Supply Chain Inventory Competition and Coordination in e-business

Transcription

1 Sixteenth Annual Conference of POMS, Chicago, IL, April 29 - May 2, Abstract Number: Supply Chain Inventory Competition and Coordination in e-business By Wei-yu Kevin Chiang Department of Information Systems University of Maryland, Baltimore County 1000 Hilltop Circle, Baltimore, MD 21250, USA Phone: wchiang@umbc.edu Abstract A game-theoretical model is developed to investigate the competitive and cooperative stocking behavior in a two-echelon supply chain in which a supplier uses both its wholly owned online channel and an independent retailer to distribute its products to customers with heterogeneous channel preference. The customers may dynamically substitute between the two sales channels. We analyze the channel inefficiencies induced by inventory competition and propose coordination contracts for the supply chain.

2 1 Introduction Over the last decade, companies have been exploring opportunities to use the Internet. The advent of e-commerce together with the rapid development of third-party logistics from highly competent companies (such as Federal Express and United Parcel Services) has signi cantly changed the way rms market and distribute their products to customers. It has made it easier for suppliers to adopt a multi-channel distribution strategy by adding an online direct sales channel alongside the traditional retail channel. A recent survey (Tedeschi 2000) indicates that about 42% of top suppliers in a variety of industries have begun to sell directly to consumers over the Internet. It appears that the trend of engaging in the Internet-based direct sales has raised serious awareness and attention, by academicians and practitioners, to the opportunities and challenges of using both integrated and non-integrated distribution channels concurrently. Indeed, there are several economical reasons of adopting a multi-channel distribution in a supply chain, one of which is its capability of reaching potential buyer segments that could not be reached by a single channel (Moriarty and Moran 1990, Rangan et al. 1992, Anderson et al. 1997). Nevertheless, while most supply chains operate as a collection of independent agents, combining the traditional retail channel with a supplier-owned direct channel in a supply chain inevitably introduces new challenges to channel operations. One of the widely recognized challenges is the existence of channel con ict (cf. Tsay and Agrawal 2004a), which occurs when a supplier-owned direct channel establishes the supply as a direct competitor to its intermediary. The resulting tension within supply chain caused by channel con ict may lead to a loss of channel pro ts if the economic advantages of adopting a multi-channel distribution fail to o set the potential damages of channel con ict. In the United States alone, there is an estimated $250 billion a year in waste attributed to ine ciency in the distribution of products from manufacturers to consumers according to New York City-based emarketer (Lawson, 2001). Channel con ict is a signi cant contributor to that. Clearly, understanding the nature and e ect of channel con ict 1

3 in a multi-channel supply chain is critical of improving channel e ciency. 1.1 Vertical and Horizontal Competitions It is interesting to point out that the type of channel con ict in a multi-channel supply chain arises from a rather unique competitive situation: a supplier and its retailer are engaged in both vertical and horizontal competitions at the same time. Speci cally, vertical competition is associated with ine cient price double marginalization (Spengler 1950), which occurs when the supplier, as a result of selling at a wholesale price above its marginal cost to an intermediary, induces its retailer to set a retail price above what it would be if it faced the true marginal cost of the channel. Double marginalization is a well-known example of supply chain ine ciency caused by the vertical competition. When the retail price is xed, the existence of double marginalization may cause the retailer to carry too little inventory relative to the optimal amount in a one-period model where a manufacturer sells to a retailer facing a newsvendor problem (Pasternack 1985, Lariviere and Porteus 2001). Similarly, in a multiple-period model where each rm uses a base stock policy, a decentralized supply chain generally understocks due to double marginalization (Cachon 1999, Cachon and Zipkin 1999, Cachon 2001, Caldentey and Wein 2003). On the other hand, horizontal competition occurs when rms sell substitutable products and competitively stock the products. In a multi-channel supply chain, a supplier and its intermediary sell essentially the same product, and thus products are di erentiated by customers channel preference. It is evident that certain customers are often willing to switch to the other channel when faced with stock-outs in their preferred channel. As a result, demand at one channel that is out of stock may be captured by the other channel with available inventory. This sort of stockoutbased substitution may create strategic interactions between a supplier and its intermediary with respect to their stocking decisions. Research studies on horizontal inventory competition are established primarily in operations literature (e.g., Parlar 1988, Lippman and McCardle 1997, 2

4 Mahajan and van Ryzin 2001, Netessine and Rudi 2003), albeit they also appear in marketing literature (e.g., Balachander and Farquhar 1994). In general, it is found that while there might be cases where it results in product understocking, horizontal competition results in overstocking for most practical situations, causing a loss in pro t relative to the monopoly case. What happens when both types of competitions co-exist in a supply chain? In addition to the literature analyzing the impact of pure-vertical and pure-horizontal competitions on supply chain performance, there is also a recent stream of literature on the e ect of combined verticalhorizontal competition (e.g., Anupindi and Bassok 1999, van Ryzin and Mahajan 2000, Netessine and Zhang 2004). Broadly speaking, it is found that when both double-marginalization and substitutability co-exist, the ine cient understocking due to vertical competition in a supply chain can be counteracted by the overstocking due to horizontal competition. In other words, the combined e ect of both vertical and horizontal competitions may help to improve supply chain performance. It should be noticed that due to a single-period (newsvendor) setting, the studies in this stream of literature generally assume that the supplier(s) carry no stock at the upper echelon. Hence, inventory competition is only among rms located within the same supply chain echelon. Essentially, these studies are unable to provide stylized insights and guidelines for managing a multi-channel supply chain in which inventory competition is among rms located both within the same echelon and in di erent echelons. Perhaps due to the mathematical di culties inherent in the problem, thus far, there is no published paper attempting to address the imperative question regarding the stocking behaviors and the performance of a supplier and a retailer under both horizontal and vertical inventory competitions with a multi-echelon multi-channel framework. It is against this backdrop that this paper makes a contribution by enhancing our understanding of competitive and cooperative inventory management of a multi-channel supply chain. 3

