A Two-Echelon Inventory Model with Lost Sales

Transcription

1 DEPARTMENT OF OPERATIONS RESEARCH UNIVERSITY OF AARHUS Publcaton no. 99/4 A Two-Echelon Inventory Model wth Lost Sales Jonas Andersson Phlp Melchors ISSN Department of Mathematcal Scences Telephone: Buldng 530, Ny Munkegade DK-8000 Aarhus C, Denmark

2 A Two-Echelon Inventory Model wth Lost Sales Jonas Andersson Dvson of Producton Management, Department of Industral Engneerng, Unversty oflund, Box 118, SE Lund,Sweden Phlp Melchors æ Department of Operatons Research, Unversty ofaarhus, Ny Munkegade, Buldng 530, DK-8000 Aarhus C, Denmark. September 1999 Abstract Almost all mult-echelon nventory models assume that demand not satsæed mmedately can be backordered. In some stuatons ths assumpton s not realstc. For example, t may be more representatve to model stockouts as lost sales when the retalers are n a compettve market and customers can easly turn to another ærm when purchasng the good. Assumng lost sales at the retalers, we consder a one warehouse several retalers nventory system. Usng the well-known METRIC-approxmaton as a framework, we present a heurstc for ændng cost eæectve base-stock polces. In a numercal study we ænd that the cost of the polces suggested by the heurstc s on average 0.40è above the cost of the ès, 1;Sè-optmal polcy. Keywords : Inventory, Mult-echelon, Lost sales, METRIC 1 Introducton Consder a two-echelon nventory system wth one central warehouse and an arbtrary number of retalers. See Fgure 1. The retalers face customer demand and replensh ther stocks from the central warehouse. The warehouse, n turn, replenshes ts stock from an outsde suppler. Evaluaton and optmzaton of control polces for such nventory systems have attracted massve nterest n the lterature. See, for example, Axsíater ë3ë for an overvew. In the exstng lterature dealng wth mult-echelon nventory control the prevalent assumpton s that complete backloggng of orders s allowed n case of stockouts. For example, Axsíater ë4ë shows how to exactly evaluate the performance for dæerent èr; nqè-polces when the retalers face compound Posson demand and nventores æ Correspondng author. Tel.: ; fax: ; e-mal: phlp@mf.au.dk 1

3 Retalers Warehouse Fgure 1: Mult-echelon nventory system are contnuously revewed. Cachon ë5ë gves an exact method for the perodc revew case wth dentcal retalers. In some stuatons the assumpton of complete backloggng may not be so realstc. For example, t may be more representatve to model stockouts as lost sales when the retalers are n a compettve market and customers can easly turn to another ærm when purchasng the good. For some reason the research dealng wth mult-echelon nventory models has focused manly on the backorder case and the number of models dealng wth lost sales s rather lmted. Anupnd and Bassok ë1ë consder a perodc revew twoechelon nventory system where a part of the unsatsæed sales at the retalers are lost. Snce the transportaton tme between the manufacturer and the retalers s zero, the optmal order polcy at each retaler s a base-stock polcy. The manufacturer carres lnear producton cost and no holdng cost. The retalers can agree to centralze ther stocks and the problem consdered s whether or not ths wll lead to an ncrease n total expected sales at the manufacturer. Nahmas and Smth ë8ë also consder lost sales n a mult-echelon envronment n a paper more closely related to ths paper. However, ther model dæers from ours n several mportant aspects. Frst, they consder perodc revew batch order polces. The model s more general snce they deal wth partal lost sales. Ths means that, wth probablty u, demand not satsæed mmedately, s lost, and wth probablty 1, u, t s satsæed later by a specal order. Moreover, for the model to be tractable they assume nstantaneous delveres from the warehouse to the retalers. For sngle-echelon nventory models the lost sales assumpton s more common. The exact cost for a sngle level nventory system facng Posson demand and æxed leadtmes was ærst gven by Hadley and Whtn ë6ë. Smth ë11ë demonstrates how to evaluate and ænd optmal ès, 1;Sè-polces for an nventory system wth zero replenshment costs and generally dstrbuted stochastc leadtmes. Recently Hll ë7ë showed that for the lost sales case the ès, 1;Sè-polcy s not necessarly optmal. 2

