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

The impact of inventory sharing on the bullwhip effect in decentralized inventory systems Logist. Res. (2013) 6:89–98 DOI 10.1007/s12159-012-0096-7 OR IGINAL PAPER The impact of inventory sharing on the bullwhip effect in decentralized inventory systems • • Dang Van Le Luong Trung Huynh Kifor Vasile Claudiu Muntean Achim Received: 12 April 2012 / Accepted: 21 November 2012 / Published online: 19 December 2012 Springer-Verlag Berlin Heidelberg 2012 Abstract The paper derives the impact of inventory stage supply chain, retailers are the parties who receive sharing policy on the bullwhip effect in two-stage supply customers’ demands directly. To satisfy customers’ ser- chains with two independent suppliers and two integrated vice, usually customer demand is estimated by using retailers. There exists an inventory sharing policy between forecasting techniques before placing order to supplier. The two retailers. Under inventory sharing policy, when lacking of information leads to fluctuation orders from all demand in one retailer exceeds its inventory, this retailer levels of the chain in term of volumes. The fluctuation of can ask for a product sharing volume from the other in customer demand through the chain is well known as the order to satisfy customer demand. With certain assump- bullwhip effect [3, 19]. tions, the bullwhip effect is quantified in both cases, with In Fig. 1, we can see the fluctuation of customers’ inventory sharing policy and without inventory sharing demands through different facility epochs in a four-stage policy. We found that inventory sharing has significant supply chain with single manufacturer, distributor, whole- impact on the bullwhip effect in the supply system. How- saler, and retailer. We may notice that demands fluctuation ever, inventory sharing policy does not synchronously increases from lower levels to higher levels in the chain. reduce or increase the bullwhip effect in both suppliers in The reasons can be explained as follows: the retailer the same period. A numerical example is given to illustrate has directly customer information. Retailer will use this the study model. information to estimate actual demands. To maintain desired service level, retailer needs to hold a certain Keywords Inventory sharing policy  Bullwhip effect  inventory in the warehouse. That leads to the wholesaler Supply chain management  Decentralized inventory  will receive higher original orders from the retailer. Simi- Order lead time larly, wholesaler receives customer information from the retailer and places an order to his supplier, the distributor. To determine the order quantities from retailer, the 1 Introduction wholesaler must forecast customer demand. Unfortunately, the wholesaler does not have access to the customer actual The information about customer demand is varied through information; so that they must use the information from the the levels of the chain due to many factors (e.g., inventory retailer to perform his forecasting. Therefore, the variation policy, forecasting method, order lead time, etc.). In two- of customer demand increases from lower epoch to higher epoch in the chain. In the supply chain system applying inventory sharing D. V. Le (&)  L. T. Huynh policy, distribution centers, wholesalers, and retailers are School of Engineering and Technology, Asian Institute of Technology, Bangkok, Thailand collaborated by sharing product in case of emergency such e-mail: haidang_math1804@yahoo.com that stock out or demand exceeds inventory. Separated inventory of parties in the same levels are virtually com- K. V. Claudiu  M. Achim bined. If one party is in stock out stage, its demands can Faculty of Engineering and Technology, be fulfilled by available inventory in the other retailers. Lucian Blaga University of Sibiu, Sibiu, Romania 123 90 Logist. Res. (2013) 6:89–98 2 Literature review The studies about bullwhip effect have been utilized for several decades. The earliest paper studied on this area is conducted by Forrester [8]. In this paper, the authors the first time pointed out the effects of information variation on production decision. This concept has been noticed and become foundation for research in the field of demand variation as we call ‘‘the bullwhip effect.’’ Main objectives of previous papers mainly focus on three aspects: (a) demonstrating the existence of the bull- whip effect, (b) identifying possible causes of the bullwhip effect, and (c) developing strategy to reduce the impact of the bullwhip effect [3]. Following the stream, our paper can be classified into the second areas, that is, identifying possible causes of the bullwhip effect. This researched area has been studying by many researchers. For instance, Lee et al. [20] pointed out in their papers four major causes of the bullwhip effect including: demand forecast updating, order batching, price fluctuation, rationing and shortage gambling. They also presented several methods to coun- Fig. 1 Variation of customer demand in a supply chain system teract the bullwhip effect such as integrating new infor- mation systems, defining new organizational relationships, This cooperation especially profits integrated supply chain and implementing new incentive and measurement system (the chain in which manufacturers and retailers are systems. dependent). For a supply chain with independent facilities, The effect of forecasting methods on the bullwhip effect inventory sharing policy sounds like a game changer and is also the main topic in many studies. Chen et al. [3] this policy would be more efficient in the centralized studied the impact of different forecasting methods on the information system. bullwhip effect in a two-stage supply chain. They con- Inventory sharing can be profited for whole supply chain cluded that exponential smoothing forecasting method system including: manufacturers, distributors, retailers, and gives higher bullwhip effect than moving average fore- customers. Manufacturers can improve brand reputation, casting method. In the same sense, Graves [25], Xu et al. increase manufacturing efficiency, and reduce unwanted [19] and Zhang [30] presented the effect of demand fore- inventory. For distributors/retailers, inventory sharing casts on the bullwhip effect in a two-stage supply chain reduces the number of lost orders and backorders, provide a system with integrated moving average demand process. new outlet for slow moving inventory, and increase Furthermore, Sun and Ren [13], Zhang [30] studied the possibility for incremental revenue (reduce inventory while impact of different forecasting methods such as MA, ES, remaining service level). Finally, customers also profit EWMA, MMSE, and suggested the forecast method which from inventory sharing in which it is an efficient strategy can mitigate the bullwhip effect. that improves product availability and reduces delivering The impact of order lead time was presented in the time. papers of Graves [25], Chen et al. [9], Zhang [30], Lee However, inventory sharing policy does not always et al. [20]. Luong and Phien [11] proved that the bullwhip guarantee making benefit for retailers. In case of tran- effect can be decreased by reducing lead time. However, shipment, reallocation, and penalty costs are too high, Duc et al. [28] showed that reducing lead time does not retailers will try to reduce the probability of stock out by always reducing the bullwhip effect. He showed that in increasing inventory level. In addition, if the ordered lead some special cases; for example, in a two-stage supply time and order duration period time from suppliers to chain with a pre-specified ARMA demand process, retailers is short, retailers may prefer waiting for new increasing the lead time may help to reduce the bullwhip delivering packages instead of receiving sharing packages effect. For more detail, we refer to [28]. When considering from other retailers. Therefore, the efficiency of inventory the impact of lead time on the bullwhip effect, the lead sharing strategy depends on geography factors, features of time can be deterministic or