5 1.2 Other Related Literature Multi-echelon inventory control problem, rst introduced by Clark and Scarf (1960), is known to be a challenging research area. Due the complexity and intractability of the problem, the adoption of single location, single echelon models for the inventory systems is recommend by Hadley and Whitin (1963). The rst multi-echelon inventory model for managing the inventory of service parts is constructed by Sherbrooke (1968). This model is capable of identifying the stock levels that minimize the expected number of backorders at the lower echelon subject to a budget constraint. Thereafter, a large set of models seeking to identify optimal lot sizes and safety stocks in a multi-echelon framework have been introduced (e.g., Deuermeyer and Schwarz 1981, Moinzadeh and Lee 1986, Svoronos and Zipkin 1988, Axsäter 1990, Axsäter 1993, Nahmias and Smith 1994, Aggarwal and Moinzadeh 1994, Grahovac and Chakravarty 2001). Besides analytical models, simulation models have also been developed (e.g., Clark and Trempe 1983, Pyke 1990, Dada 1992, Alfredsson and Verrijdt 1999) to capture the complex interactions of the multi-echelon inventory problems. In general, this body of literature focuses on centralized decision making in hierarchical inventory systems, ignoring the independence of each supply chain member. Research on multi-channel supply chain management in the setting where the upstream echelon is both a supplier to and a competitor of the downstream echelon has emerged only recently. A number of papers in this stream of literature (e.g., Rhee and Park 2000, Rhee 2001, Kumar and Ruan 2002, Chiang, et al. 2003, Tsay and Agrawal 2004b) are focused on channel competition and coordination issues. They address strategic price and/or service interactions between upstream and downstream without considering the inventory competition problem that we examine in this paper. Motivated by Anupindi and Bassok (1999) and van Ryzin and Mahajan (2001), Boyaci (2003) models multi-channel inventory competition in a newsvendor setting. Although, as in this study, he also explores the channel ine ciencies induced by the presence of simultaneous vertical and horizontal competition, his model assumes that the manufacturer s doesn t hold any stocks 4

6 at the upper echelon. In other words, inventories in the manufacturer s wholly owned channel are available only to customers (but not to its retailer), and thus inventory competition between the manufacturer and the retailer occurs only horizontally. Clearly, such problem setting is very di erent from our multi-period framework which involves both vertical and horizontal inventory competitions. From a logistics perspective, Cattani and Souza (2002) and Chiang and Monahan (2004) model multi-channel inventory problem in a multi-period setting. However, they only consider a single monopolistic company or simply assume that the multi-channel supply chain is under centralized control. As such, no channel competition and coordination issues are investigated in these papers. There are several other papers (e.g., Bell, et al. 2002, Peleg and Lee 2002, Yao and Liu 2002) that also address related issues on multi-channel supply chain. However, their foci are di erent. We refer the readers to Tsay and Agrawal (2004a) and Cattani, et al. (2004) for a recent detailed review of this stream of literature. The remainder of the paper is organized as follows. Section 2 de nes the continuous time Markov chains of the two-echelon multi-channel supply chain. Accordingly, the models that are used to measure the performance of centralized and decentralized supply chains are formulated in Section 3. The numerical study of competitive e ect is then presented in the section that follows. Section 5 contains the analysis of coordination contracts for the multi-channel supply chain. The nal section concludes the paper. 2 The Model 2.1 The Two-Echelon Dual-Channel Supply Chain Assume that the product is available for customers at both the retailer and the supplier s direct sales channel. There are two types of customers: those who prefer the retail channel (retail customers) and those who prefer the direct channel (direct customers). Demand of the retail 5

7 customers is satis ed with the on-hand inventory at the retailer, while demand of the direct customers is ful lled through direct delivery with the on-hand inventory at the supplier. Suppose the supply chain experiences demand from customers who arrive at the system in accordance with a Poisson process with mean. Suppose further that each arrival is a direct customer with probability and a retail customer with probability 1. Subsequently, it can be shown that the arrivals of direct and retail customers are both Poisson processes having respective rates d = and r = (1 ). Moreover, the two processes are independent (cf. Ross 2000). We assume that both the supplier and the retailer incur no xed ordering costs and they both implement one-for-one ordering policies to replenish their inventories. Under such replenishment policies, the inventory positions are kept constant at base-stock levels, and an order for one unit is placed each time a demand arrival is served from stock on hand. The respective base-stock levels for the supplier and the retailer are denoted by S s and S r. Note that one-for-one replenishment policies are widely recognized in the literature as e ective inventory control policies for those products that exhibit low demand rates. As the recent frequently observed trends of increasing variety, customization, and complexity of products have increased the adoption of one-for-one ordering policies, it should not be doubtful that using such replenishment policies to investigate the inventory competition and coordination problems will provide valuable insights. When a stockout occurs in the retailer, some proportion, say r, of retail customers are willing to switch to the supplier s direct channel. On the other hand, some proportion d of direct customers who incur a stockout at the supplier are willing to switch to the retailer. In case of a stockout, customers who are unwilling to shift to the other channel result in lost sales. In addition, customers are lost when both the supplier and the retailer are out of stock simultaneously. Suppose that the replenishment backorders from the retailer are allowed at the supplier, while no backorders for customers are allowed at both the direct and the retail channels. The replen- 6

