The impact of inventory sharing on the bullwhip effect in decentralized inventory systems

Transcription

1 Logist. Res. (03) 6:89 98 DOI 0.007/s ORIGINAL PAPER The imact of inventory sharing on the bullwhi effect in decentralized inventory systems Dang Van Le Luong Trung Huynh Kifor Vasile Claudiu Muntean Achim Received: Aril 0 / Acceted: November 0 / Published online: 9 December 0 Sringer-Verlag Berlin Heidelberg 0 Abstract The aer derives the imact of inventory sharing olicy on the bullwhi effect in two-stage suly chains with two indeendent suliers and two integrated retailers. There exists an inventory sharing olicy between two retailers. Under inventory sharing olicy, when demand in one retailer exceeds its inventory, this retailer can ask for a roduct sharing volume from the other in order to satisfy customer demand. With certain assumtions, the bullwhi effect is quantified in both cases, with inventory sharing olicy and without inventory sharing olicy. We found that inventory sharing has significant imact on the bullwhi effect in the suly system. However, inventory sharing olicy does not synchronously reduce or increase the bullwhi effect in both suliers in the same eriod. A numerical examle is given to illustrate the study model. Keywords Inventory sharing olicy Bullwhi effect Suly chain management Decentralized inventory Order lead time Introduction D. V. Le (&) L. T. Huynh School of Engineering and Technology, Asian Institute of Technology, Bangkok, Thailand haidang_math804@yahoo.com K. V. Claudiu M. Achim Faculty of Engineering and Technology, Lucian Blaga University of Sibiu, Sibiu, Romania The information about customer demand is varied through the levels of the chain due to many factors (e.g., inventory olicy, forecasting method, order lead time, etc.). In twostage suly chain, retailers are the arties who receive customers demands directly. To satisfy customers service, usually customer demand is estimated by using forecasting techniques before lacing order to sulier. The lacking of information leads to fluctuation orders from all levels of the chain in term of volumes. The fluctuation of customer demand through the chain is well known as the bullwhi effect [3, 9]. In Fig., we can see the fluctuation of customers demands through different facility eochs in a four-stage suly chain with single manufacturer, distributor, wholesaler, and retailer. We may notice that demands fluctuation increases from lower levels to higher levels in the chain. The reasons can be exlained as follows: the retailer has directly customer information. Retailer will use this information to estimate actual demands. To maintain desired service level, retailer needs to hold a certain inventory in the warehouse. That leads to the wholesaler will receive higher original orders from the retailer. Similarly, wholesaler receives customer information from the retailer and laces an order to his sulier, the distributor. To determine the order quantities from retailer, the wholesaler must forecast customer demand. Unfortunately, the wholesaler does not have access to the customer actual information; so that they must use the information from the retailer to erform his forecasting. Therefore, the variation of customer demand increases from lower eoch to higher eoch in the chain. In the suly chain system alying inventory sharing olicy, distribution centers, wholesalers, and retailers are collaborated by sharing roduct in case of emergency such that stock out or demand exceeds inventory. Searated inventory of arties in the same levels are virtually combined. If one arty is in stock out stage, its demands can be fulfilled by available inventory in the other retailers. 3

