Vehicle Rotation Planning for Intercity Railways

Transcription

1 Vehicle Rotation Planning for Intercity Railways Markus Reuther ** Joint work with Ralf Borndörfer, Thomas Schlechte and Steffen Weider Zuse Institute Berlin May 24, 2011 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32

2 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Outline 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

3 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

4 Introduction close cooperation with Deutsche Bahn Fernverkehr AG DB Fernverkehr AG operates trains in Europe per day well known products: ICE, IC/EC We develop an optimization module for the vehicle resources of DB Fernverkehr. Current state: We successfully modeled and implemented most of all known requirements and we are able to solve a huge subset of instances given by DB Fernverkehr. Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32

5 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Motivation Figure: ICx mega deal ( c SPIEGEL ONLINE GmbH)

6 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 DB Fernverkehr AG passengers Mio passengers/day 0.3 Mio transport service provided Mio km ICE traction units 252 IC/EC locomotives 458 IC/EC passenger cars 3108 stations and stops 8000 Table: Facts (2009) Figure: ICE network 2009

7 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

8 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Problem Given planning horizon: cyclic standard week timetabled trips: trains basic units of rail cars: vehicle groups for each train: possible vehicle configurations Thu Fri Wed Tue standard week Mon Sun Sat

9 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Problem Given planning horizon: cyclic standard week timetabled trips: trains basic units of rail cars: vehicle groups for each train: possible vehicle configurations Figure: Timetabled train ( c expired)

10 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Problem Given planning horizon: cyclic standard week timetabled trips: trains basic units of rail cars: vehicle groups for each train: possible vehicle configurations 401 (ICE I) 402 (ICE II) 415 (ICE T)

11 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Problem Given planning horizon: cyclic standard week timetabled trips: trains basic units of rail cars: vehicle groups for each train: possible vehicle configurations <401> <402#402> <415#415#415> <415#402#415>

12 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Problem Given planning horizon: cyclic standard week timetabled trips: trains basic units of rail cars: vehicle groups for each train: possible vehicle configurations <401> <402#402> <415#415#415> <415#402#415> (Main) problem Assign exactly one vehicle configuration to each timetabled trip. Each used vehicle group must rotate in a feasible rotation. Minimize the overall costs.

13 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Vehicle rotation planning for DB Fernverkehr AG (Main) constraints ensure feasible configuration assignment for each trip ensure feasible rotations for individual vehicles ensure feasible maintenance intervals for vehicles ensure feasible capacities for service locations Objective minimize vehicle cost minimize deadhead cost minimize maintenance cost minimize violating planning values for turn times (robustness) maximize regularity (Olga s talk)

14 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Maintenance constraints T-100 (short-term inspection) distance cumulative soft limit: x km hard limit: y km x km Frist (long-term inspection) time cumulative lower limit: x days upper limit: y days x days Tanken (refuel), Nachschau (inspection), Entsorgung (waste disposal), Versorgung (supply)...

15 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

16 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (raw) timetabled trips T, vehicle groups F, vehicle configurations C hypergraph H = (V, A) (cyclic, directed) node v V : a timetabled trip t T driven with a vehicle f F hyperarc a A: possible connection of multiple nodes with a configuration c C H is very dense almost complete (no timelines for idling/parking) Thu Fri Wed Tue standard week Mon Sun Sat

17 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (raw) timetabled trips T, vehicle groups F, vehicle configurations C hypergraph H = (V, A) (cyclic, directed) node v V : a timetabled trip t T driven with a vehicle f F hyperarc a A: possible connection of multiple nodes with a configuration c C H is very dense almost complete (no timelines for idling/parking)

18 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (raw) timetabled trips T, vehicle groups F, vehicle configurations C hypergraph H = (V, A) (cyclic, directed) node v V : a timetabled trip t T driven with a vehicle f F hyperarc a A: possible connection of multiple nodes with a configuration c C H is very dense almost complete (no timelines for idling/parking) (only trivial configurations)

19 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (raw) timetabled trips T, vehicle groups F, vehicle configurations C hypergraph H = (V, A) (cyclic, directed) node v V : a timetabled trip t T driven with a vehicle f F hyperarc a A: possible connection of multiple nodes with a configuration c C H is very dense almost complete (no timelines for idling/parking) (only trivial configurations)

20 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (raw) timetabled trips T, vehicle groups F, vehicle configurations C hypergraph H = (V, A) (cyclic, directed) node v V : a timetabled trip t T driven with a vehicle f F hyperarc a A: possible connection of multiple nodes with a configuration c C H is very dense almost complete (no timelines for idling/parking) (only trivial configurations)

21 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (raw) timetabled trips T, vehicle groups F, vehicle configurations C hypergraph H = (V, A) (cyclic, directed) node v V : a timetabled trip t T driven with a vehicle f F hyperarc a A: possible connection of multiple nodes with a configuration c C H is very dense almost complete (no timelines for idling/parking) 1 rotation 1 vehicle (only trivial configurations)