8 ishment lead times for the supplier and the retailer are assumed to be independent exponential random variables with means 1= s and 1= r, respectively. A customer served from stock on hand will trigger a replenishment order immediately by EDI (electronic data interchange). Therefore, it is reasonable to assume that the information lead time is zero. 2.2 The Markov Model Let x be the on-hand stock level at the supplier, and y be the on-hand stock level at the retailer. Then the Markov model that captures the model assumptions can be constructed with each state (x; y) 2, where is the state space, = f(x; y) 2 Z 2 j S r x S s, 0 y S r g. (1) The stock on hand at the supplier can be negative because we assume that replenishment backorders from the retailer are allowed at the supplier. With the base-stock levels S s and S r, the total number of states is (S s + S r + 1)(S r + 1). For each state (x; y), there are four possible events that lead to a transition from the state, and they are characterized below: h1i a customer served from stock on hand at the supplier: (x; y)! (x 1; y); Let P (Ss;Sr) h2i a customer served from stock on hand at the retailer: (x; y)! (x 1; y 1); h3i a replenishment order arrives at the supplier: (x; y)! (x + 1; y); h4i a replenishment order arrives at the retailer: (x; y)! (x; y + 1): denote the steady-state probability that x units are on hand at the supplier and y units are on hand at the retailer given S s and S r ; P (Ss;Sr) = 0 if (x; y) =2. The ow balance equations that require that for all states the input and output rates to each state be equal are given by: 4P k=1 hki P (Ss;Sr) = h1i (x+1;y) P (Ss;Sr) (x+1;y) + h2i (x+1;y+1) P (Ss;Sr) (x+1;y+1) + h3i (x 1;y) P (Ss;Sr) (x 1;y) + h4i (x;y 1) P (Ss;Sr) (x;y 1) ; 8(x; y) 2 ; (2) 7

9 where hki is the transition rate from state (x; y) for event k Transitions Due to Receiving Demands Based on the model assumptions stated previously, the two respective transition rates as a result of satisfying demand by stock on-hand at the supplier and the retailer can be modeled as h1i = (x) d + (1 (y) ) r r ; (3) h2i = (y) r + (1 (x) ) d d ; (4) where (z) is used to detect whether a stockout occurs or not. 8 >< 1 if z > 0 (z) = >: 0 otherwise Speci cally, : (5) Note that when both channels have stock available ( (x) = 1, (y) = 1), the total transition rate from state (x; y) due to receiving demand is h1i + h2i = r + d =. On the other hand, when both channels are out of stock simultaneously ( (x) = 0, (y) = 0), this total transition rate is zero. If the stock is out at the retailer ( (y) = 0) but is available at the supplier ( (x) = 1), then h1i + h2i = r r + d (some proportion of the retail customers, r, will switch to the direct channel). Likewise, it can be veri ed that the total transition rate due to receiving demand is r + d d when the stock is out at the supplier but is available at the retailer. Next, we specify the transition rates caused by receiving replenishment orders, h3i and h4i Transitions Due to Receiving Replenishment Orders The rates at which in-transit replenishment orders arrive at the supplier and the retailer are direct a ected by the following two state-related variables, respectively: V = replenishment units delayed at the supplier when the system is at state (x; y); T = in-transit orders from the supplier to the retailer when the system is at state (x; y). 8

10 Speci cally, h3i = V s and h4i = T r. Depending on the system state (x; y) and the base-stock levels, S s and S r, the values of V and T may vary. Assume that the supplier s source has in nite capacity, and the supplier ships a retailer s order immediately provided that inventory is available, i.e., x > 0. We can, subsequently, represent V and T as V = S s x; and (6) T = S r y [x] +. (7) Math notation follows: [z] + = max f0; zg, and [z] = max f0; zg. It is clear that 0 V S s + S r and 0 T S r. Based on (6) and (7), the two transition rates due to receiving replenishment orders at the supplier and the retailer, respectively, can be written as h3i = (S s x) s ; (8) h4i = S r y [x] + r : (9) Given a set of base-stock levels (S s ; S r ), the subsequent steady-state probabilities are uniquely determined and can be found by solving the following system of linear equations: A (Ss;Sr) P (Ss;Sr) = 0; (10) P 2 P (Ss;Sr) = 1; (11) where A (Ss;Sr) is the transition rate matrix, P (Ss;Sr) is the vector of steay-state probabilities, and equation (11) is the normalizing constraint. In the section that follows, the steady-state probabilities which are crucial for our investigation of the system will be used to model several measures of channel performance. 3 Optimal Base-Stock Levels and Channel Performance Suppose that the supply chain operates over an in nite horizon. The supplier sells the product to the retailer at a per unit wholesale price w and to customers at a xed marginal price d through 9

