Assortment and Inventory Planning under Dynamic Substitution with MNL Model and Capacity Constraints: A Linear Programming Approach

Abstract

We study a single-period (i.e., one replenishment cycle) joint assortment and inventory problem in a general model with Poisson arrivals, dynamic substitution, the Multinomial Logit (MNL) choice model, and an arbitrary set of capacity constraints. Motivated by business practices in the retail industry, we consider a setting where the initial inventory levels of some products (i.e., before ordering) could be positive and we allow the existence of a set of products whose inventory levels cannot be adjusted (e.g., the retailer does not wish to replenish these products because of out-of-season). The retailer makes a one-time assortment and inventory decisions at the beginning of the period and does not have a direct control over the assortment of the products throughout the remaining of the period (i.e., within the period, product availability is assumed to evolve naturally over time depending on realized sales). Computing the order quantities in the stated model at the beginning of the period is a practically relevant yet technically challenging problem. The main technical challenge here arises because customer substitution behavior is affected by product availability, which makes it difficult to characterize the impact of order quantities on the total sales of each product. In this paper, we develop a linear programming (LP)-based algorithm to compute a provably near-optimal (asymptotically optimal) solution. We start by considering the deterministic (fluid) version of the model and show that, in general, an optimal solution to this model can be computed by solving a sequence of M + 1 linear programs (LPs), where M is the number of products. We then show that the rounded version of this solution is asymptotically optimal for the original stochastic model as the market size grows large.