A Dynamic Inventory Control Policy Under Demand, Yield and Lead Time Uncertainties

Transcription

1 A Dynamc Inventory Control Polcy Under Demand, Yeld and ead Tme Uncertantes Z. Baba, Yves Dallery To cte ths verson: Z. Baba, Yves Dallery. A Dynamc Inventory Control Polcy Under Demand, Yeld and ead Tme Uncertantes. IEEE SSSM (Sevce Systems and Servce Management), Oct 2006, Troyes, France <hal > HA Id: hal Submtted on 5 Dec 2006 HA s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. archve ouverte plurdscplnare HA, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 A Dynamc Inventory Control Polcy Under Demand, Yeld and ead Tme Uncertantes Mohamed Zed Babaï, Yves Dallery aboratore Géne Industrel, Ecole Centrale Pars, France ABSTRACT In ths paper, we analyze a sngle-stage and sngle-tem nventory control system wth non-statonary demand and uncertan system parameters. We propose two extensons of a prevous wor on the dynamc reorder pont polcy (the (r,) polcy [2]). In the frst extenson, we nclude three types of uncertantes pertanng to the demand uncertanty, the lead tme uncertanty, and the yeld uncertanty. We study the mpact of these uncertantes on the (r,) polcy and we provde an approxmaton of the optmal parameters of the polcy usng a sequental approach under a cycle servce level constrant. The approxmatve parameters obtaned n ths paper are good ones for small values of the varablty of uncertantes. In the second extenson, we focus on the demand uncertanty. We determne the optmal parameters of the (r,) polcy usng a sequental approach under a fll rate servce level constrant. In the two extensons, we focus on the safety stoc parameter and we propose a method to compute t. Keywords: nventory control, forecasts, cycle servce level, fll rate, safety stoc, polcy parameters 1. INTRODUCTION There s an abundant lterature on nventory control polces whch extends snce the 30 s. The most nown polces are the reorder pont polcy, called also as the (r,) polcy and the order-up-to-level polcy, called as the (S, T ) polcy. Several other alternatves of these polces are developed such as the (s,s) polcy, the (T,r,) polcy and the (T, r,s) polcy. Most of the models gven n the lterature to analyze these polces assumes a statonary demand and a cycle servce level. Note that these polces are statc,.e. ther parameters are constant over tme. For more detals on these polces, the reader s referred to [8] and [10]. There s also some lterature that studes dynamc nventory control polces based on the nvestgatons of [4] and [6]. However, ths lterature s nterested n optmal nventory control polces that are not easy for mplementaton. In earler papers [1] and [2], we proposed a dynamc reorder pont polcy for a non statonary demand, namely, the (r,) polcy. The parameters of ths polcy are determned usng a sequental approach whch means that the value of the orderng quantty s computed gnorng the mpact upon t of the reorder pont r. Indeed, the orderng quantty s ndependent of uncertantes, whereas the reorder pont taes nto account the mpact of the uncertantes by the mean of a safety parameter. To determne the safety parameter, a cycle servce level approach s used whch means that we mpose, at each cycle (tme perod between two successve orders), that the probablty of not havng a stocout s hgher than a target cycle servce level. Our proposed polcy n these nvestgatons ncludes only the uncertanty assocated wth demand. In real nventory control systems, besdes the randomness related to demand forecasts, several other types of randomness may exst. For example, randomness n the replenshment process, randomness related to product qualty, randomness due to the unrelablty of supplers, etc. These /06/$20.00 c 2006 IEEE randomness mples several uncertantes n the system, such as the lead tme uncertanty and the yeld uncertanty. Therefore, n order to guarantee a better servce level, the computaton of the parameters of the control polcy have to tae nto account these uncertantes. There s an extensve lterature that studes nventory polces by consderng the yeld uncertanty [3], [5], [7] and [9]. There s also much wor that studes nventory systems under the lead tme uncertanty [8] and [10]. However, we remar that n these wors the mpact of each uncertanty on the polcy parameters s analyzed separately and most of the results are gven for a statonary demand and statc polces. Ths has motvated us to extend our results concernng the (r, ) polcy by ncludng yeld and lead tme uncertantes. We also remar that most of the wor on dynamc nventory control polces consders a cycle servce level to compute the parameters of the polces. So, we extend results provded n [2] by consderng a fll rate servce level whch s amongst the most useful measure of servce n nventory control systems. Ths paper s organzed as follows: n Secton 2, we descrbe the system and the assumptons we consder. In Secton 3, we brefly recall the prncple and the parameters of the (r,) polcy. We study the mpact of uncertantes on the parameters of the polcy and we provde the optmal parameters by consderng the three uncertantes smultaneously and a cycle servce level. In Secton 4, we provde the optmal parameters of the (r,) polcy for a fll rate servce level. The conclusons are gven n Secton SYSTEM DESCRIPTION AND ASSUMPTIONS We consder a sngle-stage and sngle-tem nventory system wth a non-statonary demand. The system s not capactated and the nventory replenshment requres a leadtme, as represented n Fg. 1. We assume that demand s nown by means of uncertan forecasts,.e. forecasts and forecast uncertantes are gven

3 Orderng quantty System ead tme Stoc Fg. 1: The nventory system model Demand for each forecast perod. We also assume that Forecast Uncertantes (F U) are ndependent and dentcally normally dstrbuted over all the perods of the horzon of forecasts wth parameters (0,σ FU ). We also consder that the forecast uncertanty s addtve,.e. at each perod, the probablty dstrbuton of F U s ndependent of the forecast. For more detals on other type of forecast uncertanty models, the reader s referred to [1]. The replenshment lead tme s random. Snce, we consder a dscrete tme control system, we assume that s a random varable wth a dscrete probablty dstrbuton. The lead tme taes a value wth a probablty P (.e. P( = ) = P ). We consder that the suppler s not relable whch mples a yeld uncertanty,.e. f a quantty s ordered to replensh the stoc, the receved quantty r s a functon of the quantty and the yeld uncertanty E, and s expressed as follows: r = +E (.e. addtve uncertanty model). We assume that the yeld uncertanty s also normally dstrbuted wth parameters (m E,σ E ). We also consder these notatons: F : forecast at perod : replenshment lead tme I : nventory poston at the end of perod CS: target cycle servce level A: fxed orderng cost h: holdng cost H: number of perods n the horzon of forecasts CFU R : cumulatve forecast uncertanty over an nterval R Φ CFUR (.): cumulatve dstrbuton functon of CFU R φ CFUR (.): probablty densty functon of CFU R Φ(.): standard normal cumulatve dstrbuton functon P(x): probablty of the random event x 3. THE (r,) POICY UNDER DEMAND, YIED AND EAD TIME UNCERTAINTIES In the (r, ) polcy, the system s controlled at each forecast perod. At the begnnng of each perod, f the nventory poston falls below the reorder pont r, a quantty s ordered. The quantty ordered s receved after perods. The nventory level evoluton n the (r,) polcy s represented n Fg. 2. If the lead tme and the order quantty are constant,.e. only the forecasted demand s uncertan, the reorder pont r s equal to the cumulatve forecasts over + 1 perods plus the safety stoc necessary to cover the forecast uncertanty wth the cycle servce level. The safety stoc Ss s equal to the maxmal cumulatve forecast uncertanty over the protecton nterval whch s equal to +1 perods. The reorder pont s gven by: +1 r = F +j 1 + Ss where: Ss = Φ 1 (CS)σ CFU+1 The quantty can be computed by usng the Wlson s formula, as follows: = 2A H =1 F More detals on ths polcy and ther parameters are gven n [1] and [2]. In the followng, we study the mpact of each uncertanty on the (r,) polcy. The optmal parameters under demand, yeld and lead tme uncertantes are gven n Secton Impact of the yeld uncertanty Here, we consder that there s a yeld uncertanty n the system,.e. when a quantty s ordered, the receved quantty s random. As shown n Fg. 3, a small yeld uncertanty does not have any nfluence on the stocout probablty durng the replenshment lead tme snce there s only a shft n the perod when the order s placed. Thus, the equaton of the safety stoc and the reorder pont are as the same as n the (r,) polcy wthout the yeld uncertanty: +1 r = F +j 1 + Φ 1 (CS)σ CFU+1 However, the ordered quantty changes and may be approxmated by: = 2A H =1 F m E Ths approxmaton s a good one for small values of the varablty of the yeld.

