How to Optimize Fire Station Placement Using Facility Location Models

1 Location Problem

1.1 Concept

The location problem involves selecting positions for facilities within a region to achieve optimal objectives; it is a classic mutually exclusive planning problem.

Examples include choosing plant sites given construction costs, transportation costs, and regional demand.

The problem is a classic in operations research with applications in production, logistics, and even military contexts such as factories, warehouses, emergency centers, fire stations, waste treatment sites, logistics hubs, and missile depots. It is also a frequent topic in mathematical modeling competitions.

Location decisions are long‑term and directly affect service mode, quality, efficiency, and cost, thereby influencing profit and market competitiveness, making the study economically, socially, and militarily significant.

1.2 Elements

The location problem has four basic elements: facilities, region, distance, and optimization objective.

Facilities

Facilities can be factories, warehouses, service stations, etc.

Region

The region may refer to internal layouts of a plant or a given geographical area. Regions can be continuous (any coordinate) or discrete (selected candidate sites). Most facility‑location models are discrete.

Distance

Distance denotes the travel distance, time, cost, or coverage between a facility and its service points; in graph‑theoretic formulations it corresponds to edge weights. Different metrics such as Euclidean, Manhattan, or Chebyshev may be used.

Optimization Objective

The goal is to choose facility locations that minimize weighted distance, cost, or maximize profit or minimize response time. Typical objective types include median (minimize total cost), center (minimize maximum cost), and anti‑center (maximize minimum cost).

2 Case Study (Fire Station Location)

2.1 Problem Statement

A city has eight districts, each can host at most one fire station. The longest travel times from each candidate station to each district are given. The requirement is that any fire incident be reached within 15 minutes while minimizing the number of stations. Construction costs are assumed equal.

2.2 Analysis

The task is to select a subset of the eight candidate stations such that all districts are served within the 15‑minute threshold and the number of stations is minimal.

2.3 Modeling

Variables and Parameters

Define binary parameters indicating whether station i can cover district j within the time limit. The following matrix shows coverage (1 = covered, 0 = not covered):

1 2 3 4 5 6 7 8
1  1 0 0 0 0 0 0 0
2  0 1 1 0 0 0 0 0
3  0 1 1 0 0 1 0 0
4  0 0 0 0 1 0 0 0
5  0 0 0 0 0 1 0 0
6  0 0 0 0 0 1 1 0
7  0 0 0 0 0 0 1 1
8  0 0 0 0 0 0 0 1

Decision variables x_i indicate whether station i is established (1) or not (0).

2.4 Solution

The model is solved using the Python Pulp library. The optimal solution places fire stations in districts 1 and 6, achieving coverage of all districts within the 15‑minute limit with only two stations.

3 Summary

This article uses the fire‑station placement problem to illustrate the mathematical modeling of facility location problems.