Optimal routes and timetables of a urban bus line

The bus timetables of a urban bus line are commonly found by rather laborious graphical methods, not easily applicable when there is some branching route (reentering the line at the same or at some other point). Here we present a formal approach that covers also this case, under a wide vari of operating conditions. After partitioning the line, and given the fol1owing input data for each preassigned period of the day: expected total number of passengers per hour, accomodation and average speed of each bus, line structure (with the branching points and those sections where bus service may be suspended for some part of the day), location and duration of each terminus stop, then the program which realizes this approach finds the timetable of each bus and its route (also through a branching point) so that their total number, together with the total time on duty of their personnel, is minimum.