22 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (raw) timetabled trips T, vehicle groups F, vehicle configurations C hypergraph H = (V, A) (cyclic, directed) node v V : a timetabled trip t T driven with a vehicle f F hyperarc a A: possible connection of multiple nodes with a configuration c C H is very dense almost complete (no timelines for idling/parking) 1 rotation 2 vehicles (only trivial configurations)

23 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (raw) timetabled trips T, vehicle groups F, vehicle configurations C hypergraph H = (V, A) (cyclic, directed) node v V : a timetabled trip t T driven with a vehicle f F hyperarc a A: possible connection of multiple nodes with a configuration c C H is very dense almost complete (no timelines for idling/parking) 2 rotations 3 vehicles (only trivial configurations)

24 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (detailed) trip 1 trip 3 trip 4 trip 2 trip 5 possible trivial and non-trivial vehicle configurations for each trip

25 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (detailed) trip 1 A1 trip 3 A A2 A3 trip 4 A1 trip 2 A2 A1 A2 B trip 5 B nodes departure and arrival for each vehicle of a trip

26 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (detailed) trip 1 A1 trip 3 A A2 A3 trip 4 A1 trip 2 A2 A1 A2 B trip 5 B trips with trivial configurations (single traction)

27 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (detailed) trip 1 A1 trip 3 A A2 A3 trip 4 A1 trip 2 A2 A1 A2 B trip 5 B trips with non-trivial configurations (double traction)

28 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (detailed) trip 1 A1 trip 3 A A2 A3 trip 4 A1 trip 2 A2 A1 A2 B trip 5 B trips with non-trivial configurations (triple traction)

29 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (detailed) trip 1 A1 trip 3 A A2 A3 trip 4 A1 trip 2 A2 A1 A2 B trip 5 B connections with non-trivial configurations without coupling

30 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (detailed) trip 1 A1 trip 3 A A2 A3 trip 4 A1 trip 2 A2 A1 A2 B trip 5 B connections with trivial configurations without coupling

31 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (detailed) trip 1 A1 trip 3 A A2 A3 trip 4 A1 trip 2 A2 A1 A2 B trip 5 B connections with trival configurations with coupling

32 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (detailed) trip 1 A1 trip 3 A A2 A3 trip 4 A1 trip 2 A2 A1 A2 B trip 5 B connection with coupling in between (currently not implemented)

33 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Graph model (detailed) trip 1 A1 trip 3 A A2 A3 trip 4 A1 trip 2 A2 A1 A2 B trip 5 B hypergraph

34 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Train composition is much more complicated... Incorporate: configuration dependent turn times conservation of trunk (and branch) vehicles avoiding blocking of vehicles after/before de-/coupling rules for positions of individual vehicles in configurations rules for orientations of individual vehicles in configurations (orientation is first or second class in front) rules for regularity (e.g. Wagenstandanzeiger) Figure: car position indicator

35 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Maintenance graph model maintenance possibilities are modeled by replenishment arcs only trivial configurations becoming maintained... a m m v 1 v 2... a = (v 1, v 2 ) Figure: Maintenance graph model v 1, v 2 V a A m: maintenance location

36 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Maintenance graph model T-100 (short-term inspection) distance cumulative soft limit: x km hard limit: y km x km Frist (long-term inspection) time cumulative lower limit: x days upper limit: y days x days without maintenance

37 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Maintenance graph model T-100 (short-term inspection) distance cumulative soft limit: x km hard limit: y km x km Frist (long-term inspection) time cumulative lower limit: x days upper limit: y days x days with maintenance

38 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

39 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Recapitulation Hypergraph H = (V, A) Set of timetabled trips T Set of vehicle groups F Set of vehicle configurations C

40 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Variables Figure: Maintenance arc and resource flow x am m... w a + m wa m... v 1 x a, w a v 2 x a {0, 1} x am {0, 1} [ ] w a 0, max f F Lf a A (hyperarcs) a m A (replenishment arcs) a = (v 1, v 2 ) V V x Q A is the vehicle hyperflow and w Q V V is the resource flow

41 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Mixed integer program (only main structure) min c ax a a A (objective) x a = 1 t T (covering) a A(t) x a x a = 0 v V (inflow) a δ + (v) a A(v) x a x a = 0 v V (outflow) a δ (v) a A(v) w (v,w) L x a 0 (v, w) V V (coupling) a A(v,w) w (v,w) w (w,v) r a x a = 0 v V (resource flow) (v,w) δ (v) (w,v) δ + (v) a δ (v)... (miscellaneous)

42 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Mixed integer program (only main structure) min c ax a a A (objective) x a = 1 t T (covering) a A(t) x a x a = 0 v V (inflow) a δ + (v) a A(v) x a x a = 0 v V (outflow) a δ (v) a A(v) w (v,w) L x a 0 (v, w) V V (coupling) a A(v,w) w (v,w) w (w,v) r a x a = 0 v V (resource flow) (v,w) δ (v) (w,v) δ + (v) a δ (v)... (miscellaneous)

