Train dispatching model with stochastic capacity breakdowns on an N- tracked railroad network

1. Train dispatching model with stochastic capacity breakdowns on an N-tracked railroad network Xuesong Zhou University of Utah Utah, U.S.A. Email: zhou@eng.utah.edu Lingyun Meng Beijing Jiaotong University Beijing, China Email: lymeng@bjtu.edu.cn October 15th 2012 INFORMS, Phoenix, U.S.A.

2. Outline Introduction Mathematical formulations Solution algorithms Experimental results

3. Task of Train Dispatching Goal: Recover impacted train schedules from . Measures: Re-timing Re-ordering Re-routing Re-servicing

4. Dispatching in a Dynamic & Stochastic Environment Dispatching schedules are updated when new information are available. Uncertain disturbance information: e.g. stochastic incident duration.

5. Problem Description Characteristics of disruptions in this study • 1.It refers to a blockage of one track. It’s a strong perturbation. • 2. It has a relatively longer duration compared to minor disturbances. Re-routing and Re-Servicing become strongly necessary, because Re-timing and Re-ordering are too week to deal with disruptions.

6. When can the capacity be fully restored ? • How to reschedule trains so that the system-wide performance canbe optimized ? In a word, the key question is how to generate a train dispatching plan?

7. State of the art A wide range of studies are devoted to optimization model formulation and algorithm development, e.g. Kraft (1983) Jovanovic (1989), Carey (1994) and D’Ariano (2008) . The majority of previous optimization models for train dispatching primarily assumecertain and perfect information of disruptions, e.g. Adenso-Diaz et al. (1999) and Chikara et al. (2009). Meng and Zhou (2011) has proposed an approach for robust train dispatching on a SINGLE-TRACK line. This study tries to extend the model to the N-TRACKED network context. Lingyun Meng, Xuesong Zhou, 2011. Robust single-track train dispatching model under a dynamic and stochastic environment: a scenario-based rolling horizon solution approach. Transportation Research Part B, 45(7): 1080-1102.

8. Solution approach General ideas and contributions to literature 1. Use cumulative flow count-based variables to represent train arrival/departure times at stations/blocks 2. Use lagrangian relaxation method to simultaneously re-route and re-schedule trains 3. Use capacity aggregation mechanism to capture the stochasticity of capacity

9. Mathematical model 1. Objective function Minimize the expected train exit(completion) time for all trains 2. Constraints Capacity (breakdown)/headway times constraints Departure time constraints Segment running time constraints Dwell time constraints

10. Notation 1 General subscripts Flexible path-based

11. 2 Input variables Cell capacity

12. 3 Decision variables By and At

13. Cumulative flow count-based decision variables (CFCD) Cell occupancy time: t: 5 6 7 8 9 10 11 12 13 14 15 - = 0 0 1 1 1 0 0 0 0 0 0 3 time units are used

14. Headway time constraints represented by CFCD Occupancy starting time arrival time Occupancy time shift constraints:

15. Capacity issue for multiple trains by CFCD Whether cell e isoccupied by train f along path p at time t under scenario k Avoid if-then / big “M” constraints Capacity(resource)-oriented train scheduling model, compared to conflict-oriented model

16. N-track issue Cell decomposition of one directional link l corresponding to a double track Cell decomposition of bi-directional link l corresponding to a single track Please also see Steven Harrod (2010) on block-based scheduling by Hypergraph

17. Network issue earliest starting time Ensure that one train only selects one path from the corresponding set of possible paths at its starting cell

18. Stochastic capacity issue Capacity aggregation technique to deal with Uncertainty of Disruptions. See Luh (1999) for Job shop scheduling under uncertainty. Cell capacity constraints is satisfied in an expected manner, rather than for each scenario. Note that we will further deduce the solutions to feasible solutions under each scenario

19. Lagrangian relaxation based solution algorithm Scenario case Aggregated case Cell capacity constraints (side constraints) are relaxed.

20. Lagrangian relaxation based solution algorithm Algorithm 1 Subgradient algorithm to update lagrangian multipliers Algorithm 2 Time-dependent shortest path algorithm to find optimal solution (Lower Bound) for the relaxed problem Algorithm 3 Priority rule-based algorithm to deduce solutions into problem feasible solutions under given scenario Simultaneously and flexibly rerouting and rescheduling trains on an N-tracked network

21. Numerical experiments results 7 major stations 144 trains belonging to 4 railway companies own about 40%, 30%, 20% and 10% of total trains

22. Prelimiaryresults by LR algorithm Performance of the lagrangian relaxation algorithm For RAS data set 1 within a network of 13 nodes ,14 links and 3 trains. Optimality: 83.7% Within less than 1 second Recall the modification of objective function to total completion time

23. Ongoing work (1) Algorithm fine tuning (2) Lagrangian relaxation decomposition technique (3) More experiments with larger number of trains, larger network size

24. Thanks for your attention!Any questions?