4 In ths paper we analyze a model for a one warehouse, multple retaler nventory system. Demand occurs only at the retalers and follows ndependent Posson processes. All leadtmes are assumed to be constant. All nstallatons use ès, 1;Sè-polces wth contnuous revew. It s assumed that backloggng of customer demand s not allowed. The analyss departs n one of the most wdely known mult-echelon nventory models, the METRIC-model developed by Sherbrooke ë10ë. In ts orgnal settng, t s assumed that stockouts at the retalers are completely backlogged. We demonstrate how the METRICmodel can be modæed to handle the lost sales case. Our approach gves an approxmate model whch s qute smple and eæcent from a computatonal pont of vew. Smulaton experments ndcate that the performance s very good. The outlne of ths paper s as follows: In Secton 2 we gve a detaled problem formulaton and pose all assumptons. Secton 3 gves the soluton procedure. The numercal results are gven n Secton 4, and n Secton 5 we gve some conclusons and pont out some possble drectons for future research. 2 Problem Formulaton The nventory system under consderaton conssts of one central warehouse and an arbtrary number of retalers. The retalers face Posson customer demand. No backloggng s allowed at the retalers. Consequently, the customers that arrve to a retaler that s out of stock wll become lost sales for the retaler. When stockouts occur at the warehouse, all demands from the retalers are fully backlogged and the backorders are ælled accordng to a FIFO-polcy. The transportaton tme between the warehouse and a gven retaler s assumed to be constant as well as the transportaton tme from the external suppler to the warehouse. The cost of a replenshment s assumed to be zero or neglgble compared to the holdng and stockout costs. The external suppler s assumed to have nænte capacty, whch means that the replenshment leadtme for the central warehouse s constant. All nstallatons use ès, 1;Sè-polces wth contnuous revew. Unts held n stock both at the warehouse and at the retalers ncur holdng costs per unt and tme unt. Moreover, a æxed penalty cost per lost customer s ncurred at the retalers. In ths paper we present a model for the consdered nventory system, whch can be used to evaluate the long-run average cost for dæerent polces wthn the class of ès, 1;Sè-polces. The objectve s to ænd the polcy that mnmzes the long-run average cost for the nventory system. Let us ntroduce the followng notaton: N = the number of retalers, ç = demand ntensty at retaler, =1; 2;:::;N, L = transportaton tme for the delveres from the warehouse to retaler, =1; 2;:::;N, L 0 =transportaton tme for the delveres from the external suppler to the warehouse, 3

5 S 0 = order-up-to level at the warehouse, S = order-up-to level at retaler, =1; 2;:::;N, h 0 = holdng cost rate at the warehouse, h = holdng cost rate at retaler, =1; 2;:::;N, ç = penalty cost for a lost sale at retaler, =1; 2;:::;N. We want to determne the total cost for the nventory system n steady state. Deæne TC = total cost for the nventory system per tme unt n steady state, C 0 = cost per tme unt for the warehouse n steady state, C = cost per tme unt for retaler n steady state, =1; 2;:::;N. Obvously, TC = C 0 + Our objectve s to determne a control polcy, S 0 ;S 1 ; ::; S N that mnmzes the total cost, TC. NX =1 C : è1è 3 Soluton Procedure In ths secton we ærst demonstrate how the total cost for dæerent control polces can be evaluated. For the backorder case the exact cost of the system can be derved by observng that any unt ordered by retaler s used to fulæll the S th demand. The cost can then be derved by condtonng on the arrval tme of the S th demand èwhch s Erlang dstrbutedè and the arrval of the ordered unt èsee Axsíater ë3ëè. In a lost sales envronment the correspondng observaton s that any unt ordered by retaler s used to fulæll the S + X th demand, where X s a random varable denotng the number of lost sales ncurred at the retaler durng the replenshment lead tme. X s obvously very hard to characterze and we have therefore chosen to focus on a heurstc rather than on the exact soluton. The analyss has many smlartes wth the analyss n Sherbrooke ë10ë. However, our assumpton of lost sales at the retalers destroys some of the nce propertes vald for the backorder model. The analyss of the warehouse, e.g., becomes more complex for the lost sales case. In the backorder case, all customers arrvng at the retalers generate demands at the warehouse mmedately at the arrval epoch, snce all retalers use contnuous revew ès, 1;Sè-polces. Consequently, the warehouse faces a Posson process wth ntensty ç 0 = ç 1 + ç 2 + æææ+ ç N. For the lost sales case ths s not true. When backorderng s not allowed, customer demands can be lost due to stockouts at the retalers. Therefore the demand at the warehouse s not Posson process anymore. 4