4 Inventory level Net nventory Inventory poston r r j j t Fg. 2: The (r, ) polcy 3.2. Impact of the lead tme uncertanty Snce we use a sequental approach to determne the parameters of the polcy, the optmal ordered quantty s ndependent of the lead tme uncertanty. However, under the lead-tme tme uncertanty, at each perod the equaton of the reorder pont r changes. Proposton. The reorder pont r can be computed numercally by resolvng the equaton: P Φ FD +1(r ) = CS Where FD + 1 denotes the cumulatve forecasted demand over + 1 perods and Φ FD +1(.) ts cumulatve dstrbuton functon. Ths equaton may be solved by usng, for example, an algorthm of dchotomy. Proof. The reorder pont r s computed such as: P(Cumulatve Demand over +1 r ) = CS (1) Snce the lead tme has a dscrete probablty dstrbuton. By usng the Total Probablty Theorem and equaton (1), we have: P P(Cumulatve Demand over + 1 r ) = CS Denote by FD + 1 the cumulatve forecasted demand over + 1 perods and by Φ FD +1(.) ts cumulatve dstrbuton functon. Hence, at each perod : +1 Φ FD +1 = CFU +1 + F +j 1 At each perod, the random varable FD + 1 s then normally dstrbuted wth a mean +1 F +j 1 and a standard devaton σ CFU gven by: +1 σ CFU +1 = σ FU + 1 Thus, the reorder pont r s gven by: P Φ FD +1(r ) = CS Hence, at each perod, the reorder pont r can be computed numercally by solvng ths last equaton Optmal parameters under demand, yeld and lead tme uncertantes The mpact of these uncertantes on the (r,) Polcy are studed separately n Secton 3.1 and 3.2. In ths Secton, we gve the optmal parameters of the (r,) polcy by consderng the demand, yeld and lead tme uncertantes smultaneously. We assume that uncertantes are relatvely small. The result s gven by usng a sequental approach to satsfy a target cycle servce level CS. The reorder pont r can be computed numercally by resolvng the equaton: P Φ FD +1(r ) = CS The optmal quantty to order s gven by:

5 Net nventory wthout yeld uncertanty Net nventory wth yeld uncertanty Inventory level Inventory poston E E r r j j t Fg. 3: The (r, ) polcy under the yeld uncertanty = 2A H =1 F m E 4. THE (r,) POICY UNDER A FI RATE SERVICE EVE CONSTRAINT In Secton 3, a cycle servce level approach s used, under three types of uncertantes, to determne the reorder pont. In ths secton, we are nterested n a fll rate servce level approach, and we consder only a forecast demand uncertanty. Recall that the cycle servce level may be defned as the probablty to not have a stocout durng a cycle (a cycle s defned as the perod between two successve orders). The fll rate s the proporton of demand satsfed drectly by the avalable stoc [8]. et us gve a smple example to explan the dfference between the cycle servce level and the fll rate. We consder the Tab 1. whch shows the ordered quanttes and the stocout over 10 forecast perods. In ths case, the fracton of perods wthout stocout s 8/10, thus a cycle servce level of 80 % s satsfed. In terms of quantty, only 55 demands are satsfed over a total demand of 1450, so a fll rate of % s satsfed (( )/1450=96.21%). Snce we use a sequental approach, the formula of the optmal quantty to order does not change and s gven by: = 2A H =1 F Tab 1. Example to llustrate servce level measures Perod Demand Stocout Total So, our am n ths secton s to compute the reorder pont r necessary to satsfy the target fll rate at each perod. et Fr denotes the target fll rate, n(r ) the average number of stocout durng a cycle, and Ss the safety stoc. Proposton. The optmal safety stoc Ss can be computed numercally by solvng the equaton: (1 Fr) = Ss [ 1 Φ( Ss σ CFU+1 + σ CFU+1 φ( Ss σ CFU+1 The optmal reorder pont r s then gven by: +1 r = F +j 1 + Ss ) )]

