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The probabilistic p-center problem: Planning service for potential customers

Luisa I. Martínez-Merino, Maria Albareda-Sambola, Antonio M. Rodríguez-Chía

TL;DR

This work proposes and analyzes different formulations as well as a Variable Neighborhood Search heuristic for the probabilistic p-center problem, which aims at minimizing the expected maximum distance between any site with demand and its center.

Abstract

This work deals with the probabilistic p-center problem, which aims at minimizing the expected maximum distance between any site with demand and its center, considering that each site has demand with a specific probability. The problem is of interest when emergencies may occur at predefined sites with known probabilities. For this problem we propose and analyze different formulations as well as a Variable Neighborhood Search heuristic. Computational tests are reported, showing the potentials and limits of each formulation, the impact of their enhancements, and the effectiveness of the heuristic.

The probabilistic p-center problem: Planning service for potential customers

TL;DR

This work proposes and analyzes different formulations as well as a Variable Neighborhood Search heuristic for the probabilistic p-center problem, which aims at minimizing the expected maximum distance between any site with demand and its center.

Abstract

This work deals with the probabilistic p-center problem, which aims at minimizing the expected maximum distance between any site with demand and its center, considering that each site has demand with a specific probability. The problem is of interest when emergencies may occur at predefined sites with known probabilities. For this problem we propose and analyze different formulations as well as a Variable Neighborhood Search heuristic. Computational tests are reported, showing the potentials and limits of each formulation, the impact of their enhancements, and the effectiveness of the heuristic.
Paper Structure (21 sections, 23 equations, 3 figures, 9 tables)

This paper contains 21 sections, 23 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. The problem
  3. Formulation for the homogenous case ($\mathbf{ q_i=q}$ for all $\mathbf{ i\in N}$)
  4. Formulations for the ${\mathbf K\textrm{-}PpCP\xspace}$
  5. Three index formulation
  6. Valid inequalities
  7. Variable fixing
  8. Compact 3-index formulation
  9. Valid inequalities
  10. Formulation with probability chains
  11. Lower and upper bounds
  12. Variable Neighborhood Search for the $K\textrm{-}PpCP$
  13. Computational experience
  14. Quality of the bounds
  15. Evaluation of the formulations
  16. ...and 6 more sections

Figures (3)

  • Figure 1: Solutions with different demand probabilities
  • Figure 2: Solutions of the instance in Example \ref{['ejnoCAC']} without CAC (left) and with CAC (right).
  • Figure 3: CPU times for the different variants