I Planning in Railway Transportation 11 Estimated demand for (freight) railway transportation in
Germany source Federal Transport Infrastructure Plan-ning Project Group (2003) [87] 2
4 Requested train paths at DB source Klabes (2010) [129] 135 Possible train composition for track f = (vr 14 wb 20 4) 236 Visualization of line plan for Potsdam 257 Screenshot of visualization tool for public transport net-
Klabes (2010) [129] 319 Simple conflict example and re-solution for track alloca-
tabling by Caimi (2009) [57] 3311 A partial cyclic rolling stock rotation graph visualized in
12 Crew Scheduling Graph 4613 Set of legs (above) and a set of covering pairings (below)
show as a Gant chart in the planning tool NetLine 4714 General column generation approach to solve LPs with a
II Railway Modeling 541 Idealized closed loop between railway models of different
scale for railway track allocation 552 Detailed view of station Altenbeken provided by DB Netz
3 Screenshot of the railway topology of a microscopic net-work in the railway simulator OpenTrack Signals can beseen at some nodes as well as platforms or station labels 59
4 Idea of the transformation of a double vertex graph to astandard digraph 59
5 Microscopic network of the Simplon and detailed repre-sentation of station Iselle as given by OpenTrack 61
6 Blocking time diagrams for three trains on two routesusing 6 blocks In the lower part of the diagram twosubsequent trains on route r2 and at the top one train onthe opposite directed route r1 are shown 62
7 IO Concept of TTPlib 2008 (focus on macroscopic rail-way model) 64
8 Example of macroscopic railway infrastructure 66
9 Example of aggregated infrastructure 67
10 Train types and train sets defined as a poset 68
11 Macroscopic modeling of running and headways times ontracks 72
12 Macroscopic modeling of a single way track 72
13 Representation as event-activity digraph G = (VN AN) 73
14 Implausible situation if headway matrix is not transitive 74
15 Transformation of running time on track Ararr B for timediscretizations between 1 and 60 seconds 80
16 Rounding error for different time discretizations between1 and 60 seconds comparison of ceiling vs cumulativerounding 81
17 Headway time diagrams for three succeeding trains onone single track (j1 j2) 83
18 Constructed aggregated macroscopic network by netcast
86
19 New routing possibilities induced by given routes 87
20 Macroscopic network produced by netcast visualize byTraVis 87
III Railway Track Allocation 90
1 Concept of TTPlib 2008 (focus on train demand specifi-cation and TTP) 92
2 Penalty functions for departure(left) and arrival(right)times 94
3 Profit function w() depending on basic profit and depar-ture and arrival times 94
4 Explicit and implicit waiting on a timeline inside a station 98
5 Complete time expanded network for train request 101
6 Irreducible graph for train request 102
7 Preprocessed time-expanded digraph D = (VA) of ex-ample 16 105
8 Example for maximum cliques for block occupation con-flicts 109
9 Example for an equivalence class and a hyperarc 113
10 Example for the construction of a track digraph 114
11 Example for a path which does not correspond to a validconfiguration if the headway times violate the transitiv-ity 116
12 Relations between the polyhedra of the different models 122
13 Idea of the extended formulation (PCP) for (PPP) 124
14 From fragile q1 and q2 to robust configuration q3 130
15 Robustness function r of two buffer arcs 130
16 Pareto front on the left hand and total profit objective(blue left axis) and total robustness objective (greenright axis) in dependence on α on the right hand 132
17 Flow chart of algorithmic approach in TS-OPT 134
18 Cutting plane model fPQ of Lagrangean dual fPQ 138
19 The new solution sets at iteration k source Weider(2007) [213] 146
IV Case Studies 148
1 Infrastructure network (left) and train routing digraph(right) individual train routing digraphs bear differentcolors 150
2 Reduction of graph size by trivial preprocessing for sce-narios req 36 and τ = 20 151
3 Artifical network wheel see TTPlib [208] 154
4 Solving the LP relaxation of model (PCP) with columngeneration and the barrier method 161
5 Solving the LP relaxation of model (PCP) with the bun-dle method 162
6 Testing different bundle sizes 164
7 Solving a track allocation problem with TS-OPT dual(LP) and primal (IP) stage 167
8 Solving track allocation problem req 48 with TS-OPT 1689 Auction procedure in general 17110 Micro graph representation of Simplon and detailed rep-
resentation of station Iselle given by OpenTrack 17711 Given distribution of passenger or fixed traffic in the Sim-
plon corridor for both directions 17912 Traffic diagram in OpenTrack with block occupation for
request 24h-tp-as 18513 Comparison of scheduled trains for different networks
(simplon ) for instance 24h-tp-as in a 60s discretisation 18714 Distribution of freight trains for the requests 24h-tp-as
and 24h-f15-s by using network simplon big and a round-ing to 10 seconds 188
List of Algorithms
I Planning in Railway Transportation 1
II Railway Modeling 541 Cumulative rounding method for macroscopic running time
discretization 772 Calculation of Minimal Headway Times 813 Algorithm for the Micro-Macro-Transformation in netcast
84
III Railway Track Allocation 904 Construction of D 1005 Proximal Bundle Method (PBM) for (LD) of (PCP) 1396 Perturbation Branching 145
IV Case Studies 148
xxix
Chapter I
Planning in Railway Trans-portation
The purpose of our work is to develop mathematical optimization mod-els and solution methods to increase the efficiency of future railwaytransportation systems The reasons for this is manifold liberaliza-tion cost pressure environmental and energy considerations and theexpected increase of the transportation demand are all important fac-tors to consider Every day millions of people are transported by trainsin Germany Public transport in general is a major factor for the pro-ductivity of entire regions and decides on the quality of life of people
Figure 1 shows the expected development of freight transportation inGermany from 2003 to 2015 as estimated by the Deutsche Bahn AG(DB AG) This estimate was the basis of the last German FederalTransport Infrastructure Plan 2003 (Bundesverkehrswegeplan 2003)see Federal Transport Infrastructure Planning Project Group (2003)[87] It is a framework investment plan and a planning instrumentthat follows the guiding principle of ldquodevelopment of Eastern Germanyand upgrading in Western Germanyrdquo The total funding available forroad rail and waterway construction for the period from 2001 to 2015is around 150 billion euros
