Dynamic Substitution for Selling Multiple Products under Supply and Demand Uncertainties

Substitution is a commonly adopted strategy for matching the demand mix by appropriately allocating the supplies. The recent development of information technology has allowed firms to easily swap products to meet consumers’ preferences. When formulating substitution policies, firms have to take into account the supply and demand conditions, as well as the cost and benefit of substitution. We formulate a dynamic model, in which the firm replenishes the product inventories from uncertain sources and dynamically allocates the available products to meet the uncertain demands with the flexibility of substitution. To address the analytical challenge associated with multi‐product management, we develop a heuristic algorithm that leverages the value of substitution, while allowing separability of future profit among the products. This heuristic algorithm iteratively solves a transportation problem in a network with the reverse Monge property. The application of the reverse Monge property allows us to deal with general substitution structures, which generalizes the commonly studied downward substitution models. Given the product inventories, the derived allocation policy is asymptotically optimal when the firm can frequently remix the products to fulfill the received demands. Through an extensive numerical analysis, we demonstrate that the heuristic policy yields good performance measured by the percentage profit gap from an upper bound problem. We also show that substitution can generate significant benefits when the supply capacities are moderate, the supply and demand uncertainties are high, or the replenishment cycle is short.