Bullwhip and Reverse Bullwhip Effects Under the Rationing Game

Transcription

1 Bullwhip and Reerse Bullwhip Effects Under the Rationing Game Ying Rong Antai College of Economics and Management, Shanghai Jiao Tong Uniersity, Shanghai, China , Lawrence V. Snyder Department of Industrial and Systems Engineering, Lehigh Uniersity, Bethlehem, PA 805, Zuo-Jun Max Shen Department of Industrial Engineering and Operations Research, Berkeley, CA 94720, When an unreliable supplier seres multiple retailers, the retailers may compete with each other by inflating their order quantities in order to obtain their desired allocation from the supplier, a behaior known as the rationing game. In this paper, we proide the formal condition of the existence of the bullwhip effect (BWE) under the rationing game when the mean demand changes oer time. Moreoer, when the capacity information is shared and the capacity reseration mechanism is applied, we proide the condition when the reerse bullwhip effect (RBWE) occurs upstream, a consequence of the disruption caused by the supplier. In addition, we show that the smaller unit reseration payment leads to the seere [R]BWE. Finally, we find that capacity information sharing does not necessarily mitigate the [R]BWE and that it may reduce the profitability of the supply chain as a whole. Key words : rationing game, bullwhip effect, reerse bullwhip effect, supply uncertainty, order ariance. Introduction U.S. firms hae reduced their supplier base significantly since the late 980s (Trent and Monczka 998). In many industries, a majority of the total olume of raw materials is sourced by only a few suppliers (Carbone 999). Although this consolidation may be beneficial in a stable business enironment, the associated supply uncertainty, especially when coupled with demand uncertainty, produces increasingly significant challenges as the importance of each supplier increases (Tang 2006). If a supplier seres seeral retailers (or other clients), a capacity disturbance at the supplier affects all its clients. Sheffi (2005) gies seeral examples of this, including the Taiwan Semiconductor Manufacturing Company (TSMC), which was affected by an earthquake in 999 and which Electronic copy aailable at:

2 2 Rong, Snyder and Shen: BWE & RBWE under Rationing Game proides chips for numerous leading computer companies, and Philips, whose plant fire in 2000 affected supplies for both okia and Ericsson. Another example is that Hurricane Katrina affected many chemical firms, including DuPont and Cheron, whose products are necessities for many of their customers (Storck 2005). Moreoer, the retailers must simultaneously cope with the demand uncertainty produced by their own customers. When the supplier s capacity is insufficient to meet the total demand, a natural response from the retailers is to inflate their order quantities to try to obtain a larger piece of the pie. In this paper, we proe that, under this so-called rationing game, the supply end of the system experiences the reerse bullwhip effect (RBWE), in which order ariance increases as one moes downstream in the supply chain, and that the demand end of the system experiences the classical bullwhip effect (BWE), in which order ariance increases in the opposite direction. When the BWE and RBWE both occur, the middle stages of the supply chain suffer the most from the two types of uncertainty. Our research proides a theoretical explanation to the obseration by Cachon et al. (2007) from raw industry-leel data that the middle echelon of many supply chains has the greatest order olatility, contradicting the conentional wisdom that the BWE should cause the olatility to be greatest at the upstream end. Moreoer, our research highlights the importance of treating the supply chain as an integrated system rather than as a collection of isolated players, since any type of uncertainty-supply or demand, and anywhere it occurs in the supply chain, can be magnified for the remaining parties. When supply uncertainty exists, the presence or absence of capacity information is often a key determinant of retailers order-quantity decisions. Under preailing information-sharing programs such as collaboratie planning, forecasting, and replenishment (CPFR), supply and demand information flows in both directions in many supply chains. Moreoer, real-time supply information sharing may be particularly important when capacity is a major limiting factor. Sheffi (2005, page 7) proides a iid example of okia demanding to be informed of Philips s supply status on a timely basis after its plant fire in Electronic copy aailable at:

3 Rong, Snyder and Shen: BWE & RBWE under Rationing Game 3 Furthermore, capacity reseration is another common practice when capacity is a bottleneck (Wu et al. 2005). A typical example is that Apple offered its suppliers upfront cash payments in an effort to secure its order allocation after the Japanese earthquake in 20 (AppleInsider 20). In addition, capacity reseration can sere as a compensation to the supplier for sharing its capacity information. In model-r (for reseration ), we introduce capacity information sharing coupled with a reseration payment in order to reduce oer-ordering under tight capacity. We show that there always exists a E, regardless of the size of the reseration payment (proided that it is strictly positie). We characterize the conditions under which the BWE and RBWE do occur, and proe that the smaller the reseration payment is, the more frequently and seerely the BWE and RBWE occur. We also consider a benchmark model, which we call model-l (for Lee et al. (997), and hereinafter referred to as LPW ), to ealuate the benefit of capacity information sharing and capacity reseration. This model is equialent to the seminal rationing game model studied by LPW, in which the retailers place their orders before knowing the realized capacity. LPW use this model to show that the retailers orders are inflated and to argue for the existence of the BWE when the demand mean changes oer time. They also point out, among other adocates(e.g. Chen et al. 2000), that capacity information sharing and capacity reseration are among remedies to mitigate the BWE. Howeer, by comparing model-l and model-r, we demonstrate that these two treatments may trigger een greater retailer order ariability and reduce the profit of the whole supply chain under certain circumstances. Our work also suggests that the insights deried under demand uncertainty may not always apply under supply uncertainty, which is similar in spirit to the inentory placement problem in a network with disrupted supply studied by Snyder and Shen (2006). The remainder of this paper is organized as follows. In Section 2, we relate our paper to the existing literature on the subject. In Section 3, we outline the basic assumptions for our two analytical models. Sections 4 and 5 proide analyses of the order ariability under model-l and