stochastic. Results in those system, products, and services. papers mentioned above are in case of deterministic lead 123 Logist. Res. (2013) 6:89–98 91 time, whereas in practical lead time usually behaves as a inventory sharing leads to decreased dealers’ rationing stochastic process. level rather than increasing their based stock level, (3) a Chaffied [4] and So and Zheng [18] used simulation smaller level of incentive for inventory sharing may be approaches to demonstrate the impact of lead time varia- sufficient to achieve the benefit of full inventory sharing tion and information sharing on the customer demand policy, (4) the benefit of inventory sharing increase the fluctuation in a supply chain. Other results in this area are system utilization, and (5) customer service may improve contented in [15, 16, 28]. significantly with inventory sharing. Li et al. [31] analyzed the impact of demand substitution Recently, Kutanoglu [7] considered a model to allocated on the bullwhip effect in a two-stage supply chain with a stock level in the warehouse in a service part logistic net- singer supplier, singer retailer, and two types of products A work. The network includes one supplier with the infinitive and B such that a certain fraction product A can be used to warehouse capacity and a number of local warehouses. substitute product B. They showed the relation between the Each local warehouse has independent based stock policy. bullwhip effect and the forecasting method, lead time, Moreover, local warehouses share their inventory as a way demand process, and the product substitution. The impact to increase service levels. They concluded that inventory of demand substitution has been noticed and investigated sharing can reduce based stock levels and total system cost. by some previous authors [1, 2, 5, 21, 22, 24]. As mention above, the principle of the bullwhip effect is Inventory sharing has become important perception in the variation of customer demand through the chain. some supply chain models. For example, in the supply Meanwhile, inventory sharing has significant impact on chain with singer supplier and multi-independent retailers, inventory levels and order quantities as well. That results in the delivering time from suppliers to retailers’ warehouses the change in customer information. Previous researches take a long time because of the long distance between mainly focus on various areas of the bullwhip effect and suppliers and retailers, whereas retailers may located very inventory sharing. However, there are no works directly closed in one area. There is probability that inventory of focusing on the impact of inventory sharing on the bull- one retailer exceeds customer demand, while others are in whip effect yet. In this paper, we will derive this issue and stock out state. In this situation, stock out can be satisfied examine the impact of inventory sharing policy on the by transferring products among retailers through inventory bullwhip effect. The remaining of the paper is organized as sharing policy. follows: Sect. 3 describes the problem and develops The concept of third-party warehouse is created aiming mathematical formulation; Sect. 4 gives a numerical at identifying inventory policy and customer information. example, results, and discussion. Conclusions and recom- It may help to decrease total inventory at retailers. In this mendations for further study is the content of the last area, Duc et al. [29] studied the effect of third-party section. warehouse on the bullwhip effect. They found that third- party warehouse does not always reducing the bullwhip effect. They also stated the conditions in which the utili- 3 Model development zation of third-party warehouse decreases the bullwhip effect in supply chain. 3.1 Notations and assumptions Rudi [23] studied on the relation among following fac- tors: inventory sharing, transhipment cost, and inventory The following notations are used in this paper orders in the supply chain with one supplier and two local t order period number index; retailers. They pointed out that inventory sharing and i retailer and corresponding supplier index, i = 1, 2; transhipment costs have significant effects on inventory order at each retailer. A case of study was conducted in k percentage of product that one retailer may share to other retailers in the period t; which a two-stage supply chain with singer supplier, Bosch based in Germany, and five retailers based in Norway. In L the order lead time for retailer i, i = 1, 2; p the number of demand observation periods used in this supply chain system, the ordered delivering time from Germany to Norway takes about 3 weeks, while tranship- the moving average forecast; l average demand of product at retailer i in the ment time within Norway is insignificant. This supply system is more similar with the supply chain model in our autoregressive demand model; e forecast error for product at retailer i during time paper. t,i Zhao [14] presented an optimal inventory policy for period t; q the autocorrelation coefficient of the autoregressive each dealer in decentralized dealer networks. They con- model of product at retailer i; q \1; q \1 cluded that (1) inventory sharing has big impact on the jj jj 1 2 level of inventory by the independent dealer, (2) increasing D product demand at retailer i during time period t; t,i 123 92 Logist. Res. (2013) 6:89–98 lead time demand for product at retailer i; t;i b the forecast of the lead time demand product at t;i retailer i; z normal z-score determined by the desire service level; standard deviation of forecast error of lead time t;i demand for product at retailer i; y order-up-to level inventory for retailer i at the t,i beginning of time period t; q order quantity product for retailer i at the beginning t,i of time period t; B the bullwhip effect at supplier i; C a constant function of Land q L,q The proposal model is studying under following assumptions Fig. 2 Relationship among parties in the chain As1 Order lead time of each retailer is smaller than duration time of one period ðL \rðtÞ; i ¼ 1; 2Þ: As2 Time period index t and duration of order period are equal for both retailers As3 In one order period, only one retailer agree share inventory, while the other will receive this amount of inventory As4 Inventory sharing percentage is varying from period to period As5 In every period, order from each retailer is positive 3.2 Problem description In this paper, we consider a two-stage supply chain with two suppliers and two retailers with separate markets (each retailer has their own customer). A single product is delivered from suppliers to retailers following discrete time period. The relationship among the parties in the chain is rep- resented in Fig. 2. That is, whenever one retailer is in stock out state, they can ask for transhipment from the other retailer. Whether transhipment is made depending on predetermine conditions stated in the sharing policy. The process of satisfying customer demand with inventory sharing policy is illustrated in Fig. 3. We assume that order lead times for both retailers (time from placing an order until receiving the order) are deter- Fig. 3 Progress of satisfying customer demand ministic. We support that both retailers use the same forecasting method (MA) and inventory policy (order-up-to level). Since inventory sharing policy could affect order forecast, q is order quantity from retailer i placed at sup- quantities that a retailer placing on its supplier, we will plier i, i = 1, 2. consider the impact of inventory sharing on the bullwhip effect of each retailer. The bullwhip effect can be deter- 3.3 Mathematical formulation mined by identifying the ratio of variance of retailer’s order to the supplier to the variance of customer’s demand Support at the beginning of order period t, both retailer (1) varðq Þ to the retailers , in which D is customer demand and retailer (2) estimate customer demand and place an varðD Þ 123 Logist. Res. (2013) 6:89–98 93 order to its supplier. The order cost is negligible. From 3.4 Bullwhip effect quantify Zhang [30] and Gilbert [10], end customer demand can be modeled by an autoregressive (AR). That means end cus- Since the variance of q and D is different in each retailer, t,i i tomer demand at each retailer during time period t can be the bullwhip effect at each supplier will be discussed separately. Given the equations of the order-up-to level, determined as follows: demand forecasting, and standard deviation of forecast D ¼ l þ q D þ e ; t;1 t1;1 t;1 1 1 error, q can be expressed as follows: t,1 ð1Þ D ¼ l þ q D þ e : t;2 t1;2 t;2 2 2 q ¼ y  y þ D þ kq t;1 t;1 t1;1 t1;1 t;1 From (1), we have (See ‘‘Appendix’’ for details) ¼ ½y  y þ D t;1 t1;1 t1;1 1  k 2 L L L L 1 1 1 1 b b l r ¼ ½ðD þ z b r ÞðD þ z b r Þþ D 1 1 t1;1 1 t;1 t;1 t1;1 t1;1 1  k EðD Þ¼ ; varðD Þ¼ ; t;1 t;1 1  q 1  q 1 L L 1 1 1 1 L L 1 1 ð2Þ ¼ 1 þ D  D þ z b r  b r : t1;1 tp1;1 1 t;1 t1;1 1  k p p l r 2 2 EðD Þ¼ ; varðD Þ¼ t;2 t;2 ð8Þ 1  q 1  q 2 2 Then, the variance of the order quantity q at supplier (1) At the beginning of time period t, the actual inventory level t,1 at time period t is as follows: at retailer (i)is y  D . Retailer (i) will order amount t1;i t1;i of product in order to reach the target inventory level y . 