2 90 Logist. Res. (03) 6:89 98 Literature review Fig. Variation of customer demand in a suly chain system This cooeration esecially rofits integrated suly chain system (the chain in which manufacturers and retailers are deendent). For a suly chain with indeendent facilities, inventory sharing olicy sounds like a game changer and this olicy would be more efficient in the centralized information system. Inventory sharing can be rofited for whole suly chain system including: manufacturers, distributors, retailers, and customers. Manufacturers can imrove brand reutation, increase manufacturing efficiency, and reduce unwanted inventory. For distributors/retailers, inventory sharing reduces the number of lost orders and backorders, rovide a new outlet for slow moving inventory, and increase ossibility for incremental revenue (reduce inventory while remaining service level). Finally, customers also rofit from inventory sharing in which it is an efficient strategy that imroves roduct availability and reduces delivering time. However, inventory sharing olicy does not always guarantee making benefit for retailers. In case of transhiment, reallocation, and enalty costs are too high, retailers will try to reduce the robability of stock out by increasing inventory level. In addition, if the ordered lead time and order duration eriod time from suliers to retailers is short, retailers may refer waiting for new delivering ackages instead of receiving sharing ackages from other retailers. Therefore, the efficiency of inventory sharing strategy deends on geograhy factors, features of system, roducts, and services. The studies about bullwhi effect have been utilized for several decades. The earliest aer studied on this area is conducted by Forrester [8]. In this aer, the authors the first time ointed out the effects of information variation on roduction decision. This concet has been noticed and become foundation for research in the field of demand variation as we call the bullwhi effect. Main objectives of revious aers mainly focus on three asects: (a) demonstrating the existence of the bullwhi effect, (b) identifying ossible causes of the bullwhi effect, and (c) develoing strategy to reduce the imact of the bullwhi effect [3]. Following the stream, our aer can be classified into the second areas, that is, identifying ossible causes of the bullwhi effect. This researched area has been studying by many researchers. For instance, Lee et al. [0] ointed out in their aers four major causes of the bullwhi effect including: demand forecast udating, order batching, rice fluctuation, rationing and shortage gambling. They also resented several methods to counteract the bullwhi effect such as integrating new information systems, defining new organizational relationshis, and imlementing new incentive and measurement systems. The effect of forecasting methods on the bullwhi effect is also the main toic in many studies. Chen et al. [3] studied the imact of different forecasting methods on the bullwhi effect in a two-stage suly chain. They concluded that exonential smoothing forecasting method gives higher bullwhi effect than moving average forecasting method. In the same sense, Graves [5], Xu et al. [9] and Zhang [30] resented the effect of demand forecasts on the bullwhi effect in a two-stage suly chain system with integrated moving average demand rocess. Furthermore, Sun and Ren [3], Zhang [30] studied the imact of different forecasting methods such as MA, ES, EWMA, MMSE, and suggested the forecast method which can mitigate the bullwhi effect. The imact of order lead time was resented in the aers of Graves [5], Chen et al. [9], Zhang [30], Lee et al. [0]. Luong and Phien [] roved that the bullwhi effect can be decreased by reducing lead time. However, Duc et al. [8] showed that reducing lead time does not always reducing the bullwhi effect. He showed that in some secial cases; for examle, in a two-stage suly chain with a re-secified ARMA demand rocess, increasing the lead time may hel to reduce the bullwhi effect. For more detail, we refer to [8]. When considering the imact of lead time on the bullwhi effect, the lead time can be deterministic or stochastic. Results in those aers mentioned above are in case of deterministic lead 3

