COST OPTIMAL ALLOCATION AND RATIONING IN SUPPLY CHAINS

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1 COST OPTIMAL ALLOCATIO AD RATIOIG I SUPPLY CHAIS V..A. akan a & Chrstopher C. Yang b a Department of Industral Engneerng & management Indan Insttute of Technology, Kharagpur, Inda b Department of Systems Engneerng & Engneerng Management The Chnese unversty of Hong Kong, Hong Kong ABSTRACT A supply chan wth a sngle suppler and multple retalers has been consdered for the study. Each retaler has a unque dscrete demand dstrbuton, unque determnstc lead-tme and the nventory postons of all the retalers are known to the central decson maker. All retalers have unque holdng and penalty costs. Transportaton costs are not consdered for the model n order to avod possble bas whle allocatng or ratonng nventory. We consder two cases of allocaton. In the frst case, the suppler has lmted avalablty of nventory stock to dstrbute among the retalers and n the second case, suppler has unlmted nventory n stock and lke to raton t among the retalers. Unque servce level constrants of each retaler, such as the mnmum requred non-stock-out probablty durng the perod n the ratonng and allocaton problems s also ncluded n the model. The proposed model can handle any type of dscrete demand dstrbuton of each retaler. An teratve procedure has been proposed for solvng the problem. An llustratve example s dscussed. KEY WORDS Supply chan management, ratonng, allocaton, optmzaton, non-stock-out probablty. ITRODUCTIO Ths paper focuses on allocaton and ratonng problems n supply chans. The objectve s to cost optmally allocate the avalable (lmted/unlmted nventory to a number of lower level locatons. Many determnstc as well as stochastc varables are to be consdered when tryng to solve allocaton and ratonng problems. Some of these varables are: demand, present stock and safety stock at each lower locaton, lead tmes, dfferent cost elements, servce levels requrements, number of locatons, and avalable nventory for dstrbuton. There may be more varables dependng on the system and assumptons. The rest of the paper has been organzed as follows. A bref revew of lterature of the most mportant recent publcatons on ratonng and allocaton problems has been dscussed n secton 2. The problem under nvestgaton has been clearly stated and dscussed n secton 3. The proposed methodology for modelng and solvng the stated problem has been dscussed n secton. A numercal example has been ncluded n secton 5 for llustratng the methodology. Major fndngs and conclusons are gven n secton 6 and the latest references are gven n secton 7 of the paper. 2. REVIEW OF LITERATURE Federgruen and Zpkn (98a present a method to approxmate a problem wth several retal outlets by a sngle-outlet model. They obtan a near-optmal polcy and a good approxmaton of the cost of the system. Ther results suggest that the balance assumpton, and hence the Clark (960 approach, s napproprate when coeffcent of varaton are unequal. Federgruen and Zpkn (98b address the combned problem of allocatng scarce nventory amoung several locatons and plannng delveres usng a fleet of vehcles. Demands are random, holdng, shortage and transportaton costs are consdered n the model. They extend some avalable determnstc methods of vehcle routng problem to ths case. Federgruen and Zpkn (983 present methods for solvng allocaton problems that can be stated as convex knapsack problems wth generalzed upper-bounds. Such bounds may express upper lmts on the total amount allocated to each of the several locatons. They do not consder any servce level constrant n the allocaton models. Eppen and Schrage (98 analyze a depot-warehouse system wth ndependent, normally-dstrbuted, statonery warehouse demand; dentcal proportonal costs of holdng and backorderng; and no transshpment.

