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Constrained Pricing in Choice-based Revenue Management

Qian Shao, Tien Mai, Shih-Fen Cheng

TL;DR

This work tackles constrained pricing under MNL demand, where the objective $F(\mathbf{r})=\sum_j r_j P_j(\mathbf{r})$ is non-convex in prices and price constraints complicate classical probability-space convexifications. It introduces a piecewise-linear approximation combined with a bisection (Dinkelbach) framework to transform the problem into a sequence of MILPs, enabling near-optimal solutions for static pricing under joint price and purchasing-probability constraints. The approach extends to dynamic pricing via dynamic programming decomposition, employing MILP subproblems to handle price constraints in each stage, and uses a deterministic SP$^\ast$ model to aid decomposition. Theoretical results provide $O(1/K)$ bounds on approximation errors, and extensive numerical experiments show that SP-DMIP and DP-DMIP consistently outperform standard baselines across static and dynamic settings, with $K=15$ offering a practical balance between accuracy and computation.

Abstract

We consider a dynamic pricing problem in network revenue management where customer behavior is predicted by a choice model, i.e., the multinomial logit (MNL) model. The problem, even in the static setting (i.e., customer demand remains unchanged over time), is highly non-concave in prices. Existing studies mostly rely on the observation that the objective function is concave in terms of purchasing probabilities, implying that the static pricing problem with linear constraints on purchasing probabilities can be efficiently solved. However, this approach is limited in handling constraints on prices, noting that such constraints could be highly relevant in some real business considerations. To address this limitation, in this work, we consider a general pricing problem that involves constraints on both prices and purchasing probabilities. To tackle the non-concavity challenge, we develop an approximation mechanism that allows solving the constrained static pricing problem through bisection and mixed-integer linear programming (MILP). We further extend the approximation method to the dynamic pricing context. Our approach involves a resource decomposition method to address the curse of dimensionality of the dynamic problem, as well as a MILP approach to solving sub-problems to near-optimality. Numerical results based on generated instances of various sizes indicate the superiority of our approximation approach in both static and dynamic settings.

Constrained Pricing in Choice-based Revenue Management

TL;DR

This work tackles constrained pricing under MNL demand, where the objective is non-convex in prices and price constraints complicate classical probability-space convexifications. It introduces a piecewise-linear approximation combined with a bisection (Dinkelbach) framework to transform the problem into a sequence of MILPs, enabling near-optimal solutions for static pricing under joint price and purchasing-probability constraints. The approach extends to dynamic pricing via dynamic programming decomposition, employing MILP subproblems to handle price constraints in each stage, and uses a deterministic SP model to aid decomposition. Theoretical results provide bounds on approximation errors, and extensive numerical experiments show that SP-DMIP and DP-DMIP consistently outperform standard baselines across static and dynamic settings, with offering a practical balance between accuracy and computation.

Abstract

We consider a dynamic pricing problem in network revenue management where customer behavior is predicted by a choice model, i.e., the multinomial logit (MNL) model. The problem, even in the static setting (i.e., customer demand remains unchanged over time), is highly non-concave in prices. Existing studies mostly rely on the observation that the objective function is concave in terms of purchasing probabilities, implying that the static pricing problem with linear constraints on purchasing probabilities can be efficiently solved. However, this approach is limited in handling constraints on prices, noting that such constraints could be highly relevant in some real business considerations. To address this limitation, in this work, we consider a general pricing problem that involves constraints on both prices and purchasing probabilities. To tackle the non-concavity challenge, we develop an approximation mechanism that allows solving the constrained static pricing problem through bisection and mixed-integer linear programming (MILP). We further extend the approximation method to the dynamic pricing context. Our approach involves a resource decomposition method to address the curse of dimensionality of the dynamic problem, as well as a MILP approach to solving sub-problems to near-optimality. Numerical results based on generated instances of various sizes indicate the superiority of our approximation approach in both static and dynamic settings.
Paper Structure (30 sections, 7 theorems, 55 equations, 1 figure, 3 tables, 1 algorithm)

This paper contains 30 sections, 7 theorems, 55 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Problem Formulation
  4. Static Pricing
  5. Dynamic Pricing
  6. Constrained Static Pricing
  7. Piece-wise Linear Approximation
  8. Solving the Approximate Problem
  9. Approximation Errors
  10. Constrained Dynamic Pricing
  11. Numerical Study
  12. Baselines
  13. Static Pricing:
  14. Dynamic Pricing:
  15. Instance Generation
  16. ...and 15 more sections

Key Result

Proposition 1

For any $\xi\geq 0$, Algorithm Algorithm binary search always terminates after no more than $\lceil\log\left(\frac{U-L}{\xi}\right)\rceil$ steps, and return a solution $\widehat{\textbf{r}}$ such that $\widehat{F}(\widehat{\textbf{r}}) \geq \widehat{F}(\textbf{r}^*) -\xi$, where $\textbf{r}^*$ is an

Figures (1)

  • Figure 1: For m=16, n=80, T=200, the mean and standard deviation of revenue gap/running time of 5 different random seeds

Theorems & Definitions (8)

  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Definition 1
  • Lemma 3
  • Lemma 4
  • Theorem 2