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Coordinating Inventory Control and Pricing Strategies Under Batch Ordering

In this paper we investigate joint pricing and inventory control problems in a finite-horizon, single-product, periodic-review setting with certain/uncertain supply capacities. The demands in different periods are random variables whose distributions depend on the posted price exhibiting the additive form. The order quantity in each period is required to be of integral multiples of a given specific batch size denoted by Q. Inventory replenishment incurs a linear ordering cost. Referred to as the cost-rate function, the sum of holding and backorder costs can either be convex or quasi-convex. The objective is to determine a joint ordering and pricing decision that can maximize the total expected profit over the planning horizon. 
 
We first consider the case in which the cost-rate function is convex and show that the modifiedr, Q list-price policy is optimal for the system with certain and limited capacities, a special case of which is the r, Q list-price policy when capacities become unlimited. As supply capacities become random, the optimal policy follows a new structure wherein the optimal order-up-to level and posted price must be coordinated to make the optimal safety stock level follow the r, Q policy. We further consider the case of a quasi-convex cost-rate function, which may arise when a service level constraint is used as a surrogate for the shortage cost. We demonstrate that the r, Q list-price policy is optimal for the system without supply capacity constraints. In addition, extensions to several other models are discussed. The enabling technique is based on the notion of Q-jump convexity and its variants.