Gradient-based and Gradient-free Optimization of a Naive Bus Route

Public transportation like trains, buses, and light-rail vehicles can transport more people than commuter cars while taking up minimal space on roadways. While public transportation systems are highly effective in areas with high population density in the United States, they are definitively underused. In order to increase the number of people using public transportation, public transportation must improve. One of the most common and cost-effective methods of public transit is the city bus. The time a bus takes to complete its route is heavily dependent on the amount of traffic in a given area and road signals. Researchers have applied optimization techniques to bus routes where most of these algorithms have focused on either graph based methods or genetic algorithms, since a bus route is discrete problem. We have developed two models. The first is a continuous model of a city block, which allows us to apply gradient-based methods of optimization to the bus route problem. The second being a discrete graph like the ones performed previously to compare results. This paper reports on how continuous and discrete models can find an optimal route that changes when different traffic conditions are present. The model is presented and shown to work in naive cases, through standard Manhattan square blocks. The results are shown to find the optimal path for a city bus traveling between two points within a city grid and a full path.<br>