An enhanced heuristic framework for solving the Rank Pricing Problem

TL;DR

This work tackles the Rank Pricing Problem (RPP), a challenging bilevel, NP-hard pricing problem where the upper level selects prices to maximize total benefit under budget-constrained, ranked customer choices. The authors propose a two-phase heuristic framework that combines a base search (Variable Neighborhood Search or a genetic algorithm) with four problem-informed local-improvement subroutines, designed to refine solutions without solving extra optimization problems. Computational experiments across several instance sizes show that the proposed methods often outperform standard MIP solvers in both solution quality and time, with certain variants achieving or closely approaching the true optimum in many runs. The approach demonstrates practical potential for large-scale RPP settings and suggests avenues for parallelization and extensions to variants with unknown budgets or preferences.

Abstract

The Rank Pricing Problem (RPP) is a challenging bilevel optimization problem with binary variables whose objective is to determine the optimal pricing strategy for a set of products to maximize the total benefit, given that customer preferences influence the price for each product. Traditional methods for solving RPP are based on exact approaches which may be computationally expensive. In contrast, this paper presents a novel heuristic approach that takes advantage of the structure of the problem to obtain good solutions. The proposed approach consists of two phases. Firstly, a standard heuristic is applied to get a pricing strategy. In our case, we choose to use the Variable Neighborhood Search (VNS), and the genetic algorithm. Both methodologies are very popular for their effectiveness in solving combinatorial optimization problems. The solution obtained after running these algorithms is improved in a second phase, where four different local searches are applied. Such local searches use the information of the RPP to get better solutions, that is, there is no need to solve new optimization problems. Even though our methodology does not have optimality guarantees, our computational experiments show that it outperforms Mixed Integer Program solvers regarding solution quality and computational burden.

Figures (16)

• Figure 1: Feasible solutions for the illustrative example.
• Figure 2: $N_1((50, 50)$ and $N_2((50, 50)$ for the illustrative example.
• Figure 3: Sets $\mathcal{L}$, $\mathcal{Q}$ and $\mathcal{T}$ for the first iteration of the illustrative example.
• Figure 4: Sets $\mathcal{L}$, $\mathcal{Q}$ and $\mathcal{T}$ for the second iteration of the illustrative example.
• Figure 5: Flowchart of the different heuristic methodologies.