Lines, Timetables, Delays: Algorithms in Public Transportation

Transcription

1 Lines, Timetables, Delays: Algorithms in Public Transportation Anita Schöbel Institut für Numerische und Angewandte Mathematik Georg-August Universität Göttingen March 2015 Anita Schöbel (NAM) Optimization in public transport March / 75

2 Planning in public transportation Anita Schöbel (NAM) Optimization in public transport March / 75

3 Planning in public transportation Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

4 Planning in public transportation Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

5 Planning in public transportation Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

6 Planning in public transportation Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

7 Planning in public transportation Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

8 Planning in public transportation Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

9 Planning in public transportation Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

10 Content of the talk 1 Line planning 2 Timetabling 3 Delay management 4 Current Research Directions Integrating Passengers Routes Integrating robustness issues Integrating the Planning Stages 5 The end... Anita Schöbel (NAM) Optimization in public transport March / 75

11 Line planning How to plan the lines? Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

12 Line planning How to plan the lines? Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

13 Line planning Line planning is a well known problem Many papers, starting in 1925 and still ongoing research Focus is on development of the models. Solution approaches mostly IP-based and work well for the size of real-world input data Anita Schöbel (NAM) Optimization in public transport March / 75

14 Line planning Line planning Let a public transportation network PTN with its stops V and its direct connections E be given. A line P is a path in the public transportation network The frequency of a line says how often it is operated. Anita Schöbel (NAM) Optimization in public transport March / 75

15 Line planning Line planning Let a public transportation network PTN with its stops V and its direct connections E be given. A line P is a path in the public transportation network The frequency of a line says how often it is operated. Line planning: Given a PTN and a pool of potential lines, choose a subset of lines from the pool and assign frequencies to them such that Anita Schöbel (NAM) Optimization in public transport March / 75

16 Line planning Line planning Let a public transportation network PTN with its stops V and its direct connections E be given. A line P is a path in the public transportation network The frequency of a line says how often it is operated. Line planning: Given a PTN and a pool of potential lines, choose a subset of lines from the pool and assign frequencies to them such that? Anita Schöbel (NAM) Optimization in public transport March / 75

17 Line planning Goals of Line planning? For the public transportation company: All passengers should be transported. Costs should be small. For the passengers: many direct passengers small riding times few transfers high frequencies Anita Schöbel (NAM) Optimization in public transport March / 75

18 Line planning Goals of Line planning? For the public transportation company: All passengers should be transported. Costs should be small. For the passengers: many direct passengers small riding times few transfers high frequencies We have many different models to be split into cost-oriented models and passenger-oriented models. Anita Schöbel (NAM) Optimization in public transport March / 75

19 Line planning Basic model... contained in (almost) all line planning models Anita Schöbel (NAM) Optimization in public transport March / 75

20 Line planning Basic model... contained in (almost) all line planning models Given fe min, fe max for all e E, a line concept is feasible if f min e l:e l f l f max e for all e E (Basic model) Given a line pool, find a feasible line concept. Anita Schöbel (NAM) Optimization in public transport March / 75

21 Line planning Basic model... contained in (almost) all line planning models Given fe min, fe max for all e E, a line concept is feasible if f min e l:e l f l f max e for all e E (Basic model) Given a line pool, find a feasible line concept. Note: Multi-Covering and Multi-Packing constraints already NP-hard. Anita Schöbel (NAM) Optimization in public transport March / 75

22 Line planning Cost oriented models Given costs c l for any line l L. (Cost model) min c l f l l L s.t. f min e l:e l f l IN f l f max e The lower edge frequencies make sure that enough passengers can travel. Anita Schöbel (NAM) Optimization in public transport March / 75

23 Line planning Cost oriented models Also NP-hard, but may be solved in small computation time by integer programming solvers for reasonable sizes. Heuristics neglect upper bounds and rely on multi-covering Extensions with more realistic costs exist. Literature: Claessens (1994), Zwaneveld, Claessens, van Dijk (1996), Claessens, van Dijk, Zwaneveld (1996), Bussieck and Zimmermann (1997), Claessens, van Dijk and Zwaneveld (1998), Goessens, Hoesel, and Kroon (2001, 2002), Bussieck, Lindner, and Lübbecke (2003) Anita Schöbel (NAM) Optimization in public transport March / 75

24 Line planning Passenger-oriented models Many goals are possible: Maximize number of direct travelers with respect to upper line frequency constraints: Patz (1925), Wegel (1974), Dienst (1970), Reinecke (1992) and Reinecke (1995), Bussieck, Kreuzer and Zimmermann (1996), Bussieck and Zimmermann (1997), Zimmermann, Bussieck, Krista and Wiegand (1997),Bussieck (1998) Maximize number of direct travelers w.r.t. budget constraint: Simonis (1980 and 1981) Maximize number of travelers within a reasonable amount of travel time with respect to budget constraint: Laporte, Marin, Mesa, Ortega (2004) Minimize travel time with respect to budget constraint: Schöbel and Scholl (2004), Borndörfer, Grötschel and Pfetsch (2005), Schneider (2005), Scholl (2006), Schöbel and Scholl(2006),Borndörfer, Grötschel and Pfetsch (2007), Michaelis and Schöbel (2009), Anita Schöbel (NAM) Optimization in public transport March / 75

25 Line planning Other approaches Game-theoretic approach Schöbel and Schwarze (2007), Kontogiannis and Zaroliagis (2008) Optimize modal split Laporte, Mesa and Ortega Generate lines dynamically Borndörfer and Pfetsch Survey on models and algorithms Schöbel, OR Spectrum (2012) Anita Schöbel (NAM) Optimization in public transport March / 75

