Robust Optimization of the 0-1 Knapsack Problem - Balancing Risk and Return in Assortment Optimization

Abstract

Retailers face the important but challenging task of optimizing their product assortments. The challenge is to find, for every category in every store, the assortment that maximizes (expected) category profit. Adding to the complexity of this 0-1 knapsack problem, retailers should also consider the risk associated with every assortment. While every product in the assortment offers an expected return, there is also uncertainty around its expected demand and profit contribution. Therefore, retailers face the difficult task of designing a portfolio of products that balances risk and return. In this paper, we develop a robust approach to optimize retail assortments that offers this balance. Since the dimensionality of this robust 0-1 knapsack problem in practice often precludes full enumeration, we propose a novel, efficient and real-time heuristic that solves this problem. The heuristic constructs an approximation of the risk-return Efficient Frontier of assortments. We find that the robust solutions offer the retailer a considerable reduction in risk (variance), yet only imply a small reduction in expected return. The constructed approximations contain assortments that are optimal solutions to the robust assortment optimization problem. Moreover, they represent insightful visualizations of the solution space, allowing for interactivity (“what risk premium should the retailer pay?”) in real-time (matter of seconds).

Keywords: Retailing, Assortments, Risk-Return, Efficient Frontier, Robust Optimization, Knapsack Problem

JEL Classification: M31, C11, G11, L81