6 Inventory level Net nventory Inventory poston r r j r j t Stocout durng a cycle : n(r ) Fg. 4: The (r, ) polcy wth stocout Proof. At each perod, the fll rate s defned as follows: Fr = 1 n(r ) Average demand durng a cycle As shown n Fg 4, the demand durng the cycle whch begns at perod s: + I 1 I j 1 Where j s the frst perod where an order s placed after the perod. In practce, f we consder a long horzon of forecasts, we can approxmate the average forecasted demand durng a cycle by, where: = 2A H =1 F The average number of stocout, denoted by n(r ), s defned as follows: n(r ) = We showed n [1] that: + x=r (x r )φ +1 (x)dx ( Ss n(r ) = Ss [1 Φ σ CFU+1 ( Ss + σ CFU+1 φ σ CFU+1 ) )] Thus, the optmal safety stoc Ss can be computed numercally, by usng for example an algorthm of dchotomy, to solve the followng equaton: (1 Fr) = Ss [ 1 Φ( Ss σ CFU+1 + σ CFU+1 φ( Ss σ CFU+1 Hence, the reorder pont r s gven by: +1 r = F +j 1 + Ss 5. CONCUSIONS ) )] In ths paper, we studed the (r,) polcy under demand, yeld and lead tme uncertantes. A sequental approach s consdered and a cycle servce level s used to compute the optmal parameters of ths polcy. We provded good approxmatons of the optmal parameters of the (r,) polcy for small values of the varablty of uncertantes. We showed that, under the yeld uncertanty, there s only an mpact on the ordered quantty, however, the reorder pont does not change. We also showed that, under the lead tme uncertanty, the reorder pont changes and t can be computed numercally. In the second part of the paper, we brefly explaned the dfference between the cycle servce level and the fll rate. Then, we determned the optmal parameters of the (r, ) polcy usng a sequental approach under a fll rate servce level constrant. In ths part, only the demand uncertanty s consdered. In the future, t would be nterestng to develop ths analyss by studyng the behavor of the optmal parameters for hgh

7 values of the varablty of uncertantes. More wor could also be done to conduct the same study provded n ths paper for other nventory control polces such as the orderup-to-level polces. Another nterestng further research conssts n developng the optmal parameters of the (r,) polcy for a fll rate servce level under demand, yeld and lead tme uncertantes. REFERENCES [1] M.Z. Babaï, Poltques de plotage de flux dans les chaînes logstques: Impact de l utlsaton des prévsons sur la geston de stocs, Ph.D. Thess, Ecole Centrale Pars, [2] Babaï, M.Z., Y. Dallery, An analyss of forecast based reorder pont polces: The beneft of usng forecasts, Proceedngs of the the 12th IFAC Symposum on Informaton Control Problems n Manufacturng: INCOM 06, Sant-Etenne (France),2006. [3] Heng, M., Y. Gercha, The structure of perodc revew polces n the presence of varable yeld, Operatons Research, Vol. 38, pp , [4] S. Karln, Dynamc Inventory Polcy Wth Varyng Stochastc Demands, Management Scence, Vol. 6, pp , [5] Re, Y., E. Sahn, Y. Dallery, A Comprehensve Analyss of the Newsvendor Model wth Unrelable Supply, To appear n OR Spectrum, [6] H. Scarf, The optmalty of (S,s) polces n the Dynamc Inventory Problem, Mathematcal Methods n the Socal Scences, Stanford Unversty Press, Stanford, Calforna, [7] E.A. Slver, Establshng the Reorder uantty When the Amount Receved s Uncertan, INFOR, Vol. 14, pp32-39, [8] Slver, E.A., R. Peterson, Decson Systems for Inventory Management and Producton Plannng, John Wley and Sons, New Yor, [9] Yano, C.A., H.. ee, ot szng wth random yelds: a revew, Operatons Research, Vol. 43, pp , [10] P.H. Zpn, Foundatons of Inventory Management, McGraw-Hll, USA, 2000.