Stochastic optimization models with substitution as a result of price differences and stockouts

Many firms produce a variety of products that are subject to demand uncertainty and are substitutable by customers where the potential for product substitution affects the firm's pricing and production decisions. Two common reasons for product substitution are stockouts (or inventory‐driven substitution), when a customer will substitute a product that is out of stock with a similar product, and price‐driven substitution where customers respond to price differences by substituting a lower priced product for a similar but higher priced product. In this paper, we include the potential for demand to move from one product or market segment to another into the demand model of the firm and present a series of single‐period stochastic models for finding optimal solutions for production quantity and product prices separately, as well as investigating the joint pricing and production decision model. We derive the optimal solution with and without a total production capacity limit and consider both inventory‐driven and price‐driven substitutions. We investigate how the firm should modify pricing and production decisions to take into account both price‐driven and inventory‐driven substitution and examine changes in the optimal prices and supply quantities with and without an aggregate supply limit. Our results demonstrate that both forms of substitution provide a revenue or profit opportunity if the firm is able to recognize the potential for substitution and respond in advance using an optimal pricing and/or ordering strategy. The contribution of this research is to present theoretical results that demonstrate the value of more complex demand models that include the possibility of stockouts and customer “leakage” from higher priced market segments to lower priced segments. Insights derived from these models lead to modified pricing and ordering strategies that will increase firm revenues and/or profitability.