3 Logist. Res. (03) 6: time, whereas in ractical lead time usually behaves as a stochastic rocess. Chaffied [4] and So and Zheng [8] used simulation aroaches to demonstrate the imact of lead time variation and information sharing on the customer demand fluctuation in a suly chain. Other results in this area are contented in [5, 6, 8]. Li et al. [3] analyzed the imact of demand substitution on the bullwhi effect in a two-stage suly chain with a singer sulier, singer retailer, and two tyes of roducts A and B such that a certain fraction roduct A can be used to substitute roduct B. They showed the relation between the bullwhi effect and the forecasting method, lead time, demand rocess, and the roduct substitution. The imact of demand substitution has been noticed and investigated by some revious authors [,, 5,,, 4]. Inventory sharing has become imortant ercetion in some suly chain models. For examle, in the suly chain with singer sulier and multi-indeendent retailers, the delivering time from suliers to retailers warehouses take a long time because of the long distance between suliers and retailers, whereas retailers may located very closed in one area. There is robability that inventory of one retailer exceeds customer demand, while others are in stock out state. In this situation, stock out can be satisfied by transferring roducts among retailers through inventory sharing olicy. The concet of third-arty warehouse is created aiming at identifying inventory olicy and customer information. It may hel to decrease total inventory at retailers. In this area, Duc et al. [9] studied the effect of third-arty warehouse on the bullwhi effect. They found that thirdarty warehouse does not always reducing the bullwhi effect. They also stated the conditions in which the utilization of third-arty warehouse decreases the bullwhi effect in suly chain. Rudi [3] studied on the relation among following factors: inventory sharing, transhiment cost, and inventory orders in the suly chain with one sulier and two local retailers. They ointed out that inventory sharing and transhiment costs have significant effects on inventory order at each retailer. A case of study was conducted in which a two-stage suly chain with singer sulier, Bosch based in Germany, and five retailers based in Norway. In this suly chain system, the ordered delivering time from Germany to Norway takes about 3 weeks, while transhiment time within Norway is insignificant. This suly system is more similar with the suly chain model in our aer. Zhao [4] resented an otimal inventory olicy for each dealer in decentralized dealer networks. They concluded that () inventory sharing has big imact on the level of inventory by the indeendent dealer, () increasing inventory sharing leads to decreased dealers rationing level rather than increasing their based stock level, (3) a smaller level of incentive for inventory sharing may be sufficient to achieve the benefit of full inventory sharing olicy, (4) the benefit of inventory sharing increase the system utilization, and (5) customer service may imrove significantly with inventory sharing. Recently, Kutanoglu [7] considered a model to allocated stock level in the warehouse in a service art logistic network. The network includes one sulier with the infinitive warehouse caacity and a number of local warehouses. Each local warehouse has indeendent based stock olicy. Moreover, local warehouses share their inventory as a way to increase service levels. They concluded that inventory sharing can reduce based stock levels and total system cost. As mention above, the rincile of the bullwhi effect is the variation of customer demand through the chain. Meanwhile, inventory sharing has significant imact on inventory levels and order quantities as well. That results in the change in customer information. Previous researches mainly focus on various areas of the bullwhi effect and inventory sharing. However, there are no works directly focusing on the imact of inventory sharing on the bullwhi effect yet. In this aer, we will derive this issue and examine the imact of inventory sharing olicy on the bullwhi effect. The remaining of the aer is organized as follows: Sect. 3 describes the roblem and develos mathematical formulation; Sect. 4 gives a numerical examle, results, and discussion. Conclusions and recommendations for further study is the content of the last section. 