ravindra k. ahuja university of florida, gainesville & innovative scheduling

Business Process ModelDevelopment – Phase 2 DiscussionAugust 19, 2002

Solving Large-Scale Combinatorial Optimization Problems in Railroad Scheduling

Ravindra K. AhujaRavindra K. Ahuja

University of Florida, GainesvilleUniversity of Florida, Gainesville&&

Innovative SchedulingInnovative Scheduling(www.InnovativeScheduling.com)(www.InnovativeScheduling.com)

2

Overview of US Freight Railroads Four major (Class I) US Railroads:Four major (Class I) US Railroads:

BNSF RailwayBNSF Railway

Union PacificUnion Pacific

CSX TransportationCSX Transportation

Norfolk SouthernNorfolk Southern

3

Overview of US Freight Railroads (contd.)

US railroads carry about 30 million carloads per year.

Total revenue: $40 billion per year.

Major goods transported:

Intercity freight revenue:

4

The Railroad Network

A typical rail network spans 20,000 to 50,000 track miles.

5

Role of Yards in Railroads Railroad yards act as hubs where cars change

trains.

6

Railroad Planning and Scheduling Process

Blocking ProblemBlocking Problem

Train Scheduling & Block AssignmentTrain Scheduling & Block Assignment

Service Design

Train DispatchingTrain DispatchingLocomotive Sched.Locomotive Sched. Crew SchedulingCrew Scheduling

Planning Real-Time Scheduling

7

State-of-the-Art in Railroad Planning

Railroad planning and scheduling problems are very large-scale and very difficult discrete optimization problems.

Almost all railroad planning and scheduling problems are solved manually.

There are teams of 10-20 highly experienced personnel for solving each problem.

We plan to automate most of these decision processes.

8

Railroad Blocking Problem

9

Package Delivery Problem

OriginsOrigins DestinationsDestinationsSorting StationsSorting Stations

Design the sorting network and route all packages in it to minimize the weighted sum of travel times and sortings.

10

Railroad Blocking Problem (contd.)

OriginsOrigins DestinationsDestinationsYardsYards

11

Problem DescriptionGiven:Given:

A set of shipments with different origins/destinations

Determine:Determine:

Design the network and route all shipments

Constraints:Constraints:

Maximum number of arcs we can build at a node Volume of shipments passing through a node is limited

Objective Function:Objective Function:

Minimize the weighted sum of distance traveled by shipments and their node handlings

12

Combinatorial Nature Size:

3,000 nodes 50,000 commodities

Origin-destination of each block (over 1 million possibilities)

Routing of each origin-destination shipment (hundred’s of billions of possibilities)

Two-stage problem

13

Costs involved

Substantial amount of costs involved: Cost of flow: $1,000 - $2,000 million Cost of handling: $500 - $1,000 million

Currently solved manually.

14

Very Large-Scale Neighborhood (VLSN) Search Algorithm

Start with a feasible solution. Use a simple heuristic.

Repeatedly improve the solution Use a VLSN search method.

15

Improvement Algorithm

1

2

3

4

5

6

7

8

9

10

11

Perform multiple passes over the network until convergence.

16

Convergence of the Algorithm

car miles vs. the number of passes

870

890910

930

950970

990

10101030

1050

0 1 2 3 4 5 6 7 8 9 10

Pass No

Car M

iles

Very fast convergence for car miles.

17

Convergence of the Algorithm (contd.)

car handlings vs. the number of passes

0.8

0.85

0.9

0.95

1

1.05

1.1

0 1 2 3 4 5 6 7 8 9 10

Pass No

Car

Han

dlin

gs

Very fast convergence for car handlings.

18

Sensitivity to the Starting Solution

Average Handlings per Car

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5 6 7 8 9 10 11

Pass No.

Han

dles

/Car

Insensitivity to the starting solution.

19

Computational Results for Railroads Savings in average car miles: 2% - 5%

Savings in intermediate handlings: 15% - 30%

Anticipated dollar savings: $20 - $50 million yearly

Running time: 1 - 2 hours

20

Incremental Changes to Current Solution

CSX Unrestricted Solution

New Blocks

Percentage

Savings inCar Miles

Savings in Intermediate

Handlings

1% 2.3% 9.8%

2% 2.3% 10.8%

3% 2.4% 13.4%

5% 3.2% 15.2%

8% 3.2% 17.6%

15% 3.8% 23.8%

22% 4.2% 27.4%

35% 4.4% 31.7%

Even small changes in the current solution obtain significant improvements.

