A Mixed-Integer Programming Model for Healthcare Facility Location Under Budget Constraints in Zamfara State

Equitable spatial distribution of primary healthcare facilities in Zamfara State, North West Nigeria, is critical for reducing maternal and child mortality rates in one of the states with the lowest healthcare access indices in the country. This study develops a mixed-integer programming model for optimising the selection and location of new primary health centres under a constrained state health budget, incorporating multiple objectives of population coverage, travel time equity, and cost efficiency. The model formulation extends the classical p-median problem to accommodate facility capacity constraints, minimum service level requirements, and geographic access equity constraints measured using the Gini coefficient of travel time distribution. GIS-derived ward-level population data, road network travel time matrices, and candidate facility site construction cost estimates are compiled for all 14 local government areas. The model is solved using commercial branch-and-bound with valid inequality cuts to improve computational tractability. Sensitivity analysis examines the trade-off frontier between coverage and equity objectives across a range of budget levels from 1.5 to 4.0 billion naira. Results demonstrate that at a 2.5 billion naira budget, 18 optimally located new PHCs achieve 91 percent population coverage within a 5-kilometre travel distance, compared to 73 percent under a random siting alternative. Keywords: mixed-integer programming, facility location, healthcare access, Zamfara State, coverage optimisation.