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Enhancing Movement Planner Problem Solutions via Mixed-Integer Programming Techniques

This paper presents a comprehensive analysis of the Movement Planner Problem (MPP) using Mixed-Integer Linear Programming (MILP) and heuristic approaches. We define track segments essential for train routing between nodes and develop enhanced formulations, heuristic variable-fixing procedures, and a decomposition algorithm to effectively manage scheduling complexities. Our proposed solutions significantly reduce computation times, achieving optimal outcomes in under 30 seconds for larger instances. By refining MILP formulations and solution strategies, we improve the efficiency of train scheduling in complex networks.

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Enhancing Movement Planner Problem Solutions via Mixed-Integer Programming Techniques

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  1. Mixed-integer Programming Based Approaches for the Movement Planner Problem: Model, Heuristics and DecompositionBamboo@Tsinghua ChiweiYan Department of Civil & Environmental EngineeringMassachusetts Institute of Technology LuyiYang The University of ChicagoBooth School of Business RAS Problem Solving Competition 2012

  2. Problem Formulation: Definition of Segments • Acollection of tracks (main tracks, sidings, switches, crossovers) between two adjacent nodes • A train must pass through everysegment between its origin and destination and travel on one specific track within a given segment.

  3. Notation entry (exit) time for train at segment

  4. Mixed-integer Linear Programming Model schedule deviance train delay TWT deviance unpreferred track time

  5. Mixed-integer Linear Programming Model

  6. Mixed-integer Linear Programming Model

  7. Solution Approaches • Combinatorially difficult to solve • Even the smallest test instance requires more than one hourin our implementation! • What we propose: • Formulation enhancement • Heuristic variable fixing procedure • Decomposition algorithm

  8. Solution Approaches: Formulation Enhancement • Dominance transitivity • No delays at intermediate nodes = • Fixing MOW-related variables • Fine-tuning big-M

  9. Solution Approaches: Heuristic Variable Fixing • Imposing dominance for “distant” trains If the lower bounds are too far apart, there is little chance for the later train to catch up … • Prohibiting unattractive overtakes • Entry time is no later • Type priority is no lower • Origin is no farther • Estimating what to be realized prior to the end of planning horizon

  10. Solution Approaches: Decomposition Algorithm End of Iteration 1 End of Iteration 2 End of Iteration 3 End of Planning Horizon Time Axis roll back ratio

  11. Computational Results • Implementation: C++ and ILOG CPLEX 12.1 • Platform: a PC with 2.40 GHz CPU and 4GB RAM • Maximum computational time: 1 hour

  12. Concluding Remarks • Successfully formulate the Movement Planner Problem as MILP • To solve the problem, we propose • Formulation enhancement • Heuristic variable fixing • Decomposition algorithm • Summary of computational results • Expedite the search for optimal solutions by a factor of 400 for Data Set 1 • Obtain satisficing solutions for larger instances Data Set 2: less than 30 seconds Data Set 3: less than 2.5 minutes

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