Assortment and Inventory Planning under Dynamic (Stockout-based) Substitution in the Presence of Customer Returns: A Fluid Analysis

Abstract

We consider a deterministic (fluid) multi-period assortment and inventory planning problem under the Multinomial Logit (MNL) choice model with dynamic (stockout-based) substitution, customer returns with a general return time distribution, and a cumulative capacity (storage) constraint for all products. Specifically, there is a one-time inventory decision at the beginning of the selling horizon and a customer who makes a purchase in the current period is allowed to return the purchase at a future period. The returned item will be inspected and, if it passes the inspection, it will be restocked and becomes available again for sale in the next period. The objective of the firm is to identify the initial inventories of the products that maximize the total profit throughout the horizon. Due to the dynamics created by the dynamic substitution and customer returns, this is a technically challenging problem to solve. Unlike in the setting without returns where a product will no longer be available throughout the remaining periods after it stocks out, in the presence of returns, a product could again be available due to restocked returns and may even stock out again, and this cycle might continue for several times depending on the availability of other products.

We focus on the fluid version of this problem and develop a linear program (LP)-based approach to exactly solve this problem. If the LP has a unique optimal solution, then this solution is also an optimal solution to our problem. If, however, the LP does not have a unique optimal solution, an arbitrary optimal solution of the LP may not be feasible for our problem. To deal with this scenario, we develop an algorithm using which, given any optimal LP solution, we can construct an alternative optimal LP solution that is also optimal for our problem. The algorithm uses both primal and dual LP solutions together with exploiting some key structural properties of the problem. We visualize our approach using illustrative examples.