Robust inventory management with stock-out substitution

Abstract

Stock-out substitution is a well-documented phenomenon that occurs when customers seek a different product as a substitute for their first-choice item if it runs out of stock. We consider a single-period inventory model with limited information regarding the external demands (i.e., mean, variance, and covariance) and focus on identifying the inventory levels that maximize the worst-case expected profit. We formulate a two-stage optimization model: the second stage characterizes the worst-case joint demand distribution by treating the inventory levels as input parameters, and the first stage identifies the optimal inventory levels based on the results of the second stage. The closed form solution of the second stage problem is analytically intractable except for two special cases. We develop closed form solutions for these two special cases and use them to develop a heuristic for the general case. An extensive numerical study over a wide range of parameters indicates that the performance of the heuristic that we develop is nearly optimal. We also provide insights into the effects of demand correlation, substitution rate and other parameters on the optimal worst-case expected profit of the system.