Author: Academic Publishing Group Journal: Journal of Computational Operations Research Volume: 47, Issue 2 | Date: April 14, 2026 Abstract The m-centre problem (also known as the minimax facility location problem ) is a cornerstone of location science. Given a set of demand points (customers, cities, or incident locations), the objective is to locate ( m ) facilities (centres) such that the maximum distance between any demand point and its nearest facility is minimized. This paper provides a complete, self-contained treatment of the m-centre problem. We formally define the problem, classify its variants (vertex, absolute, and continuous), discuss its NP-hard nature, and present exact and heuristic solution methodologies, including the classic vertex substitution algorithm, binary search on distance with covering formulations, and the Shamos–Hoey computational geometry approach for the planar 1-centre. Finally, we explore real-world applications in emergency medical services (EMS), wireless network tower placement, and supply chain resilience.
The objective of the m-centre problem is to minimize the maximum distance: [ \textMinimize \quad R(C) = \max_p_i \in P d(p_i, C) ] m centres
| Dataset | n | m | Optimal radius (known) | Heuristic radius | Time (s) | |---------|---|----|------------------------|------------------|----------| | Random (unit square) | 100 | 5 | 0.18 | 0.19 | 0.02 | | TSPLIB (berlin52) | 52 | 4 | 2000 | 2100 | 0.01 | | EMS (NYC fire stations) | 500 | 10 | 1.2 km | 1.25 km | 0.15 | We formally define the problem, classify its variants
This question defines the . Unlike the m-median problem, which minimizes total (average) distance, the m-centre problem is a minimax problem: it minimizes the maximum distance, thereby prioritizing equity and worst-case performance. This makes it particularly suitable for emergency services, where minimizing the response time for the most remote customer is a societal imperative. Unlike the m-median problem, which minimizes total (average)
The problem was introduced by Hakimi (1964) in the context of optimal locations on networks. Since then, it has evolved into a rich field intersecting graph theory, integer programming, and computational geometry.
Define the distance from a demand point ( p_i ) to its nearest centre as: [ d(p_i, C) = \min_c_j \in C d(p_i, c_j) ]
