Joint Assortment Planning and Inventory Control under General Choice Models

We propose a constant-factor approximation framework for the joint assortment planning and inventory management problem, where customers substitute to less preferred in-stock products if more preferred products are out of stock. The customer arrivals follow an increasing-failure-rate (IFR) distribution, and customer purchasing behavior is governed by a discrete choice model with dynamic inventory availability. Our decision involves determining the initial inventory vector, subject to a capacity constraint. Our algorithm begins with a novel fluid relaxation that identifies at most two candidate assortments with the highest potential profitability. For each assortment, we construct a static-allocation-based lower bound on the expected total revenue for an inventory vector that stocks only the products in the assortment. The final inventory decision is the vector that maximizes the stronger of these two lower bounds. We establish a (κ/4-ϵ)-approximation guarantee for a broad class of substitutable choice models, in which adding products weakly decreases the choice probabilities of existing items, and for which the cardinality-constrained assortment optimization problem admits a κ-approximation algorithm. Our result provides the first constant-factor approximation ratios for this problem under several important choice models, including the Markov chain choice model and the mixture of MNL models. Moreover, our framework also achieves the state-of-the-art approximation ratio of 0.25 for the MNL choice model. Our guarantees can be further strengthened when the capacity constraint is relatively loose or when the demand distribution is deterministic or Poisson.