ERASMUS CENTER OF OPTIMIZATION IN PUBLIC TRANSPORT A MACROSCOPIC RAILWAY TIMETABLE RESCHEDULING APPROACH LUCAS P. VEELENTURF, MARTIN P. KIDD, VALENTINA CACCHIANI, LEO G. KROON, PAOLO TOTH L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 1 / 23
OUTLINE 1. Introduction 2. Problem description 3. Computational experiments 4. Conclusions L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 2 / 23
Introduction RAILWAY SCHEDULES In railway operations there are three major schedules Timetable Rolling stock schedule Crew schedule Problem during operations: Unexpected events make the planned resource schedules infeasible. Disruption L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 3 / 23
Introduction RAILWAY SCHEDULES In railway operations there are three major schedules Timetable Rolling stock schedule Crew schedule Problem during operations: Unexpected events make the planned resource schedules infeasible. Disruption L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 3 / 23
Introduction DISRUPTIONS Infrastructure malfunctions Rails, switches, catenary, bridges Computer problems in control centers Rolling stock breakdowns Accidents with other traffic Weather conditions Crew no shows... L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 4 / 23
Introduction THE DISRUPTION MANAGEMENT PROCESS Disruption management includes three major steps 1. Update timetable according to the disruption. 2. Reschedule rolling stock to cover the new timetable. 3. Reschedule crew to operate the rolling stock. Must be solved within seconds These steps are interdependent but solved separately Several iterations of steps 1 3 may be necessary L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 5 / 23
Introduction THE DISRUPTION MANAGEMENT PROCESS Disruption management includes three major steps 1. Update timetable according to the disruption. 2. Reschedule rolling stock to cover the new timetable. 3. Reschedule crew to operate the rolling stock. Must be solved within seconds These steps are interdependent but solved separately Several iterations of steps 1 3 may be necessary L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 5 / 23
Introduction THE DISRUPTION MANAGEMENT PROCESS Disruption management includes three major steps 1. Update timetable according to the disruption. 2. Reschedule rolling stock to cover the new timetable. 3. Reschedule crew to operate the rolling stock. Must be solved within seconds These steps are interdependent but solved separately Several iterations of steps 1 3 may be necessary L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 5 / 23
Introduction EXAMPLE OF A DISRUPTION L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 6 / 23
Introduction EXAMPLE OF A DISRUPTION L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 7 / 23
Introduction EXAMPLE OF A DISRUPTION Updated timetable: L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 8 / 23
Problem description THE RAILWAY TIMETABLE RESCHEDULING PROBLEM Reschedule the trains by taking into account the reduced capacity such that: As many trains as possible can still run. As few trains as possible will incur a delay. In this research the only allowed modifications to the timetable are: Cancelling trains. Delaying trains. L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 9 / 23
Problem description THE RAILWAY TIMETABLE RESCHEDULING PROBLEM Reschedule the trains by taking into account the reduced capacity such that: As many trains as possible can still run. As few trains as possible will incur a delay. In this research the only allowed modifications to the timetable are: Cancelling trains. Delaying trains. L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 9 / 23
Problem description MATHEMATICAL FORMULATION Sets: E train set of timetabled events e (departures/arrivals) T set of trains t t e corresponding train of event e Parameters: q e scheduled time of event e d e maximum allowed delay of event e µ e cost of delaying event e λ t cost of cancelling train t Variables: x e rescheduled time of event e y t binary variable (1 if train t is cancelled) L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 10 / 23
Problem description MATHEMATICAL FORMULATION Minimize λ t y t + µ e (x e q e ) e E t T subject to x e q e 0 e E x e q e (1 y te )d e y t {0, 1} x e N e E t T e E L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 11 / 23
Problem description RESOURCES We consider three types of resources which a train may occupy at a certain moment: open track sections, stations/junctions and rolling stock compositions. Open track section Consist of a number of parallel tracks Station Consist of a number of parallel tracks Rolling stock composition Consist of a number of intercity or regional rolling stock units L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 12 / 23
Problem description RESOURCES We consider three types of resources which a train may occupy at a certain moment: open track sections, stations/junctions and rolling stock compositions. Open track section Consist of a number of parallel tracks Station Consist of a number of parallel tracks Rolling stock composition Consist of a number of intercity or regional rolling stock units L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 12 / 23
Problem description ASSUMPTIONS Open track sections from each open track section on its route a track should be assigned to the train. Between trains on the same track a certain minimum headway is considered Each track can be used in each direction Stations From each station on its route a track should be assigned to the train. Between trains on the same track a certain minimum headway is considered. Each track can be used in each direction Each track can be reached by trains arriving from the open track section L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 13 / 23