4 4 Rong, Snyder and Shen: BWE & RBWE under Rationing Game model-r, respectiely. We illustrate the impact of information sharing and capacity reseration in Section 6. Section 7 concludes with some final remarks on the BWE and RBWE. 2. Literature Reiew The BWE was first described by Forrester (958), though the term bullwhip effect was introduced to the literature by LPW. LPW s paper was the first to proide a theoretical understanding of the BWE. They suggest four mechanisms that can trigger the BWE: demand forecasting, rationing game, order batching, and price fluctuations. They introduce analytical models to study demand olatility propagation under all four settings. The subsequent theoretical research on the BWE generally concentrates on exploring additional causes of the BWE, measuring the seerity of the BWE, and analyzing mitigation strategies. The reader is referred to Lee et al. (2004) and Geary et al. (2006) for more thorough reiews of the literature on the BWE. Most of the research on the BWE considers demand uncertainty and ignores supply uncertainty. Of course, both supply and demand uncertainty are present in most supply chains, and Snyder and Shen (2006) show that the insights gained from the study of one type of uncertainty often do not apply to the other. Therefore, when we examine the order ariance at one stage of the supply chain, we need to consider the effect of both types of uncertainty. Although empirical studies conducted by Baganha and Cohen (998) and Cachon et al. (2007) indicate that the BWE does not dominate upstream, there is still ery little theoretical analysis of the impact of supply uncertainty in the context of the BWE, especially at the stage closest to the source of the supply uncertainty. Our paper explicitly considers the interaction between the two types of uncertainty in creating the BWE and RBWE. Moreoer, we demonstrate that capacity information sharing does not necessarily mitigate the [R]BWE, although demand information sharing is a common practice to mitigate the BWE. Cachon et al. (2007) explain the non-dominance of the BWE by the seasonality of the demand. By remoing the seasonality, their data show that there is still a significant portion of industries in the upstream part of the supply chain not exhibiting the BWE. Chen and Lee (2009) argue that the aggregate data at a macro-leel may underestimate the magnitude of the BWE when it exists but that using industry-leel data does not affect the estimation of the existence of the BWE.

5 Rong, Snyder and Shen: BWE & RBWE under Rationing Game 5 The present paper and the papers by Rong et al. (2009, 20) are the first to inestigate the RBWE. Rong et al. (2009) present a behaioral study that uses a ariant of the beer game in which the manufacturer faces supply uncertainty, in contrast to the ample-supply assumption in the traditional ersion of the game (e.g., Sterman 989). Rong et al. (20) inestigate how price fluctuations that result from supply uncertainty trigger the RBWE when the firm does not possess a correct model of customer behaior. The present paper demonstrates that competition for scarce resources is an operational cause of the RBWE. The roots of our models can be found in the literature on the rationing game. Our setting is closest to the rationing game model in Lee et al. (997) (LPW), who also study multiple retailers sharing a common unreliable supplier. Moreoer, we show that under LPW s original model (called model-l in our paper), the BWE may not occur. In addition, we proide the formal conditions of the existence of the BWE under both model-l and model-r as well as the existence of the RBWE under model-r. Cachon and Lariiere (999a,c) model a rationing game with two retailers, deterministic capacity at the common supplier and a linear demand function in the sales price with two demand states. Cachon and Lariiere (999c) compare the linear, proportional, and uniform allocation rules. They find that an E may not exist under the linear and proportional rules, while it always does under the uniform rule. Cachon and Lariiere (999a) extend the rationing game into two periods and study an allocation rule based on past sales ( turn-and-earn ), as opposed to a fixed allocation. They demonstrate that the supplier always benefits from turn-and-earn since the retailers increase their order quantities. Lu and Lariiere (20) extend the work by Cachon and Lariiere (999a) to an infinite-horizon setting and multiple demand states. Their numerical study shows that turnand-earn may induce the retailers to absorb their local demand ariability. Cachon and Lariiere (999b) consider two broad classes of allocation mechanisms under a more general form of the retailers profit function with multiple retailers: those that induce the retailers to increase their orders or those that induce them to tell the truth. They proide conditions under

6 6 Rong, Snyder and Shen: BWE & RBWE under Rationing Game which there exists an E for the relaxed linear allocation rule. Moreoer, they show that truthinducing mechanisms do not maximize total retailer profit and therefore may not be appealing. To this end, Cachon and Lariiere (999c) show that, if an E exists under the linear and proportional allocation rules, the total supply chain is better off on aerage compared to the uniform allocation rule (a truth-inducing mechanism). Moreoer, Rong (2008) shows that the proportional allocation rule induces lower supply chain cost compared to the uniform allocation rule and the sales based allocation rule under a multi-period simulation study. It is because the proportional allocation rule allows the supplier to delier goods based on retailers true needs (though it is inflated). Howeer, the aboe papers either assume deterministic capacity or stochastic capacity with no information sharing. We inestigate the impact of capacity information sharing on both the [R]BWE and supply chain profit when the capacity is random. Information sharing is regarded as an effectie way to improe the performance of the supply chain. Most studies concentrate on demand information sharing (Chen 2003). Chen and Yu (2005) consider upstream leadtime information sharing. Li and Gao (2008) study the benefit of sharing new product deelopment progress. Jain and Moinzadeh (2005) inestigate how information about upstream inentory affects the retailer s order decision. They allow the retailer to change from a oneleel base stock policy to a two-leel base stock policy to take adantage of the shared information. They find that upstream inentory information sharing induces the BWE under stationary Poisson demand. In these three papers, the one-supplier, one-retailer setting enables the retailer to extract a benefit from upstream information sharing. In contrast, we study the behaior of multiple retailers in a competitie enironment and find that upstream capacity information sharing may induce higher retailer order ariability as well if the reseration payment is small enough. Moreoer, the profit of the whole supply chain may be smaller when capacity information is shared. Most literature on unreliable suppliers assumes a single retailer with one or more unreliable suppliers (see Parlar and Perry (996), Swaminathan and Shanthikumar (999), Tomlin (2006), Babich et al. (2007), Wang et al. (2009), Yang et al. (2009), Feng (200) etc.). There are relatiely

