Mustafa Karakul and Lap Mui Ann Chan. Optimal Procurement of Substitutable Products over a Finite Planning Horizon with Random Demands and Discretionary Sales

Abstract. We study a multiple-period, two-product stochastic inventory control model that allows substitution among the products. Demand for each product over the finite horizon follows a nonstationary stochastic distribution that is independent of the demand of the other product and periods. We assume zero set-up cost, linear procurement, holding, shortage, and substitution costs, and lost sales in case of shortage. It is necessary to determine the optimal procurement quantity of each product at the beginning of each period. Furthermore, we incorporate discretionary sales, which means that inventory for a product can be used to substitute the demand for the other product or set aside for future demand even if some present demand is lost. Hence in each period, after realizing the demands, it is necessary to determine the fractions of the current inventory of each product to be used to satisfy its own demand, the other product's demand, and to be saved for future periods. Assuming that after paying the known substitution costs, the retailer can substitute the demand of either product with the other, we show that the objective function, the total expected profit, is concave and submodular in procurement quantities.Hence, the optimal procurement strategy follows an order-up-to policy and the substitution decision can be easily determined by solving a concave program.