Problem description ASSUMPTIONS Open track sections from each open track section on its route a track should be assigned to the train. Between trains on the same track a certain minimum headway is considered Each track can be used in each direction Stations From each station on its route a track should be assigned to the train. Between trains on the same track a certain minimum headway is considered. Each track can be used in each direction Each track can be reached by trains arriving from the open track section L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 13 / 23
Problem description ASSUMPTIONS Rolling stock compositions To each train a rolling stock composition of the corresponding type should be assigned. Between subsequential use of the same composition by different trains a certain turn around time should be considered To model these assumptions we construct an Event-Activity network. The events correspond with the timetable events and with inventories of the resources. The activities correspond with the subsequential use of the same resource. L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 14 / 23
Problem description ASSUMPTIONS Rolling stock compositions To each train a rolling stock composition of the corresponding type should be assigned. Between subsequential use of the same composition by different trains a certain turn around time should be considered To model these assumptions we construct an Event-Activity network. The events correspond with the timetable events and with inventories of the resources. The activities correspond with the subsequential use of the same resource. L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 14 / 23
Problem description MATHEMATICAL FORMULATION Sets: A: set of activities a = (e, f ) A + (e): A collection of in-activity subsets for event e E C A + (e) with C A: A subset of activities into event e E associated with a single type of resource. A (e): A collection of out-activity subsets for event e E C A (e) with C A: A subset of activities out of e E associated with a single type of resource Parameters L a : minimum duration of activity a Parameters z a : binary decision variable such that z a = 1 if activity a A is selected L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 15 / 23
Problem description MATHEMATICAL FORMULATION Additional constraints: z a + y te = 1 e E train, C A + (e) a C z a + y te 1 e E train, C A (e) a C z a i e e E inv, C A (e) a C x f x e + M(1 z a ) L a z a {0, 1} a = (e, f ) A a A L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 16 / 23
Computational experiments COMPUTATIONAL EXPERIMENTS L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 17 / 23
Computational experiments COMPUTATIONAL EXPERIMENTS The considered part of the Dutch railway network consists of: 26 stations/junctions 27 open track sections 3 single tracked 21 double tracked 1 with three tracks 2 with four tracks 16 train series (mostly twice an hour) 61 rolling stock compositions L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 18 / 23
Computational experiments COMPUTATIONAL EXPERIMENTS The considered disruptions contain a blockage of a number of tracks of one open track section For each of the 27 open track sections we consider 60 disruptions (30 for single tracked sections) 1 track blocked 9:00-11:00 1 track blocked 9:01-11:01... 1 track blocked 9:29-11:29 all tracks blocked 9:00-11:00 all tracks blocked 9:01-11:01... all tracks blocked 9:29-11:29 The cases are solved with CPLEX 12.5 L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 19 / 23
Computational experiments COMPUTATIONAL EXPERIMENTS The considered disruptions contain a blockage of a number of tracks of one open track section For each of the 27 open track sections we consider 60 disruptions (30 for single tracked sections) 1 track blocked 9:00-11:00 1 track blocked 9:01-11:01... 1 track blocked 9:29-11:29 all tracks blocked 9:00-11:00 all tracks blocked 9:01-11:01... all tracks blocked 9:29-11:29 The cases are solved with CPLEX 12.5 L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 19 / 23
Computational experiments COMPUTATIONAL EXPERIMENTS The considered disruptions contain a blockage of a number of tracks of one open track section For each of the 27 open track sections we consider 60 disruptions (30 for single tracked sections) 1 track blocked 9:00-11:00 1 track blocked 9:01-11:01... 1 track blocked 9:29-11:29 all tracks blocked 9:00-11:00 all tracks blocked 9:01-11:01... all tracks blocked 9:29-11:29 The cases are solved with CPLEX 12.5 L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 19 / 23
Computational experiments COMPUTATIONAL EXPERIMENTS Minutes of delay allowed Cancelled minutes Computation time (s) Min Avg Max Min Avg Max Complete disruption 0 28 385.29 947 5 6.53 12 5 28 361.00 887 7 8.68 15 One track blocked no balancing 0 0 155.70 487 6 8.78 22 5 0 99.33 335 8 12.50 140 One track blocked with balancing 0 0 166.19 487 6 9.51 23 5 0 107.80 420 8 14.24 196 L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 20 / 23
Computational experiments COMPUTATIONAL EXPERIMENTS Minutes of delay allowed Delayed trains Total maximum delay Min Avg Max Min Avg Max Complete disruption 0 0 0.28 3 0 1.68 35 5 0 2.70 23 0 7.09 65 One track blocked no balancing 0 0 0.19 3 0 2.28 51 5 0 5.55 27 0 15.18 85 One track blocked with balancing 0 0 0.15 3 0 1.59 30 5 0 5.10 21 0 13.69 72 L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 21 / 23
Conclusions CONCLUSION We developed a solution approach for the macroscopic railway timetable rescheduling problem in case of large scale disruptions. Results demonstrate that the approach can find optimal solutions in reasonable computation times. Not presented today, but already implemented: rerouting of trains. Further research must be made in combining this with microscopic timetable rescheduling. L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 22 / 23
Conclusions THANKS FOR YOUR ATTENTION Questions/Remarks? L.P. Veelenturf (lveelenturf@rsm.nl) Rotterdam School of Management, Erasmus University October 6, 2013 23 / 23