Mathematical Means of Determining the Sufficiency of the Functional Capacity of Fire Departments in the Local Area

Abstract

The paper addresses the topical issue of fire safety concerning optimising the number and territorial location of fire departments. The ultimate goal of this optimisation is to minimise the cost associated with creating and maintaining fire departments within a particular administrative territory while ensuring the required level of fire safety for all users, whether residential or industrial facilities. In solving this problem, we consider options for different functional capacities of fire departments involving various combinations of specialised firefighting vehicles and equipment. To determine the minimum necessary number of fire units, we have developed a mathematical model based on models of simultaneous parallel calls. The model uses a probabilistic approach to estimating the number of calls per unit of time. In our effort to optimise the deployment of fire units within local areas, the article introduces an optimisation geometry model, which searches for conditions that cover the entire range of potentially suitable deployment points. To address this complex issue, we propose a method known as the weighted p-median problem, which allows us to represent the array of potential fire incident locations and the potential points for deploying fire units using two matrices. The determining criterion is the distance between the caller and the fire department, as it is the distance that determines the time of arrival of the department to the place of call. The arrival time of the fire department to the call location is the key indicator of the quality of service per most regulatory documents from different countries worldwide. The use of the proposed mathematical models as a theoretical basis for designing new buildings in cities and administrative districts or for checking compliance with safety standards will allow the administrations of the respective territories to reduce the costs of creating and maintaining fire departments while preserving the required level of fire safety. Furthermore, to simplify the use of the obtained theoretical results, there is potential to develop an automated software package based on the proposed models. Such a tool would extend the accessibility of our results to a broad audience, including practitioners and decision-makers. Keywords: fire departments, fire safety level, call flows, optimal placement, functionality.