1 L L t,i 1 1 varðq Þ¼ var 1 þ D  D t;1 t1;1 tp1;1 Taking inventory sharing into consideration, the quantity 1  k p p for the demand at each retailer at the beginning of time L L 1 1 þz ðb r  b r Þ t;1 t1;1 period t can be determined as varðD Þ 2L 2L 1 1 p ¼ 1 þ þ ð1  q Þ q ¼ y ðy  D Þþ kq t;1 t;1 t1;1 t1;1 t;1 1 2 2 p p ð1  kÞ ¼ y  y þ D þ kq ; t;1 t1;1 t1;1 t;1 1 L L 1 1 þ varðb r  b r Þ: ð9Þ 2 t;1 t1;1 or ð1  kÞ 1 For the variance of the order quantity q at supplier (2) at t,2 q ¼ ½y  y þ D ; ð3Þ t;1 t;1 t1;1 t1;1 time period t, we have 1  k q ¼ y  y þ D  kq t;2 t;2 t1;2 t1;2 t;1 and L L L L 2 2 2 2 ^ ^ ¼ðD þ z r ^ ÞðD þ z r ^ Þþ D  kq 2 2 t1;2 t;1 t;2 t;2 t1;2 t1;2 q ¼ y ðy  D Þ kq t;2 t;2 t1;2 t1;2 t;1 L L 2 2 L L 2 2 ð4Þ ¼ 1 þ D  D þ z ðr ^  r ^ Þ kq : t1;2 tp1;2 2 t;1 t;2 t1;2 ¼ y  y þ D  kq : p p t;2 t1;2 t1;2 t;1 The target inventory y at the beginning of period t is t,i Then, the variance of the order quantity q time period t is t,2 estimated from observed demand as as follows: L L varðq Þ 1 1 t;2 y ¼ D þ z b r ; t;1 1 t;1 t;1 ð5Þ L L 2 2 L L 2 2 L2 L 2 ¼ var 1 þ D  D þ z b r  b r t1;2 tp1;2 2 y ¼ D þ z b r : t;2 t1;2 t;2 2 t;2 t;2 p p k varðq Þ t;1 Both retailers use simple moving average technique to 2 2 Li i 2L 2L 2L 2L 2 2 2 2 p estimate D and r based on the information of the past ¼ 1 þ þ  þ :q :varðD Þ t;i t;i 2 2 2 p p p p p periods. We have 2 L L 2 2 2 þ z var b r  b r  k varðq Þ: ð10Þ t;1 P 2 t;2 t1;2 tj;i L j¼1 D ¼ L ; ð6Þ t;i We denote B , B are the bullwhip effect of supplier (1) and 1 2 supplier (2) in period t, respectively. We have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 ðD  D Þ varðq Þ tj;i tj;i t;1 j¼1 B ¼ r ¼ C : ð7Þ 1 L ;q t;i i varðD Þ 1 2L 2L 1 p where D  D is the forecast error of the (t - j)th tj;i tj;i ¼ 1 þ þ ð1  q Þ 2 2 p p ð1  kÞ period at retailer (i) and C is a constant function of L ;q L and q [26].The bullwhip effect can be calculated as L L i 1 1 1 þ varðb r  b r Þ: ð11Þ 2 t;1 t1;1 varðq Þ t;i ð1  kÞ B ¼ . varðD Þ 123 94 Logist. Res. (2013) 6:89–98 varðq Þ t;2 will not focus in determine absolutely bullwhip effect in B ¼ varðD Þ each manufacturer but compare the impact of inventory sharing policy on the bullwhip effect and the variation of 2L 2L 2 p 2 L L 2 2 ¼ 1 þ þ ð1  q Þ þ z varðb r  b r Þ 2 t;2 t1;2 2 bullwhip effect with some related parameters. For conve- p p nient purpose, bullwhip effect from each retailer (com- k 2L 2L varðD Þ 1 1 1 p 1 þ þ ð1  q Þ pany) can be written as bellow: 2 2 p p varðD Þ ð1  kÞ 2 varðq Þ L L 2 2 2 t;1 varðb r  b r Þ z k B ¼ t;2 t1;2 1 1 : ð12Þ varðD Þ varðD Þ ð1  kÞ 2 1 2L 2L 1 p ¼ 1 þ þ ð1  q Þþ h ; 2 2 p p ð1  kÞ 4 Numerical example and analysis 2 2 2L 2L k 2 p B ¼ 1 þ þ ð1  q Þþ h  ½ w ; 2 2 p p ð1  kÞ From (11) and (12), we can see that B and B are 1 2 functions of the following parameters: p- the number where of observation use in MA; L , L -the order lead time; 1 2 L L 2 1 1 h ¼ z varðb r  b r Þ [ 0; q , q -first order autocorrelation coefficients of the auto- 1 1 2 1 t;1 t1;1 regressive demand process at retailer (1) and retailer (2), L L 2 2 h ¼ z varðb r  b r Þ [ 0; 2 t;2 t1;2 respectively; k-inventory sharing percentage. In this sec- 2L 2L varðD Þ tion, we will provide a numerical example to illustrate the 1 1 1 p w ¼ 1 þ þ ð1  q Þ impact of those parameters on the bullwhip effect on both p p varðD Þ retailers. L L 2 2 varðb r  b r Þ t;2 t1;2 z [ 0: varðD Þ 4.1 System description 4.2 Results and discussion Two trading companies A and B has center in North and South Vietnam, respectively. Company A imports steel 4.2.1 No inventory sharing (k = 0) from manufacturer (1) which has center in Italy while company B imports steel from manufacturer (2) which has If k = 0, that is, there is no inventory sharing policy center in Japan. Estimated order lead time of company A between the two companies, B and B are determined as 1 2 is 45 days, estimated order lead time of company B is follow: 25 days. Both company A and B conduct forecasting and 2L 2L placing order two times per year (2 periods per year, each 1 p B ¼ 1 þ þ ð1  q Þ þ h ; 1 1 p p period has duration of 6 months). The order is placed at the beginning of each period. In Vietnam, construction is 2L 2L 2 p B ¼ 1 þ þ ð1  q Þ þ h : highly depending on the weather. In addition, the weather 2 2 p p in North Vietnam and South Vietnam is quite different due to geometry position. To reduce risk of overstock, stock out In this case, we may consider the chain as two independent as well increase customer service, both companies have supply chains with single manufacturer, company, and signed an inventory sharing contract. The condition in the customer. These results are identified with results of Li contract is revised and signed before every order period. [31]. In Chen et al. [3], the authors pointed out the effects The content of the contract state that in certain period, one of common parameters in the bullwhip effect. The details company will agree to share k percentage of its inventory are as follows: to the other in case this company is in stock out state and • The bullwhip effect is a decreasing function of p, the other sharing conditions are satisfied. To enhance quality of number of observations use in MA. forecast technique, both companies use 3 demand obser- • The bullwhip effect is an increasing function of L, the vation periods in the moving average forecast (p = 3). lead time. Demand observations used in forecast are corresponded to • The bullwhip effect is a decreasing function of q when time of years. That means to forecast demand for spring q [ 0 and larger for odd values of p than for even period, only demand observations of spring period in the values of p, when q \ 0. pass is applied and similar for autumn period. The auto- correlation coefficient of the autoregressive model of With the given information and relevant data, bullwhip product each company is q = q = 0.5. In this paper, we effect in both manufacturer can be calculated as 1 2 123 Logist. Res. (2013) 6:89–98 95 B ¼ 420:875 þ h ; ð13Þ B ¼ 136:9 þ h : ð14Þ 4.2.2 Company A shares inventory to company B (k [ 0) In case of k [ 0, we mean that company A will deliver amount of inventory, kq to the warehouse of company B (if sharing conditions are satisfied) in the period t. Then, the bullwhip effect at manufacturer (1) and (2) are determined as: 1 420:875 þ h B ¼  B ¼ ð15Þ 2 2 ð1  kÞ ð1  kÞ 2 2 k k B ¼ B  ½w¼ð136:9 þ h Þ ½wð16Þ 2 2 2 2 2 ð1  kÞ ð1  kÞ • From (13) and (15), it is clear that the bullwhip effect at manufacturer (1) in case of inventory sharing is higher Fig. 4 Variation of the bullwhip effect in manufacturer (1) with than that in case of without sharing (since [ 1), ð1kÞ inventory sharing percentage and the bullwhip effect is an increasing function of k. That means, if amount of delivered inventory increase, the bullwhip effect at manufacturer (1) also increase. • From (14) and (16), the bullwhip effect at manufacturer (2) in case of sharing is smaller than that in case of without sharing. Furthermore, the bullwhip effect quantity is a deceasing function of k which means that higher transhipment company B receive from company A, smaller the bullwhip effect at manufacture (1). The reason is that the variation of customer demand at supplier is a result of changing customer information through middle parties such as retailer, wholesaler. Because those parties always desire to keep desirable service level, the order quantity placing at suppliers are usually higher than actual demands. When one retailer is expected to receive a certain amount of goods from another, the order quantity this retailer place at the supplier will reduce and closer to the actual demands. That leads to the bullwhip effect reduces. The variation of bullwhip effect with inventory sharing percentage in each manufacturer is given in Figs. 4 and 5. Herein, we assign h ¼ h ¼ w ¼ 1. 