The railway industry has to solve challenging tasks to guarantee or evenincrease their quality of service and their efficiency Besides the needto implement adequate technologies (information control and book-ing systems) and latest technology of equipment and resources (trainsrailway infrastructure elements) developing mathematical support sys-tems to tackle decision planning and in particular optimization prob-lems will be of major importance
1
2
Figure 1 Estimated demand for (freight) railway transportation in Germanysource Federal Transport Infrastructure Planning Project Group(2003) [87]
In Section 1 we will give a comprehensive introduction on the politicalenvironment and organizational structures because both directly affectthe planning and operation of railway transport In addition we willrefurbish an early publication from Charnes amp Miller (1956) [67] thatdemonstrates prominently that railway transport is one of the initialapplication areas for mathematics in particular for discrete and linearoptimization
Only recently railway success stories of optimization models are re-ported from Liebchen (2008) [149] Kroon et al (2009) [140] and Caimi(2009) [57] in the area of periodic timetabling by using enhanced inte-ger programming techniques This thesis focuses on a related planningproblem ndash the track allocation problem Thus Section 2 gives a generaloverview of an idealized planning process in railway transportationWe will further describe several other planning problems shortly in-cluding line planning in Section 5 and crew scheduling in Section 8 inmore detail Mathematical models and state of the art solution ap-proaches will be discussed as well as the differences to and similaritieswith equivalent planning tasks of other public transportation systemsMoreover in Section 6 we will depict the requirements and the processof railway capacity allocation in Europe to motivate and establish ageneral formulation for the track allocation problem
We will show how to establish a general framework that is able tohandle almost all technical details and the gigantic size of the railway
3
infrastructure network by a novel aggregation approach Thereforeand to build a bridge to railway engineering we explain the most im-portant microscopic technical details in Chapter II Furthermore weintroduce a general standard for macroscopic railway models which ispublicly available TTPlib [208] and develop a multi-scale approach thatautomatically transforms microscopic railway models from real worlddata to general macroscopic models with certain error estimations
Nevertheless the resulting macroscopic track allocation problems arestill very large and complex mathematical problems From a complex-ity point of view track allocation problems belongs to the class of NP-hard problems In order to produce high quality solutions in reasonabletime for real world instances we develop a strong novel model formu-lation and adapt a sophisticated solution approach We believe thatthis modeling technique can be also very successful for other problemsndash in particular if the problem is an integration of several combinatorialproblems which are coupled by several constraints Chapter III willintroduce and analyze this novel model formulation called rdquoconfigura-tionldquo model in case of the the track allocation problem Furthermorewe will generalize and adapt the rapid branching heuristic of Weider(2007) [213] We will see that we could significantly speed up ourcolumn generation approach by utilizing the bundle method to solvethe Lagrangean relaxation instead of using standard solvers for the LPrelaxations
Finally to verify our contributions on modeling and solving track allo-cation problems in Chapter IV we implemented several software toolsthat are needed to establish a track allocation framework
a transformation module that automatically analyses and simpli-fies data from microscopic simulation tools and provides reliablemacroscopic railway models (netcast)
an optimization module that produces high quality solutions (to-gether with guaranteed optimality gaps) for real world track al-location problems in reasonable time (TS-OPT)
and a 3d-visualization module to illustrate the track allocationproblem to discuss the solutions with practitioners and to au-tomatically provide macroscopic statistics (TraVis)
1 Introduction 4
1 Introduction
Railway systems can be categorized as either public or private Pri-vate railway systems are owned by private companies and are with afew exceptions exclusively planned built and operated by this sin-gle owner Prominent examples are the railway systems in Japan andthe US see Gorman (2009) [102] Harrod (2010) [112] White amp Krug(2005) [215] In contrast public railway systems are generally fundedby public institutions or governments In the past an integrated rail-way company was usually appointed to plan build and operate therailway system Now the efforts of the European Commission to seg-regate the integrated railway companies into a railway infrastructuremanager (network provider) and railway undertakings (train operatingcompanies) shall ensure open access to railway capacity for any licensedrailway undertaking The idea is that competition leads to a more ef-ficient use of the railway infrastructure capacity which in the long runshall increase the share of railway transportation within the Europeanmember states However even in case of an absolute monopoly theplanning of railway systems is very complex because of the technicali-ties and operational rules This complexity is further increased by thevarying requirements and objectives of different participating railwayundertakings in public railway systems
The focus of this work is capacity allocation in an arbitrary railwaysystem In a nutshell the question is to decide which train can usewhich part of the railway infrastructure at which time Chapter Iaims to build an integrated picture of the railway system and railwayplanning process ie we will illuminate the requirements of passengerand freight railway transportation In Chapter II resource models willbe developed that allow for capacity considerations Based on oneof these railway models ie an aggregated macroscopic one we willformulate a general optimization model for private and public railwaysystems in Chapter III which meets the requirements of passenger andfreight railway transportation to a large extent
Several railway reforms in Europe were intended to promote on-railcompetition leading to more attractive services in the timetable How-ever even after the reforms were implemented the railways continuedto allocate train paths on their own networks themselves Discrimi-nation was thus still theoretically possible However competition can
1 Introduction 5
only bring benefits if all railway undertakings are treated equally whenseeking access to the infrastructure