26 Timetabling How to plan the timetable? Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

27 Timetabling How to plan the timetable? Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

28 Timetabling Focus of timetabling Model is rather clear: Assume that paths of passengers are already known. Objective: minimize traveling time Feasibility constraints (e.g. technical minimal driving times) Only distinction: periodic and aperiodic timetables Main focus: Development of solution approaches. Success stories: Optimize timetable of Berlin underground Liebchen and Möhring (2008) Optimize timetable in the Netherlands L. Kroon and D. Huisman and E. Abbink and P.-J. Fioole and M. Fischetti and G. Maroti and A. Steenbeek L. Schrijver and R. Ybema (2009) Anita Schöbel (NAM) Optimization in public transport March / 75

29 Timetabling Modeling timetables Use event-activity networks (E, A) with E = E dep E arr, A = A wait A A trans A head station 1 wait station 1 departure station 2 wait station 2 departure change station 1 wait station 1 departure station 2 wait station 2 departure station 1 station 1 station 2 station 2 wait departure wait departure transfer activity Anita Schöbel (NAM) Optimization in public transport March / 75

30 Timetabling Modeling timetables Use event-activity networks (E, A) with E = E dep E arr, A = A wait A A trans A head station 1 wait station 1 departure station 2 wait station 2 departure change station 1 wait station 1 departure station 2 wait station 2 departure station 1 station 1 station 2 station 2 wait departure wait departure transfer activity Anita Schöbel (NAM) Optimization in public transport March / 75

31 Timetabling Modeling timetables Use event-activity networks (E, A) with E = E dep E arr, A = A wait A A trans A head station 1 wait station 1 departure station 2 wait station 2 departure change station 1 wait station 1 departure station 2 wait station 2 departure station 1 station 1 station 2 station 2 wait departure wait departure headway constraint transfer activity Anita Schöbel (NAM) Optimization in public transport March / 75

32 Timetabling What is a timetable? A timetable assigns a time π i to every and departure event i. Anita Schöbel (NAM) Optimization in public transport March / 75

33 Timetabling What is a timetable? A timetable assigns a time π i to every and departure event i. A timetable is feasible if for all activities a = (i, j) if aperiodic case: L ij π j π i U ij π =45 i Lij =8 π j =55 Anita Schöbel (NAM) Optimization in public transport March / 75

34 Timetabling What is a timetable? A timetable assigns a time π i to every and departure event i. A timetable is feasible if for all activities a = (i, j) if aperiodic case: L ij π j π i U ij π =55 i Lij =8 π j =05 Anita Schöbel (NAM) Optimization in public transport March / 75

35 Timetabling What is a timetable? A timetable assigns a time π i to every and departure event i. A timetable is feasible if for all activities a = (i, j) if aperiodic case: L ij π j π i U ij periodic case: L ij (π j π i ) mod T U ij π =55 i Lij =8 π j =05 Anita Schöbel (NAM) Optimization in public transport March / 75

36 Timetabling Aperiodic Timetables (no headway constraints) (Feasible Differential Problem) Rockafellar (1984) Given an event-activity network (A, E) with upper and lower bounds L a U a for all activities weights w a for all activities find an aperiodic feasible timetable π i for all i E minimizing w a (π j π i ) a=(i,j) A Anita Schöbel (NAM) Optimization in public transport March / 75

37 Timetabling Solution approach in the aperiodic case (no headway constraints) IP-formulation min s.t. w a (π j π i ) a A L a π j π i U a for all a = (i, j) A π i ZZ Algorithms Coefficient matrix is transposed of node-arc-incidence matrix and hence TU. solvable in polynomial time. solvable more efficiently by shortest path techniques. Anita Schöbel (NAM) Optimization in public transport March / 75

38 Timetabling Periodic Timetables (PESP) Serafini und Ukovich (1989) Given an event-activity network (A, E) with upper and lower bounds L a U a for all activities weights w a for all activities find a periodic feasible timetable π i for all i E minimizing w a (π j π i ) mod T. a=(i,j) A Anita Schöbel (NAM) Optimization in public transport March / 75

39 Timetabling Solution approach in the periodic case IP-formulation: min s.t. w a (π j π i + p a T ) a A L a π j π i + p a T U a for all a = (i, j) A p a ZZ, π i ZZ Anita Schöbel (NAM) Optimization in public transport March / 75

40 Timetabling Solution approach in the periodic case IP-formulation: min s.t. w a (π j π i + p a T ) a A L a π j π i + p a T U a for all a = (i, j) A p a ZZ, π i ZZ The modulo parameters make the problem hard. More precisely: Complexity of (PESP) Serafini and Ukovich (1989) (PESP) is NP-hard. Anita Schöbel (NAM) Optimization in public transport March / 75

41 Timetabling More theory is necessary! Anita Schöbel (NAM) Optimization in public transport March / 75

42 Timetabling More theory is necessary! The tension of an activity a = (i, j) is in the aperiodic case: x ij = π j π i in the periodic case: x ij = (π j π i ) mod T π =55 i Lij =8 π j =05 Anita Schöbel (NAM) Optimization in public transport March / 75

43 Timetabling More theory is necessary! The tension of an activity a = (i, j) is in the aperiodic case: x ij = π j π i in the periodic case: x ij = (π j π i ) mod T π =55 i Lij =8 π j =05 Observation In a feasible timetable: in the aperiodic case: The tension along any circle is zero. in the periodic case: The tension along any circle mod T is zero. Anita Schöbel (NAM) Optimization in public transport March / 75

44 Timetabling Circle-based IP-formulation for (PESP) min a A w a x a x a x a = Tq C a C + a C L a x a U a q C ZZ x a ZZ for all circles C in G for all a A for all circles C in G for all a A Anita Schöbel (NAM) Optimization in public transport March / 75

45 Timetabling Solution approaches in the periodic case Fix the modulo parameters and solve the resulting aperiodic problem. Exploit circle based IP formulation: It is enough to investigate circles in a cycle basis. Finding good cycle basis is hence important. Serafini und Ukovich (1989), Odijk (1996), Nachtigall (1998), Peeters (2003), Liebchen (2006), Liebchen (2008), Huisman, Kroon, Vromans (2009) Heuristic: Modulo simplex exploiting spanning tree structures. Nachtigall and Opitz (2008), Goerigk and Schöbel (2011), Goerigk and Schöbel (2013) Anita Schöbel (NAM) Optimization in public transport March / 75

