An Integrated Optimization Model and Algorithm for Train Operating ...

Research - China

An Integrated Optimization Model and Algorithm for Train Operating Plan and Locomotive Allocation IBM China Research Lab WANG Bao Hua, ZHANG Xin MO Wen Ting, WANG Feng Juan IBM Global Business Services Howard A. Rosen

© 2011 IBM Corporation

Research - China

Motivation Operational Planning

Infrastructure Maintenance Dispatcher

Train Operation Dispatcher

Loading and unloading Dispatcher

Train Formation Dispatcher

Locomotive Dispatcher

Integrated Optimization Methodology and Tool

Train Formation and Locomotive Dispatcher © 2011 IBM Corporation

Research - China

Problem Description Cars are classified at Shunting Yard A

Fueling and maintenance

Locomotive on train 12001 will be re-assigned to train 22001

Cars with same or similar destination costitute train 22001

A locomotive comes from depot to pull train 13002

 Train

 Locomotive

–

Formation (Make-up)

–

Origin and destination time

Integration

–

Assignment

–

Fueling and maintenance

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Research - China

Literature Review Locomotive Allocation Authors

Planning Level

Objective Function

Model Structure

Solution Strategy

Ziarati (1997)

Operational

Min Operating Cost

Multi-commodity

D&W decomposition

Ziarati (1997)

Tactical

Min Operating Cost

Multi-commodity

Branch & Cut

Ghoseiri (2010)

Operational

Min Operating Cost

Assignment

Genetic Algorithm

Forbes (1991)

Tactical

Min Operating Cost

Assignment

Branch & Bound

Train Operating Plan Authors

Planning Level

Objective Function

Model Structure

Solution Strategy

Haghani (1989)

Tactical

Min Operating and time cost

Non-linear MIP

Heruristic Decomposition

Keaton (1992)

Tactical

Min Operating and time cost

Linear 0-1 IP

Lagragian Relaxation

Marin (1996)

Tactical

Min Operating Cost

Non-linear IP

Local Search Heuristic

Gorman (1998)

Tactical

Min Operating Cost

Linear 0-1 IP

Genetic Algorithm

Train Operating Plan with Asset Balance Authors

Planning Level

Objective Function

Model Structure

Solution Strategy

Anderson (2007)

Tactical

Min Operating and time cost

Linear MIP

Heruristic Decomposition

Anderson (2010)

Tactical

Min Operating and time cost

Linear MIP

Branch and price

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Discrete Time-space Network Time Period

Virtual destination for all car flow destined at this yard

Loaded cars are located at these nodes initially

Each car flow corresponds a super arc from its origin to the super node of its destined yard

Virtual destination for all locomotives

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Input and Output of the Model  Input – Candidate train set (or basic train timetable) with fixed cost and variable cost – Train capacity and power-needed – Initial status of locomotives (location, fuel, etc) with power

– Car flows with required service level – Delivery and delay cost for locomotives and cars

 Output – Trip plan for car flows (including classification plan) – Trip plan for locomotive flows (including refueling plan)

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Assumptions  No empty car involved  No blocking policy involved. In other words, each yard corresponds a block  Unit flow cost of cars and locomotives are constants  Unit time cost of car flows is constant

 Power-needed of the train is determined by the infrastructure, not the weight of the train  Time between different trains is enough for the classifcation operation of the cars and transfer operation of the locomotives  Insert light travel arc as the special kind of train arc manually according to number of operating trains in each track section

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Parameter

locomotives with same location, type could be treated as a flow

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Parameter (cont'd)

The difference of these two parameters is small and does not affect powerneeded on that arc

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Parameter (cont'd)

Decision Variables

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Model Development Fixed cost of open arcs

Routing cost of car flows

Routing cost of locomotive flows

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Model Development (cont'd) M is a largest number of locomotive which could be assigned to this train

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Research - China

Tabu Search Heuristic by PATH-CHANGE Make all design variables as 1 and solve car routing prolem and locomotive routing problem

Construct Initial Feasible Solution    

Step 1 Update tabu list Step 2 Find a neighborhood Step 3 Check if the neighborhood is in the tabu list. If it is, return to Step 2; else, go to Step 4 Step 4 Move to the neighborhood

different path between two nodes with enough capacity

Randomly PATHCHANGE Tabu search

Sub-problem for car flow routing

Sub-problem for locomotive routing

No Termination Condition

Path1 Yes

Path2 Algorithm Terminate © 2011 IBM Corporation

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Branch-and-Price for Car Routing Problem Find an initial feasible solution for all car flows

Solve the restricted master problem with initial path set and relaxed decision variable

such path exists Generate an attractive path by pricing Considering service level requirement

no such path

Column generation terminates

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Initial Feasible Solution for Car Routing Problem Super arc for each car flow with infinity capacity and large enough cost Car flow origin

Car flow destination

 Step 1 Generate super arc for each car flow  Step 2 Make these super arcs as the initial path set and formulate the Restricted Master Problem (RMP)  Step 3 Each car flow will choose the super arc as their initial path, which could be an initial feasible solution of the problem

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Solution Strategy for Locomotive Routing Problem

Virtual locomotives with high cost will be located on super arc to obtain an initial feasible solution easily

Define

One feasible path for a specific locomotive. Initial feasible path set should ensure all trains' power requirement

Pl

Pl  Pl

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Initial Feasible Solution for Locomotive Routing Problem Virtual locomotives are located at the initial time period at each yard

Find a train string

Calculate the min powerneeded of the train string

Update the power of each train in the train string

Another train string exist Train String: A sequence of trains, which could be pulled by a certain locomotive

