ravindra k. ahuja university of florida, gainesville & innovative scheduling
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Solving Large-Scale Combinatorial Optimization Problems in Railroad Scheduling . Ravindra K. Ahuja University of Florida, Gainesville & Innovative Scheduling (www.InnovativeScheduling.com). Union Pacific. Norfolk Southern. CSX Transportation. BNSF Railway. - PowerPoint PPT PresentationTRANSCRIPT
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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)
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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
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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:
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The Railroad Network
A typical rail network spans 20,000 to 50,000 track miles.
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Role of Yards in Railroads Railroad yards act as hubs where cars change
trains.
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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
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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.
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Railroad Blocking Problem
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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.
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Railroad Blocking Problem (contd.)
OriginsOrigins DestinationsDestinationsYardsYards
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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
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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
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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.
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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.
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Improvement Algorithm
1
2
3
4
5
6
7
8
9
10
11
Perform multiple passes over the network until convergence.
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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.
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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.
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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.
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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
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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.
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Train Schedule Design Problem
Collaborators:
Jian Liu, Innovative SchedulingKrishna C. Jha, Innovative Scheduling
Arvind Kumar, University of FloridaPooja Dewan, BNSF Railway
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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
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Flow of Blocks on Trains
1 2 3 4 5 6 7
7
8
9
10
1 2 3 4 5 6
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Combinatorial Nature Trains
Train origins, destinations, routes,
Train frequencies
Train timings
Blocks
Train assignments
Sequencing
Timings
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Constraints and Objectives
Constraints: Node and link capacities Minimum and maximum train size
Objective Function: Total train miles Car days Locomotives Crew Block swaps
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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
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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%
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Optimizing Yard Locations
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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
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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?
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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.
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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
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NeO: Swap
Consider exchanging each potential yard location with the existing yard location to reduce the total cost.
1
2
3
5 4
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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.
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Crew Planning and Scheduling
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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
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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
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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.
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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
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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.
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Locomotive Planning
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Combinatorial Nature
Given:
Weekly train schedule Types and number of locomotives
Determine:
Assignment of locomotives to trains Determine their rotation plans
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Locomotive Assignment
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Locomotive Flow Network
Train 1
Train 4
Train 5
Train 2
Train 3
Train 6
Ground Nodes at a Station
Time
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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.