...configuration, offered back-order the car and give additional discount to make up for the delay. Despite the offer, the author decided to take his chances at another dealer, where he found the configuration of his choice and made a purchase.

Situations like this occur quite often in various retail and industrial settings: a customer who does not find a certain product at the first-choice retailer might decide to switch to another retailer selling the same product or a close substitute. A wealth of research literature addresses the problem of optimally stocking substitutable products under competition. Traditionally, though, this problem is analyzed in a single-period, newsvendor-like setting, and hence the standard assumptions include the risk of lost sales and the salvage of remaining products at a loss at the end of the period. Another feature of a single-period model that is not preserved in a more general multi-period setting is the fact that demand for each retailer depends on the competitor's but not the retailer's own inventory.

In some situations customers are willing to back-order the product in the case of a stock-out. For example, a car buying trip rarely results in an immediate purchase since there are many variations to choose from. Often, the desired car is not available, and the customer faces the choice of back-ordering the car with the first dealer or continuing the search at another dealer. Furthermore, the second dealer may be out of stock too, and the customer faces the dilemma of back-ordering the car with this second dealer or perhaps returning to the first dealer and back-ordering the car there. In such a situation, the total demand faced by each retailer generally depends on the retailer's own inventory level as well as the competitor's inventory, and thus retailers compete for customers by setting the stocking quantities of the product. A recent survey of retailers has found that "of the customers that do not find what they want on the shelf, 40% either defer the purchase or go to another store to find the item" (Andraski and Haedicke, 2003). Clearly, operational decisions about inventory control that must be made in connection with customer switching and back-ordering behavior differ from those that arise in a single-period setting. We seek to better understand the influence of customers' decisions to back-order a product on the optimal stocking policies and the resulting profits of the competing retailers, since this influence is key to a conceptual understanding as well as to generating rules for managerial decisions.

Another major issue arising under multi-period competition is giving customers incentives to back-order: it might be profitable for the retailer to offer a monetary enticement (as in the Volkswagen example at the beginning) to induce more customers to back-order the product rather than go to the competitor. In practice, customer incentives are often handled by the marketing department of a company, while stocking decisions are independently set by the operations department. Hence, it is important to understand how the marketing decision to offer a monetary incentive to back-order the product affects the operational decisions involved in selecting an optimal inventory replenishment policy under competition.

In this paper, we analyze situations in which retailers compete for customers by setting the stocking quantities of a single product with exogenously given prices. Specifically, in a multi-period setting we consider two retailers that simultaneously make inventory replenishment decisions at the beginning of each period using a periodic review base-stock policy. Each retailer's demand is a function of the retailer's own inventory as well as the competitor's inventory in the current period, but neither demand depends on any past decisions by either of the two firms. Leftover inventory at the end of the period is carried over to the next period, incurring an inventory holding cost. We begin the analysis by formulating the multi-period problem in a quite general setting and proving that under appropriate regularity conditions an infinite-horizon policy under which both retailers employ stationary base-stock inventory levels is a Nash equilibrium, i.e., a competitive equilibrium can be found by solving an appropriately defined single-period static game.

With respect to customer back-ordering behavior, we formulate four models. Demand that is unsatisfied by both retailers is either completely lost (model I), or the product is back-ordered (models II-IV), with retailers incurring penalty charges for backlogging customers. For the case of backlogging we further consider the following scenarios. In model II we assume that in the case of a stock-out, at, say, retailer i, those customers who are willing to switch to retailer j do so and are backlogged with retailer j in the case that retailer j cannot satisfy them in the same period. In model III we assume that in the case that demand is not filled initially by retailer i, those customers who are willing to switch do so only if retailer j has inventory to satisfy them in the same period; otherwise, they stay and are backlogged with retailer i. As we demonstrate, in the first three models we analyze situations in which the total (effective) demand that a retailer faces in each period (from the first-choice and the second-choice customers) is a piece-wise linear function of the inventory levels of the two retailers. In model IV we analyze a backlogging problem in which the mean of the total demand that each retailer faces is an arbitrary function of the stocking quantities of the two retailers. This last model may account for effects other than demand substitution. Examples include the stimulating effect of inventory on demand. (Wolfe (1968) provides extensive empirical data to show that weekly sales of some merchandise are strongly correlated with the weekly beginning inventory See also Balakrishnan et al. (2004).)