7 Rong, Snyder and Shen: BWE & RBWE under Rationing Game 7 few papers considering supply uncertainty in distribution networks. Tomlin and Wang (2005) analyze the alue of flexibility when multiple products share a single or dual unreliable resources. Snyder and Shen (2006) and Schmitt et al. (2008) study the best location to hold inentory in a one-warehouse, multiple-retailer setting with an unreliable warehouse. Their setting differs from ours in that it assumes a centralized system, and also because they consider a Bernoulli-type supply uncertainty for which the allocation rule does not matter, while the allocation policy plays an important rule in our paper because of the more general capacity process. 3. Model Assumptions We consider a system with identical retailers who face independent random demands from customers and replenish their inentory from a common supplier whose capacity is also random. In addition, the retailers obsere the shift of the demand distribution (see below) before their order decision. 2 Each retailer makes ordering decisions to maximize its own profit, that is, the system is managed in a decentralized manner. To be consistent with LPW s model, we consider a single-period setting in our analytical models. The [R]BWE is defined based on the ariability of retailers orders when the game is repeated with a different realization of the random capacity and demand shift. Before retailers place their orders, they obsere a public signal which is translated as a shift in the demand distribution. For example, a sudden stock market crash affects people s willingness to consume. Or, under unusual high temperatures, the sales of icecream would go up. Let X be the total demand shift (across all retailers), which is a random ariable. We assume that the total demand shift is allocated eenly among the retailers. We assume that the retailers obsere X before placing orders. After obsering X = x, the oerall demand for retailer i is D i + x, where D i represents the remaining randomness of demand with pdf g(d) and cdf G(d). We assume that G(d) = 0 when d < 0 and that G(d) is strictly increasing in this domain. That is, if 0 < G(d) <, then g(d) > 0. 2 LPW also suggests that the reaction to the shift of the demand distribution can cause the BWE.

8 8 Rong, Snyder and Shen: BWE & RBWE under Rationing Game The capacity at the supplier is random, represented by the random ariable V with pdf f() and cdf F (). We assume that X, D i and V are pairwise independent. D i is identically distributed for all the retailers. And we assume that the oerall demand for retailer i is nonnegatie. That is, D i + X 0. For a gien obsered alue x of the demand shift X, retailer i s demand, D i + x, has pdf g x (d) and cdf G x (d), which follow the equations below: ( g x (d) = g d x ) ( G x (d) = G d x ) () G x (α) = G (α) + x In the analysis that follows, if we drop the subscript x subscript, it means x = 0. from a quantity that normally has such a We introduce ζ, the wholesale price receied by the supplier from the retailers, and ϑ, a unit ariable production cost incurred the supplier. The unit sales price for each retailer is ϱ and the unit salage alue is κ. Thus, each retailer faces an underage cost of p = ϱ ζ for each unit of unmet demand and an oerage cost of h = ζ κ for each unit in inentory at the end of the period. We assume h, p > 0. If there were no capacity constraint and the total demand shift were x, then each retailer would order the newsboy quantity for the distribution G x, denoted S x : ( ) ( ) p p S x = G x = G + x p + h p + h = S + x. (2) ote that we use S to denote newsboy quantity when x = 0. In this paper, we propose two models. In model-l, we assume that the supplier does not disclose the capacity status to the retailers before their order decisions. In model-r, we assume that the supplier shares the real-time capacity information with the retailers. In return, the retailers incur a reseration payment r (r ζ) for each unit ordered, whether or not the unit is actually deliered. In addition, for each unit the supplier deliers, the retailers also pay an additional purchase cost

9 Rong, Snyder and Shen: BWE & RBWE under Rationing Game 9 (ζ r). 3 The details of each model s setting will be described in Sections 4 and 5, when it is analyzed. 3.. Definition of BWE and RBWE Let z i be the order size of retailer i. ote that z i and S x are not the same: S x is the newsboy order quantity, which the retailer would order if there were no capacity constraints, whereas z i is the retailer s order quantity taking into account both the potential capacity shortage and the other retailers actions. Let y = z i i be the total order size of all the retailers, and let z i be the ector of the other retailers order quantities. With a slight abuse of notation, we let z i i represent the total order quantity for retailers other than retailer i. Let ω be the index of the model. That is, ω = L (R) under model-l (model-r). Let πi ω (z i z i ) be the expected profit for retailer i when it orders z i and the other retailers order z i under model-ω. Let zi (z i ) be the best response mapping for retailer i when the others order z i. Let zi ω (x, ) be the E of order quantity chosen by retailer i gien the realized demand shift x and capacity. Let y ω (x, ) = i z ω i (x, ) be the total order quantity. We are interested in comparing the ariance of the total retailer orders, y ω, with that of the demand and supply processes. To measure the ariability of demand process, we use ar(x), the ariance of the demand shift, not ar( D i i + X), the ariance of the actual demand. This is because in both models we assume the retailers place orders before they know D i. Thus the realization of D i does not hae an impact on y ω. This can also be seen in LPW s analysis since 3 We assume that the supplier keeps the reseration payment een for undeliered units is not strictly required; alternately, one can simply think that the reseration payment returns to the retailers but the opportunity cost associated with the reseration payment incurs during the time between when the retailers order and when the supplier fulfills the orders. These two assumptions are equialent. All the analysis for the rationing game in this paper is applicable to both of them.

