mixed-integer programming based approaches for the movement planner problem: model , heuristics...

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

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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.

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Notationtrain 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 𝑗

0 , otherwise

ContinuousVariables

Binary Variables

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Mixed-integer Linear Programming Model

train delay schedule deviance

TWT deviance unpreferredtrack time

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Mixed-integer Linear Programming Model

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Mixed-integer Linear Programming Model

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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

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Solution Approaches: Formulation Enhancement

• Dominance transitivitysegment 𝑗 segment 𝑗+1

=• No delays at intermediate nodes

• Fixing MOW-related variables• Fine-tuning big-M

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Solution Approaches: Heuristic Variable Fixing

• Imposing dominance for “distant” trainsIf 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

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Solution Approaches: Decomposition Algorithm

End ofIteration 1

End ofIteration 2

End ofIteration 3

End ofPlanning Horizon

TimeAxis

roll back ratio

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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 9.86 844.70

6 169.57 856.165 3600 867.21

6 3600

2 4077.65 26.91 - - - - - -

3 7049.25 147.71 10935.

6 3600 - - - -

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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