3 Model develoment 3. Notations and assumtions The following notations are used in this aer t order eriod number index; i retailer and corresonding sulier index, i =, ; k ercentage of roduct that one retailer may share to other retailers in the eriod t; L i the order lead time for retailer i, i =, ; the number of demand observation eriods used in the moving average forecast; l i average demand of roduct at retailer i in the autoregressive demand model; e t,i forecast error for roduct at retailer i during time eriod t; q i the autocorrelation coefficient of the autoregressive model of roduct at retailer i; jq j\; jq j\ D t,i roduct demand at retailer i during time eriod t; 3

4 9 Logist. Res. (03) 6:89 98 D L i t;i bd L i t;i lead time demand for roduct at retailer i; the forecast of the lead time demand roduct at retailer i; z i normal z-score determined by the desire service level; r L i standard deviation of forecast error of lead time t;i demand for roduct at retailer i; y t,i order-u-to level inventory for retailer i at the beginning of time eriod t; q t,i order quantity roduct for retailer i at the beginning of time eriod t; B i the bullwhi effect at sulier i; C L,q a constant function of Land q The roosal model is studying under following assumtions As As As3 As4 As5 Order lead time of each retailer is smaller than duration time of one eriod ðl i \rðt; i ; : Time eriod index t and duration of order eriod are equal for both retailers In one order eriod, only one retailer agree share inventory, while the other will receive this amount of inventory Inventory sharing ercentage is varying from eriod to eriod In every eriod, order from each retailer is ositive Fig. Relationshi among arties in the chain 3. Problem descrition In this aer, we consider a two-stage suly chain with two suliers and two retailers with searate markets (each retailer has their own customer). A single roduct is delivered from suliers to retailers following discrete time eriod. The relationshi among the arties in the chain is reresented in Fig.. That is, whenever one retailer is in stock out state, they can ask for transhiment from the other retailer. Whether transhiment is made deending on redetermine conditions stated in the sharing olicy. The rocess of satisfying customer demand with inventory sharing olicy is illustrated in Fig. 3. We assume that order lead times for both retailers (time from lacing an order until receiving the order) are deterministic. We suort that both retailers use the same forecasting method (MA) and inventory olicy (order-u-to level). Since inventory sharing olicy could affect order quantities that a retailer lacing on its sulier, we will consider the imact of inventory sharing on the bullwhi effect of each retailer. The bullwhi effect can be determined by identifying the ratio of variance of retailer s order to the sulier to the variance of customer s demand to the retailers varðq i varðd i, in which D i is customer demand Fig. 3 Progress of satisfying customer demand forecast, q i is order quantity from retailer i laced at sulier i, i =,. 3.3 Mathematical formulation Suort at the beginning of order eriod t, both retailer () and retailer () estimate customer demand and lace an 3

5 Logist. Res. (03) 6: order to its sulier. The order cost is negligible. From Zhang [30] and Gilbert [0], end customer demand can be modeled by an autoregressive (AR). That means end customer demand at each retailer during time eriod t can be determined as follows: D t; l þ q D t; þ e t; ; D t; l þ q D t; þ e t; : From (), we have (See Aendix for details) EðD t; l ; varðd t; r q q ; EðD t; l ; varðd t; r q q ð ð At the beginning of time eriod t, the actual inventory level at retailer (i)isy t;i D t;i. Retailer (i) will order amount of roduct in order to reach the target inventory level y t,i. Taking inventory sharing into consideration, the quantity for the demand at each retailer at the beginning of time eriod t can be determined as q t; y t; ðy t; D t; þkq t; y t; y t; þ D t; þ kq t; ; or