Switzerland has been pioneer in introducing competition in the use ofthe rail networks The three different Swiss railway network providersSBB BLS and SOB outsourced the allocation of their train paths to ajoint independent body Accordingly at the beginning of 2006 and inconjunction with the Swiss Public Transport Operatorsrsquo Associationthese railways together founded the Trasse Schweiz AG (trassech)
By outsourcing train path allocation to a body which is legally in-dependent and independent in its decision making the three largestSwiss standard gauge railways together with the Swiss Public Trans-port Operators Association reinforced their commitment to fair on-railcompetition This institution ensures that the processes to prepare forthe timetable are free of discrimination Trasse Schweiz AG coordi-nates the resolution of conflicts between applications and allocate trainpaths in accordance with the legislation One of their principles is
ldquoWe increase the attractiveness of the rail mode by makingthe best use of the network and optimizing the applicationprocessesrdquo
That statement essentially summarizes the main motivation of thisthesis
An initial publication on applying linear optimization techniques comesfrom railway freight transportation Charnes amp Miller (1956) [67] dis-cussed the scheduling problem of satisfying freight demand by traincirculations The setting is described by a small example in Figure 2In a graph with nodes 12 and 3 a directed demand which hasto be satisfied is shown on each arc The goal is to determine directedcycles in that graph that cover all demands with minimal cost ieeach cycle represents a train rotation For example choosing four timesthe rotation (121) would cover all required freight movementsbetween 1 and 2 However the demand from 2 to 1 is only oneand therefore that would be an inefficient partial solution with threeempty trips called ldquolight movesrdquo in the original work Charnes andMiller proposed a linear programming formulation for the problem enu-merating all possible rotations ie five directed cycles (121)(131) (232) in Figure 2 Multiple choices of cycles thatsatisfy all demands represent a solution Thus for each rotation aninteger variable with crew and engine cost was introduced The opti-
1 Introduction 6
1
2
3
4rarr1larr
6rarr6larr
5larr9rarr
Figure 2 Simplified routing network of Charnes amp Miller (1956) [67]
mization model states that the chosen subset has to fulfill all demandsThis was one of the first approaches to solve real applications by meansof a set partitioning problem ie to represent a solution as a set ofsub-solutions here cycles Finally they manually solved the instanceby applying the simplex tableau method
After that pioneering work on modeling it took many years of improve-ment in the solution techniques to go a step further and to support morecomplex planning challenges in public transportation and in particularin railway transportation by optimization
In fact the airline industry became the driving force of the developmentOne reason is the competitive market structure which leads to a highercost pressure for aviation companies Therefore the airline industryhas a healthy margin in the implementation of automated processesand the evaluation of operations Integrated data handling measuringthe quality of service and controlling the planning and operation byseveral key performance indicators (KPI) are anchored in almost allaviation companies over the world Nowadays in the airline industrythe classical individual planning problems of almost all practical prob-lem sizes can be solved by optimization tools Integration of differentplanning steps and the incorporation of uncertainty in the input datacan be tackled A prominent example for such robust optimization ap-proaches is the tail assignment problem which is the classical problemof assigning flights to individual aircraft Nowadays robust versionscan be tackled by stochastic optimization see Lan Clarke amp Barn-hart (2006) [144] or a novel probability of delay propagation approachby Borndorfer et al (2010) [41] Suhl Duck amp Kliewer (2009) [205] usesimilar ideas and extensions to increase the stability of crew schedules
An astonishing situation happened in Berlin which somehow documentsthe challenges and problems that might result from the deregulationThe British Financial Times wrote on 27th of July 2009
1 Introduction 7
ldquoConcrete walls watch-towers barbed wire and armed bor-der guards for decades prevented Germans travelling acrossBerlin from the east to the west But as the German capi-tal gears up to celebrate 20 years since the fall of the BerlinWall leftwing commentators are claiming that capitalismnot communism is now keeping the two apart For the S-Bahn - the suburban commuter railway running into andaround Berlin that became a symbol of the cold war divide- has come grinding to a halt
More than two-thirds of the networkrsquos 550 trains werewithdrawn from service last week and the main east-westline closed after safety checks following a derailment showedthat about 4000 wheels needed replacing Hundreds ofthousands of Berliners have been forced to get on theirbikes or use alternative overcrowded routes to work whiletourists weaned on stereotypical notions of German punc-tuality and efficiency have been left inconvenienced and be-mused by the chaos Deutsche Bahn the national railwayoperator is under fire for cutting staff and closing repairworkshops at its S-Bahn subsidiary in an attempt to boostprofitability ahead of an initial public offering that hassince been postponed
For businesses dependent on the custom of S-Bahn pas-sengers the partial -suspension of services is no joke ldquoForthe past two or three days itrsquos been really bad Customersare down by more than halfrdquo said an employee at a clothing-alteration service situated below the deserted S-Bahn plat-form at Friedrichstrasse station in the former East BerlinldquoGerman trains are world famous I didnrsquot think -somethinglike this could happenrdquo
A columnist for Tagesspiegel a Berlin-based newspa-per drolely observed that the number of S-Bahn carriagesrendered unusable by management incompetence was onlyslightly less than the total number damaged by the RedArmy in 1945 Others note that even the Berlin Wall itselfdid not prevent S-Bahn passengers traveling between westand east so long as they held a West German passportThe East German authorities continued to operate the S-Bahn in West Berlin after the partition of the city followingthe second world war until the 1980s West Berliners even-tually boycotted this service in protest of the communist
1 Introduction 8
regime But now it is being claimed that capitalism is driv-ing passengers away
