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Presolving and cutting planes for the generalized maximal covering location problem

Wei Lv, Cheng-Yang Yu, Jie Liang, Wei-Kun Chen, Yu-Hong Dai

TL;DR

The GMCLP with mixed positive and negative customer weights presents large-scale, weakly-relaxed MIP challenges. The authors introduce three tailored techniques—isomorphic aggregation, dominance reduction, and two-customer inequalities—and integrate them into a branch-and-cut framework, yielding significantly tighter LP relaxations and smaller problem instances. Computational experiments show the approach solves many previously unsolved benchmarks and outperforms an extended Benders decomposition, particularly on large, imbalanced instance sets. These results indicate strong practical potential for solving GMCLPs and related variants with additional constraints.

Abstract

This paper considers the generalized maximal covering location problem (GMCLP) which establishes a fixed number of facilities to maximize the weighted sum of the covered customers, allowing customer weights to be positive or negative. Due to the huge number of linear constraints to model the covering relations between the candidate facility locations and customers, and particularly the poor linear programming (LP) relaxation, the GMCLP is extremely difficult to solve by state-of-the-art mixed integer programming (MIP) solvers. To improve the computational performance of MIP-based approaches for solving GMCLPs, we propose customized presolving and cutting plane techniques, which are isomorphic aggregation, dominance reduction, and two-customer inequalities. The isomorphic aggregation and dominance reduction can not only reduce the problem size but also strengthen the LP relaxation of the MIP formulation of the GMCLP. The two-customer inequalities can be embedded into a branch-and-cut framework to further strengthen the LP relaxation of the MIP formulation on the fly. By extensive computational experiments, we show that all three proposed techniques can substantially improve the capability of MIP solvers in solving GMCLPs. In particular, for a testbed of 40 instances with identical numbers of customers and candidate facility locations in the literature, the proposed techniques enable us to provide optimal solutions for 13 previously unsolved benchmark instances; for a testbed of 336 instances where the number of customers is much larger than the number of candidate facility locations, the proposed techniques can turn most of them from intractable to easily solvable.

Presolving and cutting planes for the generalized maximal covering location problem

TL;DR

The GMCLP with mixed positive and negative customer weights presents large-scale, weakly-relaxed MIP challenges. The authors introduce three tailored techniques—isomorphic aggregation, dominance reduction, and two-customer inequalities—and integrate them into a branch-and-cut framework, yielding significantly tighter LP relaxations and smaller problem instances. Computational experiments show the approach solves many previously unsolved benchmarks and outperforms an extended Benders decomposition, particularly on large, imbalanced instance sets. These results indicate strong practical potential for solving GMCLPs and related variants with additional constraints.

Abstract

This paper considers the generalized maximal covering location problem (GMCLP) which establishes a fixed number of facilities to maximize the weighted sum of the covered customers, allowing customer weights to be positive or negative. Due to the huge number of linear constraints to model the covering relations between the candidate facility locations and customers, and particularly the poor linear programming (LP) relaxation, the GMCLP is extremely difficult to solve by state-of-the-art mixed integer programming (MIP) solvers. To improve the computational performance of MIP-based approaches for solving GMCLPs, we propose customized presolving and cutting plane techniques, which are isomorphic aggregation, dominance reduction, and two-customer inequalities. The isomorphic aggregation and dominance reduction can not only reduce the problem size but also strengthen the LP relaxation of the MIP formulation of the GMCLP. The two-customer inequalities can be embedded into a branch-and-cut framework to further strengthen the LP relaxation of the MIP formulation on the fly. By extensive computational experiments, we show that all three proposed techniques can substantially improve the capability of MIP solvers in solving GMCLPs. In particular, for a testbed of 40 instances with identical numbers of customers and candidate facility locations in the literature, the proposed techniques enable us to provide optimal solutions for 13 previously unsolved benchmark instances; for a testbed of 336 instances where the number of customers is much larger than the number of candidate facility locations, the proposed techniques can turn most of them from intractable to easily solvable.
Paper Structure (19 sections, 11 theorems, 38 equations, 3 figures, 5 tables, 1 algorithm)

This paper contains 19 sections, 11 theorems, 38 equations, 3 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions and outlines
  3. Literature review
  4. MIP formulation and its weaknesses
  5. Problem formulation
  6. Challenges of solving the MIP formulation \ref{['mclp-nw1']}
  7. Isomorphic aggregation
  8. Dominance reduction
  9. Strengthening the LP relaxation
  10. Constraint reduction
  11. Two-customer inequalities
  12. Derived inequalities
  13. How two-customer inequalities strengthen the LP relaxation of formulation \ref{['mclp-nw1']}
  14. Separation
  15. Computational results
  16. ...and 4 more sections

Key Result

Theorem 2.3

Let $\mathcal{Y} = \left\{y \in \{0, 1\}^{|\mathcal{I}|} \,:\, y(\mathcal{I}) = p\right\}~\text{and}~\mathcal{Y}_{\text{\rm\rm L\xspace}} = \left\{y \in [0, 1]^{|\mathcal{I}|} \,:\, y(\mathcal{I}) = p\right\}$. The optimal values of formulation mclp-nw1 and its LP relaxation are given by

Figures (3)

  • Figure 1: Performance profiles of the CPU time and number of explored nodes for settings CPX, CPXC, and CPXC+IDT.
  • Figure 2: Performance profiles of the CPU time for settings BD and CPXC+IDT.
  • Figure 3: Performance profiles of the CPU time and number of explored nodes for settings NO_AGG, NO_DR, NO_TCI, and CPXC+IDT.

Theorems & Definitions (27)

  • Remark 2.1
  • Theorem 2.3
  • Example 2.4
  • Remark 2.5
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Corollary 3.4
  • Example 3.5: continued
  • ...and 17 more