Periodic Railway Timetabling with Event Flexibility

Transcription

1 Periodic Railway Timetabling with Event Flexibility G. Caimi, M. Fuchsberger, M. Laumanns, K. Schüpbach ETH Zurich ATMOS, Sevilla, 16 November 2007 ETH Zürich 16 November 2007

2 Outline Approach to the train scheduling problem Timetable generation with PESP Event flexibility in timetables (FPESP) Results Conclusions 2

3 Generating train schedules 3

4 Generating train schedules INPUT: Global train service intention (GSI) Train lines with stops and frequencies Interconnections Rolling stock Aggregated and detailed track topology Rolling stock with dynamic properties OUTPUT: Conflict-free train schedule 4

5 Two level approach Macro scheduling: Find a timetable that fulfills basic properties, such as trip times, connections and headways. Micro scheduling: Find locally a conflict free routing, fulfilling detailed safety requirements for a given macro schedule. 5

6 Scheduling on the macro level Periodic Event Scheduling Problem (Serafini Ukovich 1989) Generates cyclic timetables or gives a proof of infeasibility Train line has to be known a priori (from GSI) Simplified safety system: headway time 6

7 Periodic Event Scheduling Problem (PESP) Events: train departures and arrivals 0 π i < T Constraints: τ i (l ij, u ij ) τ j Periodicity: T l ij π j - π i + T p ij u ij Period jumps: p ij, integer variables (binary for 0 l ij u ij < T) Serafini and Ukovich

8 PESP constraints Trip time Dwell time A Departure in A Headway time Connection B Arrival at B A (22,25) B (1,5) B (10,12) C (2,58) (2,58) A (28,33) B (3,8) B (18,21) D 8

9 MIP formulations of PESP Original Cycle Periodicity 1 Problem is NP-complete Cycle Periodicity is more efficient 1 1 Peeters 03 1 Liebchen 06 9

10 Outline Approach to the train scheduling problem Timetable generation with PESP Event flexibility in timetables (FPESP) Results Conclusions 10

11 Two level interface Interaction of the macro and micro scheduling Find a feasible macro schedule check Find a conflict free routing C. et al 2007 Kroon et al 1996 Ehrgott et al 2005 reject Goal: Increase search space in micro scheduling generate time slots instead of fixed times for the events at the macro level 11

12 Recall PESP Example x = [45,49] x = 3 [2,4] [2,6] x = 4 1 [2,57] x = 5 56 May be too restrictive! 12

13 FPESP: introduce event slots [4,5] [45,49] [51,53] [2,4] [2,6] [2,57] Each choice of event times in interval corresponds to a feasible schedule [1,2] [55,57] 13

14 Introduction of event slots Assign time slots (π li, π ui ) for the events instead of fixed times t All π i 2(π li, π ui ) fulfill the PESP constraints δ j Event flexibility δ i = π ui - π li Event flexibilities are dependent δ i + δ j γ ij for all constraints δ i i j where the γ ij = u ij l ij is the constraint interval length δ i γ ij δ j 14

15 Flexible PESP model (FPESP) Event slots are modelled by constraint adaptation (l,u) π (l,u) (l,u δ) (π,π+δ) (l +δ,u) (l,u) (l +δ,u) Fits into both MIP formulations Close to the original PESP model No additional integer variables 15

16 Testcase Bi-objective problem of minimizing the total triptime and maximizing the flexibility 16

17 Test Case Zug Luzern ArthGoldau Trains: intercity, local, cargo Service Intention from SBB timetable 2007 Matlab implementation with Mosek/Cplex solver 16 November 2007 G. Caimi ETH Zurich Periodic Railway Timetabling with Event Flexibility 17

18 Test case: generated timetable Muri Rotkreuz Immensee Arth-Goldau Erstfeld 18

19 Trip time vs flexibility: Pareto line >0.5 MIXFLEX <0.5 19

20 How do we use this flexibility? Event slots are used as input for the micro scheduling Enables different time alternatives for train departure/arrival at portal Time choice should be independent for each train Dependencies for same train in same station are not problematic Improves timetable stability by better distributing trains over time 20

21 Flexbox Example 4 Arrival a 1 Departure 1 4 d a d More flexibility per event Larger search space for the micro scheduling 21

22 Flexbox Results Reference scenario, objective Maxflex Scenario FPESP FPESP + station boxes FPESP + dwell boxes FPESP + separation boxes Flexibility

23 Conclusion Generation of flexible train schedules works well with the FPESP model Reasonable performance for ~ 50 trains Computation time depends strongly on objective and scenario Flexbox approach is promising to make use of natural dependencies 23

