A Re-solving Heuristic for Dynamic Assortment Optimization with Knapsack Constraints

TL;DR

This paper introduces a novel epoch-based re-solving heuristic for multi-stage dynamic assortment optimization with knapsack constraints, addressing computational challenges and providing theoretical regret bounds.

Contribution

It develops a new linear programming reformulation for fluid approximation in assortment optimization, enabling practical and theoretical analysis of the re-solving approach.

Findings

Regret scales logarithmically with time horizon and resource capacities.

The proposed algorithm effectively handles the non-linear fluid approximation.

The method improves computational tractability for complex assortment problems.

Abstract

In this paper, we consider a multi-stage dynamic assortment optimization problem with multi-nomial choice modeling (MNL) under resource knapsack constraints. Given the current resource inventory levels, the retailer makes an assortment decision at each period, and the goal of the retailer is to maximize the total profit from purchases. With the exact optimal dynamic assortment solution being computationally intractable, a practical strategy is to adopt the re-solving technique that periodically re-optimizes deterministic linear programs (LP) arising from fluid approximation. However, the fractional structure of MNL makes the fluid approximation in assortment optimization non-linear, which brings new technical challenges. To address this challenge, we propose a new epoch-based re-solving algorithm that effectively transforms the denominator of the objective into the constraint, so that the re-solving technique is applied to a linear program with additional slack variables amenable to practical computations and theoretical analysis. Theoretically, we prove that the regret (i.e., the gap between the resolving policy and the optimal objective of the fluid approximation) scales logarithmically with the length of time horizon and resource capacities.