46 Delay management Delay Management Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

47 Delay management Delay Management Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

48 Delay management The delay management problem Train i arrives at a station with a delay. i j v j Anita Schöbel (NAM) Optimization in public transport March / 75

49 Delay management The delay management problem Train i arrives at a station with a delay. i j v j What should a connecting train j do? Anita Schöbel (NAM) Optimization in public transport March / 75

50 Delay management The delay management problem Train i arrives at a station with a delay. i j v j What should a connecting train j do? Wait but cause further delays? for the passengers in train j for passengers who wait for train j for subsequent other vehicles Anita Schöbel (NAM) Optimization in public transport March / 75

51 Delay management The delay management problem Train i arrives at a station with a delay. i j v j What should a connecting train j do? Wait but cause further delays? for the passengers in train j for passengers who wait for train j for subsequent other vehicles depart on time although passengers who wanted to transfer from train i to train j miss their connection? Anita Schöbel (NAM) Optimization in public transport March / 75

52 Delay management What are the variables? Wait or depart on time? Priorities (if capacities on tracks have to be considered) result: disposition timetable Anita Schöbel (NAM) Optimization in public transport March / 75

53 Delay management What are the variables? Wait or depart on time? Priorities (if capacities on tracks have to be considered) result: disposition timetable Goal: Minimize the passengers delays when they reach their destinations! Anita Schöbel (NAM) Optimization in public transport March / 75

54 Delay management What are the variables? Wait or depart on time? Priorities (if capacities on tracks have to be considered) result: disposition timetable Goal: Minimize the passengers delays when they reach their destinations! Anita Schöbel (NAM) Optimization in public transport March / 75

55 Delay management Model: event-activity network station 1 wait station 1 departure station 2 wait station 2 departure change station 1 wait station 1 departure station 2 wait station 2 departure station 1 station 1 station 2 station 2 wait departure wait departure transfer activity Anita Schöbel (NAM) Optimization in public transport March / 75

56 Delay management Model: event-activity network station 1 wait station 1 departure station 2 wait station 2 departure change station 1 wait station 1 departure station 2 wait station 2 departure station 1 station 1 station 2 station 2 wait departure wait departure changing activity: delete or not? Anita Schöbel (NAM) Optimization in public transport March / 75

57 Delay management Model: event-activity network station 1 wait station 1 departure station 2 wait station 2 departure change station 1 wait station 1 departure station 2 wait station 2 departure station 1 station 1 station 2 station 2 wait departure wait departure headway constraint: which one? transfer activity: delete or not? Anita Schöbel (NAM) Optimization in public transport March / 75

58 Delay management (Approximate) Integer programming formulation (DM cap) min f (x, z) = i E w i (x i π i ) + a A transfer w a Tz a such that x i π i + d i for all i E (1) x j x i L a + d a for all a = (i, j) A wait A (2) Mz a + x j x i L a for all a = (i, j) A transfer (3) Mg ij + x j x i H ij (i, j) A head (4) g ij + g ji = 1 (i, j) A head (5) x i IN for all i E z a {0, 1} for all a A transfer g ij {0, 1} for all (i, j) A head Anita Schöbel (NAM) Optimization in public transport March / 75

59 Delay management Complexity Theorem (Gatto, Jakob, Peeters, Schöbel, 2005) The delay management problem without capacity constraints is NP hard. Proof: Reduction to independent set. Theorem (Conte and Schöbel, 2007) The delay management problem without wait-depart-decisions is NP hard. Proof: Reduction to machine scheduling But: The delay management problem without wait-depart-decisions and without priority decisions is easy! Anita Schöbel (NAM) Optimization in public transport March / 75

60 Delay management Research in delay management Part 0: Assume that all wait depart and priority decisions are already fixed. Part 1: Consider changing activities. wait wait station 1 station 1 station 2 station 2 departure departure change station 1 station 1 station 2 station 2 departure departure wait wait station 1 station 1 station 2 station 2 wait departure wait departure wait wait station 1 station 1 station 2 station 2 departure departure change station 1 station 1 station 2 station 2 departure departure wait wait station 1 station 1 station 2 station 2 wait departure wait departure changing activity: delete or not? Part 2: Consider changing activities and headway activities. wait wait station 1 station 1 station 2 station 2 departure departure change station 1 station 1 station 2 station 2 departure departure wait wait station 1 station 1 station 2 station 2 wait departure wait departure headway constraint: which one? changing activity: delete or not? Anita Schöbel (NAM) Optimization in public transport March / 75

61 Delay management Part 0: Delay management with fixed decisions problem is polynomially solvable, even if network contains circles equivalent to feasible differential problem solve by Critical Path Methods (CPM) of project planning station 1 station 1 wait wait station 1 departure station 1 departure station 2 change station 2 wait wait station 2 departure station 2 departure station 1 station 1 station 2 station 2 wait departure wait departure Anita Schöbel (NAM) Optimization in public transport March / 75

62 Delay management Part 0: Delay management with fixed decisions problem is polynomially solvable, even if network contains circles equivalent to feasible differential problem solve by Critical Path Methods (CPM) of project planning station 1 station 1 wait wait station 1 departure station 1 departure station 2 change station 2 wait wait station 2 departure station 2 departure station 1 station 1 station 2 station 2 wait departure wait departure x 0 := max{π 0 + delay 0 }, x i = max{π i + delay i, max a=(j,i) x j + L a } Anita Schöbel (NAM) Optimization in public transport March / 75

