The Stability of MNL-Based Demand Under Dynamic Customer Substitution and Its Algorithmic Implications

We study the dynamic assortment planning problem under the widely utilized multinomial logit choice model (MNL). In this single-period assortment optimization and inventory management problem, the retailer jointly decides on an assortment, that is, a sub-set of products to be offered, as well as on the inventory levels of these products, aiming to maximize the expected revenue subject to a capacity constraint on the total number of units stocked. The demand process is formed by a stochastic stream of arriving customers, who dynamically substitute between products according to the MNL model. Although this dynamic setting is extensively studied, the best known approximation algorithm guarantees an expected revenue of at least 0.139 times the optimum, assuming that the demand distribution has an increasing failure rate. In this paper, we establish novel stochastic inequalities showing that, for any given inventory levels, the expected demand of each offered product is "stable" under basic algorithmic operations, such as scaling the MNL preference weights and shifting inventory across comparable products. We exploit this sensitivity analysis to devise the first approximation scheme for dynamic assortment planning under the MNL model, allowing one to efficiently compute inventory levels that approach the optimal expected revenue within any degree of accuracy. The running time of this algorithm is polynomial in all instance parameters except for an exponential dependency on log Delta, where Delta = w(max)/w(min) stands for the ratio of the extremal MNL preference weights. Finally, we conduct simulations on simple synthetic instances with uniform preference weights (i.e., Delta = 1). Using our approximation scheme to derive tight upper bounds, we gain some insights into the performance of several heuristics proposed by previous literature.