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Robust personalized pricing under uncertainty of purchase probabilities

Shunnosuke Ikeda, Naoki Nishimura, Noriyoshi Sukegawa, Yuichi Takano

TL;DR

The paper addresses robust personalized pricing for a single item when predicted purchase probabilities are uncertain. It develops a bootstrap-based uncertainty set for purchase probabilities and embeds it in a mixed-integer linear optimization (MILO) framework that can be solved exactly, complemented by a scalable Lagrangian-decomposition heuristic with golden-section search. Empirical results from synthetic and real-world grocery data show that accounting for probability uncertainty improves expected revenue and that the heuristic enables fast, high-quality solutions on large-scale problems. This work enables reliable, data-driven prescriptive pricing in single-item settings and points to extensions to multi-item pricing and cross-elasticities as promising directions.

Abstract

This paper is concerned with personalized pricing models aimed at maximizing the expected revenues or profits for a single item. While it is essential for personalized pricing to predict the purchase probabilities for each consumer, these predicted values are inherently subject to unavoidable errors that can negatively impact the realized revenues and profits. To address this issue, we focus on robust optimization techniques that yield reliable solutions to optimization problems under uncertainty. Specifically, we propose a robust optimization model for personalized pricing that accounts for the uncertainty of predicted purchase probabilities. This model can be formulated as a mixed-integer linear optimization problem, which can be solved exactly using mathematical optimization solvers. We also develop a Lagrangian decomposition algorithm combined with line search to efficiently find high-quality solutions for large-scale optimization problems. Experimental results demonstrate the effectiveness of our robust optimization model and highlight the utility of our Lagrangian decomposition algorithm in terms of both computational efficiency and solution quality.

Robust personalized pricing under uncertainty of purchase probabilities

TL;DR

The paper addresses robust personalized pricing for a single item when predicted purchase probabilities are uncertain. It develops a bootstrap-based uncertainty set for purchase probabilities and embeds it in a mixed-integer linear optimization (MILO) framework that can be solved exactly, complemented by a scalable Lagrangian-decomposition heuristic with golden-section search. Empirical results from synthetic and real-world grocery data show that accounting for probability uncertainty improves expected revenue and that the heuristic enables fast, high-quality solutions on large-scale problems. This work enables reliable, data-driven prescriptive pricing in single-item settings and points to extensions to multi-item pricing and cross-elasticities as promising directions.

Abstract

This paper is concerned with personalized pricing models aimed at maximizing the expected revenues or profits for a single item. While it is essential for personalized pricing to predict the purchase probabilities for each consumer, these predicted values are inherently subject to unavoidable errors that can negatively impact the realized revenues and profits. To address this issue, we focus on robust optimization techniques that yield reliable solutions to optimization problems under uncertainty. Specifically, we propose a robust optimization model for personalized pricing that accounts for the uncertainty of predicted purchase probabilities. This model can be formulated as a mixed-integer linear optimization problem, which can be solved exactly using mathematical optimization solvers. We also develop a Lagrangian decomposition algorithm combined with line search to efficiently find high-quality solutions for large-scale optimization problems. Experimental results demonstrate the effectiveness of our robust optimization model and highlight the utility of our Lagrangian decomposition algorithm in terms of both computational efficiency and solution quality.
Paper Structure (23 sections, 1 theorem, 15 equations, 3 figures, 5 tables, 2 algorithms)

This paper contains 23 sections, 1 theorem, 15 equations, 3 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Related work
  4. Our contribution
  5. Personalized pricing model
  6. Robust optimization model for personalized pricing
  7. Problem formulation
  8. Uncertainty estimation of purchase probabilities
  9. Heuristic algorithm for large-scale problems
  10. Lagrangian relaxation method
  11. Golden section search
  12. Algorithm procedure
  13. Experimental results
  14. Synthetic datasets
  15. Real-world datasets
  16. ...and 8 more sections

Key Result

Theorem 1

The optimization problem eq:obj_robust--eq:sum_gamma can be reformulated as the following MILO problem eq:obj_robust_dual--eq:nu.

Figures (3)

  • Figure 1: Average revenues for Dataset1--3 ($\lvert \mathcal{I}^{\rm{test}} \rvert = 500$)
  • Figure 2: Average revenues for Dataset4--6 ($\lvert \mathcal{I}^{\rm{test}} \rvert = 500$)
  • Figure 3: Average revenue for "The complete journey"

Theorems & Definitions (2)

  • Theorem 1
  • proof