Mathematical Formulation for Location of Multiple Facilities on Deterministic Network

Abstract

Efficient placement of multiple facilities within a network is a critical challenge encountered in urban planning, logistics, transportation systems, and operations research. This paper presents a mathematical approach for solving this problem on a deterministic network, where travel times and transportation costs are fixed and known in advance. The main objective is to determine optimal facility locations that ensure maximum service efficiency, minimal cost, and improved overall network performance.In order to do this, we concentrate on p-median problem and p-center problem, two well-known location models. The p-median approach is appropriate for cost-sensitive applications like distribution and logistics since it seeks to reduce the average (or total) distance among demand nodes and the facilities to which they are assigned. However, in situations like emergency services or public facilities, the p-center model reduces the maximum distance a user must travel to reach a facility, which is essential for equitable service delivery. The methodology adopted in this study is based on mathematical modeling using integer linear programming (ILP). A graph with a collection of nodes (demand points) & edges (fixed trip charges or distances) is used to depict a deterministic network. Binary decision variables that specify whether a facility is positioned at a specific node and which demand points are allocated to which facilities are then used to create the facility location problem. Every demand node is assigned to the closest available facility by the ILP formulation, which guarantees that precisely p facilities are chosen. These models are solved using both real and simulated datasets utilizing optimization solvers like CPLEX and Gurobi. A sensitivity analysis is also carried out to look at how changing the quantity of facilities & demand distribution affects the solution's robustness and quality. We draw the conclusion that the p-median & p-center models are both useful instruments for strategic facility placement choices in deterministic networks based on the outcomes of the computational trials. While the p-center approach guarantees equity and accessibility for all demand nodes, the p-median model is excellent at lowering overall service costs. The results highlight how crucial it is to select the right model depending on the particular objectives and limitations of the application. This study provides a flexible and systematic framework for facility location that can be adapted across diverse sectors including healthcare, education, retail, and municipal services. Future research may extend this work by incorporating dynamic or stochastic elements into the network to reflect real-world variability in demand and travel conditions.