63 Delay management Part 1: Consider changing activities station 1 wait station 1 departure station 2 wait station 2 departure change station 1 wait station 1 departure station 2 wait station 2 departure station 1 station 1 station 2 station 2 wait departure wait departure changing activity: delete or not? Anita Schöbel (NAM) Optimization in public transport March / 75

64 Delay management Part 1: Consider changing activities (DM cap) min f (x, z) = i E w i (x i π i ) + a A transfer w a Tz a such that x i π i + d i for all i E x j x i L a for all a = (i, j) A wait A Mz a + x j x i L a for all a = (i, j) A transfer x i IN for all i E z a {0, 1} for all a A transfer Anita Schöbel (NAM) Optimization in public transport March / 75

65 Delay management Part 1: The never-meet property Definition The delay management problem has the never-meet property if 1 for each source delay d j > 0: suc(j, A) is an out-tree 2 for each pair of source delays d j, d k > 0: suc(j, A) suc(k, A) =. Consequence: Successors of removed changing activities are always on time. Anita Schöbel (NAM) Optimization in public transport March / 75

66 Delay management Part 1: Good news Theorem The delay management problem can be solved in O( A ) time if the never-meet property holds. Idea of Algorithm: Decompose in independent subproblems and compose source delay a6 a7 a8 a1 a2 a5 a9 a3 a4 subproblem belonging to a2 Anita Schöbel (NAM) Optimization in public transport March / 75

67 Delay management Part 1: Good news 1 The delay management problem can be solved in O( A ) time if the never-meet property holds. 2 The never-meet property is in practice often almost satisfied. 3 In these cases, Branch & Bound for general case is practically possible. Anita Schöbel (NAM) Optimization in public transport March / 75

68 Delay management Part 2: Consider changing and headway activities station 1 wait station 1 departure station 2 wait station 2 departure change station 1 wait station 1 departure station 2 wait station 2 departure station 1 station 1 station 2 wait departure wait station 2 departure headway constraint: which one? changing activity: delete or not? Anita Schöbel (NAM) Optimization in public transport March / 75

69 Delay management Part 2: Consider changing and headway activities (DM cap) min f (x, z) = i E w i (x i π i ) + a A transfer w a Tz a such that x i π i + d i for all i E x j x i L a + d a for all a = (i, j) A wait A Mz a + x j x i L a for all a = (i, j) A transfer Mg ij + x j x i H ij (i, j) A head g ij + g ji = 1 (i, j) A head x i IN for all i E z a {0, 1} for all a A transfer g ij {0, 1} for all (i, j) A head Anita Schöbel (NAM) Optimization in public transport March / 75

70 Delay management Part 2: How large can the delay of an event become? In the uncapacitated case: Delays are reduced along every path. Hence: M =largest source delay is large enough in the IP formulation. Anita Schöbel (NAM) Optimization in public transport March / 75

71 Delay management Part 2: How large can the delay of an event become? In the uncapacitated case: Delays are reduced along every path. Hence: M =largest source delay is large enough in the IP formulation. Now: Delays can increase due to headway constraints Anita Schöbel (NAM) Optimization in public transport March / 75

72 Delay management Part 2: How large can the delay of an event become? In the uncapacitated case: Delays are reduced along every path. Hence: M =largest source delay is large enough in the IP formulation. Now: Delays can increase due to headway constraints Theorem (Schachtebeck and Schöbel,2010) M := max i E d i + a A train d a + is large enough in the IP formulation. Unfortunately: too large... (i,j) A head :π i >π j π i π j + L ij. Anita Schöbel (NAM) Optimization in public transport March / 75

73 Delay management Part 2: Analyzing the headway constraints Also the headway constraints can delay punctual trains. They come in pairs: forward headway constraints: same priority as in the original timetable π backward headway constraints: change the priority Anita Schöbel (NAM) Optimization in public transport March / 75

74 Delay management Part 2: Analyzing the headway constraints Also the headway constraints can delay punctual trains. They come in pairs: forward headway constraints: same priority as in the original timetable π backward headway constraints: change the priority Questions: Are all headway constraints needed? Answer: (Apart from the straightforward reduction) yes! Anita Schöbel (NAM) Optimization in public transport March / 75

75 Delay management Part 2: Analyzing the headway constraints Also the headway constraints can delay punctual trains. They come in pairs: forward headway constraints: same priority as in the original timetable π backward headway constraints: change the priority Questions: Are all headway constraints needed? Answer: (Apart from the straightforward reduction) yes! More precisely: Can all headway activities delay punctual trains? Answer: No! Backward headway activities can t. Anita Schöbel (NAM) Optimization in public transport March / 75

76 Delay management Part 2: Analyzing the headway constraints A π := A train A change { } (i, j) A head : π i < π j E mark := suc(j, A π ) suc(j, A π ). j E:d j >0 a=(i,j) A:d a>0 Theorem (Schachtebeck and Schöbel, 2010) There exists an optimal solution such that x i = π i for all i E mark. If w i > 0 for all i E arr then all optimal solutions satisfy this property. Anita Schöbel (NAM) Optimization in public transport March / 75

77 Delay management Part 2: Analyzing the headway constraints A π := A train A change { } (i, j) A head : π i < π j E mark := suc(j, A π ) suc(j, A π ). j E:d j >0 a=(i,j) A:d a>0 Theorem (Schachtebeck and Schöbel, 2010) There exists an optimal solution such that x i = π i for all i E mark. If w i > 0 for all i E arr then all optimal solutions satisfy this property. In short: Backward headway constraints cannot delay punctual trains. Anita Schöbel (NAM) Optimization in public transport March / 75

78 Delay management Part 2: Analyzing the headway constraints Consequences: 1 Reduction of the network in average: solution time including preprocessing is 19.1 % of solution time without preprocessing In 98% of all cases: reduction to less than 50%. In 30% of all cases: reduction to less than 10%. 2 Never-meet property: Definition The delay management problem has the never-meet property if 1 for each source delay d j > 0: suc(j, A π ) is an out-tree 2 for each pair of source delays d j, d k > 0: suc(j, A π ) suc(k, A π ) =. Successors of removed changing activities are always on time! Anita Schöbel (NAM) Optimization in public transport March / 75