10 0 Rong, Snyder and Shen: BWE & RBWE under Rationing Game the realization of demand does not affect y ω no matter how large the ariance of D i is. In other words, y ω is fixed if X is fixed. That is why we focus on X, the demand shift, which is the actual trigger for the BWE. This is also reflected in the argument by LPW that the BWE occurs when the mean demand changes oer time. To measure the the ariability of supply process, we use ar (V ), the ariance of aailable capacity. Ultimately, we are interested in ealuating the following ratios: θ X = ar(yω (X, V )) ar(x) θ V = ar(yω (X, V )) ar (V ) (3) If θ X >, we say that the BWE occurs between the retailers and the customers. If θ V >, we say that the RBWE occurs between the supplier and the retailers. If θ X <, rather than saying that the RBWE occurs, we say that the BWE does not occur between the retailers and the customers. Similarly, if θ V <, we do not say the BWE occurs between the supplier and the retailers. This distinction is motiated by our claim that the BWE and the RBWE are triggered by demand uncertainty and supply uncertainty, respectiely. The retailers react to supply uncertainty but their customers do not, and they also react to demand uncertainty but the supplier s capacity is independent of the retailers action. Therefore, if θ X <, it is because the BWE is not occurring, rather than because the RBWE is occurring between the retailers and the customers. Finally, we define the conditional BWE and RBWE by the following ratios. Let A Ω, where Ω is the sample space of the joint random ariable (X, V ). Then θ X A = ar(yω (X, V ) (X, V ) A) ar(x (X, V ) A) θ V A = ar(yω (X, V ) (X, V ) A) ar (V (X, V ) A) (4) These two ratios measure the BWE and the RBWE within a sub-region A of the sample space. 4. Order Variability Under Model-L We define the sequence of eents in model-l:

11 Rong, Snyder and Shen: BWE & RBWE under Rationing Game. The demand shift X is realized by the retailers and the capacity V is realized by the supplier. Thus, X = x and V =. The supplier does OT share the alue of the realized capacity with the retailers. 2. Retailer i places its order z i, for i =,...,. 3. The supplier produces up to i= z i subject to capacity constraint. The produced products are distributed among the retailers using the proportional allocation rule (see below). 4. The demand at retailer i, D i, is realized for i =,...,. 5. Customer demands are satisfied to the extent possible, excess demands are lost, and oerage and underage costs are incurred. In step 3, the proportional allocation rule means that if the total retailer order (y) exceeds the realized capacity, then retailer i receies y z i. Otherwise, it receies z i. In fact, model-l is the same as the rationing game from LPW except that we explicitly model the demand shift, which is a key to argue the existence of the BWE by LPW. In model-l, there is no capacity information sharing between the supplier and retailers. For gien demand shift x under Model-L, the expected cost function for retailer i when it orders z i and the other retailers order z i is gien by the following equation. π L i (z i z i ) = (ϱ ζ)e[d i ] [ y =0 p y z i ( F (y)) ( d ) y z y z i i dg x (d) + h [ p =z i (d z i )dg x (d) + h 0 zi ( ) ] y z i d dg x (d) df () (5) ] =0 (z i d)dg x (d) Based on (5), LPW gies the following theorem (We customize it for gien demand shift x) Theorem. The symmetric E (z L (x)) for gien demand shift x, if exist, must satisfy z L (x) 0 [ ( )] ( p + (p + h)g x z L (x) 2 (z L (x)) 2 +( F (z L (x)) [ p + (p + h)g x ( z L (x) )] = 0 ) df () (6) Moreoer, z L (x) S x. Further, if F ( ) and G x ( ) are strictly increasing, the inequality strictly holds.

12 2 Rong, Snyder and Shen: BWE & RBWE under Rationing Game ow, we start to analyze the retailers order ariability under model-l. Since the real-time capacity information is not shared with the retailers under model-l, the retailers orders are not dependent on the realized capacity. That is, the retailers cannot chase the realized capacity by adjusting their orders dynamically. Therefore, we only analyze the BWE under model-l. In this section, we proide the condition to alidate the argument by LPW that the BWE exist when the mean demand changes oer time. We first proide the following lemma to facilitate the BWE analysis. Lemma. Let (U, Ω U ) be a random ariable, where Ω U is contained in the interal [a, b] and U has a strictly positie ariance. Let f and g be bounded functions in Ω U.. u [a, b], if g (u) > and g(u) is monotone, then V ar(u) < V ar(g(u)). 2. u [a, b], if f (u) > g (u) > 0, then V ar(f(u)) > V ar(g(u)). Lemma proides the basis to compare the ariance of X and ar(y L (X)) based on the the functional relationship between y L (x) and realization of X. By Lemma, we can get the sufficient condition on the existence of the BWE. Theorem 2. Suppose that (6) has a solution for x [a, b]. If dyl (x) dx > for all x [a, b], then θ X > for all possible distributions of X within [a, b]. That is, the BWE always exists. We utilize two examples to demonstrate why we need such condition to ensure the existence of the BWE for all possible distributions of X [a, b]. Figure contains two plots, in which we set = 5, h =, p =, D i U[0, 50]. In Figure.a, we assume that, with probability 0.5, V U[20, 2], and with probability 0.5, V U[80, 8]. In Figure.b, we assume that, with probability 0.5, V U[45, 46] and with probability 0.5, V U[80, 8]. We plot X and y L (X) y L (0) in both plots. We can see that the slope of the E is larger than that of X in Figure.a. The existence of the BWE is independent on the distribution of X [ 5, 5] because the disturbance brought by X triggers an een larger change in y L (X). In contrast, the slope of E is smaller than that of X in Figure.b, which indicates the non-existence of the BWE when X aries in the same area.

13 Rong, Snyder and Shen: BWE & RBWE under Rationing Game X y L (X) 5 X y L (X) 2 Unit 0 Unit X (a: BWE) X (b: o BWE) Figure BWE Results in Model-L Thus, Theorem 2 indicates that the BWE does not always occur under the rationing game when the mean demand changes oertime. This also proides an explanation of why the BWE is not ubiquitous in real-world supply chains (Cachon et al. 2007). 5. Order Variability under Model-R The remedies to reduce the BWE gien by LPW (Table in their paper) are to ) share the capacity information between the retailers and the supplier and 2) adopt the capacity reseration mechanism. In order to examine the effectieness of these remedies, we proide model-r. We define the sequence of eents in model-r:. The demand shift X is realized by the retailers and the capacity V is realized by the supplier. Thus, X = x and V =. The supplier shares the alue of the realized capacity with the retailers. 2. Retailer i places its order z i and incur the reseration payment rz i, for i =, The supplier produces up to i= z i subject to capacity constraint. The produced products are distributed among the retailers using the proportional allocation rule. The remaining purchase ( ) cost (ζ r) incurs for the allocated order quantity to retailer i. max(y,) z i 4. The demand at retailer i, D i + x, is realized for i =, Customer demands are satisfied to the extent possible, excess demands are lost, and oerage

