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Structural and Solution Analysis for the Ordered Weber Problem under Spatial Uncertainty

Víctor Blanco, Miguel Martínez-Antón

TL;DR

This paper develops a general framework for a single-facility continuous Weber problem under spatial demand uncertainty represented by general probability measures on $\mathbb{R}^d$, using an ordered weighted operator to aggregate expected distances. It establishes convex-analytic properties, proximity bounds to the convex hull of demand supports, and an adaptive sample average approximation (SAA) method with convergence and finite-sample guarantees; it also provides explicit error bounds for spherically symmetric demands and demonstrates substantial computational advantages over discretization in numerical experiments. The work links convex analysis, stochastic programming, and ordered optimization, offering a rigorous foundation for stochastic ordered location models and practical algorithms for large-scale instances. The results have potential impact for location decisions under heterogeneous, unbounded, and distributional uncertainty, with robust performance and scalable computation.

Abstract

We propose a general analytical framework for single-facility continuous location problems under spatial demand uncertainty. In contrast to classical formulations based on discrete or regionally aggregated demands, the proposed model represents uncertainty through general probability measures on $\R^d$, thereby encompassing finite, bounded, and unbounded support distributions within a unified formulation. The objective aggregates expected distances by means of an ordered weighted averaging operator, providing a flexible mathematical structure that includes the classical Weber problem and its ordered extensions as special cases. We establish fundamental properties of this stochastic ordered Weber model, including convexity, continuity, and existence of optimal solutions, and we derive quantitative bounds on the proximity between stochastic minimizers and the convex hulls of demand supports. Building upon these results, we develop and analyze an adaptive sample average approximation scheme, proving its convergence and deriving finite-sample error estimates under mild regularity conditions. For spherically symmetric distributions, we further obtain explicit analytical expressions for the approximation error. Together, these results provide a rigorous mathematical foundation for a broad class of stochastic ordered location models and highlight new theoretical connections between convex analysis, stochastic programming, and ordered optimization.

Structural and Solution Analysis for the Ordered Weber Problem under Spatial Uncertainty

TL;DR

This paper develops a general framework for a single-facility continuous Weber problem under spatial demand uncertainty represented by general probability measures on , using an ordered weighted operator to aggregate expected distances. It establishes convex-analytic properties, proximity bounds to the convex hull of demand supports, and an adaptive sample average approximation (SAA) method with convergence and finite-sample guarantees; it also provides explicit error bounds for spherically symmetric demands and demonstrates substantial computational advantages over discretization in numerical experiments. The work links convex analysis, stochastic programming, and ordered optimization, offering a rigorous foundation for stochastic ordered location models and practical algorithms for large-scale instances. The results have potential impact for location decisions under heterogeneous, unbounded, and distributional uncertainty, with robust performance and scalable computation.

Abstract

We propose a general analytical framework for single-facility continuous location problems under spatial demand uncertainty. In contrast to classical formulations based on discrete or regionally aggregated demands, the proposed model represents uncertainty through general probability measures on , thereby encompassing finite, bounded, and unbounded support distributions within a unified formulation. The objective aggregates expected distances by means of an ordered weighted averaging operator, providing a flexible mathematical structure that includes the classical Weber problem and its ordered extensions as special cases. We establish fundamental properties of this stochastic ordered Weber model, including convexity, continuity, and existence of optimal solutions, and we derive quantitative bounds on the proximity between stochastic minimizers and the convex hulls of demand supports. Building upon these results, we develop and analyze an adaptive sample average approximation scheme, proving its convergence and deriving finite-sample error estimates under mild regularity conditions. For spherically symmetric distributions, we further obtain explicit analytical expressions for the approximation error. Together, these results provide a rigorous mathematical foundation for a broad class of stochastic ordered location models and highlight new theoretical connections between convex analysis, stochastic programming, and ordered optimization.

Paper Structure

This paper contains 11 sections, 8 theorems, 40 equations, 1 figure, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Organization
  4. Preliminaries
  5. Sample Average Approximation and Convergence Analysis
  6. Bootstrap Validation Phase
  7. Convergence Guarantees
  8. Spherically Symmetric Demands: Error Bounds
  9. Numerical Validation
  10. Conclusions
  11. Funding

Key Result

Proposition 1

If $\mathrm{D}(\cdot,X)$ is convex and coercive for every $X$ and $\lambda_1 \geq \cdots \geq \lambda_n \ge 0$, then $\rho$ is convex, coercive, and locally Lipschitz on $\mathbb{R}^d$.

Figures (1)

  • Figure 1: Solutions achieved with deterministic (d) and stochastic (u) versions of the median (top) and center (bottom) problems for different (non-symmetric non-uniform) demand distributions.

Theorems & Definitions (20)

  • Definition 1: Ordered Weighted Sum
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Example 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • ...and 10 more