79 Delay management Part 2: Heuristics Idea: station 1 wait station 1 departure station 2 wait station 2 departure change station 1 wait station 1 departure station 2 wait station 2 departure station 1 station 1 station 2 wait departure wait station 2 departure headway constraint: which one? changing activity: delete or not? Anita Schöbel (NAM) Optimization in public transport March / 75

80 Delay management Part 2: Heuristics Idea: station 1 wait station 1 departure station 2 wait station 2 departure change station 1 wait station 1 departure station 2 wait station 2 departure station 1 station 1 station 2 station 2 wait departure wait departure headway constraint: decision made. changing activity: delete or not? Anita Schöbel (NAM) Optimization in public transport March / 75

81 Delay management Part 2: Heuristics Idea: station 1 wait station 1 departure station 2 wait station 2 departure change station 1 wait station 1 departure station 2 wait station 2 departure station 1 station 1 station 2 station 2 wait departure wait departure NICE constraints changing activity: delete or not? After fixing the priorities we can only have to consider changing activities (part 1) Anita Schöbel (NAM) Optimization in public transport March / 75

82 Delay management Part 2: Heuristics Idea: Use heuristics that make use of the solution of the problem without headway constraints Four heuristics (Schachtebeck and Schöbel (2010)) First Scheduled First Served First Re-scheduled First Served First Re-scheduled First Served with early fixing First scheduled First served with priority fixing Anita Schöbel (NAM) Optimization in public transport March / 75

83 Current Research Directions Plan of talk 1 Line planning 2 Timetabling 3 Delay management 4 Current Research Directions Integrating Passengers Routes Integrating robustness issues Integrating the Planning Stages 5 The end... Anita Schöbel (NAM) Optimization in public transport March / 75

84 Current Research Directions What is left to do? Anita Schöbel (NAM) Optimization in public transport March / 75

85 Current Research Directions What is left to do? A lot of challenging questions! Integrating the routes of the passengers in the optimization Integrating robustness issues Integration of the planning stages Anita Schöbel (NAM) Optimization in public transport March / 75

86 Current Research Directions Integrating Passengers Routes Plan of talk 1 Line planning 2 Timetabling 3 Delay management 4 Current Research Directions Integrating Passengers Routes Integrating robustness issues Integrating the Planning Stages 5 The end... Anita Schöbel (NAM) Optimization in public transport March / 75

87 Current Research Directions Integrating Passengers Routes Integrating the Passengers Routes Anita Schöbel (NAM) Optimization in public transport March / 75

88 Current Research Directions Integrating Passengers Routes Integrating the Passengers Routes Transportation System Modal Split Passengers Paths Anita Schöbel (NAM) Optimization in public transport March / 75

89 Current Research Directions Integrating Passengers Routes Integrating the Passengers Routes Transportation System Modal Split Passengers Paths This holds for all planning stages! Anita Schöbel (NAM) Optimization in public transport March / 75

90 Current Research Directions Integrating Passengers Routes Integrating the Passengers Routes Transportation System Modal Split Passengers Paths This holds for all planning stages! Question: With which should we start? Anita Schöbel (NAM) Optimization in public transport March / 75

91 Current Research Directions Integrating Passengers Routes Consequence The Chicken or the Egg Causality Dilemma Anita Schöbel (NAM) Optimization in public transport March / 75

92 Current Research Directions Integrating Passengers Routes Solution Idea Do both at the same time! Anita Schöbel (NAM) Optimization in public transport March / 75

93 Current Research Directions Integrating Passengers Routes Solution Idea Do both at the same time! Transportation System Modal Split Passengers Paths Anita Schöbel (NAM) Optimization in public transport March / 75

94 Current Research Directions Integrating Passengers Routes Solution Idea Do both at the same time! Transportation System AND Modal Split Passengers Paths Anita Schöbel (NAM) Optimization in public transport March / 75

95 Current Research Directions Integrating Passengers Routes Example: Line Planning Most models use fixed passengers weights on the edges. But: Passengers choose their paths dependent on the line concept, we want to compute! Anita Schöbel (NAM) Optimization in public transport March / 75

96 Current Research Directions Integrating Passengers Routes Example: Line Planning Most models use fixed passengers weights on the edges. But: Passengers choose their paths dependent on the line concept, we want to compute! More precisely: Passengers choose a specific path, if it is short, it has few transfers, it has a high frequency. but these properties depend on the line concept to be determined! Anita Schöbel (NAM) Optimization in public transport March / 75

97 Current Research Directions Integrating Passengers Routes Integrating routing decisions in line planning Design lines in such a way that the sum of the best travel times of all passengers is minimal. I.e.: we have also to find the paths for the passengers. Anita Schöbel (NAM) Optimization in public transport March / 75

98 Current Research Directions Integrating Passengers Routes Integrating routing decisions in line planning Design lines in such a way that the sum of the best travel times of all passengers is minimal. I.e.: we have also to find the paths for the passengers. Travel time model Find lines, frequencies and paths P uv for all OD-pairs (u, v) minimizing min u,v V and keeping a budget constraint. W uv Travel(P uv ) where Travel(P)=Travel time for P + k number of transfers of P k: penalty for a transfer Anita Schöbel (NAM) Optimization in public transport March / 75

99 Current Research Directions Integrating Passengers Routes Travel time model: Complexity In general NP-hard. Anita Schöbel (NAM) Optimization in public transport March / 75

100 Current Research Directions Integrating Passengers Routes Travel time model: Complexity In general NP-hard. For only one OD-pair Schmidt and Schöbel 2014, Schmidt 2014: Surprisingly still NP-hard! Can be solved in pseudo-polynomial time if no-line-twice property holds by reduction to a resource-constrained shortest path problem. Anita Schöbel (NAM) Optimization in public transport March / 75

