Constrained Assortment Optimization under the Cross-Nested Logit Model

TL;DR

This paper addresses the complex assortment optimization problem under the Cross-Nested Logit model, proposing a discretization approach that approximates the NP-hard problem with strong performance guarantees and practical solution methods.

Contribution

It introduces a novel discretization mechanism to approximate the assortment problem under the Cross-Nested Logit model with provable guarantees and extends the approach to joint assortment and pricing problems.

Findings

The approximation method achieves a 90% performance guarantee.

The method's solution gaps are no larger than 1.2% in experiments.

Optimal solutions are obtained via mixed-integer linear programming.

Abstract

We study the assortment optimization problem under general linear constraints, where the customer choice behavior is captured by the Cross-Nested Logit model. In this problem, there is a set of products organized into multiple subsets (or nests), where each product can belong to more than one nest. The aim is to find an assortment to offer to customers so that the expected revenue is maximized. We show that, under the Cross-Nested Logit model, the assortment problem is NP-hard, even without any constraints. To tackle the assortment optimization problem, we develop a new discretization mechanism to approximate the problem by a linear fractional program with a performance guarantee of $\frac{1 - \epsilon}{1+\epsilon}$, for any accuracy level $\epsilon>0$. We then show that optimal solutions to the approximate problem can be obtained by solving mixed-integer linear programs. We further show that our discretization approach can also be applied to solve a joint assortment optimization and pricing problem, as well as an assortment problem under a mixture of Cross-Nested Logit models to account for multiple classes of customers. Our empirical results on a large number of randomly generated test instances demonstrate that, under a performance guarantee of 90%, the percentage gaps between the objective values obtained from our approximation methods and the optimal expected revenues are no larger than 1.2%.