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Solving a Real-World Train Unit Assignment Problem

Solving a Real-World Train Unit Assignment Problem. V. Cacchiani, A. Caprara, P. Toth. University of Bologna (Italy). European Project ARRIVAL. Outline. Problem description A Graph representation ILP models (arc formulation & path formulation)

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Solving a Real-World Train Unit Assignment Problem

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

  2. Outline • 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 V. Cacchiani, ATMOS 2007, Seville

  3. Train Unit Assignment Problem (TUAP) Input: • timetabled train trips • train unit types Output: Minimum Cost Assignment of the train units (TUs) to thetrips Constraints: • required number of passenger seats for each trip(possiblycombining 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 V. Cacchiani, ATMOS 2007, Seville

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

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

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

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

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

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

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

  11. 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) 3. refine the solution found add some of the corresponding primal variables to the current LP 4. if there are dual constraints violated else fixing 5. if the current LP is infeasible stop value of the incumbent solution stop if else goto 1. V. Cacchiani, ATMOS 2007, Seville

  12. 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 V. Cacchiani, ATMOS 2007, Seville

  13. 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 V. Cacchiani, ATMOS 2007, Seville

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