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RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012
Mixed-integer Programming Based Approaches for the Movement Planner
Problem: Model, Heuristics and Decomposition
Bamboo@Tsinghua
RAS Problem Solving Competition 2012
Chiwei YanDepartment of Civil & Environmental Engineering
Massachusetts Institute of Technology
Luyi YangThe University of ChicagoBooth School of Business
RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 2
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.
RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 3
Notation
train 𝑖∈𝒯 segment 𝑗∈𝒢
track 𝑡∈ℒ 𝑗
𝐃𝐞𝐜𝐢𝐬𝐢𝐨𝐧𝐬𝐕𝐚𝐫𝐢𝐚𝐛𝐥𝐞𝐬
entry (exit) time for train at segment
𝑞𝑖 , 𝑗 ,𝑡={1, if train 𝑖 uses track t of segment 𝑗0, otherwise
𝛾𝑖 ,𝑖′ , 𝑗 ,𝜆𝑖 , 𝑖′ , 𝑗={ 1 ,if train 𝑖is earilier (later ) than 𝑖′
on segment 𝑗0 , otherwise
ContinuousVariables
Binary Variables
RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 4
Mixed-integer Linear Programming Model
train delay schedule deviance
TWT deviance unpreferredtrack time
RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 5
Mixed-integer Linear Programming Model
RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 6
Mixed-integer Linear Programming Model
RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 7
Solution Approaches
• Combinatorially difficult to solve• Even the smallest test instance requires more
than one hour in our implementation!• What we propose:
► Formulation enhancement► Heuristic variable fixing procedure► Decomposition algorithm
RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 8
Solution Approaches: Formulation Enhancement
• Dominance transitivitysegment 𝑗 segment 𝑗+1
=• No delays at intermediate nodes
• Fixing MOW-related variables• Fine-tuning big-M
RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 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
…
T he lower bound of 𝑥 𝑖 , 𝑗𝑒𝑥𝑖𝑡
RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 10
Solution Approaches: Decomposition Algorithm
End ofIteration 1
End ofIteration 2
End ofIteration 3
End ofPlanning Horizon
TimeAxis
roll back ratio
RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 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
Decomposition Variable Fixing Enhanced Model Original Model
Data Set
Obj ($)
Time (s)
Obj ($)
Time (s)
Obj ($)
Time (s)
Obj ($)
Time (s)
1 844.706
9.86844.70
6169.57
856.165
3600867.21
63600
2 4077.65
26.91 - - - - - -
3 7049.25
147.7110935.
63600 - - - -
RAS Problem Solving Competition 2012
INFORMS Annual Meeting 2012, Phoenix, Oct. 14, 2012 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