble. These heurstcs provde good startng solutons and serve as bencharks for SA and SP The ean deand heurstc. The Mean Deand Heurstc (MDH) s used n a local dary product ndustry n New York. It utlses the ean deand at each retaler. The coputatons nvolved can be explaned as follows. Let T denote the average cycle te,.e., the te requred for one round trp and d denote the average deand per custoer at the th retaler. Then, the optal quantty q, for the th retaler s gven by q = Td, where denotes the ean rate of arrval of custoers at the th retaler.

5 242 G. Subraana and A. Gosav The newsvendor heurstc. The popular newsvendor odel (see e.g., Nahas, 2001) can be used to deterne the optal delvery quantty when the coodty s pershable and the deand s norally dstrbuted. By usng a noral approxaton to the Posson dstrbuton, we can also apply t to the Posson dstrbuted deand. The odel works as follows. The optal quantty for the th retaler satsfes the followng condton: Cl Fq ( ) =, C + C l s where F(.) denotes the cuulatve dstrbuton functon of the deand. If the deand s norally dstrbuted, then the optal quantty s gven by: q = µ + Zσ, where Z denotes the z-value n the noral table assocated wth F(.) obtaned fro the above, and µ s the ean and σ the standard devaton of the deand at the th retaler. Here µ = Td. For the noral approxaton to the Posson dstrbuton, σ = µ for each. 3 Coputatonal results For testng the effcacy of the sulaton-optsaton approach, we conducted experents wth a large nuber of systes. We copared the perforance wth the two heurstcs dscussed above. A popular ethod for sulaton-based optsaton uses response surfaces (Law and Kelton, 2000). For a full-blown syste wth ten or ore retalers, the response surfaces would requre too any saples. In partcular, for a 10-retaler network, one would requre at the very least 2 10 ponts. Hence we used response surfaces on saller versons of the proble wth retalers. 3.1 A 2-retaler network The central dea underlyng the Response Surface Method (RSM) s to obtan an approxate for of the objectve functon by sulatng the syste at a nuber of ponts n the soluton space and fttng a functon to the set of sapled ponts. Conventonal RSM uses regresson for fttng the objectve functon. For regresson, t s necessary to assue a etaodel. The need to assue the etaodel akes the applcaton of RSM lted because t s often dffcult to recognse an obvous closed for of the functon. In such cases, we ay need to assue a nuber of etaodels sequentally and test each of the to obtan a good ft. Ths can be te-consung. An alternatve s to use neuro-response surfaces (Gosav, 2003) n whch functon fttng s done usng artfcal neural networks (va a standard algorth lke back-propagaton), whch do not requre the eta-odel apror. For the Neuro-Response Surface Method (NRSM) the soluton space was sapled at 32 ponts per retaler. Thus, the total nuber of data ponts used for tranng the neural net becae = We traned the neural net for 3500 teratons. After tranng the neural net, we generated the cost of the runnng the syste (for a gven set of paraeters) usng the neural net. We next tested f the ft obtaned wth the neural network was good, usng the coeffcent of deternaton (Montgoery et al., 2001). The systes studed are descrbed n Table 1. The values of the coeffcent of deternaton are provded n Table 2. All values are close to 1, ndcatng good fts. Frst, SP was eployed on every proble. SP rapdly took us to very good solutons. Thereafter SA was used to prove upon the soluton provded by SP by usng the soluton of SP as the startng soluton. In all the systes, the SP-SA cobnaton outperfored the best heurstc. See Table 3 for a coparson of the perforance of NRSM and the SP-SA cobnaton. Of the two heurstcs, the newsvendor heurstc obtans the better soluton. NRSM perfors uch better than both of the heurstcs, but fares worse than the SP-SA cobnaton. Table 1 Syste The values of paraeters used for each syste 1 C s 2 C s 1 C l 2 C l Table 2 Coeffcent of deternaton Syste Coeffcent of deternaton Table 3 Cost coparson for optsng ethods (sall cases) Ip of NRSM over newsvendor heurstc (%) Ip of SP-SA over newsvendor heurstc (%) Mean-deand Newsvendor Syste heurstc heurstc A 10-retaler network In ths secton, we descrbe our coputatonal results for a ore realstc 10-retaler network. Soe odfcatons had to be ade to both SA and SP. We test 18 systes, whch are descrbed va Tables 4 6.

