Mathematical Modeling and Convergence Analysis of Swarm-Based Algorithms in Public Transport Network Optimization

Abstract

Optimizing public transport networks under dynamic demand and changing operational conditions remains a major challenge due to nonlinear cost structures, fluctuating travel times, and capacity limitations. This study proposes a mathematical optimization framework to represent dynamic public transport networks using multi-criteria objective functions, flow conservation constraints, and time-dependent parameters. Based on this framework, the convergence behavior of Particle Swarm Optimization (PSO) and the Firefly Algorithm (FA) is examined through theoretical analysis under realistic assumptions, including local Lipschitz continuity, bounded feasible regions, and appropriate parameter update schemes. Numerical simulations conducted on various scenarios with demand fluctuations, congestion-sensitive travel times, and random capacity disruptions show that both algorithms converge stably and are able to track changes in optimal solutions. Comparative experiments with other metaheuristic methods further indicate that PSO and FA provide competitive solution quality, robustness, and computational efficiency. Overall, the results demonstrate that a careful alignment between algorithmic mechanisms and network structure enhances convergence reliability and optimization performance, effectively connecting theoretical insights with practical simulation in public transport network optimization.