An Empirical Analysis of Robustness Concepts for Timetabling

Calculating timetables that are insensitive to disturbances has drawn considerable research efforts due to its practical importance on the one hand and its hard tractability by classical robustness concepts on the other hand. Many different robustness concepts for timetabling have been suggested in the literature, some of them very recently. In this paper we compare such concepts on real-world instances. We also introduce a new approach that is generically applicable to any robustness problem. Nevertheless it is able to adapt the special characteristics of the respective problem structure and hence generates solutions that fit to the needs of the respective problem

A Phase I Simplex Method for Finding Feasible Periodic Timetables

The periodic event scheduling problem (PESP) with various applications in timetabling or traffic light scheduling is known to be challenging to solve. In general, it is already NP-hard to find a feasible solution. However, depending on the structure of the underlying network and the values of lower and upper bounds on activities, this might also be an easy task. In this paper we make use of this property and suggest phase I approaches (similar to the well-known phase I of the simplex algorithm) to find a feasible solution to PESP. Given an instance of PESP, we define an auxiliary instance for which a feasible solution can easily be constructed, and whose solution determines a feasible solution of the original instance or proves that the original instance is not feasible. We investigate different possibilities on how such an auxiliary instance can be defined theoretically and experimentally. Furthermore, in our experiments we compare different solution approaches for PESP and their behavior in the phase I approach. The results show that this approach can be especially helpful if the instance admits a feasible solution, while it is generally outperformed by classic mixed-integer programming formulations when the instance is infeasible

On The Recoverable Robust Traveling Salesman Problem

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal value. A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately. While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared. Finally, an alternative recovery model is discussed, where a second-stage recovery tour is not required to visit all nodes of the graph. We show that the previously NP-hard evaluation of a fixed solution now becomes solvable in polynomial time

On the recoverable robust traveling salesman problem

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal values. A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately. While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared