Coordination of Supply Chains with Downside-risk-averse Agents

Coordinating supply chains has been a major issue in supply chain management research. This paper focuses on the coordination of supply chains with downside-risk-averse agents. Motivating by the revenue sharing contract, we developed the sufficient conditions for the coordination of the supply chain with one downside-risk-averse agent. Following the sufficient conditions, the downside protection contract thus developed also allows arbitrary allocation of the profit between the supplier and the retailer. We also develop the necessary condition for the coordination of the supply chain when both the supplier and the retailer are downside-risk averse. We hope this work sheds some insights to the study of the coordination of supply chains with downside-risk-averse agents in more general situations. INTRODUCTION Coordinating supply chains has been a major issue in supply chain management research. Much of the previous research has assumed that the agents in the supply chain are risk neutral, i.e., they maximize their respective expected profits. To characterize other types of agents, the concept of risk aversion was raised. An agent is risk averse if the agent prefers a certain profit π to a risky profit, whose expected value equals π. There are many measures of risk aversion, e.g., the mean-variance trade-off in Markowitz (1959), the average downside risk in Harlow (1991) and Arterian (1996), and the value at risk in Jorion (1997). For more discussions, please see Szegö (2004). This paper devoted to the coordination of supply chains with downside-risk-averse agents, i.e., some agents maximize their expected profits, but with a constraint that their downside risks are limited. The study on downside risk dated back to the publication of Roy (1952), and most recently Gan, Sethi, and Yan (2004) and Gan, Sethi, and Yan (2005). In this paper, we adopt the well-adopted concept of the coordination of a supply chain, which is that all the players are better off with respect to (w.r.t.) the expected profit. The rest part of the paper is organized as follows. We formulate the problem first. The objective function considered in this paper is the expected profit and the constraint is the downside risk. Then we develop two groups of sufficient conditions for the coordination of the supply chain when either the supplier or the retailer is downside-risk averse. Then we show the necessary condition for the coordination of the supply chain when both the supplier and the retailer are downside-risk averse. We briefly conclude in the end. COORDINATION OF SUPPLY CHAINS WITH ONE DOWNSIDE-RISK-AVERSE AGENT We consider a single period problem. There are only one supplier and one retailer in the supply chain. Generally a supply chain runs as follows. First, the supplier and the retailer agree on a contract. Then the retailer calculates his/her optimal order quantity qr w.r.t. the expected profit. The retailer orders qr from the supplier. The supplier receives the order, buys qr raw materials, and pays c for each unit of the material. After finishes the production, the supplier sells each of the production with price w to the retailer. The retailer then sells this production with price p. Assume the demand is stochastic and contains arbitrary distribution, which has the cumulative distribution function (cdf) F(X). At the end of the period, the profit of the supply chain and the left inventory, if any, are allocated according to the contract, if there is any such requirement. First, we consider the case when there are either one risk-neutral supplier or one downside-risk-averse retailer in the supply chain. Mathematically, the supplier solves maxq πs(q) = maxq EX{Πs(q,X)}, (1) where πs(q) is the expected profit of the supplier. The retailer solves maxq πr(q) = maxq EX{Πr(q,X)}, (2) s.t. Pr{Πr(q,X)<αr}<βr, (3) where πr(q) is the expected profit of the retailer, and Equation (3) describes the downside-risk constraint, which requires the probability that the profit of the retailer is less than αr is less than βr. We consider the revue sharing contract (RSC) in Gérard and Martin (2005). RSC requires that no matter how many the retailer orders, the retailer finally gets φ of the total profit of the supply chain, 0≤φ≤1. Thus the more profit the supply chain achieves, the more profit both the supplier and retailer achieve, qr= qs= qsc. In other words, the RSC coordinates the supply chain. Furthermore, RSC allows arbitrary allocation of the total profit between the supplier and the retailer. We modify the RSC to coordinate the supply chain defined in Equations (1)-(3). The idea is: Since the supplier is risk-neutral, as long as we ensure the expected profit of the retailer does not change, the optimal order quantity for the supplier need not change. To satisfy the downside risk constraint of the retailer, we re-allocate the profit between the supplier and the retailer. When the demand is small, the supplier pays the retailer to reduce his/her downside risk. When the demand is high, the supplier takes most of the profit of the supply chain. We discuss this idea in more details as follows. We study the downside-risk of the retailer and the supplier in the RSC. Lemma 1. The downside risk of the retailer and supplier in the RSC are