Optimum polar networks for an urban bus system with a many-to-many travel demand

The spacing and headway configuration of a bus system comprising a series of ring and radial routes which minimizes travel time is found by the use of the calculus of variations. A many-to-many travel demand is described by a continuous function of the positions of a commuter's home and workplace. It is assumed that buses travel at a constant speed and are subject to a fleet size constraint. Under optimal conditions it is found that both spacing between routes and headway between buses should be inversely proportional to the cube root of the proportion of commuters joining and leaving the route. A simple numerical example is worked in which homes and workplaces are independently and uniformly distributed.