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) ]
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) ] m centres
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
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. We formally define the problem, classify its variants