Solving Periodic Timetable Optimisation Problems by Modulo Simplex Calculations

Abstract. In the last 15 years periodic timetable problems have found much interest in the combinatorial optimization community. We will fo-cus on the optimisation task to minimise a weighted sum of undesirable slack times. This problem can be formulated as a mixed integer linear problem, which for real world instances is hard to solve. This is mainly caused by the integer variables, the so-called modulo parameter. At first we will discuss some results on the polyhedral structure of the periodic timetable problem. These ideas allow to define a modulo simplex basic solution by calculating the basic variables from modulo equations. This leads to a modulo network simplex method, which iteratively improves the solution by changing the simplex basis. Key words: periodic event scheduling problem, integer programming, modulo network simplex