24 Thank you! 24

25 FPESP results Additional upper bounds for flexibilities to improve their distribution over events 25

26 26

27 Progress of the solution quality Objective function time 27

28 Event Slot Objectives Maximal sum of flexibilities versus uniform distribution Because of δ i + δ j γ ij, a limitation of δ i δ max reduces the number of zero-flexibility events 28

29 Flexbox: Definition Introduce flexibilities for multiple events together: Flexbox Each Flexbox has a variable δ box Boxes can overlap or contain each other The events in a box can be postponed by at most δ box For each constraint 29

30 Flexbox Result 30

31 Schema of global timetable generation Global SI Local SI Delete intention in SI infeasible Find macro schedule fulfilling SI feasible Check feasibility locally on detailed topology (Micro scheduling) Detect conflict, Move flexibility no Conflict-free? yes IFOR Conflict-free timetable, constraints report Appraisal, Relaxation no Satisfies GSI? yes Production Plan 31

32 Aggregated topology Used for macroscopic scheduling Nodes are stations/station regions with overtaking possibilities (e.g. additional tracks) Nodes are connected with as many edges as parallel tracks between the stations 32

33 Cycle Periodicity Formulation (CPF) Use event time difference x ij per constraint instead of event times τ i For each cycle C, it must hold Use the periodicity jumps per cycle Add cutting planes 33

34 Cycle Periodicity Formulation (CPF) B is a Integer Cycle Basis of the PESP Graph (Peeters `03, Liebchen `06) 34

35 PESP versus CPF formulation Cycle consisting of 25 arcs Cutting plane: 1 q C 2 Integer search space size can be reduced (original PESP: 2 25 CPF: 2) 35

36 Test Case Zug Luzern ArthGoldau Trains: intercity, local, cargo Service Intention from SBB timetable 2007 Matlab implementation with Mosek solver Objective: Minimize travel time 36

37 Flexibles PESP Modell: Resultate 37

38 Adding Flexibility: Time Slot Scheduling Assign time slots in the macro level to departures and arrivals. (l,u) τ i (l,u) (l,u) Add flexibility d i to each event + : fits into the PESP framework + : no additional constraints - : additional continous variables (l,u-d i ) (τ i,τ i +d i ) (l+d i,u) (l,u) (l+d i,u) 38

39 Trip time vs flexibility: Pareto line Flexibility Trip Time 39

40 PESP Period Jumps Difficult part, binary search tree, one tree level per constraint Determine event order Can be used as non-collision constraints (22,25) d 1 a 1 (2,58) (2,58) t t (28,33) d 2 a 2 p ij = 0 A B A B 40

41 PESP Period Jumps Difficult part, binary search tree, one tree level per constraint Determine event order Can be used as non-collision constraints (22,25) d 1 a 1 (2,58) (2,58) t t (28,33) d 2 a 2 p ij = 0 p ij = 0 A B p ij = 1 A B 41

42 PESP Period Jumps The periodic ordering of events along a cycle determines the number of period jumps along C (2,58) A (2,58) C B (2,58) p ij = 0 42

43 (22,25) d 1 a 1 (2,58) (2,58) t a 2 a 1 d 2 d 1 a 2 t d 1 a 2 a 1 (28,33) d 2 a 2 d 2 d 1 a 1 d 2 d 1 d 2 a 2 a 1 A B A B (22,25) d 1 a 1 (2,58) (2,58) t d 2 d 1 p ij = 0 a 2 t a 2 d 1 p ij = 1 a 1 a 2 a 1 (28,33) d 2 a 2 d 2 d 1 A B a 1 d 2 d 1 d 2 A B a 2 a 1 43

44 a 2 d 2 d 2 d 1 a 1 a 1 d 1 a 2 A B A B d 2 a 2 a 1 a 1 a 2 a 1 a 2 d 2 d 1 a 1 d 2 d 1 d 2 d 1 d 1 a 2 A B A B A B A B 44

45 τ li τ lj τ ui τ uj τ li τ lj (l (l ij, u ij ) τ ij, u ij ) i τ j (0,r) (l ij, u ij ) τ ui τ uj (0,r) 45

46 (22,26) (2,58) a 1 (0,1) (2,58) d 1 (10,14) (20,23) a 2 (0,1) d 2 (12,15) 4 a 1 1 d a 2 1 d

47 Event slots only in stations 47

48 4 56 a d a d

49 What can be modeled with PESP Time period Trip time (with flexible time reserves) Train interconnections Headway time Dwell times Train frequencies 49

50 Outlook Discussion of the generated timetables with SBB. Refinement of the model. Test larger scenarios Further tests with Flexbox concept 50