Emergency Responder Route Optimization System: A Mathematical Framework for Building Safety Sweeps

Abstract

In emergency scenarios such as fires, earthquakes, or active shooter incidents, the rapid and safe searching of multi-story buildings is critical for saving lives. This paper presents a mathematical framework for optimizing emergency responder routes during building safety sweeps. The proposed system transforms floor plans into weighted graph representations, where nodes denote rooms, hallways, stairwells, and exits, and edges represent traversable paths with associated travel and service times. A dynamic risk assessment module is developed to prioritize rooms based on type, occupancy, floor level, and real-time hazard propagation, such as fire or smoke spread. The routing problem is formulated as a variant of the vehicle routing problem with time windows and solved using constraint programming integrated with k-means clustering for exit assignment. The model minimizes total mission completion time while ensuring coverage of all rooms and adherence to safety constraints. Case studies involving single-floor offices, two-story mixed-use buildings, and high-risk facilities demonstrate the system's effectiveness. Results show that optimal team sizes range from two to five responders depending on building complexity and emergency severity, with mission completion times varying between 1,477 seconds under normal conditions and over 3,500 seconds under dynamic fire spread. The framework also incorporates realistic extensions such as cascading failures, congestion dynamics, and technology integration. This work provides a practical, data-driven tool for emergency planners to enhance responder efficiency and occupant safety.