ldquoThe chaos in the Berliner S-Bahn is a lesson in the con-sequences of capitalism It is a graphic depiction of wheresubservience to financial markets greedy pursuit of profitultimately leadsrdquo Ulrich Maurer chief whip of the radicalLeft party said Deutsche Bahn has apologized for the in-convenience but insists that cost-cutting was not the prob-lem and blames the train manufacturer instead rdquoEven if wehad had twice as many employees and three times as manyworkshops it would not have prevented these wheels frombreakingrdquo a Deutsche Bahn spokesman said NeverthelessS-Bahn-Berlinrsquos entire senior management was forced to re-sign this month after it emerged that they had not orderedsufficient safety checks The repairs refunds and lost farescould leave Deutsche Bahn up to 100 million euros out ofthe pocket according to one estimate A full service is notexpected to resume until Decemberrdquo
The described situation documents that the railway system in Europehas to face huge challenges in implementing the liberalization In ad-dition central topics of the railway system are often politically andsocially sensitive subjects A detailed characterization of the recentpolitical situation of the German railway system future perspectivesthe role of the infrastructure and other controversial issues can befound in GIlgmann (2007) [99] All in all we hope and we believethat an innovation process in the railway system in Europe is going tostart Major railway planning decisions can be supported by mathe-matical models and optimization tools in the near future in particularthe almost manual construction of the timetables and track allocationswhich is often seen as the ldquoheartrdquo of the railway system
Due to the deregulation and the segregation of national railway com-panies in Europe the transfer of mathematical optimization techniquesto railway operations will proceed In the future competition willhopefully give rise to efficiency and will lead to an increasing use ofinformation technology and mathematical models Algorithmic deci-sion support to solve the complex and large scale planning problemsmay become necessary tools for railway transportation companies Inthe future state of the art planning systems with optimization insidewill replace the ldquomanualrdquo solution The key message is that optimiza-
2 Planning Process 9
tion ie mathematical models and solution methods are predestinedto support railway planning challenges now and in the future
In the following section we will briefly highlight several of these plan-ning problems from different transportation modes We will presentmathematical models and discuss state of the art solution approachesto tackle real world applications see Barnhart amp Laporte (2007) [17]for an overview on optimization in transportation in general We use inthis thesis the definitions and notation of Grotschel Lovasz amp Schrijver(1988) [104] and Nemhauser amp Wolsey (1988) [167] for graphs linearprograms (LPs) and mixed integer programs (MIPs) Furthermore weuse the algorithmic terminology to LP and MIP solving of Achterberg(2007) [3]
2 Planning Process
Bussieck Winter amp Zimmermann (1997) [50] divide the planning pro-cess in public transport into three major steps - strategic tactical andoperational planning Table 1 shows the goals and time horizon ofall steps Public transport especially railway transportation is sucha technically complex and large system that it is impossible to con-sider the entire system at once Also the different planning horizons ofcertain decisions enforce a decomposition Therefore a sequence of hier-archical planning steps has emerged over the years However in realitythere is no such standardization as we will explain it theoretically
Two important parties are involved in the railway transportation plan-ning process ie train operating companies and railway infrastructureproviders Following the terminology of the European commissionwe will use the terms railway undertaking (RU) and infrastructuremanager (IM) respectively Furthermore several national and inter-national institutions have a huge political influence on railway trans-portation which is on the borderline between a social or public goodand a product that can be traded on a free liberalized market Thespecial case of the changing railway environment in Europe will bediscussed in detail in Section 61
In contrast to railway undertakings fully private aviation or independ-ent urban public transport companies can perform the complete plan-ning process almost internally In the airline industry the needed infras-tructure capacity ie the slots at the airports are granted by grandfa-
2 Planning Process 10
level time horizon goal
strategic 5-15 years resource acquisitiontactical 1-5 years resource allocationoperational 24h - 1 year resource consumption
Table 1 Planning steps in railroad traffic source Bussieck Winter amp Zimmer-mann (1997) [50]
ther rights see Barnier et al (2001) [21] Castelli Pellegrini amp Pesenti(2010) [66] Borndorfer Grotschel amp Jaeger (2008) [36] BorndorferGrotschel amp Jaeger (2009) [39] and Hanne amp Dornberger (2010) [108]give recent surveys about the potential of optimization for transporta-tion systems and the differences between the planning process in theairline industry urban public transport and the railway industry Inthe case of urban public transport the planning process is discussed inWeider (2007) [213] and Borndorfer Grotschel amp Pfetsch (2007) [35] Adetailed description of the process in the airline industry can be foundin Gronkvist (2005) [103] and Barnhart amp Laporte (2007) [17] Bussieck(1997) [49] describes the use of discrete optimization in the planningprocess of public rail transport in the case of an integrated systemAnalogous considerations can be found in Liebchen (2006) [148] andLusby et al (2009) [159] There the planning steps are classified withrespect to the time horizon and their general purposes
Strategic or long-term part concerns the issues of network design andline planning (resource acquisition) see Sections 3 and 5 On the tac-tical stage the level of services usually a timetable has to be createdas well as the schedules for the needed resources (resource allocation)Finally on the operational stage the resources eg rolling stock ve-hicles aircraft and crews are monitored in real operations (resourceconsumption)
On the day of operation re-scheduling and dispatching problems haveto be faced These kind of problems have a different flavor than pureplanning tasks Decisions must be made very quickly in the real-timesetting but only limited information on the ldquoscenariordquo is availableUsually data has to be taken into consideration in a so called onlinefashion More details about this kind of problem can be found inGrotschel Krumke amp Rambau (2001) [105] Albers amp Leonardi (1999)[9] and Albers (2003) [8] Recent approaches are to establish fast meth-ods which bring the ldquorealrdquo situation back to the ldquoplannedrdquo one when
2 Planning Process 11
Railway undertakings (RU) Infrastructure manager (IM)
Network Design
Line Planning
Timetabling Track allocation