6 Another mportant dæerence compared wth the backorder case s that the order-upto level S at retaler, aæects the costs at all retalers and at the warehouse. In the backorder case S only aæects the local cost at retaler, snce the warehouse demand process s unaæected by the order-up-to levels at the retalers. For the lost sales case the order-up-to level aæects the number of lost sales and consequently, the demand process at the warehouse s not ndependent of the polces at the retalers. Therefore the order-up-to level at a certan retaler aæects the costs at all nstallatons n the nventory system. We wll ærst show how to evaluate the costs at the retalers gven a certan replenshment leadtme provded by the warehouse. We then show how to calculate the cost at the warehouse gven the demand ntensty from the retalers. Fnally we ntroduce an teratve procedure from whch we obtan the total cost for the nventory system. 3.1 Approxmate retaler cost As Sherbrooke ë10ë we use a queueng system analogy when evaluatng the costs for the retalers. For a retaler where backloggng s allowed, the number of outstandng orders towards the central warehouse s the same as the occupancy level n an M=G=1 queue. Recall that the customer demand s Posson and the replenshment leadtmes are stochastc, snce orders can be delayed due to stockouts at the central warehouse. For ths type of queue a famous theorem by Palm ë9ë states that the steady state occupancy level s Posson dstrbuted wth mean çl, where ç s the arrval rate and L s the mean servce tme. Palm's theorem holds for..d. servce tmes. The stochastc leadtmes n our case are evdently not ndependent, but f we dsregard ths correlaton we can approxmate the number of outstandng orders wth a Posson dstrbuton. Ths s the dea behnd the METRIC-approxmaton. When demand s lost, the queueng system of nterest s an M=G=S=S queue, wth S servers, each wth generally dstrbuted servce tmes and no queueng allowed. If the servce tmes are ndependent random varables wth mean ç L, Erlang's loss formula states the steady-state dstrbuton for the occupancy level as q S èjè= èçlè ç j =j! P S n=0 èç Lèn ç =n! for 0 ç j ç S where q S èjè = the probablty that j servers èout of Sè are occuped n steady state. Followng METRIC we approxmate the number of outstandng orders wth ths dstrbuton. Suppose that the mean leadtme for retaler s L ç and let q S èjè be the steady state probablty of j outstandng orders gven a desred base-stock level S. The expected number of lost sales per tme unt s clearly ç q S ès è and the expected number of unts 5

7 n stock s XS j=0 ès, jèq S èjè=s, ë1, q S ès èëç ç L : è2è The total relevant cost for retaler s therefore C ès ; L ç è=ç ç q S ès è+h çs, ë1, q S ès èëç L ç ç and the rate of demand from retaler whch s not lost s è1, q S ès èèç. The dervaton of the exact cost of a ès, 1;Sè lost sales sngle stage nventory system wth generally dstrbuted leadtmes was ærst presented by Smth ë11ë. He also proves that C ès ; ç L èsconvex n S for æxed ç L, whch means that the optmal value can be found by a local search routne. 3.2 Approxmate warehouse cost In the backorder case the demand process at the warehouse s a Posson process. In the lost sales case ths s not the case. If, for example, the base-stock level at a retaler s one, the smallest nterval between two successve demands from that retaler wll be the retaler leadtme. We wll gnore ths and approxmate the demand process at the warehouse wth aposson process wth mean æ. æ depends on how much demand s lost at the retalers and s determned as æ= NX =1 ç è1, q S ès èè è3è Snce we have a æxed determnstc leadtme L 0,we can ænd the average holdng cost ncurred at the warehouse as a functon of æ and S 0. X S 0 C 0 ès 0 ; æè = h 0 ès 0, jè èæl 0è j expè,æl 0 è j! j=0 We can also derve the mean delay due to stockouts at the warehouse by ærst calculatng B 0, the average number of backorders at the warehouse. B 0 = 1X j=s 0 +1 èj, S 0 è èæl 0è j expè,æl 0 è; j! We then apply Lttle's formula to obtan the average delvery delay, B 0 =æ. The mean leadtme for retaler s then è4è çl = L + B 0 =æ è5è Fnally, we obtan the total cost from è1è. 6

