Robust Discrete Pricing Optimization via Multiple-Choice Knapsack Reductions

TL;DR

This paper introduces a robust discrete pricing optimization method by reducing the problem to a Multiple-Choice Knapsack Problem, providing an exact LP relaxation solution and demonstrating minimal revenue loss in practical tests.

Contribution

It develops a novel reduction of the robust pricing problem to MCKP and analyzes the LP relaxation's structural properties for efficient solutions.

Findings

Exact LP relaxation solution method for the problem.

Bounded integrality gap with decay rate of O(1/n).

Less than 1% revenue loss in numerical experiments.

Abstract

We study a discrete portfolio pricing problem that selects one price per product from a finite menu under margin and fairness constraints. To account for demand uncertainty, we incorporate a budgeted robust formulation that controls conservatism while remaining computationally tractable. By reducing the problem to a Multiple-Choice Knapsack Problem (MCKP), we identify structural properties of the LP relaxation, in particular upper-hull filtering and greedy filling over hull segments, that yield an exact solution method for the LP relaxation of the fixed-parameter subproblems. For the resulting fixed-parameter subproblems, we show that the integrality gap is bounded additively by a single-item hull jump, and that the corresponding relative gap decays as O(1/n) under standard boundedness and linear-growth assumptions. Numerical experiments on synthetic portfolios and a stylized retail case study with economically calibrated parameters are consistent with these bounds and indicate that robust margin protection can be achieved with less than 1 percent nominal revenue loss on the instances tested.