Rolling Stock Planning
Crew Scheduling
Real Time Management Re-Scheduling
level
strategic
tactical
operational
Figure 3 Idealized planning process for railway transportation in Europe
possible see Potthoff Huisman amp Desaulniers (2008) [177] Rezanovaamp Ryan (2010) [182] and Jespersen-Groth et al (2009) [123]
In Klabes (2010) [129] the planning process is newly considered for thecase of the segregated European railway system In Figure 3 the novelprocess is illustrated for the segregated railway industry in Europe
2 Planning Process 12
21 Strategic Planning
The responsibilities of the planning steps refer directly to either the rail-way undertaking or the infrastructure manager on behalf of the stateNevertheless the long-term decisions in up- or downgrading the networkare highly influenced by the railway undertakings and their demandsIn case of passenger railway undertakings the desired timetable aimsto implement a given line plan The timetable itself induces train slotsrequests which is one input for the track allocation problem Theseare naturally very strict with respect to departure and arrival times inorder to offer and operate a concrete and reliable timetable Furtherdetails on line planning and periodic timetabling are given in Section 5and Section 62 respectively
The requirements of train slot requests for cargo or freight railway oper-ators differ significantly from slot requests for passenger trains becausethey usually have more flexibility ie arrival and departure are onlyimportant at stations where loading has to be performed Section 3will describe the network design problem of the major European singlewagon railway transportation system In general freight railway oper-ators need a mixture of annual and ad hoc train slots The demand isof course highly influenced by the industry customers and the freightconcept of the operating railway undertaking We collected such datafor the German subnetwork hakafu simple to estimate the demandof the railway freight transportation see Chapter IV Section 1 andSchlechte amp Tanner (2010) [189]
22 Tactical Planning
The essential connection between all train slot requests is the step todetermine the complete track allocation which is the focus of this workHowever we primarily consider the point of view of a railway infras-tructure provider which is interested in optimizing the utilization ofthe network That is to determine optimal track allocations This is incontrast to timetabling where one asks for the ideal arrival and depar-ture times to realize a timetable concept or a line plan A timetablecan be seen as a set of train slot requests without flexibility Railwayoptimization from a railway undertakingrsquos point of view for passengertraffic is discussed in Caprara et al (2007) [64] State of the art model-ing and optimization approaches to periodic timetabling which is the
2 Planning Process 13
2003 2004 2005 2006 2007 2008 20090
2
4
6
8
middot104
year
nu
mb
erof
trai
nsl
otre
qu
ests total
DB railway undertakingsnon-DB railway undertakings
2003 2004 2005 2006 2007 2008 20090
50
100
150
year
reje
cted
Figure 4 Requested train paths at DB source Klabes (2010) [129]
usual type of schedule for passenger railway traffic is at length studiedby Liebchen (2006) [148]
The induced competition for railway capacity allocation in public rail-way systems in Europe has a several impacts on the allocation pro-cedure In the past a single integrated railway company performedthe complete planning Its segregation reduces the ability of the rail-way infrastructure manager to only perform network planning capac-ity allocation and re-scheduling with respect to infrastructure aspectsThus the infrastructure manager only has limited information duringthe planning process and needs to respect the confidential informationof the railway undertakings Moreover new railway undertakings en-ter the market which increases the complexity of the planning processKlabes (2010) [129] collected the relevant numbers from the DB Netzreports On the left hand of Figure 4 the changing environment isillustrated by listing the growing number of train slot requests fromrailway undertakings independent from the former integrated railwaycompany ldquoDeutsche Bahnrdquo On the right hand of Figure 4 the numberof rejected train slot requests for the same periods are shown It canbe seen that at the start of the segregation from 2003 until 2006 a lotof requests had to be rejected by DB Netz Efforts to decrease thesenumbers by providing alternative slots were apparently successful inthe following years
The business report for the year 2009 Trasse Schweiz AG [207] of theTrasse Schweiz AG documents the new challenges for constructingtrack allocations as well In the Swiss network a lot of different railwayundertakings are operating eg in 2009 there were 29 train operat-ing companies which submitted train slot requests The geographicalposition in central Europe and the limited transportation possibilitiesthrough the Alps causes that The future challenge for Switzerland
2 Planning Process 14
will be to handle the complex track allocation process as the followingextract from the report 2009 already highlights
ldquoThe regulation of the conflicts arising in train slot ordersof the annual timetable 2010 was despite or even less be-cause of the financial or economic crisis in comparison tothe last years extensive and time-consuming Indeed thenumber of submitted train slot requests by cargo operatorsfor the annual timetable 2010 decreased up to 10 percent incomparison to the last year However railway undertakings(RM) concentrated her orders due to the cost pressure andcompetitive market conditions on the most attractive timewindows and stick much longer to their original requestsNevertheless we managed together with all infrastructureproviders1 to find for all conflicts alternative train slotswhich were accepted by the railway undertakings No trainslot request had to be rejectedrdquo (translation by the au-thor)
The competing railway undertakings should interact in a transparentand free market The creation of such a market for railway capacity isa key target of the European Commission hoping that it will lead toa more economic utilization of the railway infrastructure Even moreliberalization of the railway system should lead to a growing marketand allow for innovative trends like in other old-established industriesie aviation industry telecommunication or energy market After theacceptance of train slots each railway undertaking determines his par-tial operating timetable which acts as input for the planning of theneeded resources In case of a railway operator the rolling stock ro-tations have to be constructed which is very complex problem dueto several regularities and maintenance requirements see Fioole et al(2006) [88]Anderegg et al (2003) [12] Eidenbenz Pagourtzis amp Wid-mayer (2003) [80] and Peeters amp Kroon (2008) [176]