1 2 Fig. 5 Variation of the bullwhip effect in manufacturer (2) with Because the role of each party in the chain is equivalent, inventory sharing percentage we can refer that in case of company B agree to share its inventory to company A (k \ 0) the result is that when jj k bullwhip effect by comparing the bullwhip effects at both increases, the bullwhip effect at manufacturer (1) decrea- suppliers in case of no inventory sharing policy with that in ses, while the bullwhip effect at manufacturer (2) increases. case of using inventory sharing policy. The studying supply chain system includes two suppliers, retailers, and markets. 5 Conclusions and recommendations By constructing the bullwhip effect at each supplier for different study situations, we found that inventory sharing In this paper, we considered the impact of inventory policy has significant impact on the bullwhip effect in each sharing on decentralized warehouses system on the supplier. The variation of the bullwhip effect in each 123 96 Logist. Res. (2013) 6:89–98 supplier depends on the destination of transferring inven- Similarly, we have tory. In details, the bullwhip effect at one supplier will EðD Þ¼ ; increase if its retailer is the one that received transferring 1  q inventory. In addition, the higher amount of receiving varðD Þ¼ : inventory of the retailer, the bigger the bullwhip effect in 1  q the corresponding supplier. In other words, inventory sharing reduces the bullwhip effect at this supplier but also 2. The derivation process of the further equation of q . t,1 increases the bullwhip effect at the other supplier. q ¼ y  y þ D þ kq ; t;1 t;1 t1;1 t1;1 t;1 According to the finding of this paper, supply chain managers are helpful in forecasting amount of goods that ) q ¼ y  y þ D t;1 t;1 t1;1 t1;1 1 k need to supply for their retailers. Also supply chain man- hi L L L L 1 1 1 1 b b ¼ D þ z b r  D þ z b r þ D 1 1 t1;1 agers can adjust inventory sharing policy in order to trade- t;1 t;1 t1;1 t1;1 1  k hi off the bullwhip effect for both suppliers. In many cases, L L L L 1 1 1 1 b b ¼ D  D þ D  z b r b r t1;1 1 t;1 t1;1 t;1 t1;1 inventory sharing profits for the whole supply system, 1  k p p therefore, determining the impact of inventory sharing on X X ¼ D  D ti;1 t1i;1 the bullwhip effect will further support for the development pð1  kÞ i¼1 i¼1 of inventory sharing models in supply chain. D z t1;1 1 L L 1 1 þ þ b r b r This paper can be extended through three directions. One t;1 t1;1 1  k 1 k is that we can study the impact of inventory sharing on the L D t1;1 ¼ D  D þ t1;1 tp1;1 bullwhip effect in a supply chain with only one supplier and pð1  kÞ 1  k two or multi-retailers. With inventory sharing, total product L L 1 1 þ b r b r t;1 t1;1 quantity from retailers may differ from that in case of without 1 k 1 L L inventory sharing. Another side direction would be extending 1 1 ¼ 1 þ D  D t1;1 tp1;1 the model to multi-stage supply chain, and inventory sharing 1  k p pðÞ 1  k L L is applied in different levels in the chain. For the third 1 1 þ b r b r ; t;1 t1;1 1 k direction, we can study this model with multi-type products. 0  1 1 L L 1 1 Acknowledgments The authors would like to thank the referees for 1 þ D  D t1;1 tp1;1 B C 1  k p pð1  kÞ B C their valuable comments and suggestions. varðq Þ¼ var t;1 @ A L L 1 1 þ b r  b r t;1 t1;1 1  k 0 1 Appendix 2 1 þ varðD Þ t1;1 B C B C B C 1. The derivation process of E(D ) and varðD ÞWhen t,i t;i B C B L L C 1 1 the autoregressive demand process is stationary, we B C 2 1 þ covðD ; D Þ t1;1 tp1;1 B C p p have 1 B C B C B 2 C ð1  kÞ EðD Þ¼ EðD Þ¼ EðD Þ¼ ... ¼ EðD Þ; L t;i t1;i t2;i i B 1 C 2 L L 1 1 B þ varðD Þþ z varðb r  b r ÞC tp1;1 1 t;1 t1;1 B p C B C and B  C @ A 2L þ 2z 1 þ covðD ; b r Þ varðD Þ¼ varðD Þ¼ varðD Þ¼ ... ¼ varðD Þ; 1 t1;1 t;i t1;i t2;i i t;1 0 1 2L 2L where 1 1 B C 1 þ þ varðD Þ B C p p B C D ¼ l þ q D þ e ; t;1 t1;1 t;1 1 1 B C B C B 2L 2L C EðD Þ¼ Eðl Þþ q EðD Þþ Eðe Þ 1 t;1 1 1 t1;1 t;1 B C þ covðD ; D Þ 1 t1;1 tp1;1 B C p p ¼ : EðD Þ¼ l þ q EðD Þþ 0 2 B C 1 1 1 1 ðÞ 1  k B C B 2 L L C l 1 1 þ z varðb r  b r Þ B C 1 t;1 t1;1 ) EðD Þ¼ : B C 1  q 1 B C @ A 2L 2 L þ 2z 1 þ covðD ; b r Þ varðD Þ¼ varðl Þþ q varðD Þþ varðe Þ t;1 t1;1 t;1 1 t1;1 1 1 t;1 2 2 varðD Þ¼ 0 þ q varðD Þþ r 1 1 1 1 Now we will determine covðD ; D Þ and t1;1 tp1;1 ) varðD Þ¼ : 1 L covðD ; b r Þ We have 1  q t1;1 t;1 123 Logist. Res. (2013) 6:89–98 97 L L 2 2 covðD ; D Þ t1;1 tp1;1 ¼ 1 þ 2 þ 2 varðD Þ p p ¼ cov ðl þ q D þ e Þ; D 1 1 t2;1 t;1 tp1;1 2L L 2 2 2 þ 2ð Þ covðD ; D Þ ¼ cov l ; D þ q cov D ; D t1;2 tp1;2 1 tp1;1 1 t2;1 tp1;1 p p þ cov e ; D : t;1 tp1;1 2L 2 L L 2 L 2 2 2 þ z varðb r  b r Þþ 2z 1 þ covðD ; b r Þ 2 t1;2 2 t;2 t1;2 t;2 ðSince cov l ; D ¼ 0 and cov e ; D ¼ 0Þ; tp1;1 t;1 tp1;1 k varðq Þ: t;1 covðD ; D Þ¼ q cov D ; D t1;1 tp1;1 t2;1 tp1;1 Furthermore, we have covðD ; b r Þ¼ 0 and ... t1;2 t;2 ¼ q cov D ; D tp;1 tp1;1 covðD ; D Þ t1;2 tp1;2 ¼ q varðD Þ: ¼ covðl þ q D þ e ; D Þ t2;2 t1;2 tp1;2 2 2 ¼ covðl ; D Þþ q covðD ; D Þ tp1;2 t2;2 tp1;2 2 2 We assume that forecasting customer demands by retailers þ covðe ; D Þ t1;2 tp1;2 are random variables of the form as D ¼ l þ qD þ e , t t1 t ¼ q covðD ; D Þ and the error terms e are identically independent distribution t1;2 tp1;2 t 2 with mean 0 and variance r . Let the estimate of the standard ... deviation of forecast error of the lead time demand be ¼ q covðD ; D Þ tp1;2 tp1;2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 ¼ q varðD Þ: b 2 ðD  D Þ tj tj i¼j b r ¼ C : L;p ðNote that covðl ; D Þ¼ 0 and covðe ; D Þ tp1;2 t1;2 tp1;2 ¼ 0Þ Applying the result proved in Ryan [21], we have covðD ; b r Þ¼ 0; 8i ¼ 1; 2; .. .; p: tj t Hence, Hence, L L 2 2 varðq Þ¼ 1 þ 2 þ 2 varðD Þ t;2 2 varðq Þ p p t;1 2 3 2L L 1 1 1 þ þ 2 varðD Þ 6 1 7 2L L 2 2 p p 2 6 7 þ 2 q varðD Þ 6 7 6 7 p p 1 2 6 7 2L 2L ¼ 1 6 7 2  þ covðD ; D Þ t1;1 tp1;1 6 7 ð1  kÞ L L 2 p p 2 2 2 6 7 þ z varðb r  b r Þ k varðq Þ t;1 6 7 2 t;2 t1;2 4 5 2L "# ! 2 L L L 1 1 1 þ z varðb r  b r Þþ 2z 1 þ covðD ; b r Þ 2 1 t1;1 1 t;1 t1;1 t;1 L L p 2 2 2 3 ¼ varðD Þ 1 þ 2 þ 2 ð1  q Þ 2 2 2L L 2L 2L p p 1 1 1 1 p 1 þ þ 2 varðD Þ þ q varðD Þ 6 1 1 7 1 1 p p p p 6 7 2 L L 2 2 4 5 2 2 ð1  kÞ þ z varðb r  b r Þ k varðq Þ: t;1 2 t;2 t1;2 L L 2 1 1 þ z varðb r  b r Þ 1 t;1 t1;1 L L 2 1 1 b b varðD Þ 2L 2L z varðr  r Þ 1 1 p 1 t;1 t1;1 ¼ 1 þ þ ð1  q Þ þ : 2 2 p p ð1  kÞ ð1  kÞ References 3. The derivation process of the further expression of q : t,2 L L 2 2 1. Bassok Y, Anupindi R, Akella R (1999) Single-period multi- q ¼ 1þ D  D t;2 t1;2 tp1;2 p p product inventory models with substitution. Oper Res 47(4): L L 2 2 632–642 þ z ðb r b r Þ kq ;) varðq Þ 2 t;1 t;2 t;2 t1;2 2. Bitran G, Dasu S (1992) Ordering policies in an environment of L L 2 2 L L 2 2 ¼ var 1þ D  D þ z ðb r b r Þ stochastic yields and substitutable demands. Oper Res 40(5): t1;2 tp1;2 2 t;2 t1;2 p p 177–185 L 3. Chen F, Ryan JK, Simchi-Levi D (2000) The impact of expo- k varðq Þ¼ 1þ varðD Þ t;1 t1;2 nential smoothing forecasts on the bullwhip effect. Naval Res Logist 47(4):269–286 L L 2 2 2 1þ covðD ; D Þ t1;2 tp1;2 4. Chatfield DC, Kim JG, Harrison TP, Hayya JC (2004) The p p bullwhip effect—impact of stochastic lead time, information 2 quality, and information sharing: a simulation study. Prod Oper 2 L2 L2 þ varðD Þþ z varðb r b r Þ tp1;2 2 t;2 t1;2 p Manag 13(4):340–353 5. 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The impact of inventory sharing on the bullwhip effect in decentralized inventory systems