14 4 Rong, Snyder and Shen: BWE & RBWE under Rationing Game and underage costs are incurred. We first characterize the E of retailers order quantity. Then, we proide the condition for the existence of the BWE and RBWE. 5.. Existence of E Under Model-R When S x, it is obious that zi R (x, ) = S x. Let s consider how to derie the E of retailers order quantity when S x. If an E zi R, i =,...,, exists under model-r, then the following lemma states that the E of the retailers total order size cannot be smaller than the aailable capacity. Lemma 2. When S x, we hae i zr i (x, ). ow we only need to consider candidates for the E that satisfy Lemma 2, i.e., y = i z i. Then the proportional allocation rule will always be in force. For each of the z i y z i 0 units that retailer i orders beyond its allocation, it incurs the extra unit reseration payment r under model-r. Therefore, when i z i, the expected cost function for retailer i under model-r is π R i (z i z i ) = (ϱ ζ)e[d i ] [ p y z i ( d ) y z y z i ( ) ( i dg x (d) + h 0 y z i d dg x (d) + r z i ) ] y z i Based on Lemma 3, we obtain the following theorem, which establishes the existence of a unique symmetric E of order quantities and characterizes its magnitude. (7) Theorem 3. Suppose that S x. Let ( { 0 (x) = G p x max 0, p + h r }) p + h (8) Then a unique symmetric E exists, and it satisfies { y R (x, ) = z R (x, ) = max, ( ) [ ( )] } p + r (p + h)g x (9) r Moreoer, the former case in the maximum preails iff 0 (x).

15 Rong, Snyder and Shen: BWE & RBWE under Rationing Game 5 ote that y R is a function of x, and r, but for the sake of simplicity, we include arguments for y R only as necessary. Since 0 (x) decreases with r, for large enough r, 0 (x) < S x. In this case, we can conclude, by Theorem 3, that the E of the total order quantity, y R, can be smaller than the sum of the newsboy quantities, S x. If that is the case, the reseration payment preents oer-ordering completely. Corollary. Suppose that S x. Then we hae lim r 0 y R (r) = Corollary shows that the supplier needs to implement the capacity reseration mechanism when the real-time capacity information is shared. Otherwise, there is no E when the capacity shortage incurs unless there is a limit on the order quantity BWE & RBWE Analysis Under Model-R In this section, we characterize the conditions under which the BWE and the RBWE exist by studying the partial deriaties of y R (x, ). In addition, we analyze the impact of the reseration payment r on the conditional BWE and RBWE BWE Analysis Under Model-R Let Ω 0 = {(x, ) Ω < 0 (x)}; Ω 0 is the set of all capacities and demand shifts for which the capacity is strictly less than the total order quantity (by Theorem 3). We focus our analysis on Ω 0 since if (x, ) / Ω 0, then y R (x, ) = for S x and y R (x, ) = S x for for > S x. Both cases are triial. By () and (9), the equilibrium total order quantity satisfies y R (x, ) = r ( ) [ ( p + r (p + h)g x )] Taking a partial deriatie with respect to x, we get x yr (x, ) = ( ) ( (p + h)g r x ) > 0. (0) (0) indicates that the retailers order quantity increases in the demand shift x. But this does not guarantee that the change in the retailers order quantity is always greater than that in the

16 6 Rong, Snyder and Shen: BWE & RBWE under Rationing Game demand shift. To determine when the BWE exists, by Lemma, we need to explore the region in which x yr (x, ) >, i.e., when ( g x ) > 2 r p + h. () In the hope of extracting more insights, we restrict our attention to demand distributions (D i ) that are unimodal. This is, in fact, not an oerly restrictie assumption because a great many distribution families, including normal, uniform, and gamma, fall into this category. This assumption implies that g( x ) is unimodal in x. Let g(d ) be the maximal alue of g( ). If g(d ) > 2 then there exist γ 0 () and γ () such that γ 0 () < γ () and and () holds when γ 0 () < x < γ (). ( ) ( ) g γ0 () = g γ () = 2 r p + h r, p+h Let Ω γ() = {x (x, ) Ω and γ 0 () < x < γ ()}. Ω γ() is the set of all demand shifts for fixed for which () holds, and its closure. If γ 0 () and γ () do not exist, then Ω γ() =. The following theorem specifies a subregion of Ω γ() in which the conditional BWE occurs and says that it neer occurs if Ω γ() is empty. Theorem 4. Under model-r, when is fixed,. If Ω γ() is empty, then x yr (x, ). Thus V ar(x) > V ar(y R (X, )). That is, there is no conditional BWE (conditioned on V = ) between the retailers and the customers. 2. If Ω γ() is non-empty, let A = Ω γ() Ω 0. If (x, ) A, then x yr (x, ) >. Thus V ar(x A) < V ar(y R (X, V ) A). That is, the conditional BWE (conditioned on V = ) occurs between the retailers and the customers. ext we examine the impact of magnitude of the reseration payment on the BWE. The following proposition states that the BWE becomes more seere between the retailers and the customers as r decreases. First note that Ω γ() and Ω 0 both depend on r since 0 (x), γ 0 (), and γ () do. Let A(r ) = Ω γ() Ω 0 when r = r.