In public transport and in airline industry vehicle scheduling and air-craft rotation planning are the analogous tasks see Lobel (1997) [155]and Gronkvist (2005) [103] The major objective is to operate a re-liable timetable with minimum cost which is in general minimizingthe number of engines wagons vehicles aircrafts etc Another keyrequirement for planning railway rolling stock rotations is to provide
1There are three different railway infrastructure providers in Switzerland ieBLS SBB and SOB
2 Planning Process 15
regularity of the solutions This means that a train that runs in thesame way every day of the week will also be composed in the sameway every day of the week always using the same cars from the samepreceding trains Such a regime simplifies the operation of a railwaysignificantly However the rule can not always be followed Trains mayrun later on weekends or not at all on certain days eg in order toperform a maintenance operation Although it is intuitively clear it isnot easy to give a precise definition what regularity actually means
The output of rolling stock planning is to assign trains ie specifictrain configurations to each passenger trip to select deadhead tripsie ldquoemptyrdquo movements of the trains given by the constructed rollingstock rotation and to schedule maintenances and turn around activitiesof trains Passenger trips that are trips of the published timetable anddeadhead trips need to be assigned to crews which have to executethem We will describe this planning step in more detail in Section 8in case of an aviation company This demonstrates the power of generalmathematical modeling and methodology to different applications andthat the authors experience about that planning step comes from airlinecrew scheduling ie pairing optimization However recent work onrailway crew scheduling can be found in Abbink et al (2005) [1] andBengtsson et al (2007) [24]
23 Operational Planning
As already mentioned real time problems on the day of operation havequite different requirements even if these problems can be formulatedvery similar from a mathematical modeling point In railway trans-portation disruption and delay management is very difficult becauselocal decisions have a huge influence on the complete timetable systemNevertheless easy and fast rules of thumb are used to decide whichtrains have to be re-routed have to wait or even have to be canceledDrsquoAriano et al (2008) [72] and Corman Goverde amp DrsquoAriano (2009)[71] presented a real-time traffic management system to support localdispatching in practice On the basis of this renewed timetable rollingstock rosters and crew schedules have to be adopted see Clausen et al(2010) [69] Jespersen-Groth et al (2007) [122] Potthoff Huisman ampDesaulniers (2008) [177] Rezanova amp Ryan (2010) [182]
Every single step in this idealized sequential planning process is a diffi-cult task by itself or even more has to be further divided and simplified
2 Planning Process 16
into subproblems We will discuss several of them in the following sub-sections see how they can be modeled as combinatorial optimizationproblems and solved by state of the art solution approaches
The main application of track allocation is to determine the best opera-tional implementable realization of a requested timetable which is themain focus of this work But we want to mention that in a segregatedrailway system the track allocation process directly gives informationabout the infrastructure capacity Imaging the case that two trains ofa certain type ie two train slots are only in conflict in one stationA potential upgrade of the capacity of that station allows for allocat-ing both trains This kind of feedback to the department concerningnetwork design is very important Even more long-term infrastructuredecisions could be evaluated by applying automatically the track allo-cation process ie without full details on a coarse macroscopic levelbut with different demand expectations Even if we did not devel-oped our models for this purpose it is clear that suitable extensionsor simplifications the other way around of our models could supportinfrastructure decisions in a quantifiable way For example major up-grades of the German railway system like the high-speed route fromErfurt to Nurnberg or the extension of the main station of Stuttgartcan be evaluated from a reliable resource perspective The billions ofeuros for such large projects can then be justify or sorted by reason-able quantifications of the real capacity benefit with respect to thegiven expected demand
An obvious disadvantage of the decomposition is that the in some senseldquooptimalrdquo solution for one step serves as fixed input for the subsequentproblem Therefore one cannot expect an overall ldquooptimalrdquo solutionfor the entire system In the end not even a feasible one is guaran-teed In that case former decisions have to be changed and a partor the complete process has to be repeated Prominent examples areregional scenarios for urban public transportation where traditional se-quential approaches are not able to produce feasible schedules Weider(2007) [213] demonstrates in case of vehicle and duty scheduling howintegrated models can cope with that and even more can increase theoverall planning efficiency Nevertheless hierarchic planning partitionsthe traffic planning problem into manageable tasks Tasks lead directlyto quantifiable optimization problems and can be solved by linear andinteger programming to optimality or at least with proven optimal-ity gaps Problem standardization automatization organizing datacomputational capabilities mathematical modeling and sophisticated
3 Network Design 17
algorithmic approaches on a problem specific but also on a generallevel form the basis of optimization success stories in practice As aprominent example for this we refer to the dutch railway timetable -the first railway timetable which was almost constructed from scratchIn fact the entire planning process was decomposed and each planningproblem at Netherlands Railways (NS) was solved by the support of ex-act or heuristic mathematical approaches and sophisticated techniquesin particular linear integer and constraint programming More detailscan be found in the prizewinning work Kroon et al (2009) [140] whichwas honored with the Franz Edelman Award 2008 A prize which is re-warded to outstanding examples of management science and operationsresearch practice in the world
3 Network Design
Network design is the question of construction or modification of exist-ing railway infrastructure Railway infrastructure managers take theresponsibility for that planning step in close cooperation with publicauthorities