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Springer Journals
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Copyright © 2012 by Springer-Verlag Berlin Heidelberg
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Engineering; Engineering Economics, Organization, Logistics, Marketing; Logistics; Industrial and Production Engineering; Simulation and Modeling; Operation Research/Decision Theory
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1865-035X
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1865-0368
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10.1007/s12159-012-0096-7
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Abstract

Logist. Res. (2013) 6:89–98 DOI 10.1007/s12159-012-0096-7 OR IGINAL PAPER The impact of inventory sharing on the bullwhip effect in decentralized inventory systems • • Dang Van Le Luong Trung Huynh Kifor Vasile Claudiu Muntean Achim Received: 12 April 2012 / Accepted: 21 November 2012 / Published online: 19 December 2012 Springer-Verlag Berlin Heidelberg 2012 Abstract The paper derives the impact of inventory stage supply chain, retailers are the parties who receive sharing policy on the bullwhip effect in two-stage supply customers’ demands directly. To satisfy customers’ ser- chains with two independent suppliers and two integrated vice, usually customer demand is estimated by using retailers. There exists an inventory sharing policy between forecasting techniques before placing order to supplier. The two retailers. Under inventory sharing policy, when lacking of information leads to fluctuation orders from all demand in one retailer exceeds its inventory, this retailer levels of the chain in term of volumes. The fluctuation of can ask for a product sharing volume from the other in customer demand through the chain is well known as the order to satisfy customer demand. With certain assump- bullwhip effect [3, 19]. tions, the bullwhip effect is quantified in both cases, with In Fig. 1, we can see the fluctuation of customers’ inventory sharing policy and without inventory sharing demands through different facility epochs in a four-stage policy. We found that inventory sharing has significant supply chain with single manufacturer, distributor, whole- impact on the bullwhip effect in the supply system. How- saler, and retailer. We may notice that demands fluctuation ever, inventory sharing policy does not synchronously increases from lower levels to higher levels in the chain. reduce or increase the bullwhip effect in both suppliers in The reasons can be explained as follows: the retailer the same period. A numerical example is given to illustrate has directly customer information. Retailer will use this the study model. information to estimate actual demands. To maintain desired service level, retailer needs to hold a certain Keywords Inventory sharing policy  Bullwhip effect  inventory in the warehouse. That leads to the wholesaler Supply chain management  Decentralized inventory  will receive higher original orders from the retailer. Simi- Order lead time larly, wholesaler receives customer information from the retailer and places an order to his supplier, the distributor. To determine the order quantities from retailer, the 1 Introduction wholesaler must forecast customer demand. Unfortunately, the wholesaler does not have access to the customer actual The information about customer demand is varied through information; so that they must use the information from the the levels of the chain due to many factors (e.g., inventory retailer to perform his forecasting. Therefore, the variation policy, forecasting method, order lead time, etc.). In two- of customer demand increases from lower epoch to higher epoch in the chain. In the supply chain system applying inventory sharing D. V. Le (&)  L. T. Huynh policy, distribution centers, wholesalers, and retailers are School of Engineering and Technology, Asian Institute of Technology, Bangkok, Thailand collaborated by sharing product in case of emergency such e-mail: haidang_math1804@yahoo.com that stock out or demand exceeds inventory. Separated inventory of parties in the same levels are virtually com- K. V. Claudiu  M. Achim bined. If one party is in stock out stage, its demands can Faculty of Engineering and Technology, be fulfilled by available inventory in the other retailers. Lucian Blaga University of Sibiu, Sibiu, Romania 123 90 Logist. Res. (2013) 6:89–98 2 Literature review The studies about bullwhip effect have been utilized for several decades. The earliest paper studied on this area is conducted by Forrester [8]. In this paper, the authors the first time pointed out the effects of information variation on production decision. This concept has been noticed and become foundation for research in the field of demand variation as we call ‘‘the bullwhip effect.’’ Main objectives of previous papers mainly focus on three aspects: (a) demonstrating the existence of the bull- whip effect, (b) identifying possible causes of the bullwhip effect, and (c) developing strategy to reduce the impact of the bullwhip effect [3]. Following the stream, our paper can be classified into the second areas, that is, identifying possible causes of the bullwhip effect. This researched area has been studying by many researchers. For instance, Lee et al. [20] pointed out in their papers four major causes of the bullwhip effect including: demand forecast updating, order batching, price fluctuation, rationing and shortage gambling. They also presented several methods to coun- Fig. 1 Variation of customer demand in a supply chain system teract the bullwhip effect such as integrating new infor- mation systems, defining new organizational relationships, This cooperation especially profits integrated supply chain and implementing new incentive and measurement system (the chain in which manufacturers and retailers are systems. dependent). For a supply chain with independent facilities, The effect of forecasting methods on the bullwhip effect inventory sharing policy sounds like a game changer and is also the main topic in many studies. Chen et al. [3] this policy would be more efficient in the centralized studied the impact of different forecasting methods on the information system. bullwhip effect in a two-stage supply chain. They con- Inventory sharing can be profited for whole supply chain cluded that exponential smoothing forecasting method system including: manufacturers, distributors, retailers, and gives higher bullwhip effect than moving average fore- customers. Manufacturers can improve brand reputation, casting method. In the same sense, Graves [25], Xu et al. increase manufacturing efficiency, and reduce unwanted [19] and Zhang [30] presented the effect of demand fore- inventory. For distributors/retailers, inventory sharing casts on the bullwhip effect in a two-stage supply chain reduces the number of lost orders and backorders, provide a system with integrated moving average demand process. new outlet for slow moving inventory, and increase Furthermore, Sun and Ren [13], Zhang [30] studied the possibility for incremental revenue (reduce inventory while impact of different forecasting methods such as MA, ES, remaining service level). Finally, customers also profit EWMA, MMSE, and suggested the forecast method which from inventory sharing in which it is an efficient strategy can mitigate the bullwhip effect. that improves product availability and reduces delivering The impact of order lead time was presented in the time. papers of Graves [25], Chen et al. [9], Zhang [30], Lee However, inventory sharing policy does not always et al. [20]. Luong and Phien [11] proved that the bullwhip guarantee making benefit for retailers. In case of tran- effect can be decreased by reducing lead time. However, shipment, reallocation, and penalty costs are too high, Duc et al. [28] showed that reducing lead time does not retailers will try to reduce the probability