Robust Maximum Capture Facility Location under Random Utility Maximization Models

TL;DR

This work studies 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 model, and shows that the roust model preserves the monotonicity and submodularity from its deterministic counterpart.

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.

Figures (20)

• Figure 1: Value of robustness for MNL instances; the performances of DET1 and DET2 are almost identical.
• Figure 2: Price of robustness for MNL instances.
• Figure 3: Comparison of the distributions of the objective values given by solutions from RO, SA, DET1 and DET2 approaches, under the MNL choice model and instances of size $|I|=100$ and $m = 50$.
• Figure 4: Value of robustness for nested logit instances; the performances of DET1 and DET2 are almost identical.
• Figure 5: Price of robustness for nested logit instances.

Theorems & Definitions (12)

• Remark 1: Basic properties of GEV's CPGF
• Proposition 1: Some additional properties of GEV's CPGF
• Proposition 2
• Proposition 3: Convexity of the adversary's minimization problems
• Theorem 1: Robustness Preserves the Monotonicity
• Theorem 2: Robustness Preserves the Submodularity
• Lemma 1: First and second order derivatives of $\phi^i(\textbf{x})$
• Lemma 2
• Corollary 1: Performance guarantee for a greedy heuristic
• Proposition 4: Rectangular uncertainty sets