Infrastructure decisions are long term and very cost intensive especiallyin railway systems Typically an existing infrastructure has to be mod-ified due to changes of the travel demand capacity requirements andnew technologies The usual objective is to minimize the constructioncost while still ensuring the expected travel demand Nevertheless thisis a highly political planning step relying on uncertain future demandestimations The resolution of such problems is carried out in close co-operation with senior management of the infrastructure owner due tothe obviously high capital investment and the long lasting implicationsnot only for the entire company even for the (national) railway systemand for the affected cities as well Prominent example is the recentproject Stuttgart 21 that remains a subject of dispute in the publicrsquosview see Kopper (2010) [137]
Standard approaches for the travel demand estimations are interviewsof customers evaluation of ticket sales and various statistical meth-ods based on automated passenger counts All these methods are verycostly and time consuming But of course in the future more and moreof these data will be collected automatically and available for analysisHowever this can only be done for passenger traffic the estimation of
3 Network Design 18
future demand of cargo traffic is even more difficult and needs differentapproaches Furthermore in a segregated railway system this is con-fidential information of the railway undertakings see Figure 3 Never-theless the information that a railway infrastructure manager collectsduring the allocation process for the annual timetable can be used toidentify congested parts of the network or downsizing potential
A somehow exceptional and remarkable approach to railway networkdesign was realized in the project Rail2000 in Switzerland see Krauchiamp Stockli (2004) [138] and Caimi (2009) [57] There the sequential ap-proach was re-ordered the initial step was to define a service intentionie finish line planning and passenger timetabling at first to determinethe required infrastructure The major advantage is of course that therailway infrastructure matches perfectly to the explicit given serviceintention and is not based on coarse and aggregated demand forecastThe logical drawback is that the Swiss railway timetable at least forthe passenger traffic is a very stable entity for the future years Thecrucial assumption is that the demand is almost constant and the givenservice intention will change only slightly
To the best of the authors knowledge only network design approaches tointegrated railway systems can be found in the literature The complexsituation for a segregated railway system ie for an infrastructuremanager dealing with a lot of railway undertakings using the sameinfrastructure is not considered on a general optimization level Onlyseveral individual cases are discussed and analyzed as in Niekerk ampVoogd (1999) [168] and Romein Trip amp de Vries (2003) [184] Basicapproaches are using simulation tools to evaluate to analyze and tocompare some infrastructure possibilities as in Middelkoop amp Bouwman(2000) [161] and Klima amp Kavicka (2000) [133]
A framework for a general class of network design problems is presentedin Kim amp Barnhart (1997) [126] and applied to the blocking problemin railroad traffic in the US see Barnhart Jin amp Vance (2000) [19]Integrated service network design for rail freight transportation in theUS is considered in Ahuja Jha amp Liu (2007) [6] Jha Ahuja amp Sahin(2008) [124] Zhu Crainic amp Gendreau (2009) [218] In the next sectionwe will explain and discuss the network design problem for freighttransportation for the German case in more detail
Concluding we want to point out that future developments and re-quirements of a railway infrastructure network ie passenger or freightservice networks are very difficult to anticipate and highly political
4 Freight Service Network Design 19
driven A huge system knowledge and experience are preconditions forthese crucial long-term design decisions which are hard to quantify apriori as well as a posteriori However the models developed in thiswork can support railway companies in evaluating possible networkmodifications and measure their impact from a quantifiable capacitypoint of view - even if this is not the main focus of our work
4 Freight Service Network Design
Deutsche Bahn the largest German railway company primarily offerstwo products to industrial customers that want to transport freightvia rail Typically large customers order block-trains of about 20 to40 cars In this case Deutsche Bahn ie DB Schenker as the op-erator can pull such a complete train by a locomotive from origin todestination That is a direct freight transportation offer with a fixedtrain composition Small customers on the other hand order only 1to 5 cars In such case it is too expensive to pull this group of carsby a single locomotive through the network Instead the cars are onlypulled to the next classification yard There they are grouped with thecars from other customers and then as new trains pulled to the nextclassification yard There the trains are disassembled and the carsare again re-grouped with others until each car has reached its finaldestination This second freight transportation product of DB givesrise to a natural network design question ie where are the classifica-tion yards located and how to route between them Fugenschuh et al(2008) [95] and Fugenschuh Homfeld amp Schulldorf (2009) [96] discussthe whole system of single wagon freight transportation show the pos-itive effect of bundling cars and compare the problem to other freighttransportation concepts mentioned in the literature eg the railroadblocking problem in the US or Canada
The railroad blocking problem can be formulated as a very large-scalemulti-commodity flow-network-design and routing problem with bil-lions of decision variables see Jha Ahuja amp Sahin (2008) [124] andBarnhart Jin amp Vance (2000) [19] Ahuja Jha amp Liu (2007) [6] pre-sented an algorithm using an emerging technique known as very large-scale neighborhood search to support major US railway companies thattransfers millions of cars over its network annually The authors re-port that their heuristic approach is able to solve the problem to near
4 Freight Service Network Design 20
optimality using one to two hours of computer time on a standardworkstation computer
Due to some similarities to our modeling approach for railway trackallocation we want to explain the whole problem in more detail Theversion which we will present in the next paragraphs describes theoperational situation faced at DB Schenker Rail the largest Europeancargo railway transportation company We want to thank AlexanderBelow and Christian Liebchen for several discussions on that topic andsystem
41 Single Wagon Freight Transportation
