Assortment and Inventory Planning Under Dynamic Substitution with MNL Model: An LP Approach and an Asymptotically Optimal Policy

Abstract

We revisit the uncapacitated single-period joint assortment and inventory problem in the presence of dynamic (stockout-based) substitution behavior (i.e., the so-called dynamic assortment problem). This is a very important practical problem; at the same time, it is also a very difficult analytical problem. The key technical challenge here is due to the fact that customer substitution behavior may change depending on product availability, which requires us to keep track the stockout times of all products. In this paper, we consider a general version of this problem under the Multinomial Logit (MNL) choice model. We first consider the setting with deterministic fluid demand and deterministic choice in which customers are infinitesimal, arrive into the system at a constant rate, and can simultaneously purchase fractional amounts of different products, which we call the ``DD" model for brevity. We show that an optimal solution for this model can be computed by solving a sequence of simple Linear Programs (LPs). We further show that there exists an optimal solution to each LP that satisfies a so-called ``quasi gain-ordered" property, which generalizes the well-known ``revenue-ordered" property of optimal assortment in the static assortment problem under MNL. Next, we consider a more realistic setting with random Poisson arrivals and random choice, which we call the ``RR" model. We show that the optimal solution for the DD model is asymptotically optimal in the RR model when the expected number of customers is large, which justifies studying the DD model as an approximation of the RR model. Unlike the analysis of similar asymptotic optimality results in the literature, the proof of this result in our setting is non-trivial due to the dynamic substitution behavior. Beyond analyzing the uncapacitated single-period problem, we also consider two extensions: the capacitated single-period problem and a capacitated multi-period problem with stationary demand, both with a general set of storage capacity constraints. We show that, for each of them, the corresponding DD model can be solved using a modified LP sequence and that its optimal solution is asymptotically optimal in the corresponding RR model.