q t; k ½y t; y t; þ D t; ; ð3 and q t; y t; ðy t; D t; kq t; y t; y t; þ D t; kq t; : ð4 The target inventory y t,i at the beginning of eriod t is estimated from observed demand as y t; bd L t; þ z br L t; ; y t; bd L t; þ z br L t; : ð5 Both retailers use simle moving average technique to estimate D Li t;i and r L i t;i based on the information of the ast eriods. We have P D L i t;i L j D tj;i i ; ð6 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P r L i t;i C j ðd tj;i bd tj;i L i ;q : ð7 where D tj;i bd tj;i is the forecast error of the (t - j)th eriod at retailer (i) and C Li ;q is a constant function of L i and q [6].The bullwhi effect can be calculated as B i varðq t;i varðd i. 3.4 Bullwhi effect quantify Since the variance of q t,i and D i is different in each retailer, the bullwhi effect at each sulier will be discussed searately. Given the equations of the order-u-to level, demand forecasting, and standard deviation of forecast error, q t, can be exressed as follows: q t; y t; y t; þ D t; þ kq t; k ½y t; y t; þ D t; k ½ð bd L t; þ z br L t; ð bd L t; þ z br L k þ L D t; L t; þd t; D t; þ z br L t; brl t; : ð8 Then, the variance of the order quantity q t, at sulier () at time eriod t is as follows: varðq t; var þ L D t; L D t; k i þz ðbr L t; brl t; varðd ð k þ L þ L ð q þ z ð k varðbrl t; brl t; : ð9 For the variance of the order quantity q t, at sulier () at time eriod t, we have q t; y t; y t; þ D t; kq t; ð^d L t; þ z ^r L L t;ð^d t; þ z ^r L t; þd t; kq t; þ L D t; L D t; þ z ð^r L t; ^rl t; kq t;: Then, the variance of the order quantity q t, time eriod t is as follows: varðq t; var þ L k varðq t; þ L D t; L D t; þ z þ L L þ L þ z var brl t; brl t; k varðq t; : :q :varðd br L t; brl t; ð0 We denote B, B are the bullwhi effect of sulier () and sulier () in eriod t, resectively. We have B varðq t; varðd ð k þ L þ L ð q þ z ð k varðbrl t; brl t; : ð 3

6 94 Logist. Res. (03) 6:89 98 B varðq t; varðd þ L þ L ð q þ z varðbrl t; brl t; k ð k þ L þ L ð q varðd varðd z k L varðbr t; brl t; ð k : ð varðd 4 Numerical examle and analysis From () and (), we can see that B and B are functions of the following arameters: - the number of observation use in MA; L, L -the order lead time; q, q -first order autocorrelation coefficients of the autoregressive demand rocess at retailer () and retailer (), resectively; k-inventory sharing ercentage. In this section, we will rovide a numerical examle to illustrate the imact of those arameters on the bullwhi effect on both retailers. 4. System descrition Two trading comanies A and B has center in North and South Vietnam, resectively. Comany A imorts steel from manufacturer () which has center in Italy while comany B imorts steel from manufacturer () which has center in Jaan. Estimated order lead time of comany A is 45 days, estimated order lead time of comany B is 5 days. Both comany A and B conduct forecasting and lacing order two times er year ( eriods er year, each eriod has duration of 6 months). The order is laced at the beginning of each eriod. In Vietnam, construction is highly deending on the weather. In addition, the weather in North Vietnam and South Vietnam is quite different due to geometry osition. To reduce risk of overstock, stock out as well increase customer service, both comanies have signed an inventory sharing contract. The condition in the contract is revised and signed before every order eriod. The content of the contract state that in certain eriod, one comany will agree to share k ercentage of its inventory to the other in case this comany is in stock out state and other sharing conditions are satisfied. To enhance quality of forecast technique, both comanies use 3 demand observation eriods in the moving average forecast ( = 3). Demand observations used in forecast are corresonded to time of years. That means to forecast demand for sring eriod, only demand observations of sring eriod in the ass is alied and similar for autumn eriod. The autocorrelation coefficient of the autoregressive model of roduct each comany is q = q = 0.5. In this aer, we will not focus in