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Mathematical formulations and a Relax-and-Fix heuristic algorithm for capacitated reliable fixed-charge facility

Abdolreza Roshani¹, Glenn Parry², Philip Walker-Davies³

¹Centre for Business Analytics in Practise, Surrey Business School, University of Surrey, Guildford, UK.

Journal of Heuristics

|January 1, 2026

Summary

This study introduces a new capacitated reliable fixed-charge facility location model with site-dependent failures. A novel Relax-and-Fix heuristic efficiently improves solutions for this complex supply chain problem.

Keywords:

Capacity Optimisation Relax-and-Fix heuristic Reliable facility location Supply Chain Disruption

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Area of Science:

Operations Research
Supply Chain Management
Optimization

Background:

The fixed-charge facility location problem is extended to include facility unreliability.
Existing models lack site-dependent failure probabilities and facility capacity constraints.

Purpose of the Study:

To address a novel capacitated version of the reliable fixed-charge facility location problem.
To incorporate site-dependent facility failure probabilities and limited capacities with upgrade options.

Main Methods:

Development of a non-linear mixed-integer programming formulation and its linear version.
Proposal of a novel Relax-and-Fix heuristic algorithm to solve the NP-hard problem.
Evaluation of the heuristic's efficiency on various experimental instances.

Main Results:

The Relax-and-Fix heuristic improves upper bounds for problem instances.
The proposed heuristic achieves solutions in shorter computational times compared to existing methods.
Sensitivity analysis on capacity and failure probabilities provides relevant insights.

Conclusions:

The developed Relax-and-Fix heuristic is an effective method for the capacitated reliable fixed-charge facility location problem.
The model and heuristic offer practical improvements for supply chain network design under uncertainty.
Further analysis confirms the heuristic's performance across different network configurations.