Mallows-Smoothed Distribution over Rankings Approach for Modeling Choice

Assortment optimization is an important problem arising in many applications, including retailing and online advertising. The goal in such problems is to determine a revenue-/profit-maximizing subset of products to offer from a large universe of products when customers exhibit stochastic substitution behavior. We consider a mixture of Mallows model for demand, which can be viewed as a “smoothed” generalization of sparse, rank-based choice models, designed to overcome some of their key limitations. In spite of these advantages, the Mallows distribution has an exponential support size and does not admit a closed-form expression for choice probabilities. We first conduct a case study using a publicly available data set involving real-world preferences on sushi types to show that Mallows-based smoothing significantly improves both the prediction accuracy and the decision quality on this data set. We then present an efficient procedure to compute the choice probabilities for any assortment under the mixture of Mallows model. Surprisingly, this finding allows us to formulate a compact mixed integer program (MIP) that leads to a practical approach for solving the assortment-optimization problem under a mixture of Mallows model. To complement this MIP formulation, we exploit additional structural properties of the underlying distribution to propose several polynomial-time approximation schemes (PTAS), taking the form of a quasi-PTAS in the most general setting, which can be strengthened to a PTAS or a fully PTAS under stronger assumptions. These are the first algorithmic approaches with provably near-optimal performance guarantees for the assortment-optimization problem under the Mallows or the mixture of Mallows model in such generality.