11 its own direct channel. The supplier incurs a xed cost c for each unit of the product sold. In addition, the supplier incurs a per unit inventory holding cost at rate h s. The retailer purchases the product from the supplier and sells the product to customers at a xed marginal retail price r. The inventory holding cost per item per time unit at the retailer is h r. To avoid trivial problems, we assume c d and c w r. Let m d and m r be the respective margins of direct and retail sales, m d = d c and m r = r c: With the steady-state probabilities, we can represent the steady-state expected direct and retail sales volumes as Q (Ss;Sr) PS s d = d PS r x=1 y=0 P Q r (Ss;Sr) S s = r x= S r y=1 P (Ss;Sr) PS r PS s + r r x=1 P (Ss;Sr) + d d 0P P (Ss;Sr) (x;0) ; (12) PS r x= S r y=1 P (Ss;Sr) : (13) Also, the steady-state average inventories for the supplier and the retailer, respectively, depend on the steady-state probabilities in the following way: P I s (Ss;Sr) = Ss I (Ss;Sr) r = PS r x=1 y=0 PS s x= S r y=1 xp (Ss;Sr) ; (14) PS r yp (Ss;Sr) : (15) 3.1 System Optimal Solution Assume that all the prices are competitively determined, and all the cost-related parameters are exogenous. Then the only decision variables in the system are the base-stock levels, S s and S r. The steady-state expected pro t for the whole supply chain, which is a function of the two decision variables, can be modeled as: (S s ; S r ) = m d Q (Ss;Sr) d + m r Q (Ss;Sr) r h s I (Ss;Sr) s h r I (Ss;Sr) r. (16) When the supplier and the retailer are coordinated, the objective is to nd base-stock levels that maximize the total supply chain pro t speci ed in (16). 10

12 Lemma 1 Suppose h s > 0 and h r > 0. There exist stock levels, S s and S r, such that (S s ; S r ) 0, 8S s S s and 8S r S r. Lemma 1, which is intuitively true, implies that the optimal base-stock levels are bounded. Due to the complexity of the problem, it is intractable to derive the analytical solutions for the optimal base-stock levels. However, Lemma 1 ascertains that with su ciently large values as heuristic upper bounds, the optimal base-stock levels can be identi ed by applying complete enumeration within the bounds. Since the main objective of our study is to investigate the qualitative system behavior, computational issues are tangential and thus are not addressed here. 3.2 The Decentralized Supply Chain and the Inventory Game When the supply chain is decentralized, the supplier and the retailer are independent decisionmakers, and each looks at its own welfare when making stocking decisions, ignoring the collective impact on the supply chain as a whole. Under the decentralized system, it is straightforward to verify that the steady-state expected pro ts for the supplier and the retailer, respectively, can be represented as functions of S s and S r as follows: s (S s ; S r ) = m d Q (Ss;Sr) d + m r Q (Ss;Sr) r h s I (Ss;Sr) s ; (17) r (S s ; S r ) = (1 )m r Q (Ss;Sr) r h r I (Ss;Sr) r ; (18) where is a measure of the degree of double marginalization (Spengler 1950), and is de ned as = w c r c : (19) Since c w r, it follows that 0 1. Assume that, with the objective of optimizing their own pro ts, the supplier and the retailer simultaneously choose their respective base-stock levels, S s and S r, in the game s only move, and the stocking decisions are continuously committed by the two rms over an in nite horizon. Let 11

13 (a) Base Parametric Values (b) Best Reaction Functions Replenishment Rates: s = 5 r = 10 Demand Rates: = 15 = 0 Unit Holding Costs: h s = 150 h r = 200 Unit Margins: m r = m d = 100 Double Marginalization: = 0:5 Substitution Rates: r = d = 0 S r Optimal S s Nash Equilibrium Supplier s Best Reaction Function Retailer s Best Reaction Function Table 1: System Optimal Solution vs. Nash Solution s be the supplier s strategy pro le and r be the retailer s strategy pro le, and let s = r = fs 2 Z j 0 S Sg; where S is a very large constant. Then, for the inventory game de ned above, the best reaction mappings for the supplier and the retailer, respectively, are given by r (S s ) = s (S r ) = S r 2 r j r (S s ; S r ) = max r (S s ; S) ; (20) s2 r S s 2 s j s (S s ; S r ) = max s (S; S r ) : (21) s2 s A pure strategy Nash equilibrium is a pair of base-stock levels, ( b S s ; b S r ), such that each rm s stocking decision is a best response to the other s: bs s 2 s ( b S r ) and b S r 2 r ( b S s ): For illustrative purposes, in Table 1 the two rms best reaction functions are plotted using a set of parametric values, and the resulting Nash solution is compared to the system optimal solution. Can we always nd a unique pure strategy Nash equilibrium for the inventory game? The discrete nature of the problem and its intricacy make it di cult to analytically ascertain the existence and the uniqueness of Nash equilibrium. To address this question, we performed an extensive numerical analysis. After examining a variety of parametric values considered to be re- 12

