Optimizing decisions for a dual-channel retailer with service level requirements and demand uncertainties: A Wasserstein metric-based distributionally robust optimization approach

Abstract

This study explores a dual-channel management problem of a retailer selling multiple products to customers through a traditional retail channel and an online channel to maximize expected profit. The prices and order quantities of both the online and the retail channels and the delivery times of the online channel are the decision variables. The demand for each product and each channel is assumed to be random and dependent on the prices of both channels and on the online delivery time. In addition, to ensure an adequate performance, service level requirements are considered and are modeled as joint chance constraints. Wasserstein uncertainty sets using the Wasserstein metric for demand probability distributions centered at the empirical distributions on the observed demands from the historical data are constructed in a data-driven approach. Accordingly, a data-driven distributionally robust joint chance constrained model is developed based on the data-driven Wasserstein uncertainty sets. A conservative CVaR approximation is used for the distributionally robust joint chance constraints. Through mathematical manipulations, the developed model is transformed into a bilinear program, which can be approximated by a mixed integer quadratic programming model using piecewise affine relaxations of the bilinear terms and can be solved efficiently. Numerical experiments are performed to illustrate the effectiveness and practicality of the proposed data-driven distributionally robust optimization approach to deal with demand uncertainties. The effects of the key parameters such as delivery time sensitivity, price sensitivity and customer channel preference are analyzed and managerial insights are provided. The results show that the decisions obtained by the proposed approach are robust to hedge against demand uncertainties. The proposed model and solution approach can provide effective decision supports for retailers selling products through an online channel and a traditional retail channel without reliable demand distribution information. Furthermore, compared with the L1-norm and the L2-norm, the L∞-norm is verified to perform better when used in the Wasserstein metric for constructing the Wasserstein uncertainty sets.