Optimal assignment of buses to bus stops in a loop by reinforcement learning

Bus systems involve complex bus-bus and bus-passengers interactions. We study the problem of assigning buses to bus stops minimise average waiting time passengers, W. An analytical theory for two specific cases interactions is formulated: normal situation where all board passengers from every stop, versus novel express disjoint subsets non-interacting serve stops. Our formulation allows exact calculation W general loops in examined. Compared with regular buses, we present scenarios show improvement Useful insights are obtained our theory: 1) there a minimum number needed, 2) splitting crowded stop into less ones always increases 3) changing destination location do not influence In second part, introduce reinforcement-learning platform that overcomes limitations method search better allocations previous cases, any possible interaction between allowed, unlocking emergent strategies. apply this tool simple toy model three empirically-motivated loops, based on data collected Nanyang Technological University shuttle system. simplified model, observe an unexpected strategy emerging could be analysed mathematical displays chaotic behaviour. The configurations approximately 10^11, 10^11 10^20, so brute-force approach impossible. algorithm reduces by 12% 32% compared 29% buses. This has practical applications because it works independently characteristics loop.