For all four models, we derive tractable analytical solutions, and, whenever we are able to, determine conditions that guarantee the existence/uniqueness of a stationary equilibrium. We show that model I results in higher inventories and lower profits than model II. We also conduct a sensitivity analysis on equilibrium solutions to changes in the problem parameters. Numerical experiments suggest that different customer back-ordering behavior may result in drastically different inventory decisions and profits. In addition to making an analytical comparison between models I and II, we demonstrate numerically that model II results in higher inventories and lower profits than model III. Therefore, for certain problem parameters it might be in retailers' interest to invest in customer service so as to transition from model I to model II. To transition from model II to model III, it might be worthwhile to invest in an information system that makes the competitor's inventory visible to customers (if doing so is practical). This result is somewhat counterintuitive: while in practice competitors often tend to limit the information exchange, we find that inventory visibility mitigates competitive overstocking by reducing customer switching, which in turn results in lower inventories and higher profits.

As we noted above, it is sometimes reasonable to expect that the number of customers willing to back-order a product is a function of a monetary incentive that accompanies a back-order. We therefore analyze the impact of offering a monetary incentive on the optimal inventory policy by introducing an appropriate relationship between the proportion of backlogging customers and the incentive to back-order. Our main result is that, under some technical assumptions, in models II and IV the competitors' optimal inventory policies are monotone in the amount of the incentive offered. Specifically, an increase in the incentive offered by retailer i leads to an increase in retailer i's inventory and a decrease in retailer j's inventory. Numerical experiments show that, if these technical assumptions are not satisfied, this monotonicity is not necessarily preserved. Moreover, if retailers' inventories are visible to competitors' customers, as in model III, then offering any incentive at all may be detrimental.

The contributions of this paper are twofold. First, we analyze the inventory policies of two competing retailers and make progress in considering four different nonlinear back-ordering scenarios that arise as a result of competition and customer switching behavior. As such, our paper extends the stream of research on static inventory competition with lost sales by considering a multi-period duopolistic environment and analyzing the impact of customer backlogging behavior, phenomena that previous papers have not studied. As we show, these issues have important implications for firms' profitability. Second, we address the issue of giving customers an incentive to back-order the product and provide conditions that guarantee monotonicity of equilibrium inventory levels under such an incentive.

2. Literature survey

A large body of operations literature studies the common phenomenon whereby customers substitute one product with another or switch from one retailer to another when their first-choice product or retailer is stocked out. The stream of literature most relevant to our work is the one that considers substitution under competition, i.e., when substitutable products are sold by different companies that compete for customers. In a single-period (newsvendor) setting, Parlar (1988) models the inventory decisions of two competing retailers selling substitute products and shows the existence and uniqueness of the Nash equilibrium. Wang and Parlar (1994) extend the model to three retailers. Karjalainen (1992), Lippman and McCardle (1997), Mahajan and van Ryzin (1999, 2001), Netessine and Rudi (2003) and Netessine and Zhang (2005) further study this problem for an arbitrary number of retailers. Anupindi and Bassok (1999), Avsar and Baykal-Gursoy (2002) and Nagarajan and Rajagopalan (2003) analyze the impact of substitution in a multi-period setting with lost sales. To the best of our knowledge, this line of research has thus far been constrained within the single-period framework (or a multi-period framework with an assumption of lost sales), where the modeling of demand backlogging is not an issue and hence differs from our multiple-period problem. The papers by Parlar (1988), Karjalainen (1992), Wang and Parlar (1994), Anupindi and Bassok (1999) and Netessine and Rudi (2003) model customer switching behavior similarly to our model I, where we assume that there is no back-ordering. Lippman and McCardle (1997) have a more general model with several rules for allocating demand to competing retailers. Mahajan and van Ryzin (1999, 2001) model demand as a stochastic sequence of heterogeneous customers who choose dynamically among available products based on utility maximization criteria. The closest to our work is a recent paper by Li and Ha (2003) in which the authors consider a two-period variant of the inventory competition problem and allow back-ordering with the first-choice retailer only. However, a common feature of all of these papers is that the effective demand for each retailer depends only on the competitors' inventory. As we show in models II-IV, in a more general case of multi-period competition with backlogging, effective demand should also depend on the retailer's own inventory (due to back-ordering) so that additional complexity is introduced into the analysis and optimality conditions. Hence, single-period inventory competition papers do not capture some of the effects that we analyze. A large portion of research on demand substitution focuses on centralized inventory management decisions. We refer interested readers to Mahajan and van Ryzin (1999) for a comprehensive review of this stream of literature.

