Multiple period optimization of bus transit systems

Analytic models are developed for optimizing bus services with time dependence and elasticity in their demand characteristics. Some supply parameters, i.e. vehicle operating costs and speeds are also allowed to vary over time. The multiple period models presented here allow some of the optimized system characteristics (e.g. route structure) to be fixed at values representing the best compromise over different time periods, while other characteristics (e.g. service headways) may be optimized within each period. In a numerical example the demand is assumed to fluctuate over a daily cycle (e.g. peak, offpeak and night), although the same models can also be used for other cyclical or noncyclical demand variations over any number of periods. Models are formulated and compared for four types of conditions, which include steady fixed demand, cyclical fixed demand, steady equilibrium demand and cyclical equilibrium demand. When fixed demand is assumed, the optimization objective is minimum total system cost, including operator cost and user cost, while operator profit and social welfare are the objective functions maximized for equilibrium demand. The major results consist of closed form solutions for the route spacings, headways, fares and costs for optimized feeder bus services under various demand conditions. A comparison of the optimization results for the four cases is also presented. When demand and bus operating characteristics are allowed to vary over time, the optimal functions are quite similar to those for steady demand and supply conditions. The optimality of a constant ratio between the headway and route spacing, which is found at all demand densities if demand is steady, is also maintained with a multi-period adjustment factor in cyclical demand cases, either exactly or with a relatively negligible approximation. These models may be used to analyze and optimize fairly complex feeder or radial bus systems whose demand and supply characteristics may vary arbitrarily over time.