Assortment Optimization Under History-Dependent Effects

TL;DR

This paper studies multi-period assortment planning considering how past customer choices influence current utility, introducing models and algorithms for optimal solutions under various historical effects.

Contribution

It formulates a nonlinear integer programming model for history-dependent assortment optimization and develops solution methods including a mixed-integer exponential cone reformulation and polynomial-time policies.

Findings

The problem is NP-hard with negative history effects.

A lifting-based reformulation enables solving the problem with advanced solvers.

Sequential revenue-ordered policy is optimal for positive history effects.

Abstract

This paper examines how to plan multi-period assortments when customer utility depends on historical assortments. We formulate this problem as a nonlinear integer programming model and show it is NP-hard in the presence of a negative history-dependent effect (such as a satiation effect). We build solution methodologies for obtaining global optimal solutions under a general setting that the history-dependent effects could be a mixture of positive and negative. We propose using a lifting-based framework to reformulate the problem as a mixed-integer exponential cone program that state-of-the-art solvers can solve. We also design a sequential revenue-ordered policy and show that it solves our problem to optimality in polynomial time when historical assortments positively affect customer utility (such as an addiction effect). Additionally, we identify an optimal cyclic policy for an asymptotic regime, and we also relate its length to the customer's memory length. Finally, we present a case study using a catering service dataset, showing that our model demonstrates good fitness and can effectively balance variety and revenue.