Competitive and Cooperative Inventory Management in a Two-Echelon Supply Chain with Lost Sales¤

This paper studies inventory management in a two echelon supply chain with stochastic demand and lost sales. The optimal policy is evaluated and compared with the competitive solution, the outcome of a game between a supplier and a retailer in which each ...rm attempts to maximize its own pro...t. It is shown that supply chain pro...t in the competitive solution is always less than the optimal pro...t. However, the magnitude of the competition penalty is sometimes a tri‡e, sometimes enormous. Several contracts are considered to align the ...rms’ incentives so that they choose supply chain optimal actions. These contracts contain one or more of the following elements: a retailer holding cost subsidy (which acts like a buy-back/return policy), a lost sales transfer payment (which acts like a revenue sharing contract) and inventory holding cost sharing. With the latter each ...rm incurs a ...xed fraction of the total supply chain holding cost. It is found that the retailer holding cost subsidy is generally not su¢cient to coordinate the supply chain. The most e¤ective contract combines a lost sales transfer payment with inventory holding cost sharing: it always coordinates the supply chain, both players are always better o¤ and it is simple to evaluate. ¤Thanks is extended to Martin Lariviere for his many helpful comments. Since I will soon present this work at the Kenan-Flagler School, University of North Carolina and the Kellogg School of Management, Northwestern University, I thank those seminar participants in advance. Essentially all supply chains operate as a collection of independent agents, each responsible for managing a subset of the supply chain, each with its own objectives. While it is unlikely that the agents’ objectives are entirely orthogonal, one should not expect harmoniously aligned incentives either. But, when do those di¤erence lead to sub-optimal supply chain performance? As in a good marriage, some disagreement is natural, relatively harmless and certainly does not justify costly intervention/counseling. Only when the supply chain operates signi...cantly below its optimal performance should the ...rms seek therapy, i.e., align their incentives. But what kinds of agreements align the ...rms’ incentives so that (1) each ...rm can be reasonably certain that the other ...rm will choose a supply chain optimal action (i.e., uphold its part of the deal) and (2) each ...rm is better o¤ with the agreement than without it, i.e., it is Pareto improving? This paper studies those questions with a two ...rm supply chain inventory model. The two ...rms, a supplier and a retailer, operate over an in...nite horizon. The supplier produces a single product at a ...nite production rate and at a ...xed marginal production cost. The retailer purchases the product at a ...xed wholesale price and sells the product at a ...xed retail price. Both the supplier and the retailer incur holding costs on their inventory. The inter-arrival times of customers are stochastic and each customer purchases exactly one unit if a unit is available at the retailer. If a unit is not available, that potential sale is lost. Each ...rm attempts to maximize its expected pro...t per unit time knowing that the other ...rm has the same objective. The ...rms could choose actions that lead to optimal supply chain performance: the optimal policy belongs to their feasible set of actions and all information is common knowledge (so the ...rms are able to evaluate the optimal policy). Nevertheless, the ...rms may choose sub-optimal policies because their incentives are not aligned with the supply chain’s incentives. It is not obvious whether the ...rms will carry too much or too little inventory relative to the optimal amount. Increasing supply chain inventory bene...ts both ...rms, so the supply chain enjoys a larger bene...t from increasing inventory than either ...rm, which suggests the ...rms will not stock enough. But neither ...rm incurs the full cost of increasing supply chain inventory (the increase may be divided between the ...rms), which suggests the ...rms will stock too much. If those contradictory incentives balance each other, the ...rms may in fact choose policies that are reasonably close to optimal, if not optimal. If not, it is important