Ravindra K. Ahuja University of Florida, Gainesville & Innovative Scheduling

1. Solving Large-Scale Combinatorial Optimization Problems in Railroad Scheduling Ravindra K. Ahuja University of Florida, Gainesville & Innovative Scheduling (www.InnovativeScheduling.com)

2. Union Pacific Norfolk Southern CSX Transportation BNSF Railway Overview of US Freight Railroads • Four major (Class I) US Railroads:

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. Real-Time Scheduling Planning Railroad Planning and Scheduling Process Blocking Problem Service Design Train Scheduling & Block Assignment Train Dispatching Locomotive Sched. Crew 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. Origins Destinations Sorting Stations Package Delivery Problem Design the sorting network and route all packages in it to minimize the weighted sum of travel times and sortings.

10. Yards Destinations Origins Railroad Blocking Problem (contd.)

11. Problem Description Given: • A set of shipments with different origins/destinations Determine: • Design the network and route all shipments • Constraints: • Maximum number of arcs we can build at a node • Volume of shipments passing through a node is limited • 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. Perform multiple passes over the network until convergence. Improvement Algorithm 8 1 5 9 2 7 10 3 6 11 4

16. Convergence of the Algorithm Very fast convergence for car miles.

17. Convergence of the Algorithm (contd.) Very fast convergence for car handlings.

18. Sensitivity to the Starting Solution 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 Even small changes in the current solution obtain significant improvements.

21. Train Schedule Design ProblemCollaborators:Jian Liu, Innovative SchedulingKrishna C. Jha, Innovative SchedulingArvind Kumar, University of FloridaPooja Dewan, BNSF Railway

22. Train Schedule Design Problem Trains Train Schedule Design Problem Blocks Origins, Destinations, Routes Timings and Frequencies Blocks-to-Train Assignments Train Paths for each block Shipments Trip Plan Origin, Destination, Release Time Train Path for each Shipment

23. 7 8 9 10 1 2 3 4 5 6 6 5 1 2 3 4 7 Flow of Blocks on Trains

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 Schedule Design Train Route Design Determine train frequencies and arrival/departure times Determine train origins, destinations, routes, and block-to-train assignments Outputs Inputs

27. Computational Results 27

28. Optimizing Yard Locations

29. Seattle Washington Pullman Salt Lake City Dallas Jacksonville Arlington Gainesville Atlanta Austin Houston Orlando Optimizing Yard Locations Yards are like hubs in an airline flight network where cars are reclassified into new blocks and switch trains.

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 1 2 3 5 4 2 3 5 4 NeO: Straight Drop Consider each yard for potential deletion one by one and compute the change in transportation cost.

33. 2 1 3 5 4 NeO: Swap Consider exchanging each potential yard location with the existing yard location to reduce the total cost.

34. NeO: Computational Results We can remove several yard locations with very little impact on transportcosts.

35. Crew Planning and Scheduling

36. 1 2 3 4 5 6 7 1 2 3 4 5 6 7 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.

37. Rest Deadhead Rest 7 4 3 5 1 6 2 Time Understanding Crew Scheduling Home terminal Away terminal Train 1 Train 2 Drawback: • Deadheading cost Train 3

38. Rest in hotel 1 2 3 7 Combinatorial Decisions For each crew at the away location, we need to decide whether to deadhead the crew or put in the hotel. Home terminal Away terminal Train 1 Deadhead by taxi

39. Train arcs Deadhead arcs Rest arcs 10 1 3 5 7 9 12 14 2 16 4 6 8 11 13 15 A Network Flow Formulation Home Terminal Away Terminal Network 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 Time

40. Crew Scheduling: Computational Results • 50 crew districts, 52 weeks per year • Expected savings are in tens of millions of dollars annually. 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 Ground Nodes at a Station Time Train 1 Train 4 Train 2 Train 5 Train 3 Train 6

45. Computational Results for the Planning Tool • The method uses linear programming, integer programming, network flows, and local-search heuristics. We obtained a savings of over 400 locomotives.

46. Questions ahuja@ufl.edu