17 Rong, Snyder and Shen: BWE & RBWE under Rationing Game 7 Proposition. If A(r ), then for all r 2 < r, we hae V ar(y R (X, V ) A(r ), r = r ) < V ar(y R (X, V ) A(r ), r = r 2 ) RBWE Analysis Under Model-R ow we consider the impact of the ariability of V on y R (x, V ). We hae yr (x, ) = ( ) [ [ ( (p + r) + (p + h) G r x ) + ( g x )]]. (2) The partial deriatie yr (x, ) can be either positie or negatie. If it is positie, then the increased capacity induces the retailers to increase their order size. It can happen if the risk of paying too much in the reseration payments of undeliered units is outweighed by the benefit attained by ordering more when the capacity is tighter. On the other hand, when yr (x, ) is negatie, an increase in capacity causes a decrease in the retailers order quantity, i.e., the gaming behaior among the retailers is mitigated. Therefore, the sign of yr (x, ) is determined by the tradeoff between the penalty for oer-ordering (and incurring the extra reseration payment) and the shortage risk from under-ordering (and receiing too few units because of rationing). Howeer, we will show next that it is the magnitude of yr (x, ), and not its sign, that determines whether the RBWE occurs. For any fixed x, there is a (possibly empty) collection of non-oerlapping interals in the domain of V such that, on a gien interal, y R (x, ) is monotone in and yr (x, ) >. Let I k (x) be the kth such interal. Theorem 5. Let x be fixed and let A k = I k (x) Ω 0. Then V ar(v A k ) < V ar(y R (X, V ) A k ). That is, the conditional RBWE (conditioned on X = x) occurs between the supplier and the retailers on the interal A k, for each k. Similar to Proposition, we let B k (r ) = I k (x) Ω 0 when r = r. Proposition 2. If B k (r ), then for all r 2 < r, we hae V ar(y R (X, V ) B k (r ), r = r ) < V ar(y R (X, V ) B k (r ), r = r 2 ).

18 8 Rong, Snyder and Shen: BWE & RBWE under Rationing Game Proposition 2 states that a smaller r triggers a larger RBWE. This implies that the supplier s preferred choice of reseration payment aggraates the RBWE, just as it does the BWE. We proide numerical examples to demonstrate the occurrence of the BWE and RBWE in Figure 2. We set r = 2. We use = 3, D i (00, 5 2 ), p = 0 and h =. In addition, V aries from 250 to 300 and X aries from 2 to X 0 X V V min(, y 0 (x,)/ x) min(, y 0 (x,)/ ) (a: BWE) (b: RBWE) Figure 2 When the BWE and RBWE Occur Under Model-R In Figure 2.a, in the white area, the conditional BWE occurs ( x yr (x, ) > ), while in the black and gray areas, it does not ( x yr (x, ) < ). If the mismatch between demand and supply is exceptionally seere, the BWE does not occur since the reseration payment tends to reduce the benefit of oer-ordering. Therefore, the retailers gaming behaior is mitigated. In Figure 2.b, in the white areas, the conditional RBWE occurs ( yr (x, ) > ), while in black and gray areas, it does not ( yr (x, ) < ). The two separate white areas represent the two possible signs of yr (x, ), as discussed at the start of Section The retailers needs

19 Rong, Snyder and Shen: BWE & RBWE under Rationing Game 9 to balance between the reseration payment and the competition among the retailers. When the capacity is exceptionally seere, the retailers are more concern about the reseration payment. Therefore, the decreasing capacity leads to the lower retailers order quantity. On the other hand, if the capacity is not tight enough, the retailers are more concern about the competition among the retailers. Therefore, the increasing capacity reliees the competition which leads to the lower retailers order quantity. When the shortage of capacity is moderate, the two driing forces cancel each other. Therefore, the change of capacity does not trigger een larger change in the retailers order quantity. 6. Impact of Capacity Information Sharing and Capacity Reseration In this section, we analyze how capacity information sharing and capacity reseration affect the profits of the retailers, the supplier and the whole supply chain. In addition, we closely examine the relationship between the retailers order ariance and their profit or the supplier s profit. Under model-l, the profit of retailer i under E is gien by π L i = E [ π L i (z L i (X) z L i(x)) ] (3) The supplier gains ζ ϑ for each deliered product. Therefore, the supplier s profit under E follows π L s = (ζ ϑ)e [ min(v, y L (X)) ] (4) And the profit of the whole supply chain under E follows π L = π L s + π L i (5) i Similarly, under model-r, the profit of retailer i, the supplier and the whole supply chain under E is respectiely gien by π R i = E [ π R i (z R i (X, V ) z R i(x, V )) ] (6) π R s = (ϑ ζ)e[min(v, y R (X, V ))] + re[(y R (X, V ) V ) + ] (7) π R = π R s + π R i (8) i=

20 20 Rong, Snyder and Shen: BWE & RBWE under Rationing Game In (7), the supplier s profit function under E contains both the gain of each deliered product and the reseration payment it receies. ote that all the profit components can be affected by parameters such as the unit product cost, the wholesale price and the unit reseration payment. For the sake of simplicity, we include arguments for these profit components only as necessary. Then the following proposition categorizes the impact of the magnitude of the reseration payment. Proposition 3. For r < r 2, we hae. π R (r ) = π R (r 2 ) 2. πs R (r ) πs R (r 2 ). 3. πi R (r ) πi R (r 2 ). Proposition 3. reeals that the whole supply chain profit does not change as the alue of r does. But how the supplier and retailers split the profit differs at aried r since the total reseration payment, together with the purchase payment, constituting the transfer payment from the retailers to the supplier, depends on the magnitude of r and the degree of oer-ordering, which is also a function of r. Moreoer, Propositions 3.2 and 3.3 indicate that the decrease of r is oerwhelmed by the increase cost of oer-ordering. ext, we proide the following proposition to examine the benefit of capacity information sharing and capacity reseration. Proposition 4. There exists a critical alue ϑ 0. When ϑ < ϑ 0, π L (ϑ) π R (ϑ). Otherwise, π L (ϑ) π R (ϑ) Proposition 4 shows that capacity information sharing and capacity reseration do not necessarily lead to a higher supply chain profit. In fact, it can reduce the total profit when the marginal profit is sufficiently high for the supplier (i.e. when the production cost is low enough). In such a situation, the oer-ordering behaior among the retailers under model-l can actually lessen

