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Relative Explanations for Contextual Problems with Endogenous Uncertainty: An Application to Competitive Facility Location

Jasone Ramírez-Ayerbe, Emma Frejinger

TL;DR

The paper addresses contextual stochastic optimization under endogenous uncertainty, where decisions alter the underlying distributions. It introduces relative counterfactual explanations with a Wasserstein-regularized objective that minimizes $\mathcal{J}(\boldsymbol{x}^0,\boldsymbol{x})$ plus $\lambda \mathcal{W}_2^2(\mathbb{P}^0,\mathbb{P})$, applied to a choice-based CFLP under a Multinomial Logit model. Key contributions include a mixed-integer bilinear reformulation for the explanations, a model-free lower bound for fast warm starts, and numerical evidence that the Wasserstein term yields sparser, smoother explanations and reduces computation time. This work enables interpretable, trustworthy decision-making in contextual optimization with endogenous uncertainty and offers scalable applicability to facility location and related domains.

Abstract

In this paper, we consider contextual stochastic optimization problems under endogenous uncertainty, where decisions affect the underlying distributions. To implement such decisions in practice, it is crucial to ensure that their outcomes are interpretable and trustworthy. To this end, we compute relative counterfactual explanations that provide practitioners with concrete changes in the contextual covariates required for a solution to satisfy specific constraints. Whereas relative explanations have been introduced in prior literature, to the best of our knowledge this is the first work focusing on problems with binary decision variables and endogenous uncertainty. We propose a methodology that uses the Wasserstein distance as a regularization term, which leads to a reduction in computation times compared to its unregularized counterpart. We illustrate the method using a choice-based competitive facility location problem and present numerical experiments that demonstrate its ability to efficiently compute sparse and interpretable explanations.

Relative Explanations for Contextual Problems with Endogenous Uncertainty: An Application to Competitive Facility Location

TL;DR

The paper addresses contextual stochastic optimization under endogenous uncertainty, where decisions alter the underlying distributions. It introduces relative counterfactual explanations with a Wasserstein-regularized objective that minimizes plus , applied to a choice-based CFLP under a Multinomial Logit model. Key contributions include a mixed-integer bilinear reformulation for the explanations, a model-free lower bound for fast warm starts, and numerical evidence that the Wasserstein term yields sparser, smoother explanations and reduces computation time. This work enables interpretable, trustworthy decision-making in contextual optimization with endogenous uncertainty and offers scalable applicability to facility location and related domains.

Abstract

In this paper, we consider contextual stochastic optimization problems under endogenous uncertainty, where decisions affect the underlying distributions. To implement such decisions in practice, it is crucial to ensure that their outcomes are interpretable and trustworthy. To this end, we compute relative counterfactual explanations that provide practitioners with concrete changes in the contextual covariates required for a solution to satisfy specific constraints. Whereas relative explanations have been introduced in prior literature, to the best of our knowledge this is the first work focusing on problems with binary decision variables and endogenous uncertainty. We propose a methodology that uses the Wasserstein distance as a regularization term, which leads to a reduction in computation times compared to its unregularized counterpart. We illustrate the method using a choice-based competitive facility location problem and present numerical experiments that demonstrate its ability to efficiently compute sparse and interpretable explanations.

Paper Structure

This paper contains 13 sections, 19 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Statement
  4. Cost Function and Wasserstein Regularization
  5. Model-free Distributions to Compute Lower Bounds
  6. Application to Choice-based CFLP
  7. Explanations for Choice-based CFLP
  8. Lower Bound and Warm Start
  9. Computational Experiments
  10. Illustrative Example
  11. Larger Instances
  12. Conclusion
  13. Formulation from haase2009discrete

Figures (2)

  • Figure 1: Illustration showing candidate (triangles), competitor (solid dimonds) and customer (solid circles) locations
  • Figure 2: Probability distributions for each customer. Each bar color represents the distribution induced by a a solution to one of the three problems: the factual, and the counterfactual explanations with and without Wasserstein regularization.

Theorems & Definitions (3)

  • Definition 1.1
  • Example 3.1
  • Definition 3.1