Demand Management and Inventory Control for Substitutable Products

This paper studies dynamic inventory and pricing decisions for a set of substitutable products over a finite planning horizon with full backlogging. The objective is to maximize the total expected discounted profit. We present a general stochastic, price-dependent demand model that unifies many commonly used demand models in the literature. The original formulation is not jointly concave in the decision variables and is therefore intractable. One key observation is that the problem becomes jointly concave if we work with the inverse of the price vector -- the market share vector. With this variable transformation, we are able to characterize the optimal policy and develop algorithms to compute it. In each period, the optimal policy first identifies a set of the products that are overstocked and hence need no replenishment and then specifies the optimal order-up-to levels for under-stocked products and the optimal market shares (and prices) for all products, both of which are functions of the overstocking levels. We further establish conditions under which the optimal policy demonstrates certain monotonicity property that can greatly enhance computation. Finally, we develop several computationally efficient heuristic policies and numerically compare them and discuss the interplay of pricing and inventory decisions