A distributionally robust optimization approach for multi-product inventory decisions with budget constraint and demand and yield uncertainties

Abstract

This study develops a distributionally robust optimization approach for inventory decisions for a retailer with limited budget ordering multiple products from multiple suppliers. The demand of a product is assumed to be dependent on the current inventory level. Uncertainties are involved in the demands and yields of the products with their means and standard deviations as the only known information. Using the distributionally robust optimization approach, the problem is formulated as a worst-case expected profit maximization model with a budget constraint. Through mathematical deduction, the developed model is transformed into a tractable convex programming model which can be solved efficiently. The closed-form solution for the order quantity using a Lagrange multiplier is proposed and the corresponding algorithm is presented in the case with one reliable or unreliable supplier. The retailer’s ordering decisions are investigated when there are multiple identical or different suppliers, and the effects of yield uncertainties are also assessed. Numerical experiments are performed to illustrate the effectiveness and practicality of the proposed models and the solution approaches in dealing with demand and yield uncertainties. Furthermore, the impacts of parameters such as the budget, means and standard deviations of yields and demands, and the correlations between yields on retailer’s ordering policies and performance are analyzed. Managerial insights in selecting suppliers are also provided. In particular, a bootstrapping method is used to estimate the means and standard deviations of the demand and yield distributions in a practical application, and the out-of-sample performance of the resulting optimal inventory policy is evaluated.