14 alistic, we nd that a pure strategy Nash equilibrium exists uniquely in most scenarios. Although exceptional scenarios (which, we suspect, are caused by the discrete nature of the problem) with no equilibrium or multiple equilibria are possible, they are very rare. Moreover, we nd that those exceptional scenarios appear to have no signi cant impact on our qualitative investigation of the system since most results and behaviors are generalizable through parametric sensitivity analyses. 4 Numerical Study of Competitive E ect When the manufacturer and retailer act independently, the supplier concurrently acts as a suppler and a competitor of its retailer. The competitions between the two rms occur both vertically and horizontally. In this section, we conduct numerical studies to investigate the impact of the vertical and horizontal competitions on the rms stocking behaviors and the supply chain e ciencies. While there are numerous scenarios for the numerical experiments, we found that certain parametric values generate similar qualitative results in terms of system behavior. Therefore, we avoid redundancy by choosing the set of parametric values given in Table 1(a) as the base parametric values for our numerical study 1. Note that due to the complicated setting, theoretical proofs of insightful results obtained from the model are often intractable. Thus, we resort to numerical studies to generalize managerial implications like most of the related papers cited above. 4.1 E ect of Vertical Competition The supply chain ine ciency as a result of double marginalization occurs when the manufacturer and retailer act independently and each only receives a portion of the total contribution margin. As mentioned above, it is recognized in the supply chain literature that vertical competition in- 1 A FORTRAN program was written to perform the numerical analysis. Additionally, Routines LSLRG and LFSRG in the IMSL MATH/LIBRARY were utilized to solve the system of linear algebraic equations for the steady-state probabilities using Gaussian elimination method. 13

15 duced by double marginalization may cause understocking behavior within a distribution channel. In this section, we examine the e ect of vertical competition under the multiple-channel context. Degree of α = 0.00 α = 0.25 α = 0.50 α = 0.75 α = 1.00 Double Marginalization Equilibrium Equilibrium Equilibrium Equilibrium Equilibrium  Competition Competition Competition Competition Competition      Â   δ S s S r Penalty S s S r Penalty S s S r Penalty SSs r Penalty S s S r Penalty % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Optimal: Note. Unless otherwise noted, the base parametric values in Table 1(a) are used. Table 2: Impact of Double Marginalization on Channel E ciency Table 2 reports the computational results of the analysis based on the base parametric values. Di erent values of (the proportion of direct customers) ranging from 0 to 1 with step value 0:25, and di erent values of (the degree of double marginalization) ranging from 0 to 1 with step value 0:1 are used in the analysis. Note that = 0 implies no double marginalization (w = c), while = 1 indicates the highest degree of double marginalization (w = p r ). In order to measure the channel e ciency, we de ne the competition penalty (as in Cachon 1999), as the di erence in supply chain pro t between a Nash solution and the system optimal solution, measured as a percentage of the optimal pro t. In Table 2, the equilibrium stock levels, the optimal stock levels and the corresponding competition penalty are reported for each and, and accordingly, we make some generalizable observations. Observation 1 The high degree of double marginalization may induce severe understocking behavior at both the supplier and the retailer. 14

16 Observation 2 Increasing the degree of double marginalization may intensify the competition penalty. Observation 3 For a given degree of double marginalization, the system with a higher proportion of direct customers results in a lower competition penalty. We remark that that when = 0 (no direct customer in the system), the problem reduces to the traditional two-echelon supply chain and the results are consistent with the ndings in the vertical competition literature (e.g., Cachon 1999). Though the supplier may overstock when the degree of double marginalization is moderate, the total inventory level of the supply chain as a whole understocks in most cases. On the other hand, when = 1, the system can be considered to be under centralized control as all customers are direct customers. Not surprisingly, there is no competition penalty for all in this extreme case. In this section, we focus our analysis on the e ect of vertical competition induced by double marginalization by assuming without considering channel substitutions. What happens if stockout based substitutions occur in the system? Next, we examine the combined e ect of vertical competition and horizontal competition. 4.2 Combined E ect of Vertical and Horizontal Competitions The model speci es that when a stockout occurs in either channel, customers shift to the other channel with a known probability. Recall that we de ned r as the proportion of the retail customers who will switch to the direct channel when a stockout occurs at the retail store, and d as the proportion of the direct customers who will switch to the retail store when stockout occurs at the direct channel. Now we investigate the e ect of stockout-based substitution on the system behavior and e ciency when it coexists with double marginalization. We construct three scenarios with di erent degrees of double marginalization (low: = 0:25; moderate: = 0:50; high: = 0:75). For each scenario, Tables 3 and 4 summarize the impact of 15

17 Competition Penalty 50% 40% 30% 20% 10% (a) Competition Penalty δ = 0.50 δ = 0.75 δ = % Substitution of Retail Customers: K r (b) Base Stock Levels Proportion of Direct Customers α = 0.5 Substitution Equilibrium Rate Optimal δ = 0.25 δ = 0.50 δ = 0.75 Â Â Â Â Â Â K D D r S s S r S s S r S s S r S s S r Table 3: Impact of Substitution Rate r increasing substitution of retail customers r and the impact of increasing substitution of direct customers d, respectively, on the supply chain e ciency. There are some insightful observations. Observation 4 Increasing substitution rate r reduces the competition penalty when the degree of double marginalization is high, while on the other hand, it intensi es the competition penalty when the degree of double marginalization is low. Observation 5 Increasing substitution rate d intensi es the competition penalty. When the degree of double-marginalization is high, the decentralized supply chain generally su ers from the ine ciency caused by competitive understocking. Our results show that, in this situation, the e ect of horizontal competition on the system e ciency in a multi-channel supply chain is somewhat ambiguous. Depending on the type of asymmetry of customers stockout based substitution, interestingly, the e ect of combined vertical-horizontal competition may increase or decrease the supply chain e ciency. Speci cally, horizontal competition may help 16

