1. INOC 2013 May 2013, Tenerife, Spain Train unit scheduling with bi-level capacity requirements Zhiyuan Lin, Eva Barrena, Raymond Kwan School of Computing, University of Leeds, UK 1 CASPT 23 July 2015, Rotterdam

2. 2  Motivation  Problem description - Capacity levels  Model  Computational experiments  Conclusions and further work Outline

3. Motivation Minimize operating costs Satisfy capacity requirements Train unit scheduling problem Imprecise definition Under-utilized train units Imbalanced demands Re-balance Various sources Best representation How? 3

4.  Motivation  Problem description – Capacity levels  Model  Computational experiments  Conclusions and further work Outline 4

5. Train units 5 Class 171/7 (2-car), diesel Class 375 (4-car), electric

6. Train unit scheduling Train unit scheduling problem Train IDOriginDestinationDep timeArr timedemands 2E59AB09:0510:15125 2G15BC10:3012:25206 2G71CD15:0017:35196 2E59 A  B source s 2G15 B  C 2G71 C  D sink t Sign-on arc Empty-running connection arc Path (source to sink) Scheduled work for a train unit Train node Sign-off arc 6 Station connection arc

7. 2E32 1E06 2E11 2E03 1E09 source sink 1E06 2E11 2E32 2E11 2E03 1E09 Train unit scheduling Integer multicommodity flow representation 7 Paths may overlap for coupling Coupled units may be of different but compatible types x1

8. Outline Capacity requirement can be inferred from: – Mandatory minimum provision – Historic provision – Passenger count surveys (PAX) – Future growth expectation Problems of a single level: – Requirements not precisely defined / unknown – Under-utilized train units as a result of optimization techniques 8 Train capacity requirements

9. Under-utilized train units 9

10. Outline Implicit information – Pattern of unit resource distribution – Agreements/expectations with transport authorities Potential problems – Capacity strengthening could be used for unit resource redistribution: didn’t reflecting the real level – Unreasonable pattern may stay in past schedules for years 10 Historic capacity provision

11. Outline Capacity strengthening for unit resource redistribution in historic provision: an example 11 Historic capacity provision i i j j m Requires 1 unit n Requires 2 units AB CB B D DE

12. Outline Actual passenger counts Only a subset of trains surveyed Might contradict with historic provisions Frequency and scale of surveys vary among operators 12 PAX surveys

13. Outline Over-provided (OP): if historic capacity > PAX in terms of number of train units Under-provided (UP): if historic capacity < PAX – No place available for coupling/decoupling – Result of under-optimized schedules – OP: Used for redistributing train unit resources – UP: May be inevitable due to limited fleet size and/or coupling upper bound 13 OP and UP trains

14. Outline A desirable level r’ j – Will be satisfied as much as possible – max {historic, PAX, …} A target level r j – Must be strictly satisfied – min {historic, PAX, …} 14 Bi-level capacity requirement (per train j)

15. Outline 15 Bi-level capacity requirement Historic capacity PAX Future growth Mandatory minimum … Desirable capacity Target capacity Model Scheduled capacity informationInput data Output data

16.  Motivation  Problem description – Capacity levels  Model  Computational experiments  Conclusions and further work Outline 16

17. Outline Objective function – Minimize operating costs, including Fleet size, mileage, empty-running – Reflect preferences on, e.g., long idle gaps for maintenance – Achieve the desirable capacity requirements level as much as possible 17 The integer multicommodity flow formulation

18. Outline Constraints – Fleet size bounds 18 The integer multicommodity flow formulation – Target capacity requirement – Coupling of compatible types – Complex coupling upper bounds – Target capacity requirement – Coupling of compatible types – Complex coupling upper bounds combined into “train convex hulls”

19. Outline Variables 19 The formulation Path variable number of units used for path p of type k Capacity provision variable The capacity provided by the solver at train j

20. Outline Realized in the objective 20 Desirable level r′ Get the capacity provision in constraints Minimize the deviation between y and r′ Deal with the absolute values Operating cost Desirable capacity level

21. Outline (1) Objective 21 The ILP formulation (3) Convex hulls for all trains (4) Calculate capacity provision variables (5)(6) Variable domain (2) Fleet size upper bound

22.  Motivation  Problem description – Capacity levels  Model  Computational experiments  Conclusions and further work Outline 22

23. Objective function - Competing terms Deviation from desirable level Operating costs: Fleet size, mileage,... Weights Computational experiments: Objective function terms 23

24. Computational experiments Purposes Calibrate the objective function weights Satisfy as much as possible the desirable capacity level for a given fleet size Compare with manual schedules Experiments E1: Varying weights in the objective function E2: Fix fleet size & solely minimize r’ deviation 24

25. Computational experiments Central Scotland railway network; December 2011 timetable 25

26. Actually operated schedule: 64 OP trains out of 156 If use PAX, solver = 29 units If use historic capacity, solver = 33 units Computational experiments: Input data 26

27. Computational experiments: Results on E1 27

28. Computational experiments: Results on E1 28

29. Computational experiments: Results on E2 E2: Fix fleet size & solely minimize OP deviation 29

30. Computational experiments: Comparison between E1 & E2 E2 E1 Actually operating schedule: Fleet size= 33, OP=64 30

31. Conclusions Train unit scheduling with bi-level capacity requirements:  Target: PAX; Desirable: historic provisions  Schedules with more reasonable/controlled capacities Improvements w.r.t. manual schedules:  12% reduction of fleet size  Maintaining nearly 60% OP trains 31

32. Further work UP trains & limited fleet size Multicriteria optimization Trade-offs between depot returns and maximizing capacity provision More problem contexts in train unit resource planning, e.g. – franchise bidding – maintenance scheduling 32

33. Thank you!