V. Cacchiani, A. Caprara and P. Toth DEIS, University of Bologna TIMETABLING FOR CONGESTED CORRIDORS

TRAIN TIMETABLING A railway network with (one way and double way) single and double tracks is considered. The Train Timetabling Problem is aimed at determining: –how many trains can be scheduled on a given “corridor” in a given time interval –a “good” timetable for the scheduled trains

Train Timetabling Given a set of “requests” for train paths (possibly from several Train Operators), specifying for each train: –Departure time from the first station –Arrival time at the last station –Arrival and departure times for the intermediate stations in which the train has to stop Define the trains to be scheduled and the corresponding actual paths (possibly “adjusting” the requested paths)

Separate timetabling problems are generally solved for distinct corridors in the network. The trains are assumed to have different speeds. For each train the travel time for any pair of consecutive stations is assumed to be fixed.

Basic Operational Constraints required to guarantee safety and regularity margin: Minimum time between two consecutive arrivals (departures) in each station. Overtaking between trains can occur only within a station. Maximum shift and stretch allowed for each train path.

If all the requests cannot be satisfied (because of possible path conflicts) three kinds of adjustments of the requested paths are allowed in order to obtain the actual feasible timetable: 1.change the departure time of some trains from their first station (shift), 2.increase the stopping time of the trains in some of the intermediate stations (stretch), 3.cancel a subset of the requested paths.

Requested Departure Time Requested train path Actual train path Station 1 Station 4 Station 3 Station 2 stop shift stretch 8:108:05 8:358:30 8:38 8:50 8:40 8:52 5 min. 7 min.

Main Objective: service quality –Maximum number of satisfied requests (scheduled trains) –Minimum deviation of the actual train paths with respect to the requested ones (minimum global shift and stretch) –Robustness of the solution with respect to random disturbances and failures

Two main scenarios can be considered: 1. No trains scheduled (basic problem) 2.Possibility to add new paths to an existing timetable - planning of new requests for train paths - operational scenario

Literature: –Szpigel (1973) –Jovanovic and Harker (1991) –Cai and Goh (1994) –Schrijver and Steenbeck (1994) –Carey and Lockwood (1995) –Nachtigall and Voget (1996) –Odijk (1996) –Higgings, Kozan and Ferreira (1997) –Brannlund, Lindberg, Nou, Nilsson (1998) –Lindner (2000) –Oliveira and Smith (2000)

–Caprara, Fischetti and T. (2002) –Peeters (2003) –Kroon and Peeters (2003) –Mistry and Kwan (2004) –Barber, Salido, Ingolotti, Abril, Lova, Tormas (2004) –Caprara, Monaci, T. and Guida (2005) –Semet and Schoenauer (2005) –Kroon, Dekker and Vromans (2005) –Vansteenwegen and Van Oudheusden (2006) –Cacchiani, Caprara, T. (2006) –Caprara, Kroon, Monaci, Peeters, T. (2006)

Optimization Algorithms Graph Formulation Integer Linear Programming Formulation Constructive Heuristics Algorithms based on Lagrangian and LP Relaxations Local Search Procedures

Mathematical Models A. Variables corresponding to arcs B. Variables corresponding to paths Constraints for arrival, departure and overtaking (clique constraints)

Solving Relaxation of Model A 1. Relax in a Lagrangian way all the operational constraints and use a subgradient optimization procedure to obtain “good” Lagrangian multipliers. 2. At each iteration of the subgradient procedure, for each train find the path having the maximum Lagrangian profit.

Solving Relaxation of Model B 1. Start with a reduced problem, with no operational constraints and only the ideal paths for the trains. 2. Solve the LP-relaxation of the reduced problem (CPLEX). 3. Find new paths with positive reduced profit (column generation) and find violated constraints (constraint separation). If no new paths and no violated constraints can be found, stop. Else add them to the problem and goto 2.

Heuristic Algorithms Lagrangian Heuristic Algorithm LP-based Heuristic Algorithm

Lagrangian Heuristic Algorithm 1. Solve the Lagrangian relaxation of model A. 2. Apply a constructive Heuristic Algorithm based on the use of the Lagrangian profits.

LP-based Heuristic Algorithm 1. Solve the LP relaxation of model B. 2. Apply a constructive Heuristic Algorithm based on the LP solution.