8 3.3 Overall soluton procedure We can now establsh the soluton procedure. The procedure enumerates over S 0. It can be shown that for a cost mnmzng soluton, S 0 can not be negatve. See, for example, Axsíater ë2ë. Consequently our procedure starts wth S 0 =0. Moreover, S 0 s bounded from above by an aborton crtera. We need the followng new notaton: C mn = mn S C ès ;L è = mnmum cost per tme unt for retaler n steady state when the leadtme, ç L, s equal to the transportaton tme, L, =1; 2;:::;N. S èkè = order-up-to level at retaler n teraton k. TC æ ès 0 è = mnmum value of TC gven a æxed value of S 0. Let us ærst consder two smple lemmas. The proofs can be found n the Appendx. The ærst lemma gves a lower bound for the retaler costs, and the second establshes two mportant propertes for the warehouse cost. Lemma 1. C mn s a lower bound for the retaler cost, C ès ; ç L è for all S and any çl él. Lemma 2. C 0 ès 0 ;ç 0 è ç C 0 ès 0 ; æè, for all S 0 and all æ ç ç 0. Moreover, C 0 ès 0 ;ç 0 è s convex n S 0. To construct an aborton crtera for the procedure, consder the cost functon TC lb ès 0 è=c 0 ès 0 ;ç 0 è+ NX =0 C mn : By Lemma 1 and Lemma 2, TC lb ès 0 èsalower bound for the cost functon, TC æ ès 0 è. Moreover, snce the cost functon TC lb ès 0 è s convex n S 0 the search over S 0 can be aborted when S 0 satsæes mn xçs 0 TC æ èxè ç TC lb ès 0 è: The aborton crtera s llustrated n Fgure 2. 7

9 TC æ ès 0 è TC lb ès 0 è mn xçs0 TC æ èxè S max 0 S 0 Fgure 2: Illustraton of the aborton crtera. The search for the optmal S 0 s aborted at S max 0. The soluton procedure can now be establshed as: STEP 0: Set S 0 =0andTC mn = 1. STEP 1: Set k = 0 and æ = ç 0. STEP 2: For each =1; 2;:::;N calculate ç L by è4è and è5è Let z æ = mn S C ès; ç L è and set S èkè =mnfsjc ès; ç L è=z æ g STEP 3: If ké0ands èkè =S èk, 1è for all =1; 2;:::;N then goto STEP 4, STEP 4: else calculate æ by è3è, set k P := k + 1 and goto STEP 2. Set TC æ n ès 0 è=c 0 ès 0 ; æè + =1 C ès èkè; L ç è. If TC æ ès 0 è étc mn then set TC mn = TC æ ès 0 è and let S opt 0 = S 0 and S opt = S èkè for =1; 2;:::;N. If TC mn étc lb ès 0 è then STOP, else set S 0 = S and goto STEP 1. 4 Numercal Results In order to examne the eæectveness of the presented methodology we have performed a small numercal study. In total we consder 36 dæerent test problems wth æve dentcal retalers. For each test problem we ænd the best order-up-to levels accordng to our method. We also obtan the approxmate total holdng and stock out costs for the nventory system. The accuracy of these results are then evaluated by smulaton. Each smulaton conssts of 10 runs, each wth a run length of tme unts. The result s a conædence nterval for the exact cost. We express the conædence lmts on a 95è sgnæcance level. A comparson between the total cost gven by our method and the total cost for the smulaton gves an ndcaton of how accurate our method s when estmatng the total cost for the nventory system. We also use smulaton to determne the optmal polcy for the system. The cost for ths polcy can then be compared wth the cost for the polcy determned by our technque, 8

10 to obtan an estmate for the performance of the method when optmzng the orderng polces. The polcy that we report as the optmal polcy s the polcy wth the lowest average cost. However, ths polcy does not necessarly domnate all the other polces when takng conædence ntervals nto consderaton. Moreover, we only search wthn polces where the order-up-to levels are dentcal for the retalers. The problem data and results can be found n Table 1. We only report the optmal polcy when t s dæerent from the one obtaned from our algorthm. From Table 1 we can see that our method performs rather well for all the consdered problems. It seems that we mostly tend to underestmate the total cost, especally n the problems wth hgh stockout costs at the retalers. Ths s due to the METRIC-approxmaton, where the stochastc leadtmes are replaced by ther averages when evaluatng the costs for the retalers. On average the method underestmates the costs wth 1.1 è. In 13 problems we can observe èon a sgnæcance level of 95èè that the method fals to ænd the optmal polcy. In 9 more problems the polcy suggested by our method does not have the lowest average cost accordng to the smulaton runs. However, n these cases the devatons are not sgnæcant on a 95è conædence level. In comparson to the optmal polces obtaned by smulaton, the ncrease n costs by usng the polces obtaned by our method s only 0.40 è, on average. In 16 of the 22 problems where we fal to ænd the true optmal polcy, the method merely underestmates the order-up-to level at the warehouse by a sngle unt. In one problem the warehouse order-up-to level s underestmated by two unts. In the other 5 problems where the optmal polcy s not found, the method tends to allocate more stock to the retalers and less stock to the warehouse than what s optmal from a cost perspectve. Fnally t seems that our methodology performs better f the warehouse leadtme s small compared to the transportaton tme from the warehouse to the retalers. In the 12 problem nstances wth L =1:5 the average cost ncrease, SCèCC s only 0.07è, whereas n the problems wth L =0:5, the correspondng ægure s consderable hgher, 0.67è. Ths behavor s due to the METRIC-approxmaton, where the stochastc replenshment leadtme facng a retaler s replaced by ts mean value. If the constant transportaton tme to the retalers s large compared to the warehouse leadtme, the stochastc delvery delays tend to have less relatve varaton and consequently the mpact of the METRICapproxmaton wll be smaller. 5 Conclusons and drectons for future research Ths paper presents a heurstc method for evaluaton and optmzaton of ès,1;sè-polces for a one warehouse, multple retalers nventory system. The evaluaton technque uses the well-known METRIC-approxmaton as a framework. From a computatonal pont of 9

11 vew the presented technque s very eæcent and smple. Numercal results also ndcate that the performance s qute good. Up to our knowledge, no paper s yet publshed, whch deals wth lost sales n a contnuous revew mult-echelon nventory settng. Moreover, the orgnal backorder METRICmodel ë10ë s one of the most wdely used mult-echelon nventory models. Our lost sales generalzaton makes the polcy evaluaton a bt more complex, snce we have to use an teratve procedure to obtan the cost. Stll, the model s rather smple and easy to mplement. Moreover, n many practcal stuatons lost sales s a reasonable way to model stockouts. Therefore our technque s also relevant for practtoners. In a research perspectve our model can form a framework n whch dæerent generalzatons can be consdered as optons for future research. For example, batch orderng polces and more general demand processes may be analyzed, stll usng the deas presented n ths paper. Generalzatons to perodc revew polces s also mportant. The dervaton of an exact evaluaton of costs seems to be a very dæcult problem to solve. Ths s a real challenge for future research. Appendx Proof for Lemma 1 We need to show that mn S C ès ;L è ç mn C ès ; L ç èforl ç L ç : è6è S Let l be an arbtrarly chosen leadtme, where L ç l ç ç L. Consder the cost C ès ;l è, where S s set to ts optmal value for each l. Obvously, C ès ;l è ç C ès, 1;l è for each l such that L ç l ç ç L. Start wth l = ç L and let l be contnuously lowered untl we reach l =L, whle S s set to ts optmal value for each l. Snce C ès ;l ès acontnuous functon of l for æxed S, t also s a contnuous functon of l when S s optmally chosen. Moreover, the fact that S mnmzes the cost C ès ;l è, mples that C ès ;l è ç C ès, 1;l è. Consequently, è6è follows f C ès ;l è ç C ès, 1;l è ès ;l ç 0 for L ç l ç ç L : è7è For notatonal reasons we omt the ndex from all varables. It can be shown =,hçè1, q S èsèè + çèhçl + ççèèq S ès, 1è, q S èsè+q S èsè 2 è: è8è Moreover, CèS; lè ç CèS, 1;lè mples that h hçl + çç ç qs,1 ès, 1è, q S èsè è9è 10

