Assortment and Inventory Planning Under Dynamic Substitution with MNL Model : Structural Results and a Near-Optimal Heuristic

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 faced by many retailers; at the same time, it is also a very difficult analytical problem. Indeed, despite recent progresses in the literature, little is known about the structure of its optimal solution. The key technical challenge here is due to the fact that customer substitution behavior may change depending on product availability. In this paper, we consider the version of this problem under 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 study the structures of its optimal solution. In particular, we show that there exists an optimal solution that satisfies a property similar to the well-known revenue-ordering property of optimal assortment in the "static assortment problem" under MNL. The running time complexity of our algorithm is at most O(M^2) where M is the number of products. We then consider a more realistic setting with random Poisson arrivals and random choice, which we call the "RR" model. We show that the optimal value of the DD model is not always an upper bound for the optimal value of the RR model. However, we are able to show that the optimal solution for the DD model is asymptotically optimal in the RR model when the expected number of customers is large. Thus, overall, our results contribute to the literature by providing both deeper insights into the structure of the optimal solution, in some settings, and an asymptotically optimal heuristic