43 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Mixed integer program (only main structure) min c ax a a A (objective) x a = 1 t T (covering) a A(t) x a x a = 0 v V (inflow) a δ + (v) a A(v) x a x a = 0 v V (outflow) a δ (v) a A(v) w (v,w) L x a 0 (v, w) V V (coupling) a A(v,w) w (v,w) w (w,v) r a x a = 0 v V (resource flow) (v,w) δ (v) (w,v) δ + (v) a δ (v)... (miscellaneous)

44 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Mixed integer program (only main structure) min c ax a a A (objective) x a = 1 t T (covering) a A(t) x a x a = 0 v V (inflow) a δ + (v) a A(v) x a x a = 0 v V (outflow) a δ (v) a A(v) w (v,w) L x a 0 (v, w) V V (coupling) a A(v,w) w (v,w) w (w,v) r a x a = 0 v V (resource flow) (v,w) δ (v) (w,v) δ + (v) a δ (v)... (miscellaneous)

45 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Mixed integer program (only main structure) min c ax a a A (objective) x a = 1 t T (covering) a A(t) x a x a = 0 v V (inflow) a δ + (v) a A(v) x a x a = 0 v V (outflow) a δ (v) a A(v) w (v,w) L x a 0 (v, w) V V (coupling) a A(v,w) w (v,w) w (w,v) r a x a = 0 v V (resource flow) (v,w) δ (v) (w,v) δ + (v) a δ (v)... (miscellaneous)

46 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Mixed integer program (only main structure) min c ax a a A (objective) x a = 1 t T (covering) a A(t) x a x a = 0 v V (inflow) a δ + (v) a A(v) x a x a = 0 v V (outflow) a δ (v) a A(v) w (v,w) L x a 0 (v, w) V V (coupling) a A(v,w) w (v,w) w (w,v) r a x a = 0 v V (resource flow) (v,w) δ (v) (w,v) δ + (v) a δ (v)... (miscellaneous)

47 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Mixed integer program (only main structure) min c ax a a A (objective) x a = 1 t T (covering) a A(t) x a x a = 0 v V (inflow) a δ + (v) a A(v) x a x a = 0 v V (outflow) a δ (v) a A(v) w (v,w) L x a 0 (v, w) V V (coupling) a A(v,w) w (v,w) w (w,v) r a x a = 0 v V (resource flow) (v,w) δ (v) (w,v) δ + (v) a δ (v)... (miscellaneous)

48 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Model structure x w conservation covering T = 1000 V = 3000 F = 2 coupling C = 4 A = resource flow miscellaneous

49 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Model discussion flexible, exact and integrated modeling of train composition maintenance constraints regularity polynomial in number of rows/columns only one resource flow for all fleets (vehicle groups) sufficient lp bound highly fractional lp solutions

50 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Model discussion flexible, exact and integrated modeling of train composition maintenance constraints regularity polynomial in number of rows/columns only one resource flow for all fleets (vehicle groups) sufficient lp bound highly fractional lp solutions

51 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Algorithm overview start initialize static graph

52 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Algorithm overview start initialize static graph initialize model (re)solve LP yes new variables found? price variables LP: column generation with parallel Cplex Barrier

53 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Algorithm overview start initialize static graph end initialize model do static IP (SCIP,Cplex,Gurobi) (re)solve LP yes new variables found? price variables do decomposition heuristics do local search heuristics do relaxation heuristics do rapid branching LP: column generation with parallel Cplex Barrier IP: whatever helps

54 Algorithm overview start initialize static graph end initialize model do static IP (SCIP,Cplex,Gurobi) (re)solve LP yes new variables found? price variables do decomposition heuristics do local search heuristics do relaxation heuristics do rapid branching LP: column generation with parallel Cplex Barrier Our algorithm has some exact parts, but the overall procedure is heuristic. IP: whatever helps Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32

55 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Solution progress lower bound primal lp value integer solution objective time in seconds 10 4

56 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Algorithm progress number of columns time in seconds 10 4

57 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

58 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Instances We got from DB Fernverkehr: very many, high quality, very realistic, complete, meaningful, well structured, large scale and very interesting instances.

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61 trains F C main. cap. V A yes yes yes no no yes yes no yes yes yes yes yes no no no no yes no no yes no no yes no no no no yes no no no no yes no no trains F C main. cap. V A yes no no yes no no no yes yes yes yes yes no yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes no yes Table: 71 test scenarios (may 2011)

62 Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32 Computations id trains F C maint. cap. V A gap time objective cores yes yes yes yes no no no yes yes yes yes yes yes yes yes yes yes no no yes Table: results for 20 test scenarios (may 2011)

63 Thank you for your attention! Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, / 32