of stock out by always reducing the bullwhip effect. He showed that in increasing inventory level. In addition, if the ordered lead some special cases; for example, in a two-stage supply time and order duration period time from suppliers to chain with a pre-specified ARMA demand process, retailers is short, retailers may prefer waiting for new increasing the lead time may help to reduce the bullwhip delivering packages instead of receiving sharing packages effect. For more detail, we refer to [28]. When considering from other retailers. Therefore, the efficiency of inventory the impact of lead time on the bullwhip effect, the lead sharing strategy depends on geography factors, features of time can be deterministic or stochastic. Results in those system, products, and services. papers mentioned above are in case of deterministic lead 123 Logist. Res. (2013) 6:89–98 91 time, whereas in practical lead time usually behaves as a inventory sharing leads to decreased dealers’ rationing stochastic process. level rather than increasing their based stock level, (3) a Chaffied [4] and So and Zheng [18] used simulation smaller level of incentive for inventory sharing may be approaches to demonstrate the impact of lead time varia- sufficient to achieve the benefit of full inventory sharing tion and information sharing on the customer demand policy, (4) the benefit of inventory sharing increase the fluctuation in a supply chain. Other results in this area are system utilization, and (5) customer service may improve contented in [15, 16, 28]. significantly with inventory sharing. Li et al. [31] analyzed the impact of demand substitution Recently, Kutanoglu [7] considered a model to allocated on the bullwhip effect in a two-stage supply chain with a stock level in the warehouse in a service part logistic net- singer supplier, singer retailer, and two types of products A work. The network includes one supplier with the infinitive and B such that a certain fraction product A can be used to warehouse capacity and a number of local warehouses. substitute product B. They showed the relation between the Each local warehouse has independent based stock policy. bullwhip effect and the forecasting method, lead time, Moreover, local warehouses share their inventory as a way demand process, and the product substitution. The impact to increase service levels. They concluded that inventory of demand substitution has been noticed and investigated sharing can reduce based stock levels and total system cost. by some previous authors [1, 2, 5, 21, 22, 24]. As mention above, the principle of the bullwhip effect is Inventory sharing has become important perception in the variation of customer demand through the chain. some supply chain models. For example, in the supply Meanwhile, inventory sharing has significant impact on chain with singer supplier and multi-independent retailers, inventory levels and order quantities as well. That results in the delivering time from suppliers to retailers’ warehouses the change in customer information. Previous researches take a long time because of the long distance between mainly focus on various areas of the bullwhip effect and suppliers and retailers, whereas retailers may located very inventory sharing. However, there are no works directly closed in one area. There is probability that inventory of focusing on the impact of inventory sharing on the bull- one retailer exceeds customer demand, while others are in whip effect yet. In this paper, we will derive this issue and stock out state. In this situation, stock out can be satisfied examine the impact of inventory sharing policy on the by transferring products among retailers through inventory bullwhip effect. The remaining of the paper is organized as sharing policy. follows: Sect. 3 describes the problem and develops The concept of third-party warehouse is created aiming mathematical formulation; Sect. 4 gives a numerical at identifying inventory policy and customer information. example, results, and discussion. Conclusions and recom- It may help to decrease total inventory at retailers. In this mendations for further study is the content of the last area, Duc et al. [29] studied the effect of third-party section. warehouse on the bullwhip effect. They found that third- party warehouse does not always reducing the bullwhip effect. They also stated the conditions in which the utili- 3 Model development zation of third-party warehouse decreases the bullwhip effect in supply chain. 3.1 Notations and assumptions Rudi [23] studied on the relation among following fac- tors: inventory sharing, transhipment cost, and inventory The following notations are used in this paper orders in the supply chain with one supplier and two local t order period number index; retailers. They pointed out that inventory sharing and i retailer and corresponding supplier index, i = 1, 2; transhipment costs have significant effects on inventory order at each retailer. A case of study was conducted in k percentage of product that one retailer may share to other retailers in the period t; which a two-stage supply chain with singer supplier, Bosch based in Germany, and five retailers based in Norway. In L the order lead time for retailer i, i = 1, 2; p the number of demand observation periods used in this supply chain system, the ordered delivering time from Germany to Norway takes about 3 weeks, while tranship- the moving average forecast; l average demand of product at retailer i in the ment time within Norway is insignificant. This supply system is more similar with the supply chain model in our autoregressive demand model; e forecast error for product at retailer i during time paper. t,i Zhao [14] presented an optimal inventory policy for period t; q the autocorrelation coefficient of the autoregressive each dealer in decentralized dealer networks. They con- model of product at retailer i; q \1; q \1 cluded that (1) inventory sharing has big impact on the jj jj 1 2 level of inventory by the independent dealer, (2) increasing D product demand at retailer i during time period t; t,i 123 92 Logist. Res. (2013) 6:89–98 lead time demand for product at retailer i; t;i b the forecast of the lead time demand product at t;i retailer i; z normal z-score determined by the desire service level; standard deviation of forecast error of lead time t;i demand for product at retailer i; y order-up-to level inventory for retailer i at the t,i beginning of time period t; q order quantity product for retailer i at the beginning t,i of time period t; B the bullwhip effect at supplier i; C a constant function of Land q L,q The proposal model is studying under following assumptions Fig. 2 Relationship among parties in the chain As1 Order lead time of each retailer is smaller than duration time of one period ðL \rðtÞ; i ¼ 1; 2Þ: As2 Time period index t and duration of order period are equal for both retailers As3 In one order period, only one retailer agree share inventory, while the other will receive this amount of inventory As4 Inventory sharing percentage is varying from period to period As5 In every period, order from each retailer is positive 3.2 Problem description In this paper, we consider a two-stage supply chain with two suppliers and two retailers with separate markets (each retailer has their own customer). A single product is delivered from suppliers to retailers following discrete time period. The relationship among the parties in the chain is rep- resented in Fig. 2. That is, whenever one retailer is in stock out state, they can ask for transhipment from the other retailer. Whether transhipment is made depending on predetermine conditions stated in the sharing policy. The process of satisfying customer demand with inventory sharing policy is illustrated in Fig. 3. We assume that order lead times for both retailers (time from placing an order until receiving the order) are deter- Fig. 3 Progress of satisfying customer demand ministic. We support that both retailers use the same forecasting method (MA) and inventory policy (order-up-to level). Since inventory sharing policy could affect order forecast, q is order quantity