In markets where customers tend to purchase baskets of products rather than single products, assortment optimization is a major challenge for retailers. Removing a product from a retailer's assortment can result in a severe drop in aggregate demand if this product is a complement to other products. Therefore, accounting for the complementarity effect is essential when making assortment decisions. In this paper, we develop an analytical framework designed to address this problem. We model customers' choices using the binary Ising model, which is a special case of a Markov random field. This model captures pairwise demand dependencies as well as the individual attractiveness of each product, with binary variables indicating whether products are included in the purchased basket. For estimation, we utilize existing methods developed for the spin formulation of the Ising model and then apply a parameter transformation to express the fitted model in the binary form used for assortment optimization. We show that the assortment optimization problem under the Ising model is NP-hard. We also provide multiple theoretical insights into the structure of the optimal assortments based on the graphical representation of the Ising model, and propose several heuristic algorithms that can be used to obtain high-quality solutions to the assortment optimization problem. Our numerical analysis demonstrates that the developed simulated annealing procedure leads to an expected profit gain of 15% compared to offering an unoptimized assortment (where all products are included) and around 5% compared to using a revenue-ordered heuristic algorithm.

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