determine absolutely bullwhi effect in each manufacturer but comare the imact of inventory sharing olicy on the bullwhi effect and the variation of bullwhi effect with some related arameters. For convenient urose, bullwhi effect from each retailer (comany) can be written as bellow: B varðq t; varðd ð k þ L þ L ð q þh B þ L þ L ð q þh k where h z varðbrl t; brl t; [ 0; h z varðbrl t; brl t; [ 0; w þ L þ L z varðbr L t; brl t; varðd 4. Results and discussion ð q varðd varðd [ 0: 4.. No inventory sharing (k = 0) ; ð k ½w; If k = 0, that is, there is no inventory sharing olicy between the two comanies, B and B are determined as follow: B þ L þ L ð q þ h ; B þ L þ L ð q þ h : In this case, we may consider the chain as two indeendent suly chains with single manufacturer, comany, and customer. These results are identified with results of Li [3]. In Chen et al. [3], the authors ointed out the effects of common arameters in the bullwhi effect. The details are as follows: The bullwhi effect is a decreasing function of, the number of observations use in MA. The bullwhi effect is an increasing function of L, the lead time. The bullwhi effect is a decreasing function of q when q [ 0 and larger for odd values of than for even values of, when q \ 0. With the given information and relevant data, bullwhi effect in both manufacturer can be calculated as 3

7 Logist. Res. (03) 6: B 0 40:875 þ h ; B 0 36:9 þ h : ð3 ð4 4.. Comany A shares inventory to comany B (k [ 0) In case of k [ 0, we mean that comany A will deliver amount of inventory, kq to the warehouse of comany B (if sharing conditions are satisfied) in the eriod t. Then, the bullwhi effect at manufacturer () and () are determined as: B ð k B0 40:875 þ h ð k ð5 k B B 0 ð k ½wð36:9 þ h ð k ½w k ð6 From (3) and (5), it is clear that the bullwhi effect at manufacturer () in case of inventory sharing is higher than that in case of without sharing (since ðk [ ), and the bullwhi effect is an increasing function of k. That means, if amount of delivered inventory increase, the bullwhi effect at manufacturer () also increase. From (4) and (6), the bullwhi effect at manufacturer () in case of sharing is smaller than that in case of without sharing. Furthermore, the bullwhi effect quantity is a deceasing function of k which means that higher transhiment comany B receive from comany A, smaller the bullwhi effect at manufacture (). The reason is that the variation of customer demand at sulier is a result of changing customer information through middle arties such as retailer, wholesaler. Because those arties always desire to kee desirable service level, the order quantity lacing at suliers are usually higher than actual demands. When one retailer is exected to receive a certain amount of goods from another, the order quantity this retailer lace at the sulier will reduce and closer to the actual demands. That leads to the bullwhi effect reduces. The variation of bullwhi effect with inventory sharing ercentage in each manufacturer is given in Figs. 4 and 5. Herein, we assign h h w. Because the role of each arty in the chain is equivalent, we can refer that in case of comany B agree to share its inventory to comany A (k \ 0) the result is that when jj k increases, the bullwhi effect at manufacturer () decreases, while the bullwhi effect at manufacturer () increases. 5 Conclusions and recommendations In this aer, we considered the imact of inventory sharing on decentralized warehouses system on the Fig. 4 Variation of the bullwhi effect in manufacturer () with inventory sharing ercentage Fig. 5 Variation of the bullwhi effect in manufacturer () with inventory sharing ercentage bullwhi effect by comaring the bullwhi effects at both suliers in case of no inventory sharing olicy with that in case of using inventory sharing olicy. The studying suly chain system includes two suliers, retailers, and markets. By constructing the bullwhi effect at each sulier for different study situations, we found that inventory sharing olicy has significant imact on the bullwhi effect in each sulier. The variation of the bullwhi effect in each 3