101 Current Research Directions Integrating Passengers Routes Travel time model: Complexity In general NP-hard. For only one OD-pair Schmidt and Schöbel 2014, Schmidt 2014: Surprisingly still NP-hard! Can be solved in pseudo-polynomial time if no-line-twice property holds by reduction to a resource-constrained shortest path problem. Further restrictions are necessary to obtain polynomial solvability! Anita Schöbel (NAM) Optimization in public transport March / 75

102 Current Research Directions Integrating Passengers Routes Travel time model: Complexity Schmidt and Schöbel 2015, Schmidt 2014 For linear networks: line transfer one OD-pair same origin general OD-pairs speed penalties equal no penalties poly poly poly equal equal penalties poly pseudo-poly NP-hard equal station-independent pseudo-poly NP-hard NP-hard equal arbitrary pseudo-poly NP-hard NP-hard arbitrary no penalties pseudo-poly pseudo-poly pseudo-poly arbitrary equal penalties pseudo-poly NP-hard NP-hard arbitrary station independent pseudo-poly NP-hard NP-hard arbitrary arbitrary pseudo-poly NP-hard NP-hard Anita Schöbel (NAM) Optimization in public transport March / 75

103 Current Research Directions Integrating Passengers Routes Solution approach: Change & go graph s7 l1 s1 s2 s3 s4 l2 l3 s5 s6 s8 Anita Schöbel (NAM) Optimization in public transport March / 75

104 Current Research Directions Integrating Passengers Routes Solution approach: Change & go graph l1 s1,l1 s2,l1 s3,l1 s4,l1 Anita Schöbel (NAM) Optimization in public transport March / 75

105 Current Research Directions Integrating Passengers Routes Solution approach: Change & go graph s1,l1 s2,l1 s3,l1 s4,l1 l1 l2 l3 s1,l2 s4,l2 s5,l2 s6,l2 s7,l3 s3,l3 s6,l3 s8,l3 Anita Schöbel (NAM) Optimization in public transport March / 75

106 Current Research Directions Integrating Passengers Routes Solution approach: Change & go graph s1,l1 s2,l1 s3,l1 s4,l1 l1 l2 l3 s1,l2 s4,l2 s5,l2 s6,l2 s7,l3 s3,l3 s6,l3 s8,l3 Anita Schöbel (NAM) Optimization in public transport March / 75

107 Current Research Directions Integrating Passengers Routes Solution approach: Change & go graph l1 l2 s1,l1 s2,l1 s3,l1 s4,l1 l3 s1, 0 s1,l2 s4,l2 s5,l2 s6,l2 s7,l3 s3,l3 s6,l3 s8,l3 Anita Schöbel (NAM) Optimization in public transport March / 75

108 Current Research Directions Integrating Passengers Routes Solution approach: Change & go graph s2,0 s3,0 l1 l2 s1,l1 s2,l1 s3,l1 s4,l1 l3 s1, 0 s4,0 s1,l2 s4,l2 s5,l2 s6,l2 s5,0 s7,l3 s3,l3 s6,l3 s8,l3 s7,0 s6,0 s8,0 Anita Schöbel (NAM) Optimization in public transport March / 75

109 Current Research Directions Integrating Passengers Routes Travel time model: IP-Formulation = min u,v V a A W uvc a xuv a s.t. xuv a y l 0, u, v V, l L, a l Θx u1 v 1 b u1 v Θx ur v r b ur v r l L y lcost l B xuv, a y l {0, 1} Anita Schöbel (NAM) Optimization in public transport March / 75

110 Current Research Directions Integrating Passengers Routes Travel time model: IP-Formulation = min u,v V a A W uvc a xuv a s.t. xuv a y l 0, u, v V, l L, a l Θx u1 v 1 b u1 v Θx ur v r b ur v r l L y lcost l B xuv, a y l {0, 1} Anita Schöbel (NAM) Optimization in public transport March / 75

111 Current Research Directions Integrating Passengers Routes Travel time model: IP-Formulation = min u,v V a A W uvc a xuv a s.t. xuv a y l 0, u, v V, l L, a l Θx u1 v 1 b u1 v Θx ur v r b ur v r l L y lcost l B xuv, a y l {0, 1} Structure: one block for each OD-pair u, v and a y-variable block. Use decomposition approaches for solving the model Schöbel and Scholl (2007) Use column generation to find passengers routes for minimizing the riding time Borndörfer and Pfetsch (2009) for minimizing the number of transfers Harbering (2015) Anita Schöbel (NAM) Optimization in public transport March / 75

112 Current Research Directions Integrating Passengers Routes Integrating the routing decisions Flow constraints can be added to most programs to integrate the passengers routing. Line planning: travel time models Timetabling: first approaches Schmidt (2012), Schmidt and Schöbel (2015) Delay management: Dollevoet, Huisman, Schmidt, Schöbel (2012), Schmidt (2013), Schmidt (2014) But the questions are: efficient approaches? Heuristics? How much can we gain by adding the routing decision? Challenging questions when capacity constraints are present since passengers are not interested in minimizing sums of travel times! Anita Schöbel (NAM) Optimization in public transport March / 75

113 Current Research Directions Integrating robustness issues Plan of talk 1 Line planning 2 Timetabling 3 Delay management 4 Current Research Directions Integrating Passengers Routes Integrating robustness issues Integrating the Planning Stages 5 The end... Anita Schöbel (NAM) Optimization in public transport March / 75

114 Current Research Directions Integrating robustness issues Why Robustness? Small changes... Anita Schöbel (NAM) Optimization in public transport March / 75

115 Current Research Directions Integrating robustness issues Why Robustness? Small changes may change a solution a lot! Anita Schöbel (NAM) Optimization in public transport March / 75