18 Competition Penalty 60% 50% 40% 30% 20% 10% 0% (a) Competition Penalty δ = 0.50 δ = 0.75 δ = Substitution of Direct Customers: β d (b) Base Stock Levels Proportion of Direct Customers α = 0.5 Substitution Equilibrium Rate Optimal δ = 0.25 δ = 0.50 δ = 0.75 D D Â Â Â Â Â Â K d S s S r S s S r S s S r S s S r Table 4: Impact of Substitution Rate d to improve supply chain performance by counterbalancing the understocking behaviors at both the upper and lower echelons if the substitution of retail customers dominates the substitution of direct customers. However, conversely, if the substitution of direct customers dominates the substitution of retail customers, the e ect of combined vertical-horizontal competition is generally detrimental to the supply chain e ciency since the understocking behaviors caused by the e ect of vertical competition may be aggravated by the e ect of horizontal competition. On the other hand, when the degree of double-marginalization is low, the decentralized supply chain, relatively speaking, is more e cient. In this case, our results indicate that, regardless of the type of asymmetry, increasing customers stock-out based substitution typically leads to a decrease in the supply chain e ciency. 5 Supply Chain Contracts The result of the numerical study from the previous section indicates that, in equilibrium, the competitive base-stock levels of the two-echelon dual-channel supply chain rarely agree with the 17

19 system optimal base-stock levels. In other words, the decentralized supply chain is ine cient in most cases. Can the incentives of the two supply chain entities be aligned through a contract such that the supply chain performs at the optimal level in equilibrium? 2 In this section, we address this question by investigating a set of supply chain contracts. 5.1 Retailer Holding Cost Subsidy Intuitively, when the retailer understocks, a holding cost subsidy will induce the retailer to carry more inventory. As pointed out by Cachon (1999), a holding cost subsidy contract is analogous to a buy-back contract (Pasternack 1985) under which the supplier agrees to buy back some portion of the retailer s unsold inventory. Pasternack (1985) shows that, in a hierarchical supply chain with no direct channel distribution, channel coordination can be achieved by a buy-back contract in a single period setting. Can a holding cost subsidy contract help to attain channel coordination if the supply chain with dual-channel distribution operates over an in nite horizon? With a holding cost subsidy contract, the supplier pays the retailer a per unit holding cost of retail inventory at rate h r, where 0 < < 1. Therefore, the holding cost of retail inventory incurs by the retailer per item per time unit is (1 )h r. With the contract, the respective pro ts of the supplier and the retailer are s(s s ; S r ) = m d Q (Ss;Sr) d + m r Q (Ss;Sr) r h s I (Ss;Sr) s h r I (Ss;Sr) r ; (22) r (S s ; S r ) = (1 )m r Q (Ss;Sr) r (1 )h r I (Ss;Sr) r : (23) We conduct a numerical study to better understand the supply chain s stocking behavior with a holding cost subsidy contract. We nd that, in general, when the retailer understocks, a holding cost subsidy contract can motivate the retailer to stock more aggressively. Although the contract is able to improve the overall channel performance, it cannot always coordinate the supply chain. 2 When the supply chain performs at the optimal level in equilibrium through a contract, we say that the contract coordinates the supply chain. 18

20 Equilibrium  S s Case 1 (No Direct Customers) Case 2 ( α 0.5, β = 0.8)  S r Channel Profits γ π s γ π r Competition Penalty No Subsidy: % % % % % % % % γ % % % % % % % % % % % % Optimal: Holding Cost Subsidy Note. Unless otherwise noted, the base parametric values in Table 1(a) are used. = d Equilibrium Channel Profits Competition   γ γ S s S r π s π r Penalty Table 5: Stocking Behavior with Holding Cost Subsidy Table 5 examines two cases with a holding cost subsidy contract. The two cases have the same parametric values as the base case given in Table 1(a), except that in Case 2 the proportion of direct customers is 0:5 and the direct channel substitution rate d is 0:8. For both cases, the supplier overstocks and the retailer understocks without a retailer holding cost subsidy. It is clear that when there are no direct customers (Case 1), the supply chain can be coordinated by a holding cost subsidy contract with = 0:5. However, with direct customers and channel substitution (Case 2), a holding cost subsidy contract is unable to coordinate the supply chain although it may help to improve the overall supply chain pro t. The contract motivates the retailer to stock at the e cient level, but it fails to restrain the supplier s overstocking behavior. Note that when the supply chain is coordinated by a holding cost subsidy contract, the optimal supply chain pro ts can not be arbitrarily redistributed between the two supply chain entities. Besides, as shown in Case 1, the contract may not even result in Pareto improvement. Therefore, in order to be able to implement the contract, periodic xed transfer payments, such as franchise fees, may be required to reallocate the gains from coordination. 19

21 5.2 Lost Sales Transfer Payment A holding cost subsidy contract can be viewed as a carrot treatment that alleviates the retailer s understocking behavior. Now we consider a stick treatment that compels the retailer to carry more inventory, a lost sales transfer payment contract. This opposite approach presumes that the retailer will stock more aggressively if it incurs an extra charge for lost sales. With a lost sales transfer payment contract, the retailer agrees to transfer to the supplier per expected lost sale per unit time. As a result, the respective pro ts of the supplier and the retailer are s(s s ; S r ) = m d Q (Ss;Sr) d + m r Q (Ss;Sr) r h s I (Ss;Sr) s + r L (Ss;Sr) r ; (24) r(s s ; S r ) = (1 )m r Q (Ss;Sr) r h r I (Ss;Sr) r r L (Ss;Sr) r ; (25) where L (Ss;Sr) r is the probability that a stockout occurs at the retail store. It depends upon the steady-state probabilities in the following way P L (Ss;Sr) r = Ss P (S s;s r) x= S r (x;0). (26) Note that to be exible, we do not impose a constraint that forces to be positive. When < 0, it corresponds to a transfer payment made by the supplier to the retailer. With the exibility, the contract may help to coordinate the supply chain in the situation when the retailer overstocks: to discourage the retailer from carrying too much inventory. Equilibrium   S s S r Case 1 (No Direct Customers) Case 2 ( α 0.5, β = 0.8) Channel Profits γ π s γ π r Competition Penalty Note. Unless otherwise noted, the base parametric values in Table 1(a) are used. = d Equilibrium Channel Profits Competition   γ γ S S Penalty s r π s π r No Payment : % No Payment : % % Lost Sales % Lost Sales τ % Transfer Transfer % % % Optimal: 1 5 Optimal: 3 2 Table 6: Stocking Behavior with Holding Cost Subsidy 20

22 Can a lost sales transfer payment contract coordinate the supply chain in any scenario? Table 6 shows that, similar to a holding cost subsidy contract, a lost sales transfer payment contract is able to coordinate the supply chain in Case 1, but is still unable to do so in Case 2. Apparently, the contract is not robust. 5.3 Revenue Transfer Payment with Inventory Sharing Based on the linear transfer payments suggested by Cachon and Zipkin (1999) and Cachon (1999), now we introduce a new contract that can coordinate the supply chain in any scenario, a direct sales revenue transfer payment with inventory sharing contract, or a (; )-contract for brevity. Under a (; )-contract, the supplier agrees to transfer some fraction of the direct sales revenue to the retailer, and moreover, the two rms agree to share the total supply chain inventory holding cost by making transfer payments such that the supplier incurs some fraction of the cost, and the retailer incurs the remaining fraction 1 of the cost. To attain this agreement in practice, the supplier transfers s to the retailer per unit time, while the retailer transfers r to the supplier per unit time, where s = m d Q (Ss;Sr) d + (1 )h s I (Ss;Sr) s ; (27) r = h r I (Ss;Sr) r : (28) In e ect, the pro t functions for the supplier and the retailer with a (; )-contract, respectively, are s(s s ; S r ) = (1 )m d Q (Ss;Sr) d + m r Q (Ss;Sr) r h s I (Ss;Sr) s + h r I r (Ss;Sr) ; (29) r (S s ; S r ) = m d Q (Ss;Sr) d + (1 )m r Q (Ss;Sr) r (1 ) h s I s (Ss;Sr) + h r I r (Ss;Sr) : (30) The choice of and can be arbitrary. However, if we set = 1 and =, then the 21

23 (; )-contract modi es the pro t functions in (29) and (30), respectively, to s(s s ; S r ) = (S s ; S r ); (31) r (S s ; S r ) = (1 )(S s ; S r ); (32) where (S s ; S r ) is the centralized pro t function de ned in (16). Apparently, the two rms face a pro t function that is proportional to the centralized one under the arrangement. Since both rms have the same objective of maximizing the total supply chain pro t, there must exist a Nash equilibrium that corresponds to the optimal base-stock levels of the supply chain. Note that although an appropriately designed (; )-contract guarantees the coordination of the supply chain in any scenario, it doesn t always result in Pareto improvement since the contract can not provide a degree of freedom in splitting the total supply chain pro t. Therefore periodic xed transfer payments may still be required to reallocate the gains from coordination. 6 Concluding Remarks The objective of this paper is to investigate the combined e ect of vertical competition (due to double-marginalization) and horizontal competition (due to stock-out based substitution) on the stocking behaviors and the channel e ciency of a multi-channel supply chain. Based upon the echelon inventory game in this study, we analytically develop measures of channel performance to explore the channel ine ciency induced by inventory competitions. In the echelon inventory game, it is analytically intricate to prove the existence and the uniqueness of Nash equilibrium. However, through an intensive numerical study, we nd that a pure strategy Nash equilibrium exists uniquely in most scenarios, though exceptional scenarios with no equilibrium or multiple equilibria are possible. Moreover, since most results and behaviors are generalizable through parametric sensitivity analyses, those exceptional scenarios appear to have no signi cant impact on the qualitative investigation of the system. 22

24 Albeit advancing our understanding of competitive and cooperative inventory management of a multi-channel supply chain, our model is limited in many respects due to the complex nature of the problem. For example, our model applies one-for-one replenishment policies, which are e ective only for those products (including electronics, furniture, appliances, spare parts, and fashion-oriented products etc.) that exhibit low demand rates. While we conjecture that our qualitative results should be robust, studies seeking to tackle the problem with di erent inventory policies will be warranted. For tractability, we consider only one supplier and one retailer in the multi-channel supply chain. While such a setting provides an appropriate starting point for investigating the problem, it should be valuable to extend the analysis by exploring di erent supply chain structures. Another potentially restrictive assumption of our model is that we take prices as exogenous and only focus our analysis on inventory competition. Although it is a rather standard approach in all related inventory competition papers cited above, it should be obvious that explicitly incorporating price competition into the model would make a valuable contribution to the literature. 23