2 Ballou (999 explans a method for allocaton of nventory to stockng ponts on the bass of the average demand rate, that s, the forecasted demand rate. Costs are not consdered for the allocaton. Dks and Kok (999 consder a cost functon (holdng + penalty costs n allocaton. The allocaton s based on the balanced poston assumpton. Ths allocaton need not be optmum n the sense of cost mnmzaton. Verrjdt and Kok (995 nstead of defnng a cost structure, apply a servce level approach where the man goal s to realze pre determned servce levels n the fnal stock ponts. Chung, Flynn, and Stalsk (200 propose an allocaton technque for a seral supply chan to maxmze the expected proft. Dks, Kok, and Lagodmos (996 dscuss Far Share (FS, Approprate Share (AS, and Consstent Approprate Share (CAS Ratonng Polces, whch are used n push nventory systems. Prorty Ratonng (PR s a polcy used n Pull nventory system. They do not consder cost optmzaton, but concentrate on servce levels such as non-stock-out probablty, fll rate etc. Cachon and Larvere (999 dscuss a method of mappng from retaler orders to capacty assgnments. The authors show that a broad class of mechansms s prone to manpulaton. Ths wll lead to demand amplfcaton, popularly known as bullwhp effect. Vercourt et.al. (2002 and Ha (2000 employ queung based approach and the models follow a sngle-server, sngle-product, make-to-stock queue wth multple demand classes. Ha (997 consders the producton control and stock ratonng n a make-to-stock producton system wth two prorty customer classes and back orderng. Kukreja et.al.(200 propose a model to fnd the nventory stockng levels at all locatons that mnmze the sum of nventory holdng costs and transshpment costs subject to a specfed mnmum system-servce level, n a -locaton contnuous revew system. Axsater et.al.(2002 propose a two-step allocaton heurstc rule for allocatng stock from warehouse to the retalers. You (2003 propose a dynamc ratonng polcy that decdes whether or not to accept each request from a customer for any combnaton of demand class, remanng decson perod, and remanng stock. A crtcal analyss of the avalable lterature reveals that even for two-stage systems, there s no welldemonstrated procedure, whch s wdely acceptable and can be easly and effectvely used for general purpose ratonng. Most of the related works assume varous condtons, and solve the problem usng ntutve, heurstc, or other approxmate methods. Therefore, t appears that there s a need for developng an effectve, easy-to-use, and practcal methodology for solvng allocaton and ratonng problems n supply chans. Ths research work s an attempt n that drecton. We propose to use the basc prncples of statstcal methods and an teratve procedure to formulate and solve the problem. 3. PROBLEM DEFIITIO There s one suppler and retalers. Two cases of nventory dstrbuton have been consdered. In the frst case, the suppler has only lmted quantty of nventory for dstrbuton among the retalers. There are unque determnstc lead tmes for reachng materal from the suppler to each retaler. All retalers have a polcy of safety stockng to meet demand durng the lead tmes. A central decson maker has the real tme nformaton about nventory and the demand pattern at each retaler and based on ths the avalable nventory s allocated perodcally. Each retaler has a unque dscrete demand dstrbuton, holdng cost, and shortage cost. The problem s optmal ratonng of avalable nventory wth the suppler to all the retalers. The objectve functon s the total holdng and stock-out costs of all retalers, and the constrant s the avalable stock and the servce levels. In the second case, the suppler has unlmted nventory n stock to dstrbute among the retalers.. PROPOSED METHODOLOGY The expected cost functon of each retaler s the sum of expected holdng cost and expected out-ofstock (penalty cost of each retaler. Functons for expected number of n-stock (unsold nventory at the end of perod and expected number of out-of-stock (back order nventory at the end of the perod are derved from the unque dscrete demand dstrbuton of each retaler. The total cost functon for each retaler s derved from ths. The optmzaton problem cannot be solved by usual analytcal methods, because the objectve functon conssts of nfnte number of terms. Therefore, t s solved usng an teratve algorthm dscussed n more detals n subsequent sectons. For the ratonng problem, the teraton starts wth assgnng zero nventores for each retaler. Inventory level s ncreased from ths