8 96 Logist. Res. (03) 6:89 98 sulier deends on the destination of transferring inventory. In details, the bullwhi effect at one sulier will increase if its retailer is the one that received transferring inventory. In addition, the higher amount of receiving inventory of the retailer, the bigger the bullwhi effect in the corresonding sulier. In other words, inventory sharing reduces the bullwhi effect at this sulier but also increases the bullwhi effect at the other sulier. According to the finding of this aer, suly chain managers are helful in forecasting amount of goods that need to suly for their retailers. Also suly chain managers can adjust inventory sharing olicy in order to tradeoff the bullwhi effect for both suliers. In many cases, inventory sharing rofits for the whole suly system, therefore, determining the imact of inventory sharing on the bullwhi effect will further suort for the develoment of inventory sharing models in suly chain. This aer can be extended through three directions. One is that we can study the imact of inventory sharing on the bullwhi effect in a suly chain with only one sulier and two or multi-retailers. With inventory sharing, total roduct quantity from retailers may differ from that in case of without inventory sharing. Another side direction would be extending the model to multi-stage suly chain, and inventory sharing is alied in different levels in the chain. For the third direction, we can study this model with multi-tye roducts. Acknowledgments The authors would like to thank the referees for their valuable comments and suggestions. Aendix. The derivation rocess of E(D t,i ) and varðd t;i When the autoregressive demand rocess is stationary, we have EðD t;i EðD t;i EðD t;i... EðD i ; and varðd t;i varðd t;i varðd t;i... varðd i ; where D t; l þ q D t; þ e t; ; EðD t; Eðl þq EðD t; þeðe t; EðD l þ q EðD þ0 ) EðD l : q varðd t; varðl þq varðd t;þvarðe t; varðd 0þq varðd þr ) varðd r q : Similarly, we have EðD l q ; varðd r q :. The derivation rocess of the further equation of q t,. q t; y t; y t; þ D t; þ kq t; ; ) q t; k y t; y t; þ D t; h i bd L t; k þ z br L t; bd L t; þ z br L t; þ D t; h i bd L t; k bd L t; þ D t; z br L t; brl t;! L X D ti; X D ti; ð k i i þ D t; k þ z k brl t; brl t; L ð k D D t; t; D t; þ k þ z k brl t; brl t; k þ L D t; L ð k D t; þ z k brl t; brl t; ; 0 varðq t; var k þ L D t; L ð k D t; B þ z A k brl t; brl t; 0 þ L varðd t; L þ L covðd t; ; D t; ð k þ L varðd t;þz varðbrl t; brl t; B þ z þ L covðd t; ; br L t; A 0 þ L þ L! varðd L þ L covðd t; ; D t; ð k : þ z varðbrl t; brl t; B þ z þ L covðd t; ; br L t; A Now we will determine covðd t; ; D t; and covðd t; ; br L t; We have 3

9 Logist. Res. (03) 6: covðd t; ; D t; cov ðl þ q D t; þ e t; ; D t; cov l ; D t; þ q cov D t; ; D t; þ cov e t; ; D t; : ðsince cov l ; D t; 0 and cov et; ; D t; 0; covðd t; ; D t; q cov D t; ; D t;... q cov D t;; D t; q varðd : We assume that forecasting customer demands by retailers are random variables of the form as D t l þ qd t þ e t, and the error terms e t are identically indeendent distribution with mean 0 and variance r. Let the estimate of the standard deviation of forecast error of the lead time demand be sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P br L ij t C ðd tj bd tj L; : Alying the result roved in Ryan [], we have covðd tj ; br L t 0; 8i ; ;...; : Hence, varðq t; þ L þ L! 3 varðd ð k L þ L covðd t; ; D t; 6 4 þ z varðbrl t; brl t; þ z þ L 7 covðd t; ; br L t; 5 L þ þ L! varðd L 3 þ L q varðd 6 7 ð k 4 5 þ z varðbrl t; brl t; varðd ð k þ L þ L ð q þ z varðbrl t; brl t; ð k : 3. The derivation rocess of the further exression of q t, : q t; þ L D t; L D t; þz ðbr L t; brl t; kq t;;) varðq t; var þ L D t; L D t; þz ðbr L t; brl t; k varðq t; þ L varðd t; L þ L covðd t; ;D t; þ L varðd t;þz varðbrl t; brl t; þz þ L covðd t; ; br L t; k varðq t; þ L þ L! varðd L þ ðl covðd t; ; D t; þ z varðbrl t; brl t; þz þ L covðd t; ; br L t; k varðq t; : Furthermore, we have covðd t; ; br L t;0 and covðd t; ; D t; covðl þ q D t; þ e t; ; D t; covðl ; D t; þq covðd t; ; D t; þ covðe t; ; D t; q covðd t; ; D t;... q covðd t;; D t; q varðd : ðnote thatcovðl ;D t; 0 and covðe t; ;D t; 0 Hence, varðq t; þ L þ L! varðd L þ L! q varðd References þ z varðbrl t; brl t; k varðq t; " varðd þ L þ L! # ð q þ z varðbrl t; brl t; k varðq t; :. Bassok Y, Anuindi R, Akella R (999) Single-eriod multiroduct inventory models with substitution. Oer Res 47(4): Bitran G, Dasu S (99) Ordering olicies in an environment of stochastic yields and substitutable demands. Oer Res 40(5): Chen F, Ryan JK, Simchi-Levi D (000) The imact of exonential smoothing forecasts on the bullwhi effect. Naval Res Logist 47(4): Chatfield DC, Kim JG, Harrison TP, Hayya JC (004) The bullwhi effect imact of stochastic lead time, information quality, and information sharing: a simulation study. Prod Oer Manag 3(4): Drezner Z, Gurnani H, Pasternack BA (995) An eoq model with substitutions between roducts. J Oer Res Soc 46(7):

10 98 Logist. Res. (03) 6: Evers PT (00) Heuristics for assessing emergency transshiments. Eur J Oer Res 9: Kutanoglu E (009) An inventory sharing and allocation method for a multi-location service arts logistics network with timebased service levels. Eur Oer Res 94(3): Forrester J (958) Industrial dynamics, a major breakthrough for decision makers. Harvard Bus Rev 36: Chen F, Drezner Z, Ryan JK, Levi DS (000) Quantifying the bullwhi effect in a simle suly chain. Manag Sci 46(3): Gilbert K (005) An ARIMA suly chain model. Manag Sci 5(): Luong HT, Phien NH (00) Measure of bullwhi effect in suly chains: the case of high order autoregressive demand rocess. Eur J Oer Res 83(): Luong HT (007) Measure of the bullwhi effect in suly chains with autoregressive demand rocess. Eur J Oer Res 80(3): Sun HX, Ren YT (005) The imact of forecasting methods on bullwhi effect in suly chain management. Proc Eng Manage Conf : Zhao H, Deshande V, Ryan K (005) Inventory sharing and rationing in decentralized dealer networks. Manage Sci 5(4): Kim JG, Chatfield D, Harrison TP, Hayya JC (006) Quantifying the bullwhi effect in a suly chain with stochastic lead time. Eur J Oer Res 73: Heydari J, Karemzadeh RB, Chaharsooghi SK (009) A study of lead time variation imact on suly chain erformance. Int J Adv Manuf Technol 40: Karaesmen I, van Ryzin G (004) Overbooking with substitutable inventory classes. Oer Res 5(): So KC, Zheng X (003) Imact of sulier s lead time and forecast demand udating on retailer s order quantity variability in a two-level suly chain. Int J Prod Econ 86: Xu K, Dong Y, Evers PT (000) Towards better coordination of the suly chain, Transort. Res Part E Logist Trans Rev 37(): Lee HL, Padmanabhan V, Whang S (997) Information distortion in a suly chain: the bullwhi effect. Manag Sci 43(4): McGillivray AR, Silver EA (978) Some concets for inventory control under substitutable demand. Inf Syst Oer Res 6(): Netessine S, Rudi N (003) Centralized and cometitive inventory models with demand substitution. Oer Res 5(): Rudi N, Kaur S, Pyke DF (00) A two-location inventory model with transshiment and local decision making. Manage Sci 47(): Parlar M, Goyal S (984) Otimal ordering decisions for twosubstitutable roducts with stochastic demands. Osearch : 5 5. Graves SC (999) A single-item inventory model for a non-stationary demand rocess. Manuf Serv Oer Manage (): Ryan JK (997) Analysis of inventory models with limited demand information. PhD thesis, Northwestern University, Evanston 7. Tagaras G (989) Effects of ooling on the otimization and service levels of two-location inventory systems. IIE Trans : Duc TTH, Luong HT, Kim YD (008) A measure of bullwhi effect in suly chains with a mixed autore-gressive-moving average demand rocess. Eur J Oer Res 87(): Duc TTH, Luong HT, Kim YD (00) Investigate the imact of third-party warehouse on bullwhi effect in suly chains with autoregressive demand rocess. SIMTech technical reorts (STR_V_N3_08_POM). (3) 30. Zhang X (004) The imact of forecasting methods on the bullwhi effect. Int J Prod Econ 88: Li X, Song L, Zhao Z (0) Quantifying the imact of demand substitution on the bullwhi effect in a suly chain. Logist Res 3: 3 3