116 Current Research Directions Integrating robustness issues We need robust solutions! But: What does robust mean? Anita Schöbel (NAM) Optimization in public transport March / 75

117 Current Research Directions Integrating robustness issues We need robust solutions! But: What does robust mean? Answers are given by stochastic and robust optimization. Classic robust optimization requires feasibility in the worst case. Does this make sense for planning of public transport? Anita Schöbel (NAM) Optimization in public transport March / 75

118 Current Research Directions Integrating robustness issues We need robust solutions! But: What does robust mean? Answers are given by stochastic and robust optimization. Classic robust optimization requires feasibility in the worst case. Does this make sense for planning of public transport? Examples: Line planning: the frequencies offered have to be high enough to allow (rarely) special events. Timetabling: Every departure time at every station has always to be correct. Consequence: We need less conservative concepts for robust planning of public transport. Anita Schöbel (NAM) Optimization in public transport March / 75

119 Current Research Directions Integrating robustness issues Suitable concepts Light Robustness applied to timetabling Fischetti and Monaci (2009) Idea: Relax the constraints and require a certain nominal quality. In many cases same complexity as original problem Schöbel (2015) Recovery Robustness applied to timetabling and line planning Liebchen et al (2009) Idea: Look for a solution together with a recovery strategy that can make the solution feasible for any possible scenario. Hard to compute Variation: Recovery to Optimality applied to timetabling Goerigk and Schöbel (2011, 2015) Idea: Minimize the costs to change the solution to an optimal one median version: in the average center version: in the worst case Can be computed scenario-based by location theory Anita Schöbel (NAM) Optimization in public transport March / 75

120 Current Research Directions Integrating robustness issues Robustness concepts for aperiodic timetabbling Goerigk and Schöbel (2013) 3e e e+09 nominal strict light buffer recovery 2 l1 center l1 median centroid 2.1e e e Anita Schöbel (NAM) Optimization in public transport March / 75

121 Current Research Directions Integrating robustness issues Robustness concepts for aperiodic timetabbling Goerigk and Schöbel (2013) nominal strict light buffer recovery 2 l1 center l1 median centroid Anita Schöbel (NAM) Optimization in public transport March / 75

122 Current Research Directions Integrating robustness issues Open questions Which robustness models fit to which public transport problem? E.g. for timetabling: Find a timetable such that for each delay scenario the solution of the resulting delay management problem is good enough. How can robust solutions be efficiently computed? Simulate the effects of robust solutions using real-world settings, talks of Marc and Rolf on robustness! Anita Schöbel (NAM) Optimization in public transport March / 75

123 Current Research Directions Integrating the Planning Stages Plan of talk 1 Line planning 2 Timetabling 3 Delay management 4 Current Research Directions Integrating Passengers Routes Integrating robustness issues Integrating the Planning Stages 5 The end... Anita Schöbel (NAM) Optimization in public transport March / 75

124 Current Research Directions Integrating the Planning Stages Integrating the Planning Stages Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

125 Current Research Directions Integrating the Planning Stages Integrating the Planning Stages Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

126 Current Research Directions Integrating the Planning Stages Integrating the Planning Stages Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

127 Current Research Directions Integrating the Planning Stages Integrating the Planning Stages Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

128 Current Research Directions Integrating the Planning Stages Integrating the Planning Stages Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

129 Current Research Directions Integrating the Planning Stages Integrating the Planning Stages Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

130 Current Research Directions Integrating the Planning Stages Integrating the Planning Stages Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

131 Current Research Directions Integrating the Planning Stages Integrating the Planning Stages So far, most of the research is focused on optimizing each of the planning stages for itself. Question: Does this lead to good solutions for the whole system? Anita Schöbel (NAM) Optimization in public transport March / 75

132 Current Research Directions Integrating the Planning Stages Integrating the Planning Stages So far, most of the research is focused on optimizing each of the planning stages for itself. Question: Does this lead to good solutions for the whole system? No! The result of one stage could be a bad input for the next stage. The objective used in a former stage is maybe not a good approximation for what we really want. An integrated solution is always as least as good as optimizing stage by stage. Anita Schöbel (NAM) Optimization in public transport March / 75

133 Current Research Directions Integrating the Planning Stages What could we dream of? Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

134 Current Research Directions Integrating the Planning Stages What could we dream of? Stops/Stations Line Concept Timetable Vehicle Schedules/ Rolling Stock Planning Tariff System Crew Scheduling Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

135 Current Research Directions Integrating the Planning Stages What could we dream of? Stops/Stations and Line Concept and Timetable and Vehicle Schedules/ Rolling Stock Planning and Crew Scheduling Tariff System Disposition Anita Schöbel (NAM) Optimization in public transport March / 75

136 Current Research Directions Integrating the Planning Stages What has been done towards integration? Anita Schöbel (NAM) Optimization in public transport March / 75

137 Current Research Directions Integrating the Planning Stages What has been done towards integration? There are a few papers integrating two consecutive planning stages. Goossens (2004), Liebchen und Möhring (2008), Barber et al (2009), Claessens et al (1998), Israeli and Ceder (1995), Quak (2003), Michaelis and Schöbel (2007) Liebchen and Möhring (2006), Lindner (2000) We implemented the software and data library LinTim in which the effects of integration can be studied. Some first ideas and approaches for integrated optimization Anita Schöbel (NAM) Optimization in public transport March / 75

138 Current Research Directions Integrating the Planning Stages An example from LinTim Goerigk and Schöbel (2013) game cost-heuristic cost-exact direct passenger delay e e e e e+06 number of transfers Anita Schöbel (NAM) Optimization in public transport March / 75

139 Current Research Directions Integrating the Planning Stages An iterative approach timetabling given a line plan vehicle scheduling given a line plan and a timetable line planning Anita Schöbel (NAM) Optimization in public transport March / 75