3 poston n subsequent teratons so as to maxmze the decrease of total cost compared wth the total cost of the prevous allocaton. The procedure contnues n ths way tll the constrant of avalable stock wth suppler s satsfed. When all the retalers are excluded or the avalable nventory s fully allocated, the teraton stops. If the avalable nventory wth the suppler s unlmted, the teraton procedure s contnued tll the total cost functon starts ncreasng. The allocaton matrx n the just prevous teraton s the optmum allocaton n ths case. otatons: A : Avalable stock of nventory at the Suppler λ : Mean of the Posson demand at retaler L : Lead tme for retaler x : Expected number of out-of-stocks at retaler, per perod c(x : Penalty cost functon for retaler (functon of number of out-of stock tems per perod y : Expected nventory poston at the end of a perod at retaler h(y : Holdng cost at retaler per unt perod(functon of nventory poston at the end of a perod h : Holdng cost per unt per perod C : Sellng prce of a unt at retaler c : penalty cost fracton of C S : Echelon Inventory poston at the begnnng of a perod at retaler d : Demand at retaler durng the perod z : The allocated nventory to retaler from the suppler for the perod Z : Row matrx representng allocaton of each retaler Z 0 : Intal allocaton matrx f ( d : Probablty densty (mass functon of the demand dstrbuton at retaler TC : Total holdng and penalty cost for retaler TC s : Total system cost : umber of retalers SL : % Servce Level at retaler I SLR : % Servce level target of retaler SS : Safety Stock at retaler to deal wth demand durng lead tmes R : Assessment Perod Development of Model Followng equatons can be used to estmate the expected safety stock, servce level, holdng cost, penalty cost, and total cost of the system. Safety Stock Requred at retaler : L SS = * f ( d * d R ( x Servce Level at retaler (on-stock-out probablty, SL = * 00 (2 z + S SS Expected cost of out-of-stock at retaler : c ( x = c * C * x c * C * [ d ( z + s ] f ( d 0 Expected holdng cost at retaler : ( The expected total cost for retaler : TC h y = + d = z + s z s = h y = h + z + s d f d ( d = 0 [( ] ( * C * [ d ( z + s ] f ( d h [( z + s d ] f ( d = c d = z + s + z + s + d = 0 (5

4 Snce the avalable stock wth the suppler has been completely allocated to the retalers, total cost of the system s the sum of all penalty and holdng costs for all retalers. z + s TC s = TC = c * C * {[ d ( z + s ]* f ( d } + h * { ( z + s d * f ( d } (6 = = d = z + s + d = 0 Ths s the objectve functon of the allocaton problem. The total cost functon of the system s to be mnmzed under the constrant of the avalable nventory wth the suppler. That means, Mnmze: TCs = = c * C * {[ d ( z + s ]* f ( d } + h * { ( z + s d * f ( d } d = z + s + z + s Subject to: z A; SL = SLR, for all (8 = It can be seen from the above equatons that the objectve functon wll conssts of very large (nfnte number of terms. Therefore, t wll not be feasble to fnd out an exact analytcal soluton to ths problem. We propose an teratve procedure to solve ths problem. The advantage of ths procedure s that t can handle any type of dscrete demand dstrbuton and cost functons: lnear as well as non-lnear. Followng algorthm s developed for ths the teratve allocaton and ratonng: 0 j= Calculate TCmn = TC s ( for ntal allocaton Z 0 20 = 30 { allocate z = z + ; calculate TC s wth the present matrx Z; assgn TC s ( = TC s ; remove the allocaton z = z ; = + IF ( < GO TO 30 ELSE Fnd MI [TC s (; =..]; assgn ths to TC s (mn Fnd mn for whch TC s ( s mnmum IF (TC s (mn TCmn { TCmn = TC s (mn Confrm the allocaton for z = z + mn d = 0 Update the matrx Z j by ncludng ths change Calculate x SL = * 00 z + S mn SS IF SL SLR Exclude mn from the set of from further teratons IF z < A {j = j + GO TO 20}} ELSE Z j of the prevous teraton s optmum } 5. umercal Example: Consder a supply chan wth one suppler and retalers. A central decson maker dstrbutes the avalable nventory of the suppler to the four retalers every week. The objectve s to mnmze the total holdng and stock-out costs of all locatons. Followng data are avalable to the central decson maker: (7

5 Retalers 2 3 Mean Posson Demand per week Lead Tmes (Hours Holdng Cost per unt per week ($ Stock-out Penalty Cost per unt per week ($ Maxmum Retal Prce of a unt ($ Inventory Poston at Startng of the perod Table : Data for umercal Example The avalable nventory s to be allocated or ratoned optmally among the four retalers under the followng stuatons: Case : Avalable stock wth suppler, A = 3 unts Case 2: Unlmted stock s avalable wth the suppler Soluton: Case : Lmted Stock at Suppler The constrant n ths case s the number of unts avalable wth the suppler for ratonng. z = Mathematcally, 3 Let us assume the strategy that 60% of demand of each retaler s allocated ntally. Based on ths, we get the followng startng matrx, correspondng total cost, and the total unts allocated: Startng Soluton: Z 0 = { } z = TC s (0 = $ 8.60 and = 22 After 