from retailer i placed at sup- quantities that a retailer placing on its supplier, we will plier i, i = 1, 2. consider the impact of inventory sharing on the bullwhip effect of each retailer. The bullwhip effect can be deter- 3.3 Mathematical formulation mined by identifying the ratio of variance of retailer’s order to the supplier to the variance of customer’s demand Support at the beginning of order period t, both retailer (1) varðq Þ to the retailers , in which D is customer demand and retailer (2) estimate customer demand and place an varðD Þ 123 Logist. Res. (2013) 6:89–98 93 order to its supplier. The order cost is negligible. From 3.4 Bullwhip effect quantify Zhang [30] and Gilbert [10], end customer demand can be modeled by an autoregressive (AR). That means end cus- Since the variance of q and D is different in each retailer, t,i i tomer demand at each retailer during time period t can be the bullwhip effect at each supplier will be discussed separately. Given the equations of the order-up-to level, determined as follows: demand forecasting, and standard deviation of forecast D ¼ l þ q D þ e ; t;1 t1;1 t;1 1 1 error, q can be expressed as follows: t,1 ð1Þ D ¼ l þ q D þ e : t;2 t1;2 t;2 2 2 q ¼ y  y þ D þ kq t;1 t;1 t1;1 t1;1 t;1 From (1), we have (See ‘‘Appendix’’ for details) ¼ ½y  y þ D t;1 t1;1 t1;1 1  k 2 L L L L 1 1 1 1 b b l r ¼ ½ðD þ z b r ÞðD þ z b r Þþ D 1 1 t1;1 1 t;1 t;1 t1;1 t1;1 1  k EðD Þ¼ ; varðD Þ¼ ; t;1 t;1 1  q 1  q 1 L L 1 1 1 1 L L 1 1 ð2Þ ¼ 1 þ D  D þ z b r  b r : t1;1 tp1;1 1 t;1 t1;1 1  k p p l r 2 2 EðD Þ¼ ; varðD Þ¼ t;2 t;2 ð8Þ 1  q 1  q 2 2 Then, the variance of the order quantity q at supplier (1) At the beginning of time period t, the actual inventory level t,1 at time period t is as follows: at retailer (i)is y  D . Retailer (i) will order amount t1;i t1;i of product in order to reach the target inventory level y . 1 L L t,i 1 1 varðq Þ¼ var 1 þ D  D t;1 t1;1 tp1;1 Taking inventory sharing into consideration, the quantity 1  k p p for the demand at each retailer at the beginning of time L L 1 1 þz ðb r  b r Þ t;1 t1;1 period t can be determined as varðD Þ 2L 2L 1 1 p ¼ 1 þ þ ð1  q Þ q ¼ y ðy  D Þþ kq t;1 t;1 t1;1 t1;1 t;1 1 2 2 p p ð1  kÞ ¼ y  y þ D þ kq ; t;1 t1;1 t1;1 t;1 1 L L 1 1 þ varðb r  b r Þ: ð9Þ 2 t;1 t1;1 or ð1  kÞ 1 For the variance of the order quantity q at supplier (2) at t,2 q ¼ ½y  y þ D ; ð3Þ t;1 t;1 t1;1 t1;1 time period t, we have 1  k q ¼ y  y þ D  kq t;2 t;2 t1;2 t1;2 t;1 and L L L L 2 2 2 2 ^ ^ ¼ðD þ z r ^ ÞðD þ z r ^ Þþ D  kq 2 2 t1;2 t;1 t;2 t;2 t1;2 t1;2 q ¼ y ðy  D Þ kq t;2 t;2 t1;2 t1;2 t;1 L L 2 2 L L 2 2 ð4Þ ¼ 1 þ D  D þ z ðr ^  r ^ Þ kq : t1;2 tp1;2 2 t;1 t;2 t1;2 ¼ y  y þ D  kq : p p t;2 t1;2 t1;2 t;1 The target inventory y at the beginning of period t is t,i Then, the variance of the order quantity q time period t is t,2 estimated from observed demand as as follows: L L varðq Þ 1 1 t;2 y ¼ D þ z b r ; t;1 1 t;1 t;1 ð5Þ L L 2 2 L L 2 2 L2 L 2 ¼ var 1 þ D  D þ z b r  b r t1;2 tp1;2 2 y ¼ D þ z b r : t;2 t1;2 t;2 2 t;2 t;2 p p k varðq Þ t;1 Both retailers use simple moving average technique to 2 2 Li i 2L 2L 2L 2L 2 2 2 2 p estimate D and r based on the information of the past ¼ 1 þ þ  þ :q :varðD Þ t;i t;i 2 2 2 p p p p p periods. We have 2 L L 2 2 2 þ z var b r  b r  k varðq Þ: ð10Þ t;1 P 2 t;2 t1;2 tj;i L j¼1 D ¼ L ; ð6Þ t;i We denote B , B are the bullwhip effect of supplier (1) and 1 2 supplier (2) in period t, respectively. We have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 ðD  D Þ varðq Þ tj;i tj;i t;1 j¼1 B ¼ r ¼ C : ð7Þ 1 L ;q t;i i varðD Þ 1 2L 2L 1 p where D  D is the forecast error of the (t - j)th tj;i tj;i ¼ 1 þ þ ð1  q Þ 2 2 p p ð1  kÞ period at retailer (i) and C is a constant function of L ;q L and q [26].The bullwhip effect can be calculated as L L i 1 1 1 þ varðb r  b r Þ: ð11Þ 2 t;1 t1;1 varðq Þ t;i ð1  kÞ B ¼ . varðD Þ 123 94 Logist. Res. (2013) 6:89–98 varðq Þ t;2 will not focus in determine absolutely bullwhip effect in B ¼ varðD Þ each manufacturer but compare the impact of inventory sharing policy on the bullwhip effect and the variation of 2L 2L 2 p 2 L L 2 2 ¼ 1 þ þ ð1  q Þ þ z varðb r  b r Þ 2 t;2 t1;2 2 bullwhip effect with some related parameters. For conve- p p nient purpose, bullwhip effect from each retailer (com- k 2L 2L varðD Þ 1 1 1 p 1 þ þ ð1  q Þ pany) can be written as bellow: 2 2 p p varðD Þ ð1  kÞ 2 varðq Þ L L 2 2 2 t;1 varðb r  b r Þ z k B ¼ t;2 t1;2 1 1 : ð12Þ varðD Þ varðD Þ ð1  kÞ 2 1 2L 2L 1 p ¼ 1 þ þ ð1  q Þþ h ; 2 2 p p ð1  kÞ 4 Numerical example and analysis 2 2 2L 2L k 2 p B ¼ 1 þ þ ð1  q Þþ h  ½ w ; 2 2 p p ð1  kÞ From (11) and (12), we can see that B and B are 1 2 functions of the following parameters: p- the number where of observation use in MA; L , L -the order lead time; 1 2 L L 2 1 1 h ¼ z varðb r  b r Þ [ 0; q , q -first order autocorrelation coefficients of the auto- 1 1 2 1 t;1 t1;1 regressive demand process at retailer (1) and retailer (2), L L 2 2 h ¼ z varðb r  b r Þ [ 0; 2 t;2 t1;2 respectively; k-inventory sharing percentage. In this sec- 2L 2L varðD Þ tion, we will provide a numerical example to illustrate the 1 1 1 p w ¼ 1 þ þ ð1  q Þ impact of those parameters on the bullwhip effect on both p p varðD Þ retailers. L L 2 2 varðb r  b r Þ t;2 t1;2 z [ 0: varðD Þ 4.1 System description 4.2 Results and discussion Two trading companies A and B has center in North and South Vietnam, respectively. Company A imports steel 4.2.1 No inventory sharing (k = 0) from manufacturer (1) which has center in Italy while company B imports steel from manufacturer (2) which has If k = 0, that is, there is no inventory sharing policy center in Japan. Estimated order lead time of company A between the two companies, B and B are determined as 1 2 is 45 days, estimated order lead time of company B is follow: 25 days. Both company A and B conduct forecasting and 2L 2L placing order two times per year (2 periods per year, each 1 p B ¼ 1 þ þ ð1  q Þ þ h ; 1 1 p p period has duration of 6 months). The order is placed at the beginning of each period. In Vietnam, construction is 2L 2L 2 p B ¼ 1 þ þ ð1  q Þ þ h : highly depending on the weather. In addition, the weather 2 2 p p in North Vietnam and South Vietnam is quite different due to geometry position. To reduce risk of overstock, stock out In this case, we may consider the chain as two independent as well increase customer service, both companies have supply chains with single manufacturer, company, and signed an inventory sharing contract. The condition in the customer. These results are identified with results of Li contract is revised and signed before every order period. [31]. In Chen et al. [3], the authors pointed out the effects The content of the contract state that in certain period, one of common parameters in the bullwhip effect. The details company will agree to share k percentage of its inventory are as follows: to the other in case this company is in stock out state and • The bullwhip effect is a decreasing function of p, the other sharing conditions are satisfied. To enhance quality of number of observations use in MA. forecast technique, both companies use 3 demand obser- • The bullwhip effect is an increasing function of L, the vation periods in the moving average forecast (p = 3). lead time. Demand observations used in forecast are corresponded to • The bullwhip effect is a decreasing function of q when time of years. That means to forecast demand for spring q [ 0 and larger for odd values of p than for even period, only demand observations of spring period in the values of p, when q \ 0. pass is applied and similar for autumn period. The auto- correlation coefficient of the autoregressive model of With the given information and relevant data, bullwhip product each company is q = q = 0.5. In this paper, we effect in both manufacturer can be calculated as 1 2 123 Logist. Res. (2013) 6:89–98 95 B ¼ 420:875 þ h ; ð13Þ B ¼ 136:9 þ h : ð14Þ 4.2.2 Company A shares inventory to company B (k [ 0) In case of k [ 0, we mean that company A will deliver amount of inventory, kq to the warehouse of company B (if sharing conditions are satisfied) in the period t. Then, the