140 Current Research Directions Integrating the Planning Stages An iterative approach vehicle scheduling given a line plan and a timetable timetabling given a line plan line planning timetabling given a line plan and vehicle schedules Anita Schöbel (NAM) Optimization in public transport March / 75

141 Current Research Directions Integrating the Planning Stages An iterative approach vehicle scheduling given a line plan and a timetable timetabling given a line plan line planning line planning given a timetable and vehicle schedules timetabling given a line plan and vehicle schedules Anita Schöbel (NAM) Optimization in public transport March / 75

142 Current Research Directions Integrating the Planning Stages An iterative approach vehicle scheduling given a line plan and a timetable timetabling given a line plan line planning line planning given a timetable and vehicle schedules timetabling given a line plan and vehicle schedules Anita Schöbel (NAM) Optimization in public transport March / 75

143 Current Research Directions Integrating the Planning Stages An iterative approach vehicle scheduling given a line plan and a timetable timetabling given a line plan line planning line planning given a timetable and vehicle schedules timetabling given a line plan and vehicle schedules line planning given vehicle schedules vehicle scheduling Anita Schöbel (NAM) Optimization in public transport March / 75

144 Current Research Directions Integrating the Planning Stages An iterative approach timetabling line planning given a timetable vehicle scheduling given a line plan and a timetable vehicle scheduling given a timetable timetabling given a line plan line planning line planning given a timetable and vehicle schedules vehicle scheduling given a line plan timetabling given a line plan and vehicle schedules line planning given vehicle schedules timetabling given vehicle schedules vehicle scheduling Anita Schöbel (NAM) Optimization in public transport March / 75

145 Current Research Directions Integrating the Planning Stages An iterative approach timetabling line planning given a timetable vehicle scheduling given a line plan and a timetable vehicle scheduling given a timetable timetabling given a line plan line planning line planning given a timetable and vehicle schedules vehicle scheduling given a line plan timetabling given a line plan and vehicle schedules line planning given vehicle schedules timetabling given vehicle schedules vehicle scheduling Anita Schöbel (NAM) Optimization in public transport March / 75

146 Current Research Directions Integrating the Planning Stages An iterative approach timetabling line planning given a timetable vehicle scheduling given a line plan and a timetable vehicle scheduling given a timetable timetabling given a line plan line planning line planning given a timetable and vehicle schedules vehicle scheduling given a line plan timetabling given a line plan and vehicle schedules line planning given vehicle schedules timetabling given vehicle schedules vehicle scheduling Anita Schöbel (NAM) Optimization in public transport March / 75

147 Current Research Directions Integrating the Planning Stages An iterative approach timetabling line planning given a timetable vehicle scheduling given a line plan and a timetable vehicle scheduling given a timetable timetabling given a line plan line planning line planning given a timetable and vehicle schedules vehicle scheduling given a line plan timetabling given a line plan and vehicle schedules line planning given vehicle schedules timetabling given vehicle schedules vehicle scheduling Anita Schöbel (NAM) Optimization in public transport March / 75

148 Current Research Directions Integrating the Planning Stages An iterative approach timetabling line planning given a timetable vehicle scheduling given a line plan and a timetable vehicle scheduling given a timetable timetabling given a line plan line planning line planning given a timetable and vehicle schedules vehicle scheduling given a line plan timetabling given a line plan and vehicle schedules line planning given vehicle schedules timetabling given vehicle schedules vehicle scheduling Anita Schöbel (NAM) Optimization in public transport March / 75

149 Current Research Directions Integrating the Planning Stages An iterative approach timetabling line planning given a timetable vehicle scheduling given a line plan and a timetable vehicle scheduling given a timetable timetabling given a line plan line planning line planning given a timetable and vehicle schedules vehicle scheduling given a line plan timetabling given a line plan and vehicle schedules line planning given vehicle schedules timetabling given vehicle schedules vehicle scheduling Anita Schöbel (NAM) Optimization in public transport March / 75

150 Current Research Directions Integrating the Planning Stages An iterative approach Properties of the Eigenmodel Name in reference to Möhring et al (1995) A lot of new challenging problems to be defines with interesting combinatorial structures Every path through the Eigenmodel represents a heuristic Anita Schöbel (NAM) Optimization in public transport March / 75

151 Current Research Directions Integrating the Planning Stages An iterative approach Properties of the Eigenmodel Name in reference to Möhring et al (1995) A lot of new challenging problems to be defines with interesting combinatorial structures Every path through the Eigenmodel represents a heuristic Research questions: Find good paths Convergence results? Finiteness? Optimality? Approximation results for special structures (exact algorithms for matroids) Application for public transport Anita Schöbel (NAM) Optimization in public transport March / 75

152 The end... You made it! 1 Line planning 2 Timetabling 3 Delay management 4 Current Research Directions Integrating Passengers Routes Integrating robustness issues Integrating the Planning Stages 5 The end... Anita Schöbel (NAM) Optimization in public transport March / 75

153 The end... There is still a lot to do... Robustness, integrating passengers paths Anita Schöbel (NAM) Optimization in public transport March / 75

154 The end... There is still a lot to do... Robustness, integrating passengers paths research started Anita Schöbel (NAM) Optimization in public transport March / 75

155 The end... There is still a lot to do... Robustness, integrating passengers paths Integration research started Anita Schöbel (NAM) Optimization in public transport March / 75

156 The end... There is still a lot to do... Robustness, integrating passengers paths research started Integration research is just about to start Anita Schöbel (NAM) Optimization in public transport March / 75

157 The end... There is still a lot to do... Robustness, integrating passengers paths research started Integration research is just about to start Indian perspective (together with Narayan Rangaraj, IIT Mumbai) objective is to maximize capacity microscopic view is needed Anita Schöbel (NAM) Optimization in public transport March / 75

158 The end... There is still a lot to do... Thank you for your attention! Anita Schöbel (NAM) Optimization in public transport March / 75