9 teratons, we get the followng optmal allocaton: Z 0 = { 7 9 } z = TC s (0 = $ 9.7 and = 3 Expected Servce Level (non-stock-out probablty of ths optmal ratonng at each retaler are SL = 90.3 %, SL 2 = %, SL 3 = %, and SL = 98.0 % respectvely. Case 2: Unlmted Stock at Suppler In ths case, the suppler has unlmted number of unts avalable for allocaton. As n case, let us assume the ntal soluton matrx Z 0 = { } After 2 teratons from ths poston, we get: Z 2 = { } z = TC s (2 = $ 6.96, and = 3 The 3 th teraton from startng poston, we get: Z 3 = { } z = TC s (3 = $ 7.79 and = 35 Snce TC s (3 > TC s (2, Z 2 s the optmal allocaton n ths case. Ths shows that eventhough the suppler has unlmted number of unts avalable, t s better to allocate only 3 unts among the four retalers to mnmze the total cost. Servce levels correspondng to ths optmal allocaton at each retaler are: SL = 95.7 %, SL 2 = %, SL 3 = 98. %, and SL = 98.96% 6. Conclusons The proposed algorthm s capable of fndng cost-optmal solutons for ratonng and allocaton problems under the constrants of lmted or unlmted stock avalablty of the suppler and unque servce level of each retaler. The algorthm assumes an ntal soluton matrx and subsequently uses an teratve procedure. Selecton of ntal soluton matrx has no effect on the optmal soluton, f at least one unt s allocated by the subsequent teratons to each retaler. Strateges proposed and tested n the

6 paper for decdng the ntal startng soluton, can be effectvely used for reducng the number of teratons requred to reach the fnal soluton. The proposed algorthm has several advantages over exstng models. Frst of all t ensures optmal allocaton. Ths s because, startng from the ntal soluton matrx, each teraton mproves upon the prevous one and convergng to a mnmum cost allocaton. The algorthm can be used evenf each retaler has a dfferent demand dstrbuton. Allocaton and ratonng problems of two-stage supply chans wth any number of retalers can be handled. The algorthm ensures an exact, unbased cost optmal soluton. It s also accurate, easy to mplement and communcate. 7. References. Axsater Sven, Johan Marklund, and Edward A. Slver(2002. Heurstc methods for centralzed control of one-warehouse, -retaler nventory systems. Manufacturng & Servce Operatons Management, Vol., o., pp Ballou Ronald H. (999. Busness Logstcs management. Chapter 0, Fourth Edton, Prentce-Hall, pp 39-32, 3. Cachon Gerard P., and Paul H. Zpkn (999. Compettve and cooperatve nventory polces n a two-stage supply chan. Management Scence, Vol 5, o.7, pp Cachon Gerard P., and Martn A. Larvere (999. Capacty choce and allocaton: strategc behavour and supply chan performance. Management Scence, Vol.5, o.8, pp Chung Cha-Shn, James Flynn, and Potr Stalsk (200. A sngle-perod nventory placement problem for a seral supply chan. aval Research Logstcs (8, pp Clark A.J. and Scraf Herbert (960. Optmal Polces for a Mult-Echelon Inventory Problem. Management Scence, Vol.6, pp Dks E.B. and A.G. de Kok(999, Computaton results for the control of a dvergent n-echelon nventory system. Internatonal journal of Producton economcs, (59, pp , 8. Dks E.B., A.G. de Kok, and A.G. Lagodmos(996. Mult-echelon systems: A servce measure perspectve. European Journal Operatonal Research (95, pp Eppen G. and Schrage L.Centralzed Orderng Polces n a Mult-warehouse System wth Lead Tmes and Random Demand, Mult-Level Producton/Inventory Control Systems: Theory and Practce, pp 5-67, (98 0. Federgruen A. and Zpkn P., Soluton Technques for Some Allocaton Problems, Mathematcal Programmng, (25, pp 3-2, (983. Federgruen A. and Zpkn P., Computatonal Issues n an Infnte-Horzon, Mult-echelon Inventory Model, Operatons Research, Vol.32, o., pp , (98a 2. Federgruen A. and Zpkn P., A Combned Vehcle Routng and Inventory Allocaton Problem, Operatons Research, Vol.32, o.5, pp , (98b 3. Ha A.Y, Stock ratonng n an M/E k / make-to-stock queue, Management Scence, Vol. 6, pp77-87, (2000. Ha A.Y, Stock ratonng polcy for a Make-to-stock producton system wth two prorty classes and backorderng, avel Research Logstcs, Vol., pp 57-72, ( Kukreja Anl, Charls P. Schmdt, and Davd M. Mller, Stockng decsons for low-usage tems n a mult-echelon nventory system, Management Scence,Vol. 7, o.0, pp37-383, ( Vercourt Francs de, Fkr Karaesmen, and Yves Dallery, Optmal stock allocaton for capactated supply system, Management Scence, Vol.8,o., pp 86-50, ovember Verrjdt J.H.C.M. and A.G. de Kok, Dstrbuton plannng for a dvergent -echelon network wthout ntermedate stocks under servce restrctons, Internatonal journal of Producton economcs, (38, pp , You, Peng-Sheng, Dynamc ratonng polces for product wth ncremental upgradng demands, European Journal of Operatonal Research, Vol., pp 28-37, (2003