Yes

No According to train string, generate corresponding path with given format

Assign (virtual) locomotives to paths Path: A sequence of arcs © 2011 IBM Corporation

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Pricing for Locomotive Routing Problem Define

Find

Reduced Cost

Update

Pl  Pl

p*

This path should satisfy all the business rules. If one does not exists, the algorithm terminates with current solution

Solve

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Column Generator for Locomotive Routing Problem

Modified arc cost may be negative due to the dual variable. We can change it to positive by multiply -1 to original constraint

If the shortest path of locomotive l  L relating to modified arc cost is larger than , no path need to be inserted; Else, the shortest path should be inserted to feasible path © 2011 IBM Corporation

Research - China

Testing Condition  Part of the operating network of a Chinese railway container carrier –

12 yard involved

 Tabu List –

10 iterations to free after inserting to the list Large yard level

 Neighborhood Search Strategy –

Large yard first, small yard second small yard level

 Termination Condition –

The optimum does not change in 60 iterations

 Comparison between New Methodology and Classic Methodology –

Sequential Optimization with Feedback

–

Integrated Optimization

Input

Train Plan

Highly depend on the experience of dispatcher

Locomotive Plan Feedback

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Research - China

Testing Results No.

Size

Time for SO1

Time for IO2

Gap3

1

10 yards, 2 days, 102 trains, 20 car flows, 25 locomotives

3.5 min

2.4 min

12%

2

10 yards, 2 days, 122 trains, 20 car flows, 30 locomotives

6.5 min

5.2 min

14%

3

12 yards, 2 days, 135 trains, 25 car flows, 35 locomotives

10 min

12 min

20%

4

12 yards, 2 days, 135 trains, 30 car flows, 30 locomotives

9.6 min

12 min

11%

5

12 yards, 2 days, 150 trains, 40 car flows, 40 locomotives

25 min

22 min

21%

1

SO: Sequential Optimization with Feedback IO: Integrated Optimization 3 Gap = (Optimum of SO - Optimum of IO) / Optimum of SO 2

The gap is high partly because of the high fixed cost of train and locomotive © 2011 IBM Corporation

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Testing Results (Con'd) Exp. KPI

No.1-SO

Num. of Trains

11

Num. of Locos

6

Train· kilometers Train· hours

No.2-IO

No.2-SO

10

No.2-IO

11

-9% 5

2724

7

37

3396

9042

Car· hours

1478

8187

40

-22%

10

3396

40

133440

133440

4927

1478 0%

16

22

7

3611

58

42

13

141372

7262

2164 -7%

No.5-IO

31

12

6926

72

62

17

267216

10312

2868

8536 -17%

110

86 -22%

414231

-8% 2932

15 -12%

-14% 289162

27 -13%

-5%

-19% 2320

21

No.5-SO

-8%

-28% 174282

No.4-IO

-5%

-27%

0% 1478

No.4-SO

-30%

0%

-9% 1150

5

No.3-IO

-11%

0%

-8% Car· kilometers

18

-29%

-8% 40

11 0%

-17% 2976

No.3-SO

349354 -16%

4082

-2%

3518 -14%

* Different trains may deliver different number of cars © 2011 IBM Corporation

Research - China

Testing Results (Con'd)  Effect of varying train tonnage (using example No.2)

 Higher train tonnage will increase the total operating cost. The slope is higher when train tonnage is larger than 6000 because fixed cost is increasing dramatically  Higher train tonnage may not need more locomotives. When train tonnage is less than 5000, one train needs only one locomotive. Otherwise, one train needs two locomotives. When train tonnage is 6500, less trains are operated, which decreases the number of locomotives  Higher train tonnage may decrease the service level because the frequency and speed of operating train is lower. It depends on the operating strategy of decision-maker © 2011 IBM Corporation

Research - China

Testing Results (Con'd)  Adding Cut for maximum number of trains at one node in a time period No.

Size

Running time without cut

Running Time with cut

1

10 yards, 2 days, 102 trains, 20 car flows, 25 locomotives

3.5 min

3.5 min

2

10 yards, 2 days, 122 trains, 20 car flows, 30 locomotives

6.5 min

6.1 min

3

12 yards, 2 days, 135 trains, 25 car flows, 35 locomotives

10 min

10 min

4

12 yards, 2 days, 135 trains, 30 car flows, 30 locomotives

9.6 min

9.2 min

5

12 yards, 2 days, 150 trains, 40 car flows, 40 locomotives

25 min

22.1 min

* Both two methods obtain same optimum © 2011 IBM Corporation

Research - China

Testing Results (Con'd)  Relaxing fueling and connection constraint of locomotive path No.

Size

Running Time without relaxation

Running time with relaxation

1

10 yards, 2 days, 102 trains, 20 car flows, 25 locomotives

3.5 min

3.1 min

2

10 yards, 2 days, 122 trains, 20 car flows, 30 locomotives

6.5 min

5.9 min

3

12 yards, 2 days, 135 trains, 25 car flows, 35 locomotives

10 min

8.6 min

4

12 yards, 2 days, 135 trains, 30 car flows, 30 locomotives

9.6 min

8.6 min

5

12 yards, 2 days, 150 trains, 40 car flows, 40 locomotives

25 min

24 min * Both two methods obtain same optimum © 2011 IBM Corporation

Research - China

Future Research  Smarter algorithm to solve larger-size problem  More business rules could be considered, such as maintenance, locomotive turn-around at depot (many cases in China), etc.  Continuous time-space network could be considered instead of discrete time-space network  Identify the difference between random generated parameter and practical parameter

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Research - China

Thank you!

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