Robust Maximum Capture Facility Location under Random Utility Maximization Models

TL;DR

This paper introduces a robust approach to the maximum capture facility location problem under RUM models, ensuring solutions are effective in worst-case scenarios with uncertain parameters, and demonstrates the model's theoretical properties and practical advantages.

Contribution

The paper develops a robust optimization framework for facility location under RUM models, preserving key properties and providing efficient algorithms with proven approximation guarantees.

Findings

Robust model maintains monotonicity and submodularity, enabling greedy approximation.

Concavity of the objective under MNL allows for optimal solutions via outer-approximation.

Experiments show the robustness approach outperforms deterministic and sampling methods.

Abstract

We study a robust version of the maximum capture facility location problem in a competitive market, assuming that each customer chooses among all available facilities according to a random utility maximization (RUM) model. We employ the generalized extreme value (GEV) family of models and assume that the parameters of the RUM model are not given exactly but lie in convex uncertainty sets. The problem is to locate new facilities to maximize the worst-case captured user demand. We show that, interestingly, our robust model preserves the monotonicity and submodularity from its deterministic counterpart, implying that a simple greedy heuristic can guarantee a (1-1/e) approximation solution. We further show the concavity of the objective function under the classical multinomial logit (MNL) model, suggesting that an outer-approximation algorithm can be used to solve the robust model under MNL to optimality. We conduct experiments comparing our robust method to other deterministic and sampling approaches, using instances from different discrete choice models. Our results clearly demonstrate the advantages of our roust model in protecting the decision-maker from bad-case scenarios.