21 Rong, Snyder and Shen: BWE & RBWE under Rationing Game 2 the double marginalization since the system-wide optimal order quantity is much larger than the newsendor order quantity. ow we inestigate the relationship between the retailers order ariability and their profit or the supplier s profit under both model-l and model-r through a simulation test. We assume ζ = 5, ϑ = 4.6, ϱ = 0 and κ = 4. We also set = 5, D i U[0, 50], V U[80, 250] and X U[ 25, 0]. In Figure 3, we plot the retailers order standard deiation, the retailers profit, and the supplier s profit under both models. total order std Model R Model L total retailers profit Model R Model L supplier profit Model R Model L r r r (a: Retailers Order Std) (b: Retailers Profit) (c: Supplier s Profit) Figure 3 Impacts of Capacity Information Sharing and Reseration Payment Figure 3.a shows that the retailers order ariability increases significantly under model-r compared to that under model-l when r is small enough. Howeer, the result is reersed when r is large. The reason is that in model-r, the retailers adjust their order decision dynamically based on real-time capacity information and demand shift information, and the magnitude of r determines how they react to these information. When r is small, the retailers oerreact to the supply shortage, while they just chase the capacity when r is large. As a result, the order ariability decreases when r increases under model-r. One may expect that capacity information sharing will improe the retailers profit. Howeer, this is not always the case, as shown in Figure 3.b. Excluding the reseration payment, the retailers always obtain no-less-than-newsboy-type profit in model-r since their allocated products neer

22 22 Rong, Snyder and Shen: BWE & RBWE under Rationing Game exceed S x. Howeer, when r is small, the reseration payment that is introduced to compensate the supplier for sharing its capacity information outweighs the newsendor-type profit gain for the retailers under model-r. In contrast, as shown in Figure 3.c, when r is small, the supplier can collect more money from the reseration payment under model-r than the additional sales profit it earns from the retailers oer-ordering under model-l. Lastly, we examine the three plots collectiely and make a few more obserations regarding the relationship between the retailer s order ariability and their profit. If we confine our attention only to model-r, then the higher the retailers order ariability, the lower profit they gain. But if we allow the system to shift between model-l and model-r, such relationship may break down. For example, when the system shifts from model-l to model-r for r [2, 3], then both the retailers order ariability and profit drop at the same time. In sum, the changing pattern of the order ariability can be a good indicator of that of the profit only for the retailers who react to both uncertainties in one particular model. If there is a significant change in the supply chain settings, then the relationship between the retailers order ariability and their profit becomes loose. 7. Conclusions In this paper, we find that the retailers order size changes in response to both demand and supply uncertainty. Depending on the settings, the likelihood and seerity of the occurrence of the BWE and RBWE ary. We also document the inability of capacity information sharing to reduce the [R]BWE. Moreoer, capacity information sharing may also hurts the retailers as well as the whole supply chain. Howeer, the supplier is willing to disclose its capacity information in order to collect an additional reseration payment from the retailers for sufficiently small unit reseration payment. When both the BWE and RBWE happen, the whole supply chain exhibits an umbrella pattern. But this does not necessarily occur in eery supply chain. It does in ours because the supply and demand ariances are exogenous, whereas the retailers decisions are endogenous. In other words, the retailers react to both the supply and demand processes, but the upstream echelon does not

23 Rong, Snyder and Shen: BWE & RBWE under Rationing Game 23 react to the demand uncertainty created by the retailers, nor does the downstream end react to the supply uncertainty created by the retailers. If, instead, we assumed that the customers, too, game the system as a reaction to supply shortages, then the RBWE would occur between the retailers and the customers. Similarly, if the supplier could control its capacity in response to the retailers order ariability, then the BWE would occur between the supplier and the retailers. Put another way, in a system with random capacity and demand shifts, the BWE originates from downstream and the RBWE originates from upstream, and both effects propagate through the supply chain until they reach a stage that does not react to the uncertainty that created it. Technical Appendix In this appendix, we drop the subscript x and superscript L and R to simplify the notation wheneer it causes no confusion. Lemma Part : WLOG, we proe this part for g (u) > for all u Ω U. First, we show that it holds for the discrete distribution with finite mass points. Then we use the Riemann sum to extend this result to the continuous distribution. Let u k, k =,...K be the mass points in Ω U. Let h(u k ) be the pmf at u k. Then ar(g(u)) = E[g 2 (U)] E[g(U)] 2 [ K K = g 2 (u k )h(u k ) g(u k )h(u k ) k= k= = g 2 (u )h(u ) + g 2 (u 2 )h(u 2 )+,..., +g 2 (u K )h(u K ) [g(u )h(u ) + g(u 2 )h(u 2 )+,..., +g(u K )h(u K )] 2 = g 2 (u )h(u )( h(u ))+,..., g 2 (u K )h(u K )( h(u K )) 2 g(u k )g(u j )h(u k )h(u j ) k<j = g 2 (u k )h(u k )h(u j ) 2 g(u k )g(u j )h(u k )h(u j ) k j k<j ] 2

24 24 Rong, Snyder and Shen: BWE & RBWE under Rationing Game = k<j(g(u k ) g(u j )) 2 h(u k )h(u j ) > k<j(u k u j ) 2 h(u k )h(u j ) = ar(u) The inequality is due to f (u) >. Thus g(u k ) g(u j ) > u k u j for u k u j. Then let U be a continuous ariable and H(u) be its cdf. We define a sequence of new random ariables U n which hae the mass points u n (i) := a + b a n i, for i =,...n, with pmf h n(u n (i)) = H ( a + b a n we hae i) H ( a + b a(i )). Since E[ ] is continuous and g is bounded, by the Riemann sum, n lim ar(u n) = ar(u) n lim ar(g(u n)) = ar(g(u)) n Since we hae ar(u n ) < ar(g(u n )), we only need to show that ar(u) is strictly less than ar(g(u)). Because lim n ar(u n ) = ar(u), s.t. n >, ar(u n ) > ar(u) 2 > 0. Moreoer, g (u) > implies that ε, e > such that the set {u [a, b] : g (u) < e} has Lebesgue measure less than ε. Then u, [a, b], g(u) g() > e u (e )ε, which helps to establish the following inequality. ar(g(u n )) = k<j(g(u n (k)) g(u n (j))) 2 h n (u n (k))h n (u n (j)) > e 2 k<j(u n (k) u n (j)) 2 h n (u n (k))h n (u n (j)) 2e(e )ε u n (k) u n (j) h n (u n (k))h n (u n (j)) k<j > e 2 ar(u n ) 2e(e )(b a)ε The last inequality holds since u n (k) u n (j) < b a. Then, we hae ar(g(u n )) ar(u n )