bullwhip effect at manufacturer (1) and (2) are determined as: 1 420:875 þ h B ¼  B ¼ ð15Þ 2 2 ð1  kÞ ð1  kÞ 2 2 k k B ¼ B  ½w¼ð136:9 þ h Þ ½wð16Þ 2 2 2 2 2 ð1  kÞ ð1  kÞ • From (13) and (15), it is clear that the bullwhip effect at manufacturer (1) in case of inventory sharing is higher Fig. 4 Variation of the bullwhip effect in manufacturer (1) with than that in case of without sharing (since [ 1), ð1kÞ inventory sharing percentage and the bullwhip effect is an increasing function of k. That means, if amount of delivered inventory increase, the bullwhip effect at manufacturer (1) also increase. • From (14) and (16), the bullwhip effect at manufacturer (2) in case of sharing is smaller than that in case of without sharing. Furthermore, the bullwhip effect quantity is a deceasing function of k which means that higher transhipment company B receive from company A, smaller the bullwhip effect at manufacture (1). The reason is that the variation of customer demand at supplier is a result of changing customer information through middle parties such as retailer, wholesaler. Because those parties always desire to keep desirable service level, the order quantity placing at suppliers are usually higher than actual demands. When one retailer is expected to receive a certain amount of goods from another, the order quantity this retailer place at the supplier will reduce and closer to the actual demands. That leads to the bullwhip effect reduces. The variation of bullwhip effect with inventory sharing percentage in each manufacturer is given in Figs. 4 and 5. Herein, we assign h ¼ h ¼ w ¼ 1. 1 2 Fig. 5 Variation of the bullwhip effect in manufacturer (2) with Because the role of each party in the chain is equivalent, inventory sharing percentage we can refer that in case of company B agree to share its inventory to company A (k \ 0) the result is that when jj k bullwhip effect by comparing the bullwhip effects at both increases, the bullwhip effect at manufacturer (1) decrea- suppliers in case of no inventory sharing policy with that in ses, while the bullwhip effect at manufacturer (2) increases. case of using inventory sharing policy. The studying supply chain system includes two suppliers, retailers, and markets. 5 Conclusions and recommendations By constructing the bullwhip effect at each supplier for different study situations, we found that inventory sharing In this paper, we considered the impact of inventory policy has significant impact on the bullwhip effect in each sharing on decentralized warehouses system on the supplier. The variation of the bullwhip effect in each 123 96 Logist. Res. (2013) 6:89–98 supplier depends on the destination of transferring inven- Similarly, we have tory. In details, the bullwhip effect at one supplier will EðD Þ¼ ; increase if its retailer is the one that received transferring 1  q inventory. In addition, the higher amount of receiving varðD Þ¼ : inventory of the retailer, the bigger the bullwhip effect in 1  q the corresponding supplier. In other words, inventory sharing reduces the bullwhip effect at this supplier but also 2. The derivation process of the further equation of q . t,1 increases the bullwhip effect at the other supplier. q ¼ y  y þ D þ kq ; t;1 t;1 t1;1 t1;1 t;1 According to the finding of this paper, supply chain managers are helpful in forecasting amount of goods that ) q ¼ y  y þ D t;1 t;1 t1;1 t1;1 1 k need to supply for their retailers. Also supply chain man- hi L L L L 1 1 1 1 b b ¼ D þ z b r  D þ z b r þ D 1 1 t1;1 agers can adjust inventory sharing policy in order to trade- t;1 t;1 t1;1 t1;1 1  k hi off the bullwhip effect for both suppliers. In many cases, L L L L 1 1 1 1 b b ¼ D  D þ D  z b r b r t1;1 1 t;1 t1;1 t;1 t1;1 inventory sharing profits for the whole supply system, 1  k p p therefore, determining the impact of inventory sharing on X X ¼ D  D ti;1 t1i;1 the bullwhip effect will further support for the development pð1  kÞ i¼1 i¼1 of inventory sharing models in supply chain. D z t1;1 1 L L 1 1 þ þ b r b r This paper can be extended through three directions. One t;1 t1;1 1  k 1 k is that we can study the impact of inventory sharing on the L D t1;1 ¼ D  D þ t1;1 tp1;1 bullwhip effect in a supply chain with only one supplier and pð1  kÞ 1  k two or multi-retailers. With inventory sharing, total product L L 1 1 þ b r b r t;1 t1;1 quantity from retailers may differ from that in case of without 1 k 1 L L inventory sharing. Another side direction would be extending 1 1 ¼ 1 þ D  D t1;1 tp1;1 the model to multi-stage supply chain, and inventory sharing 1  k p pðÞ 1  k L L is applied in different levels in the chain. For the third 1 1 þ b r b r ; t;1 t1;1 1 k direction, we can study this model with multi-type products. 0  1 1 L L 1 1 Acknowledgments The authors would like to thank the referees for 1 þ D  D t1;1 tp1;1 B C 1  k p pð1  kÞ B C their valuable comments and suggestions. varðq Þ¼ var t;1 @ A L L 1 1 þ b r  b r t;1 t1;1 1  k 0 1 Appendix 2 1 þ varðD Þ t1;1 B C B C B C 1. The derivation process of E(D ) and varðD ÞWhen t,i t;i B C B L L C 1 1 the autoregressive demand process is stationary, we B C 2 1 þ covðD ; D Þ t1;1 tp1;1 B C p p have 1 B C B C B 2 C ð1  kÞ EðD Þ¼ EðD Þ¼ EðD Þ¼ ... ¼ EðD Þ; L t;i t1;i t2;i i B 1 C 2 L L 1 1 B þ varðD Þþ z varðb r  b r ÞC tp1;1 1 t;1 t1;1 B p C B C and B  C @ A 2L þ 2z 1 þ covðD ; b r Þ varðD Þ¼ varðD Þ¼ varðD Þ¼ ... ¼ varðD Þ; 1 t1;1 t;i t1;i t2;i i t;1 0 1 2L 2L where 1 1 B C 1 þ þ varðD Þ B C p p B C D ¼ l þ q D þ e ; t;1 t1;1 t;1 1 1 B C B C B 2L 2L C EðD Þ¼ Eðl Þþ q EðD Þþ Eðe Þ 1 t;1 1 1 t1;1 t;1 B C þ covðD ; D Þ 1 t1;1 tp1;1 B C p p ¼ : EðD Þ¼ l þ q EðD Þþ 0 2 B C 1 1 1 1 ðÞ 1  k B C B 2 L L C l 1 1 þ z varðb r  b r Þ B C 1 t;1 t1;1 ) EðD Þ¼ : B C 1  q 1 B C @ A 2L 2 L þ 2z 1 þ covðD ; b r Þ varðD Þ¼ varðl Þþ q varðD Þþ varðe Þ t;1 t1;1 t;1 1 t1;1 1 1 t;1 2 2 varðD Þ¼ 0 þ q varðD Þþ r 1 1 1 1 Now we will determine covðD ; D Þ and t1;1 tp1;1 ) varðD Þ¼ : 1 L covðD ; b r Þ We have 1  q t1;1 t;1 123 Logist. Res. (2013) 6:89–98 97 L L 2 2 covðD ; D Þ t1;1 tp1;1 ¼ 1 þ 2 þ 2 varðD Þ p p ¼ cov ðl þ q D þ e Þ; D 1 1 t2;1 t;1 tp1;1 2L L 2 2 2 þ 2ð Þ covðD ; D Þ ¼ cov l ; D þ q cov D ; D t1;2 tp1;2 1 tp1;1 1 t2;1 tp1;1 p p þ cov e ; D : t;1 tp1;1 2L 2 L L 2 L 2 2 2 þ z varðb r  b r Þþ 2z 1 þ covðD ; b r Þ 2 t1;2 2 t;2 t1;2 t;2 ðSince cov l ; D ¼ 0 and cov e ; D ¼ 0Þ; tp1;1 t;1 tp1;1 k varðq Þ: t;1 covðD ; D Þ¼ q cov D ; D t1;1 tp1;1 t2;1 tp1;1 Furthermore, we have covðD ; b r Þ¼ 0 and ... t1;2 t;2 ¼ q cov D ; D tp;1 tp1;1 covðD ; D Þ t1;2 tp1;2 ¼ q varðD Þ: ¼ covðl þ q D þ e ; D Þ t2;2 t1;2 tp1;2 2 2 ¼ covðl ; D Þþ q covðD ; D Þ tp1;2 t2;2 tp1;2 2 2 We assume that forecasting customer demands by retailers þ covðe ; D Þ t1;2 tp1;2 are random variables of the form as D ¼ l þ qD þ e , t t1 t ¼ q covðD ; D Þ and the error terms e are identically independent distribution t1;2 tp1;2 t 2 with mean 0 and variance r . Let the estimate of the standard ... deviation of forecast error of the lead time demand be ¼ q covðD ; D Þ tp1;2 tp1;2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 ¼ q varðD Þ: b 2 ðD  D Þ tj tj i¼j b r ¼ C : L;p ðNote that covðl ; D Þ¼ 0 and covðe ; D Þ tp1;2 t1;2 tp1;2 ¼ 0Þ Applying the result proved in Ryan [21], we have covðD ; b r Þ¼ 0; 8i ¼ 1; 2; .. .; p: tj t Hence, Hence, L L 2 2 varðq Þ¼ 1 þ 2 þ 2 varðD Þ t;2 2 varðq Þ p p t;1 2 3 2L L 1 1 1 þ þ 2 varðD Þ 6 1 7 2L L 2 2 p p 2 6 7 þ 2 q varðD Þ 6 7 6 7 p p 1 2 6 7 2L 2L ¼ 1 6 7 2  þ covðD ; D Þ t1;1 tp1;1 6 7 ð1  kÞ L L 2 p p 2 2 2 6 7 þ z varðb r  b r Þ k varðq Þ t;1 6 7 2 t;2 t1;2 4 5 2L "# ! 2 L L L 1 1 1 þ z varðb r  b r Þþ 2z 1 þ covðD ; b r Þ 2 1 t1;1 1 t;1 t1;1 t;1 L L p 2 2 2 3 ¼ varðD Þ 1 þ 2 þ 2 ð1  q Þ 2 2 2L L 2L 2L p p 1 1 1 1 p 1 þ þ 2 varðD Þ þ q varðD Þ 6 1 1 7 1 1 p p p p 6 7 2 L L 2 2 4 5 2 2 ð1  kÞ þ z varðb r  b r Þ k varðq Þ: t;1 2 t;2 t1;2 L L 2 1 1 þ z varðb r  b r Þ 1 t;1 t1;1 L L 2 1 1 b b varðD Þ 2L 2L z varðr  r Þ 1 1 p 1 t;1 t1;1 ¼ 1 þ þ ð1  q Þ þ : 2 2 p p ð1  kÞ ð1  kÞ References 3. 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Logistics Research – Springer Journals

Published: Dec 19, 2012

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