1. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 Bamboo@Tsinghua Mixed-integer Programming Based Approaches for the Movement Planner Problem: Model, Heuristics and Decomposition Bamboo@Tsinghua RAS Problem Solving Competition 2012 Chiwei Yan Department of Civil & Environmental Engineering Massachusetts Institute of Technology Luyi Yang The University of Chicago Booth School of Business

2. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 Problem Formulation: Definition of Segments A collection of tracks (main tracks, sidings, switches, crossovers) between two adjacent nodes A train must pass through every segment between its origin and destination and travel on one specific track within a given segment. 2

3. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 Notation 3

4. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 Mixed-integer Linear Programming Model 4 train delay schedule deviance TWT deviance unpreferred track time

5. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 Mixed-integer Linear Programming Model 5

6. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 Mixed-integer Linear Programming Model 6

7. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 Solution Approaches 7

8. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 Solution Approaches: Formulation Enhancement Dominance transitivity 8 = No delays at intermediate nodes Fixing MOW-related variables Fine-tuning big-M

9. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 Solution Approaches: Heuristic Variable Fixing Imposing dominance for “distant” trains 9 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. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 Solution Approaches: Decomposition Algorithm 10 End of Iteration 1 End of Iteration 2 End of Iteration 3 End of Planning Horizon Time Axis roll back ratio

11. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 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 11 DecompositionVariable FixingEnhanced ModelOriginal Model Data SetObj ($)Time (s)Obj ($)Time (s)Obj ($)Time (s)Obj ($)Time (s) 1 844.7069.86844.706169.57856.1653600867.2163600 2 4077.6526.91------ 3 7049.25147.7110935.63600----

12. RAS Problem Solving Competition 2012 INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 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 12