21

Train Schedule Design Problem

Collaborators:

Jian Liu, Innovative SchedulingKrishna C. Jha, Innovative Scheduling

Arvind Kumar, University of FloridaPooja Dewan, BNSF Railway

22

Train Schedule Design Problem

Train Schedule Design Problem

BlocksTrains

Blocks-to-TrainAssignments

Trip PlanShipments

Origins, Destinations, RoutesTimings and Frequencies

Train Paths for each block

Train Path for each ShipmentOrigin, Destination, Release Time

23

Flow of Blocks on Trains

1 2 3 4 5 6 7

7

8

9

10

1 2 3 4 5 6

24

Combinatorial Nature Trains

Train origins, destinations, routes,

Train frequencies

Train timings

Blocks

Train assignments

Sequencing

Timings

25

Constraints and Objectives

Constraints: Node and link capacities Minimum and maximum train size

Objective Function: Total train miles Car days Locomotives Crew Block swaps

26

Train Schedule Design: Solution Technique

Train Route Design

Determine train origins, destinations, routes, and

block-to-train assignments Inputs Outputs

Train Schedule Design

Determine train frequencies andarrival/departure

times

27

Computational Results

BNSF Railway Norfolk Southern Corporation

Statistics BNSF Innovative % Improv NS Innovative % Improv

Cars per week 60,198 90,500

Number of train starts 1,498 1,067 29% 1,995 1306 35%Average car days 3.74 3.57 5% 2.84 2.60 8%Average car miles 904 885 2% 482 482 0%Average car block swaps 0.19 0.18 3% 0.12 0.12 0%Total train miles 760,420 621,823 18% 667,960 556,091 17%

Train travel time 43,111 37,937 12% 43,687 37,570 14%

27

28

Optimizing Yard Locations

29

Optimizing Yard Locations

Yards are like hubs in an airline flight network where cars are reclassified into new blocks and switch trains.

PullmanPullman

Salt Lake CitySalt Lake City

DallasDallas

GainesvilleGainesville

OrlandoOrlando

JacksonvilleJacksonville

AtlantaAtlanta

WashingtonWashingtonSeattleSeattle

ArlingtonArlington

AustinAustin

HoustonHouston

30

Role of Yard Locations

Yard locations have been determined historically.

Yard operating costs: $10-$20 million annually.

Can we delete some yard locations with minimal impact on cost?

31

Combinatorial Nature

Select the best yard locations:

20 locations out of a given 40 locations 40C20 possibilities

For each set of candidate locations, solve a blocking problem to assess its goodness.

32

1

NeO: Straight Drop

Consider each yard for potential deletion one by one and compute the change in transportation cost.

1

2

3

5 4

2

3

5 4

33

NeO: Swap

Consider exchanging each potential yard location with the existing yard location to reduce the total cost.

1

2

3

5 4

34

NeO: Computational Results

Number of Hubs Deleted Vs Total Cost

$15

$20

$25

$30

$35

$40

$45

$50

0 5 10 15 20 25 30

Mill

ions

Number of Hubs Deleted

Tota

l Cos

t

We can remove several yard locations with very little impact on transportcosts.

35

Crew Planning and Scheduling

36

Crew Scheduling at US Railroads

Each train requires a crew and changes crew at several locations at it travels from its origin to its destination.

1 2 3 4 5 6 7

1 2 3 4 5 6 7

37

Understanding Crew Scheduling

1

Home terminal Away terminal

2

3

4

5

6

Train 1

Train 2

Train 3

7

Drawback: Deadheading cost

RestDeadhead

Rest

Time

38

Combinatorial Decisions

1

Home terminal Away terminal

2

3

Train 1

7

Rest in hotelDeadhead by taxi

For each crew at the away location, we need to decide whether to deadhead the crew or put in the hotel.

39

A Network Flow FormulationHome Terminal Away Terminal

TimeNetwork Construction: One-to-one correspon.

between crew flow and arc flow

Union Rules: Minimum rest rule Detention wage rule FIFO rule

Objectives: Minimize deadhead Minimize detention Minimize train delay

1

3

5

7

9

10

12

14

15

2

16

4

6

8

11

13

Train arcs

Deadhead arcs

Rest arcs

40

Crew Scheduling: Computational Results

50 crew districts, 52 weeks per year

Expected savings are in tens of millions of dollars annually.

Criteria Current Solution Our Solution Improvement

Train delay 48 hours 0 hours 100 %

Dead-headings 322 hours 200 hours 37.9 %

Crew detention 88 hours 53 hours 39.8 %

Results of one crew district and for one week.

41

Locomotive Planning

42

Combinatorial Nature

Given:

Weekly train schedule Types and number of locomotives

Determine:

Assignment of locomotives to trains Determine their rotation plans

43

Locomotive Assignment

44

Locomotive Flow Network

Train 1

Train 4

Train 5

Train 2

Train 3

Train 6

Ground Nodes at a Station

Time

45

Computational Results for the Planning Tool

Locomotive type used

CSX Our method

AC60 99 99AC44 172 171C40 651 639SD40 533 280SD50 154 19Total: 1609 1208

We obtained a savings of over 400 locomotives.

The method uses linear programming, integer programming, network flows, and local-search heuristics.

46

Questions

ahuja@ufl.edu