1. Dynamic Freight Train Rerouting Alborz Parcham-Kashani Dr. Alan Erera Georgia Institute of Technology H. Milton Stewart School of Industrial & Systems Engineering INFORMS Annual Meeting 2014 – San Francisco

2. Agenda Introduction Problem background Methodology Preliminary computational results Next steps

3. Introduction and Motivation Train re-routing is an invaluable task for freight train companies Re-routes considered on an ad-hoc basis Reason: changes arising in train network capabilities Weather-related impact Equipment-related impact Work-force-related impact Volume-related impact Congested areas can be relieved via re-routing trains!

4. Motivation – Management’s Objectives 1.Consider re-routing potentially several trains at a time 2.Choose re-route options that has minimal cost impact (labor, fuel, etc.) 3.Reduce overall-system car delay Specifically: consider impact of decision on other trains Case-specific Consideration: Is not re-routing the train away from the congested area a viable option?

5. Introduction – Value of OR Opportunity: Currently, re-routing assignments are not performed with advanced OR techniques. Management ObjectiveAs-Is TechnologyProposed OR Capability Cost MinimizationEstimated time-in-system cost Accurate time-in-system cost Delay Minimization___Accounted for Impact of Joint Decisions___Accounted for

6. Background Commodity = block of rail-cars Commodities originate with origin-destination pairs Trip plan – A-priori sequence of a rail-car’s route through the train network – Function of origin-destination pair, among other things

7. Background

9. Each option corresponds to a new a-priori trip plan for commodity #1: Option #1 Option #2

10. Background Also known a-priori: Most up-to-date weekly train schedule All present and short-term future cars in the system Processing capacity of each terminal – Number of rail-cars per hour MWF 8:30 – 15:45 7 days/week 11:00 – 16:15 MWF 12:25 – 23:35

11. Challenge #1

12. Challenge #2 Direct Impact: Indirect Impact:

13. Modeling Approach Event-based Directed Time-space Network Nodes correspond to train arrivals and departures “Train arcs” represent trains “Terminal storage arcs” connect nodes that correspond to the same terminal at “consecutive” points in time.

14. Modeling Approach Event-based Time-space Network - Example … … … … … … Terminal storage arcs Train arcs …

15. Modeling Approach Most updated train schedule with Example Re-route Choice … … … … … … Terminal storage arcs (recurring) … Train arcs (recurring) Re-route candidates (non- recurring) Re-route choices (non- recurring) “do-nothing” option Potential Reroute option

16. Modeling Approach How to explicitly model processing capacity of each terminal?  Arrival tracks v.s. departure tracks  Number of cars processed per hour  Augment time-space network with “processing arcs”  Processing arcs connect two nodes which are positioned ɛ minutes prior to a train departure/arrival node

17. Modeling Approach Augmented Event-based Time-space Network - Example … … … Terminal storage arcs Train arcs … … … … … … … … … Terminal processing arcs Arrival Tracks Departure Tracks Arrival Tracks Departure Tracks Arrival Tracks Departure Tracks

18. Modeling Approach Two more arc types: 1.Exit arcs – Directed time-space graph has one sink-node – Connect heads of all train arcs to sink – Represent commodities exiting the network 2.End-of-horizon arcs – connect last time-space node at each terminal to sink node – With a penalty cost

19. Modeling Approach – Arc-based IP  Multi-commodity network design problem  Commodities = blocks of cars which share the same origin-destination pair  Induced on augmented event-based time-space network  Design aspect allows for modeling the re-routing choice

20. Bundled Arc Capacities … Terminal storage arcs Train arcs … … … … Terminal processing arcs Processing capacity at terminal Amount of time passed since prior processing arc

21. Setting Objective Coefficients Able to capture delay as a linear function of the arc flow variables as follows

22. Utilization of Problem Structure

23. Data Arc-based IP Formulation Designated based on the a-priori trip plan for each commodity

24. Appendix: Multi-commodity Network Flow bundling Flow balance Decision variables Formulation

25. Additional Data Appendix: Multi-Commodity Network Design Additional Decision Variables

26. Appendix: Multi-commodity Network Design Bundling Flow balance Formulation “Future” re-routing decisions “Present” re-routing decisions Choose one reroute option A-priori trip plan allows a smaller set of decision variables

27. Multi-commodity network flow problem (not the design problem) Commodity size = 1 rail-car Actual physical terminal network used Time horizon = 2 weeks Preliminary Computational Results # of CommoditiesLP Optimal ?Number of ConstraintsNumber of Variables 500Yes46K155K 650Yes84K290K 950Yes220K600K 1600Yes770K1,400K 2150IP Solution within 0.01% gap 1,700K2,700K 2200IP Solution within 0.01% gap 1,700K2,700K 2400Yes1,900K2,900K

28. 1) Run instances of the network design problem – Reasonable instance: 7 – 10 trains to be re-routed – Solution space: 3 7 – 3 10 2) If IP is not quick enough, attempt heuristic methods – Initial solution: Re-route each train to alternate destination geographically closest to its original destination – Greedy heuristic: optimize for one train at a time – 2-exchange – Modified 3-exchange Next Steps

29. Questions?