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The Value of Robust Assortment Optimization Under Ranking-based Choice Models

Bradley Sturt

Abstract

We study a class of robust assortment optimization problems that was proposed by Farias, Jagabathula, and Shah (2013). The goal in these problems is to find an assortment that maximizes a firm's worst-case expected revenue under all ranking-based choice models that are consistent with the historical sales data generated by the firm's past assortments. We establish for various settings that these robust optimization problems can either be solved in polynomial-time or can be reformulated as compact mixed-integer optimization problems. To establish our results, we prove that optimal assortments for these robust optimization problems have a simple structure that is closely related to the structure of revenue-ordered assortments. We use our results to show how robust optimization can be used to overcome the risks of estimate-then-optimize and the need for experimentation with ranking-based choice models in the overparameterized regime.

The Value of Robust Assortment Optimization Under Ranking-based Choice Models

Abstract

We study a class of robust assortment optimization problems that was proposed by Farias, Jagabathula, and Shah (2013). The goal in these problems is to find an assortment that maximizes a firm's worst-case expected revenue under all ranking-based choice models that are consistent with the historical sales data generated by the firm's past assortments. We establish for various settings that these robust optimization problems can either be solved in polynomial-time or can be reformulated as compact mixed-integer optimization problems. To establish our results, we prove that optimal assortments for these robust optimization problems have a simple structure that is closely related to the structure of revenue-ordered assortments. We use our results to show how robust optimization can be used to overcome the risks of estimate-then-optimize and the need for experimentation with ranking-based choice models in the overparameterized regime.
Paper Structure (26 sections, 24 theorems, 31 equations)

This paper contains 26 sections, 24 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. A Robust Optimization Approach
  3. Main Questions
  4. Contributions
  5. Notation and Terminology.
  6. Problem Setting
  7. Answer to Question \ref{['question:1']}
  8. Characterization of Optimal Assortments for \ref{['prob:robust']}
  9. Answer to Question \ref{['question:2']}
  10. Upper Bounds
  11. Discussion of Upper Bound
  12. Overview of Proof of Theorem \ref{['thm:main']}
  13. Algorithms
  14. Polynomial-Time Algorithm for Two Past Assortments
  15. Polynomial-Time Algorithm for Fixed Number of Past Assortments
  16. ...and 11 more sections

Key Result

Theorem 1

There exists an assortment $S \in \widehat{\mathcal{S}}$ that is optimal for prob:robust.

Theorems & Definitions (32)

  • Theorem 1
  • Lemma 1
  • Corollary 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 1
  • Lemma 5
  • Theorem 2
  • Definition 2
  • ...and 22 more