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


Chiwei YanDepartment of Civil & Environmental Engineering

Massachusetts Institute of Technology

Luyi YangThe University of ChicagoBooth School of Business


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.



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



Mixed-integer Linear Programming Model

train delay schedule deviance

TWT deviance unpreferredtrack time



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



Solution Approaches: Formulation Enhancement

• Dominance transitivitysegment 𝑗 segment 𝑗+1

=• No delays at intermediate nodes

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



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



Solution Approaches: Decomposition Algorithm

End ofIteration 1

End ofIteration 2

End ofIteration 3

End ofPlanning Horizon

TimeAxis

roll back ratio



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 - - - -



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

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

Documents

annual meeting

problem formulation

tsinghua ras problem

instance2notation3 ras

ras mixedinteger programming

movement planner problem

train i segment j

later train