Our work fits within the stream of research on stochastic multi-period games that Shapley (1953) initiated with his seminal paper. While a number of papers model single-period inventory competition, an analysis of multi-period stochastic games involving inventory decisions by competing retailers is scarce: except for work that includes Kirman and Sobel (1974) almost 30 years ago and recent work by Avsar and Baykal-Gursoy (2002) and Bernstein and Federgruen (2004), the literature has been rather silent on the issues specific to multi-period oligopolies with inventories. Kirman and Sobel (1974) consider an oligopoly in which retailers set prices and inventory levels but compete on price only (that is, the demand faced by each of two retailers is a function of both retailers' prices but not their inventories). They show that the stationary mixed-pricing policy in which firms randomize their prices is a Nash equilibrium. Bernstein and Federgruen (2004) analyze a similar model. They recognize that randomized policies are undesirable in practice, determine the conditions for the existence of stationary pure-strategy equilibrium policies, and further analyze the game under more specific assumptions about the nature of competition. The major difference between these two papers and our work is that in their models retailers compete on price (even though inventory decisions are made as well), whereas we take prices as exogenous (a rather standard approach in operations literature with the same assumption being made in all the related single-period competition papers cited above) and focus on competition for inventory (product availability). One standard justification for taking prices as exogenous is that in many situations prices are fixed for long periods of time, whereas inventory replenishment decisions are made much more frequently.

Competition for inventory among firms located in different echelons of the supply chain has attracted significant attention among researchers. Representative publications in this stream include Cachon and Zipkin (1999), Chen (1999), Lee and Whang (1999), and Porteus (2000). In all of these papers the supplier and the buyer in a two-stage serial supply chain independently choose base-stock policies resulting in suboptimal decisions from a supply chain perspective. Clearly, the setting for such a problem differs greatly from ours, where competition among retailers takes place within the same supply chain echelon. With regard to customer back-ordering behavior under competition, we are aware only of previous work that considers forms of back-ordering that are no different from noncompetitive back-ordering, i.e., the customer either back-orders the product or leaves without making a purchase (as in Kirman and Sobel (1974), Cachon and Zipkin (1999) and Bernstein and Federgruen (2004)). To the best of our knowledge there is no previous work that considers situations where the customer can switch to a competitor and back-order the product there. A large body of literature in marketing extensively studies the customer choice process (see, for example, Chapter 2 in Lilien et al. (1992)) and hence is related to our models of different back-ordering behavior. However, the marketing literature rarely accounts for inventory issues and does not explicitly model back-ordering.

With regard to incentives to back-order, the closest work is DeCroix and Arreola-Risa (1998), who also assume that the number of customers willing to back-order the product can be influenced by monetary incentives. They, however, consider only a single monopolistic company. Furthermore, our modeling technique differs from theirs and, unlike DeCroix and Arreola-Risa (1998), we do not...

NOTE: All illustrations and photos have been removed from this article.



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