6 Sulaton-based optsaton for ateral dspatchng n Vendor-Managed Inventory systes 243 Table 4 Level defntons Syste paraeters Retaler 1 Retaler 2 Retaler 3 Retaler 4 Retaler 5 C s C s C s C l C l C l λ λ Table 5 Level defntons (contnued fro prevous table) Syste paraeters Retaler 6 Retaler 7 Retaler 8 Retaler 9 Retaler 10 C s C s C s C l C l C l λ λ Table 6 Syste paraeters (large cases) Syste C s -level C l -level λ-level We were able to prove the perforance of SP by usng a reactve step sze. In such a schee, when the algorth strkes a worse soluton, t s returned to the orgnal soluton, and the step-sze-reducton rate for that step s ncreased. The rate s restored to ts orgnal value only after an proved soluton s found. Table 7 shows the behavour of regular S and our reactve SP on Syste. Table 7 Coparson of cost fro the two SP ethods for syste 1 Regular SP Reactve SP Optsed cost The results obtaned for the 10-retaler network are provded n Table 8. Lke n the 2-retaler case, an SP-SA cobnaton was used. In all the cases but 5, 13 and 18 SA was able to prove argnally upon the soluton of SP. The SP-SA cobnaton outperfored the proble-specfc heurstcs n all cases. The SP-SA cobnaton took no ore than ten nutes for the 10-retaler case on a UNIX Sunblade achne. The percentage proveent of the algorth over the heurstc s denoted by Ip %. MDH, although prevalent n a local ndustry, s not lkely to perfor well because of ts nablty to take all the syste varablty nto account. On the other hand, the robust perforance of the newsvendor odel s an encouragng fndng; after all, the newsvendor odel s well-understood, and can be easly adapted nto exstng systes. Thus, anagers who prefer not to use black-box ethods could use the newsvendor odel nstead. Table 8 Cost-perforance coparson (equaton (1)) of heurstcs and SP-SA Syste Mean-deand heurstc Newsvendor heurstc Ip of SP-SA over best heurstc (%) Non-convexty of the objectve functon We wanted to deterne f the cost functon s convex. An portant reason for ths s that convex functons are easer to optse. Hence, we perfored experents

7 244 G. Subraana and A. Gosav wth a two-retaler syste n whch the quantty delvered to Retaler 2 was vared and that delvered to Retaler 1 was fxed at a constant value. Table 9 provdes cost values at soe values of the quantty delvered to Retaler 2 (Q); the table clearly shows the presence of two local opta. It s thus clear that even for the two-retaler proble, the functon s not convex. Table 9 Non-convexty of the objectve functon Q Cost A senstvty analyss A full factoral experent was desgned to deterne whch factors affect the optal objectve functon obtaned fro our sulaton-optsaton approach. The three factors that we studed are: the nventory-holdng costs at three levels the stock-out costs at three levels the rate of arrval of custoers at two levels. The values of the factors are enuerated n Table 6. The results of the ANOVA are presented n Table 10. A nuber of two-way nteractons and one three-way nter acton were also consdered. The nventory-holdng cost and the stock-out cost were found to be sgnfcant. Ths ples that anagers nterested n usng the sulaton-optsaton approach should exercse great care n estatng these eleents of the cost. Table 10 Source An analyss of varance Degrees of freedo Su of squares Mean squares F P-value C s C l λ C s C l C s λ C l λ C s C l λ Error Total Conclusons Deternng the optal quanttes to be dspatched to each retaler fro the warehouse s a long-standng proble n the ndustry. In the past, custoers placed orders and warehouses et the as and when needed. Ths led to a peak of deands on certan days of the week. Wth the advent of vendor anaged nventory systes, the warehouse anager sends ateral at regular te ntervals on the bass of average deand at each retaler. The proble s coplcated by randoness arsng fro custoer behavour and transportaton delays. There are three ajor costs assocated wth anagng ths syste: the transportaton cost, the nventory-holdng cost and the cost due to lost sales. Accoodatng all the governng rando varables and costs n one atheatcal odel s a challengng task, and hence, uch of the lterature akes splfyng assuptons to develop tractable atheatcal odels. In ths paper, we have developed a sulaton odel that accounts for ost features of a real-world syste, and have used