25 Rong, Snyder and Shen: BWE & RBWE under Rationing Game 25 By choosing ε = ar(u), we get 8(b a) > (e 2 )ar(u n ) 2e(e )(b a)ε > (e 2 )( ar(u) 2 2(b a)ε) ar(g(u n )) ar(u n ) > e2 ar(u) > 0. 4 This implies that ar(g(u n )) ar(u n ) is bounded below by a strictly positie number, if n. Then we hae ar(u) < ar(g(u)). Part 2: Let y = g(u). We want to show that [f(g (y))] >. Applying the result of part, we get ar([f(g (Y ))]) > ar(y ), which implies ar[f(u)] > ar[g(u)]. Since [f(g (y))] = f (g (y)) g (g (y)) >, the proof is complete. Theorem 2 It follows directly from Lemma. Lemma 2 We use contradiction to proe this lemma. Suppose that i zr i = τ <. Then let j = arg min i z R i. Obsere that z R j <. If retailer j orders ẑ j = min(z R j + τ, ), its allocation is equal to ẑ j since τ z R j +ẑ j. Obsere that z R j < ẑ j < S. Therefore, retailer j s cost by ordering ẑ j is strictly lower than that by ordering z R j because the newsboy cost function is strictly conex. This contradicts the fact that {z R i } i= is a E. Therefore, i zr i. Theorem 3 The E of maximizing the newsendor profit is equialent to the E of minimizing the newsendor cost. Based on (7), the newsendor cost function follows ( C R i (z i z i ) = p d ) y z y z y z i ( ) ( i dg x (d) + h i 0 y z i d dg x (d) + r z i ) y z i. (9) Before we proe Theorem 3, we need the following lemma.

26 26 Rong, Snyder and Shen: BWE & RBWE under Rationing Game Lemma 3. Suppose that S x. Then. When i z i, ( C R i (z i z i ) = z i y z ) [ ( )] i (p + r) + (p + h)g y 2 x y z i + r; (20) 2. C R i (z i z i ) is strictly pseudoconex; 3. there is a unique z i (z i ) such that either (a) z i C R i (z i, z i ) = 0 or (b) z i = 0 and z i Ci R (zi, z i ) > 0. Proof of Lemma 3 It is straightforward to obtain Part by taking the deriatie of (9). Part 2: It is easy to see that C R i (z i z i ) is strictly pseudoconex if no saddle points. Then suppose that there exists ẑ i satisfying z i C R i (z i z i ) zi =ẑ i C R i (z i z i ) is strictly pseudoconex, we need to show ) if z i > ẑ i, z i Ci R (z i z i ) < 0. This implies that ẑ i is unique. Let ŷ = ẑ i + i z i. Obsere that y z i y 2 0 since z i y. Then (p + r) + (p + h)g( ŷ ẑi) < 0. As G(d) is strictly increasing, when z i > ẑ i, ( ) ( ) (p + r) + (p + h)g y z i > (p + r) + (p + h)g ŷ ẑi There are two cases. First, (p + r) + (p + h)g Since y z i y 2 ( ) z y i 0; then z i Ci R (z i z i ) is monotone and has = 0. In order to show that z i Ci R (z i z i ) > 0, 2) if z i < ẑ i, z i C i (z i z i ) zi =ẑ i = 0 implies z i C i (z i z i ) > 0. Second, ( ) ( ) 0 > (p + r) + (p + h)g y z i > (p + r) + (p + h)g ŷ ẑi. is a decreasing function in z i, 0 < y z i y 2 < ŷ ẑi ŷ 2. Thus we hae ( y z ) [ ( )] i (p + r) + (p + h)g y 2 y z i + r ( ) [ ( )] > ŷ ẑi (p + r) + (p + h)g ŷ 2 ŷ ẑi + r = 0

27 Rong, Snyder and Shen: BWE & RBWE under Rationing Game 27 Similarly, we can proe that if z i < ẑ i, Part 3: Set z i high enough so that y z i > S. By (20), z i Ci R (z i z i ) < 0. Therefore, we hae part 2. ( C R i (z i z i ) > z i y z ) i r + r > 0. y 2 The first inequality is due to the fact that G( y z i) > p for z p+h y i > S. The last inequality is due to the fact that y. Then we can get part 3 directly from part 2 since eentually z i C i (z i z i ) will hae positie alues. Proof of Theorem 3 If < 0 (x), then z i C i (z i, z i ) zi =z 0,z i =z 0 = 0 for ( ) [ ( z0 = r p + r (p + h)g x )]. Therefore, z R i = z 0 is a E due to the pseudoconexity of C i (z i z i ). If 0 (x), then z i C i (z i, z i ) zi =,z i= 0. zr = x than and yr by Lemma 2. The uniqueness of y R relies on the condition that Corollary is a E since no retailer will order higher z i C i (z i, z i ) zi =z i is strictly increasing in y. It directly comes from Theorem 3. Theorem 4 It directly comes from Lemma and the definition of γ 0 () and γ (). Proposition When r 2 < r, we hae ( )(p + h)g( x ) > ( )(p + h)g( x r 2 r ) > 0. Applying part 2 of Lemma, we get the result directly. Theorem 5 It directly comes from Lemma and definition of I k (x). Proposition 2 If for any (x, ) B k (r ), < ( ) [ [ ( (p + r ) + (p + h) G r x ) ( + g x )]]