Vehicle Rotation Planning for Intercity Railways

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, 2011 1 / 32

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 2 / 32 Outline 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 3 / 32 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

Introduction close cooperation with Deutsche Bahn Fernverkehr AG DB Fernverkehr AG operates 1.298 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, 2011 4 / 32

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 5 / 32 Motivation Figure: ICx mega deal ( c SPIEGEL ONLINE GmbH)

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 6 / 32 DB Fernverkehr AG passengers 2009 122.7 Mio passengers/day 0.3 Mio transport service provided 149.3 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 7 / 32 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 8 / 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.

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 9 / 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)

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 10 / 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)...

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 11 / 32 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 12 / 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 13 / 32 Graph model (detailed) trip 1 trip 3 trip 4 trip 2 trip 5 possible trivial and non-trivial vehicle configurations for each trip

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 13 / 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 13 / 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)

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 13 / 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)

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 13 / 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 13 / 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 13 / 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)

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 13 / 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 14 / 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 15 / 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 16 / 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 17 / 32 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 18 / 32 Recapitulation Hypergraph H = (V, A) Set of timetabled trips T Set of vehicle groups F Set of vehicle configurations C

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 19 / 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 20 / 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)

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 21 / 32 Model structure x w conservation covering T = 1000 V = 3000 F = 2 coupling C = 4 A = 9000000 resource flow miscellaneous

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 22 / 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 23 / 32 Algorithm overview start initialize static graph

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 23 / 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

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 23 / 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

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, 2011 23 / 32

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 24 / 32 Solution progress 4 3 10 7 lower bound primal lp value integer solution objective 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time in seconds 10 4

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 25 / 32 Algorithm progress 10 6 1 number of columns 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time in seconds 10 4

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 26 / 32 1 Introduction 2 Problem 3 Hypergraph model 4 Model and algorithm 5 Computations

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 27 / 32 Instances We got from DB Fernverkehr: very many, high quality, very realistic, complete, meaningful, well structured, large scale and very interesting instances.

trains F C main. cap. V A 126 1 1 yes 0 617 582086 126 1 1 yes 0 617 563309 126 1 1 yes 0 617 574530 126 1 1 no 0 617 429172 165 1 1 no 0 884 878234 165 1 1 yes 0 884 1105810 126 1 1 yes 0 617 563309 43 1 1 no 0 267 80149 43 1 1 yes 0 267 103308 410 8 8 yes 0 10913 23474708 61 1 1 yes 0 310 130258 288 4 6 yes 0 2433 2264022 298 6 6 yes 0 7379 12835881 298 6 6 no 0 7379 10702287 298 6 6 no 0 7379 9088577 298 24 24 no 0 26396 34399642 298 24 24 no 0 26396 29210321 298 2 2 yes 0 2753 5188184 298 2 2 no 0 2753 4325979 298 2 2 no 0 2753 3692070 298 8 8 yes 0 9896 16983277 298 8 8 no 0 9896 14010210 298 8 8 no 0 9896 11951200 298 18 18 yes 0 7474 9843134 298 18 18 no 0 7474 8074672 298 18 18 no 0 7474 6857506 298 8 8 no 0 3619 3930759 298 8 8 no 0 3619 3340476 298 7 7 yes 0 2913 4036117 298 7 7 no 0 2913 3311258 298 7 7 no 0 2913 2824802 443 16 16 no 0 13538 24963128 443 16 16 no 0 13538 21842737 443 16 16 yes 0 13538 30849232 443 16 16 no 0 9275 10299936 443 16 16 no 0 9275 9012444 trains F C main. cap. V A 443 16 16 yes 0 9275 12864771 252 1 1 no 0 406 167130 252 1 1 no 0 406 158698 252 1 1 yes 0 406 240130 443 24 24 no 0 30446 52768895 443 24 24 no 0 20124 24248940 443 24 24 no 0 20124 20754314 19 2 4 yes 0 534 63624 19 2 4 yes 0 534 63624 19 1 2 yes 0 534 122136 11 2 4 yes 0 323 22150 8 2 4 yes 0 288 17301 19 2 4 no 0 534 47535 61 1 1 yes 0 310 129988 61 1 1 yes 8 310 129988 61 1 1 yes 32 310 129988 288 4 6 yes 10 2435 2257924 288 4 6 yes 40 2435 2257924 137 3 6 yes 21 2339 1864894 137 3 6 yes 63 2339 1864894 137 3 6 yes 0 2333 1865198 5 2 5 yes 0 235 11838 137 3 7 yes 22 2373 1817638 19 2 5 yes 22 486 53588 19 1 2 yes 16 486 97856 11 2 5 yes 22 305 19638 8 2 5 yes 22 270 15319 137 3 7 yes 22 2373 1817638 19 2 5 yes 22 486 53588 19 2 5 yes 22 486 53588 556 10 19 yes 102 6145 6306459 556 10 19 yes 102 6145 6306459 556 10 19 yes 0 6145 6306459 556 10 19 no 0 6145 4650852 135 3 6 yes 66 1848 2486160 Table: 71 test scenarios (may 2011)

Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 31 / 32 Computations id trains F C maint. cap. V A gap time objective cores 01 126 1 1 yes 0 617 582086 0.037 75190 1865.77 1 02 165 1 1 yes 0 884 1105810 0.018 94807 5534.04 1 03 126 1 1 yes 0 617 563309 0.018 95024 4071.86 1 04 43 1 1 yes 0 267 103308 0.010 13424 1653.83 1 05 298 24 24 no 0 26396 34399642 0.008 75227 5924846.09 1 06 443 16 16 no 0 13538 24963128 0.002 22670 26338265.01 1 07 443 16 16 no 0 13538 24963128 0.003 38165 26359612.68 1 08 61 1 1 yes 8 310 129988 0.045 96860 1161441.72 1 09 61 1 1 yes 32 310 129988 0.012 94365 1122472.81 1 10 137 3 6 yes 63 2339 1864894 0.095 80837 4448118.83 1 11 137 3 7 yes 22 2373 1817638 0.097 121212 4447930.22 4 12 137 3 7 yes 22 2373 1817638 0.103 77423 4477971.24 1 13 137 3 7 yes 22 2373 1817638 0.080 121351 4363668.59 4 14 19 2 5 yes 22 486 53588 0.006 41 800665.43 4 15 19 2 5 yes 22 486 53588 0.006 60 800665.43 4 16 556 10 19 yes 102 6145 6306459 0.097 121717 23043657.35 4 17 556 10 19 yes 0 6145 6306459 0.014 121741 21102957.35 4 18 556 10 19 no 0 6145 4650852 0.002 10218 20737819.44 4 19 556 10 19 no 0 6145 4407395 0.001 26389 20734316.96 4 20 135 3 6 yes 66 1848 2486160 0.010 47751 7183109.41 4 Table: results for 20 test scenarios (may 2011)

Thank you for your attention! Markus Reuther (Zuse Institute Berlin) Vehicle Rotation Planning May 24, 2011 32 / 32