optsaton technques that can be cobned effcently wth sulaton to generate solutons. On the bass of our experentaton, we fnd that a cobnaton of SP and SA s able to outperfor a heurstc used n the real world, naely, MDH. An nterestng fndng s that the newsvendor odel can be adapted to develop an effcent heurstc for solvng ths proble. Although t was outperfored by sulaton-optsaton ethods, ts encouragng perforance ndcates that anagers who do not ntend to sulate the entre network can resort to the newsvendor odel for obtanng relable solutons. The solutons developed by the sulaton-optsaton approach can be drectly ncorporated nto the decsonakng technology of warehouse anagers. Sulaton s a wdely used tool n the ndustry, and hence the soluton ethodology presented here should fnd ready acceptance n the ndustry. Furtherore, the sulaton-optsaton approach can be run on any personal coputer wth great ease. Several extensons to ths research can be envsaged. Frstly, a possble alternatve s to develop an approxate Markov chan odel for ths syste, and use t for optsaton. Secondly, one can consder the entre network and ncorporate the effects of transportaton fro the regonal warehouse to the local warehouses nto the odel. Acknowledgeents Ths research was partally supported by a research grant to the second author fro the Natonal Scence Foundaton. The authors are grateful to the anonyous revewers of ths paper and to the specal-ssue edtor, Dr. Sanjay Jan, who ade any suggestons for provng ths paper. References Andradóttr, S. (2002) Sulaton optzaton: ntegrated research and practce, INFORMS Journal on Coputng, Vol. 14, No. 3, pp Beran, O. and Larson, R. (2001) Delveres n a nventory/routng proble usng stochastc dynac prograng, Transportaton Scence, Vol. 35, No. 2, pp

8 Sulaton-based optsaton for ateral dspatchng n Vendor-Managed Inventory systes 245 Carson, Y. and Mara, A. (1997) Sulaton optzaton: ethods and applcatons, Proceedngs of the 1997 Wnter Sulaton Conference, Atlanta, GA, pp Cauchy, A. (1847) Méthode Générate pour la Résoluton des Systés d Équatons Sultanées, Cop. Rend. Acad. Sc., Pars, pp Clark, A. and Scarf, H. (1960) Optal polces for a ult-echelon nventory proble, Manageent Scence, Vol. 6, No. 4, pp Eppen, G. and Schrage, L. (1981) Centralzed orderng polces n a ult-warehouse syste wth lead tes and rando deand, Mult-Level Producton/Inventory Control Systes: Theory and Practce, Asterda, North Holland, p.169. Federgruen, A. and Zpkn, P. (1984) Approxatons of dynac, ult-locaton producton and nventory probles, Manageent Scence, Vol. 30, No. 1, pp Fu, M. and Hu, J. (1997) Condtonal Monte Carlo: Gradent Estaton and Optzaton Applcatons, Kluwer Acadec Publshers, Boston, MA. Gosav, A. (2003) Sulaton-based Optzaton: Paraetrc Optzaton Technques and Renforce Ment Learnng, Kluwer Acadec Publshers, Massachusetts, USA. Haer, M. (2001) The supereffcent copany, Harvard Busness Revew, Septeber October Issue, pp Krkpatrck, S., Gelatt, C.D. and Vecch, M.P. (1983) Optzaton by sulated annealng, Scence, Vol. 220, pp Kleywegt, A.J., Nor, V.S. and Savelsergh, M.W.P. (2002) The stochastc nventory routng proble wth drect delveres, Transportaton Scence, Vol. 36, No. 1, pp Kuar, A., Schwarz, L.B. and Ward, J.E. (1995) Rsk-poolng along a fxed delvery route usng a dynac nventory-allocaton polcy, Manageent Scence, Vol. 41, No. 2, pp Law, A. and Kelton, W. (2000) Sulaton Modelng and Analyss, 3rd ed., McGraw-Hll, New York, NY. Lundy, M. and Mees, A. (1986) Convergence of the annealng algorth, Matheatcal Prograng, Vol. 34, pp McGavn, E.J., Schwarz, L.B. and Ward, J.E. (1993) Two nventory allocaton polces n a one warehouse N-dentcal retaler dstrbuton syste, Manageent Scence, Vol. 39, No. 9, pp Mnkoff, A.S. (1993) A Markov decson odel and decoposton heurstc for dynac vehcle dspatchng, Operatons Research, Vol. 41, No. 1, pp Montgoery, D., Ruer, G. and Hubele, N. (2001) Engneerng Statstcs, 2nd ed., John Wley and Sons, New York, NY, USA. Nahas, S. (2001) Producton and Operatons Analyss, McGraw-Hll, Irwn, USA. Nahas, S. and Sth, S.A. (1994) Optzng nventory levels n a two-echelon retaler syste wth partal lost sales, Manageent Scence, Vol. 40, No. 5, pp Spall, J.C. (1992) Multvarate stochastc approxaton usng a sultaneous perturbaton gradent approxaton, IEEE Transactons on Autoatc Control, Vol. 37, pp