25 References Aggarwal, P. K., K. Moinzadeh Order Expedition in Multi-Echelon Production/Distribution Systems. IIE Transactions. 26, Alfredsson, P., J. Verrijdt Modeling Emergency Supply Flexibility in a Two-Echelon Inventory System. Management Science Anderson, E., G. S. Day, V. K. Rangan Strategic Channel Design. Sloan Management Review. 38(4) Anupindi, R., Y. Bassok Centralization of Stocks: Retailers vs. Manufacturer. Management Science, 45, Axsäter, S Simple Solution Proceres for a Class of Two-Echelon Inventory Problems. Operations Research. 38, Exact and Approximate Evaluation of Batch Ordering Policies for Two-Level Inventory Systems. Operations Research. 41, Balachander, S., P. F. Farquhar Gaining More by Stocking Less: A Competitive Analysis of Product Availability. Marketing Science. 13, Bell, D. R., Y. Wang, V. Padmanabhan An Explanation for Partial Forward Integration: Why Manufacturers Become Marketers. Working Paper, The Wharton School of Business, University of Pennsylvania. Boyaci, T Manufacturer-Retailer competition and Coordination in a Dual Distribution System. Working paper, McGill University. Cachon, G. P Competitive and Cooperative Inventory Management in a Two-Echelon Supply Chain with Lost Sales. Working paper, The Wharton School of Business, University 24

26 of Pennsylvania, Philadelphia, PA. Cachon, G. P Stock Wars: Inventory Competition in a Two Echelon Supply Chain with Multiple Retailers. Operations Research 49(5) Cachon, G. P, P. H. Zipkin Competitive and Cooperative Inventory Policies in a Two- Stage Supply Chain. Management Science. 45(7) Caldentey, R., L. M. Wein Analysis of a Decentralized Production-Inventory System. Manufacturing & Service Operations Management. 5(1) Cattani, K., G. C. Souza Inventory Rationing and Shipment Flexibility Alternatives for Direct Market Firms. Production & Operations Management. 11(4) Cattani, K., W. G. Gilland, J. M. Swaminathan Coordinating Traditional and Internet Channels. Supply Chain Analysis in the ebusiness Era. Kluwer Academic Publishers Chiang, W. K., D. Chhajed, J. D. Hess Direct Marketing, Indirect Revenues: A Strategic Analysis of Dual-Channel Supply Chain Design. Management Science. 49(1) Chiang, W. K., G. E. Monahan Managing Inventories in a Two-Echelon Dual-Channel Supply Chain. Working Paper, University of Maryland, Baltimore County. Clark, A., H. Scarf Optimal Policies for a Multi-Echelon Inventory Problem. Management Science Clark, T. D., R. E. Trempe, H. E. Trichlin Complex Multi-echelon Inventory System Management Using a Dynamic Simulation Model. Decision Sciences. 14(3) Dada, M., A Two-Echelon Inventory System with Priority Shipments. Management Science. 38,

27 Deuermeyer, B. L., L. B. Schwarz A Model for the Analysis of System Service Level in Warehouse Retailer Distribution Systems: the Identical Retailer Case. Multi-level Production/Inventory Control Systems: Theory and Practice. Studies in Management Science. 16, Grahovac, J., A. Chakravarty Sharing and Lateral Transshipment of Inventory in a Supply Chain with Expensive Low-Demand Items. Management Science. 47(4) Hadley, G., T. M. Whitin Analysis of Inventory Systems. Englewood Cli s, N.J.: Prentice-Hall. Kumar, N., R. Ruan On Strategic Pricing and Complementing the Retail Channel with a Direct Internet Channel. Working Paper, University of Texas at Dallas. Lariviere, M., E. Porteus Selling to the Newsvendor: An Analysis of Price-Only Contracts. Manufacturing & Service Operations Management. 3(4) Lawson, R Integrating Multiple Channels. Chain Store Age. 77(4) 58. Lippman, S. A., K. F. McCardle The Competitive Newsboy. Operations Research. 45(1) Mahajan, S., G. van Ryzin Inventory Competition Substitution. Operations Research. 49(5) Moinzadeh, K., H. L. Lee Batch Size and Stocking Levels in Multi-Echelon Repairable Systems. Management Science. 32, Moriarty, R. T., U. Moran Managing Hybrid Marketing Systems. Harvard Business Review. 90(6)

28 Nahmias, S., S. A. Smith Optimizing Inventory Levels in a Two-Echelon Retailer System with Partial Lost Sales. Management Science. 40, Netessine, S., N. Rudi Centralized and Competitive Inventory Models with Demand Substitution. Operations Research. 51(2) Netessine, S. and F. Zhang Positive Vs. Negative Externalities in Inventory Management: Implications for Supply Chain Design. Forthcoming in Manufacturing &Service Operations Management. Parlar, M., Goyal, S. K Optimal Ordering Decisions for Two Substitutable Products with Stochastic Demand. OPSEARCH. 21(1) Pasternack, B Optimal Pricing and Returns Policies for Perishable Commodities. Marketing Science. 4(2) Peleg, B., H.L. Lee Secondary Markets for Product Diversion with Potential Manufacturer s Intervention, Working Paper, Department of Management Science and Engineering, Stanford University. Pyke, D. F Priority Repair and Dispatch Policies for Repairable-Item Logistics Systems. Naval Research Logistics. 37, Rangan, V. K., M. A. J. Menezes, E. P. Maier Channel Selection for New Industrial Products: A Framework, Method, and Application. Journal of Marketing. 56(3) Rhee, B A Hybrid Channel System in Competition with Net-Only Direct Marketers. Working Paper, The Hong Kong University of Science & Technology. Rhee, B., S. Park Online Store as a New Direct Channel and Emerging Hybrid Channel System. Working Paper, The Hong Kong University of Science & Technology. 27