12 Let From è8è we have that A = è9è and è11è now gve h hçl + çç A = 1 çèhçl + ès; lè : q S èsè, 1 æ + q S ès, 1è, q S èsè+q S èsè 2 è: è11è A ç èq S,1 ès, 1è, q S èsèèèq S èsè, 1è + q S ès, 1è, q S èsè+, q S èsè æ 2 = q S,1 ès, 1èq S èsè, q S,1 ès, 1è + q S ès, 1è = q S,1 ès, 1èq S èsè, èq S,1 ès, 1èq S èsèè = 0: Note that q S èsè ç 1. complete. Consequently, A ç 0 and therefore è7è holds and the proof s Proof for Lemma 2 Snce æ ç ç 0,we only need to show 0èS 0 ;æè ç 0. The convexty 0 ès 0 ;ç 0 èns 0 follows, for example, from Axsíater 0 ès 0 ; 0 =,h 0 L 0 expè,æl 0 è@ S0 + X S 0,1 =,h 0 L 0 expè,æl 0 è ç 0: j=0 S X 0 j=1 èæl 0 è j j! ès 0, jè æ èèæl 0 è j j!, èæl 0è èj,1è èj, 1è!!1 A References ë1ë R. Anupnd and Y. Bassok. Centralzaton of stocks: Retalers vs. manufacturer. Management Scence, 45:178í191, ë2ë S. Axsíater. Smple soluton procedures for a class of twoíechelon nventory problems. Operatons Research, 38:64í69, ë3ë S. Axsíater. Contnuous revew polces for multílevel nventory systems wth stochastc demand. In S.C. Graves, A.H.G. Rnnooy Kan, and P. Zpkn, edtors, Handbooks n OR & MS, Vol. 4, pages 175í197. Elsever Scence Publshers B.V., NorthíHolland,

13 ë4ë S Axsíater. Exact analyss of contnuous revew èr; Qèípolces n twoíechelon nventory systems wth compound Posson demand. Lund Unversty, Sweden, ë5ë G.P. Cachon. Exact evaluaton of batch-orderng nventory polces n twoíechelon supply chans wth perodc revew. Fuqua School of Busness, Duke Unversty, ë6ë G. Hadley and T.M. Whtn. Analyss of Inventory Systems. PrentceíHall, Englewood Clæs, NJ, ë7ë R.M. Hll. On the suboptmalty of ès,1;sè lost sales nventory polces. Internatonal Journal of Producton Economcs, 59è1-3è:377í385, ë8ë S. Nahmas and S.A. Smth. Optmzng nventory levels n a two-echelon retaler system wth partal lost sales. Management Scence, 40:582í596, ë9ë C. Palm. Analyss of the Erlang traæc formula for busy sgnal assgnment. Ercson Techncs, 5:39í58, ë10ë C.C. Sherbrooke. METRIC: A mult-echelon technque for recoverable tem control. Operatons Research, 16:122í141, ë11ë S.A. Smth. Optmal nventores for an ès, 1;Sè system wth no backorders. Management Scence, 23:522í528,

14 ç ç L S Calc cost Sm cost Spread Opt pol Cost Spread Cost dev , , , , , , è , , è , , è , , è , , , è , , è , , è , , è , , è , , è , , è , , , , , , , è , , è , , , è , , è , , , è , , è , , , è , , è , , , è , , è , è Table 1: Numercal results. The optmal polcy s only reported when t s dæerent than the polcy suggested by our algorthm. 13