1. V. Cacchiani, ATMOS 2007, Seville1 Solving a Real-World Train Unit Assignment Problem V. Cacchiani, A. Caprara, P. Toth University of Bologna (Italy) European Project ARRIVAL

2. V. Cacchiani, ATMOS 2007, Seville2 Problem description A Graph representation ILP models (arc formulation & path formulation) Strong inequalities for the Capacity Constraints Maintenance Constraints An LP-based heuristic algorithm Experimental Results Conclusions Outline

3. V. Cacchiani, ATMOS 2007, Seville3 Train Unit Assignment Problem (TUAP) Input: timetabled train trips train unit types Output: Minimum Cost A ssignment of the train units (TUs) to the trips required number of passenger seats for each trip (possibly combining TUs ) maximum number of TUs that can be assigned to each trip (generally 2) maximum number of TUs available sequencing constraints between trips maintenance constraints Constraints:

4. V. Cacchiani, ATMOS 2007, Seville4 SV 352 MI 389 MI 356 MD 396 SA 437BA 464 BA 540DAT 552 BA 720DAT 732 360 720 516 876 Trips TUs 516 2 360 2 Output: A B C D E A A B CD E E Input:

5. V. Cacchiani, ATMOS 2007, Seville5 Graph Representation node  trip arcs for TUs of type k iff a TU of type k can be assigned to i and then to j within the same day A E D C B (and dummy nodes 0 and n+1) n+1 0 dummy start node dummy end node arc n # of trips p # of TU types

6. V. Cacchiani, ATMOS 2007, Seville6 Arc Formulation number of times that arc a is selected in the solution max # of paths for TU of type k capacity constraints min of the total cost of the paths max # of TU for trip j # of times that arc a is selected in the solution flow constraints

7. V. Cacchiani, ATMOS 2007, Seville7 Path Formulation number of times that path P is selected in the solution subcollection of paths in that visit trip j, max # of paths for TU of type k capacity constraints min of the total cost of the paths max # of TU for trip j # of times that path P is selected in the solution collection of paths from 0 to n+1 in

8. V. Cacchiani, ATMOS 2007, Seville8 “Strong” capacity constraints An example number of TUs of type k assigned to a trip j weak strong

9. V. Cacchiani, ATMOS 2007, Seville9 “Strong” capacity constraints The described capacity constraints are the following: Assuming For a trip j define: such that The derived strong inequalities describe the convex hull of the capacity constraints.

10. V. Cacchiani, ATMOS 2007, Seville10 ILP Formulation number of paths for a TU of type k which contain a maintenance arc maintenance every days maintenance constraints collection of maintenance paths from 0 to n+1 in

11. V. Cacchiani, ATMOS 2007, Seville11 An LP-based Heuristic Algorithm 0. initialize the current LP as the described model (“strong” capacity constraints, a subset of variables) 1. solve the current LP (CPLEX) 2. constructive heuristic (based on the optimal dual solution) 4. if there are dual constraints violated add some of the corresponding primal variables to the current LP else stop5. if the current LP is infeasible if value of the incumbent solution fixing stop else goto 1. 3. refine the solution found

12. V. Cacchiani, ATMOS 2007, Seville12 Experimental Results on the Case Study Methods implemented in C, computational tests on a Pentium IV, 3.2 GHz, 1 Gb Ram, Cplex 9.0 instArc LP Arc LP Time (sec) Path LPPath LP Time (sec) Strong Arc LP Strong Arc LP Time (sec) Strong Path LP Strong Path LP Time (sec) A (528,8) 57624571366212016250 B (662,10) 414724241174532690753150 C (660,10) 402384140179532735053177 instActual Solution LP Maint LP Maint Time (sec) Heur Time (sec) A (528,8) 7262130633544 B (662,10) 7656196595471 C (660,10) 7455295588875

13. V. Cacchiani, ATMOS 2007, Seville13 Conclusions ILP formulations based on a Graph representation of the TUAP LP-relaxation with “strong” capacity constraints LP-based Heuristic Signifcant improvement over the practitioners’ solution. Current Research (cooperation with Erasmus University Rotterdam) Build a “robust” solution based on the possibility of exchanging TUs in case of delays