The single wagon network N = (BR) is a graph that describes thelocal transport possibilities of single wagons in a railway system Allinbound tracks and sorting sidings on satellite terminals junction sta-tions and classification yards induce a node b isin B An arc r = (u v)with u v isin B exists if a train trip from u to v is possible
A shipment is an accepted order that consists of a number of singlewagons (with different weight length type etc) departure station andinterval (freight pickup definition) arrival station and interval (freightdelivery definition) and a measure of the service quality of the trans-shipment in terms of penalties for the deviation of the requirementsThe set of all shipments is denoted by S
A routing is an unique path in N for each origin and destination pairgiven as a routing matrix ie in some places depending on the wagontypes or time of the day The routing can equivalently be characterizedby a set of in-trees An in-tree is a directed graph with a so-called rootnode such that there exists exactly one directed path from each nodeto the root
A train slot denotes a concrete temporal allocation of an arc in N bya standard freight train with a given number of wagons maximumlength and maximum weight ie each slot f has a discrete departuretime df and an arrival time af T denote the set of all given slots Inthe German case we have to distinguish between three different typesof slots
1 safe slots with fixed timing eg by master contracts
2 optional slots with relatively safe timing eg system slots
4 Freight Service Network Design 21
3 (vague) requested slots with desired timing eg chartered orextra train (slots)
The network design part at DB Schenker consist of deciding whichof these timed slots should be requested from the network providerin order to run the system with a certain shipment quality and withminimal cost
A freight train trip or shortly trip denotes an allocation of a slot withan ordered set of at most k shipments Z denotes the set of all feasibletrips The set of all trips for slot f is denoted by Zf
In classification yards all single wagons will be rearranged with respectto the routing matrices ie they will be sorted and shelfed in thecorresponding siding Classification yards are made of three partsentry tracks sorting tracks and exit tracks There the freight train isdisassembled and the individual shipments are pushed over the humpentering the sorting tracks behind Each sorting track is assigned to anunique successor b isin B As soon as enough shipments are gathered onone sorting track this new train is pulled into the exit group Thereit waits until it can leave the yard and re-enter the network
The nodes of N represent a simplified model of these yards eg witha maximum shunting capacity per time interval In practice the shunt-ing procedure at the special yards is more restricted eg minimumtransition times minimum distances between arrivals and departuresfixed downtimes maximum operations per periods etc
A production schedule is an assignment of all shipments to feasibletrips such that the pickup and delivery definitions of all shipments canbe guaranteed In addition the production schedule ie the set oftrips has to respect the routing principles and all operation rules andcapacities at the classification yards
42 An Integrated Coupling Approach
The problem of finding a production schedule can be modeled as aninteger program with an exact representation of the given degrees offreedom The main challenge is to adhere to the FIFO principle Infact each trip that arrives in a yard has to be disassemble immediatelyEach shipment will arrive as fast as possible at their unique sortingyard and will depart directly with the next trip
4 Freight Service Network Design 22
The model belongs to a broad class of integer programs where a set ofpath systems are meaningfully coupled In that application transporta-tion paths of the shipments are linked with additional ldquoconfigurationrdquovariables ie variables for trip construction in the yards
The model is based on a trip scheduling digraph D = (VA) inducedby N that describes the transportation of the individual shipmentsin place time and position within a trip Each classification yard binduces an arrival track that models a waiting queue in front of theshunting hump
For each yard b isin B we associate an additional node b+ and severaladditional nodes bminuslowast that represents the different directions and sortingtracks to control the queue in front of the humping yard b Each arcr = (b x) isin R of the single wagon network N is also considered astwo arcs to handle sorting ie an arc from (b x) isin R induces (b+ bminusx )and (bminusx x
+) Let G = B+ cup Bminus the set of all those expanded nodesassociated with sorting on railway tracks [T ] = 0 T minus1 denotesa set of discrete times and [m] = 0 m minus 1 a set of possiblepositions of shipments within a trip
Thus a node
v = (g t i) isin V sube Gtimes [T ]times [m]
is a possible event modeling that a shipment arrives at track g time tand position i within a trip Moreover it is an arrival event if g isin B+ orotherwise a departure event The position of a shipment is relevant dueto the fact that we have to follow the FIFO principle at the classificationyards A larger position in a trip could result in a later departure fromthis classification yard The set V contains all these events as well asthe pickup and delivery of a shipment
Arcs of D model the transport of shipments at precise positions withinthe trip and the transition of shipments from the incoming track ofa yard to the sorting yards with all potential position changes Inaddition all local rules eg time restrictions can be incorporated inthat arc construction as well as the routing requirements
Figure 5 shows a possible block (train) composition q for slot f =(b 14 y 20 4) ie a train slot that departs at bminusx (b) and time 14 andarrives at y+ (y) at time 20 with a maximum of 4 shipments Two trainsarrive from xminusb at b+ within the considered interval and reach the siding
4 Freight Service Network Design 23
xminusb b+ bminusy y+
8
10 10
12
14
20
3-4 1-2
5-6
1-2-5-6
1-2
--5-6
time
Figure 5 Possible train composition for track f = (vr 14 wb 20 4)
to y via bminusy In the course of this the position of shipments changes egshipments 1 and 2 from position 3 and 4 in the first train trip to 1 and2 in the second The arcs associated with b+ and bminusy control the sortingwith respect to the routing matrix and the potential position changesof the shipments ie lowast denotes wild cards for first positions Theshipments 3 and 4 are not routed via y and therefore are not sortedon (b+ bminusy ) The proposed trip composition networks can obviouslybecome very large due to the ordering However the degree of freedomis somehow limited due to the fixed slots and routing principles ieonly certain positions are possible for the shipments
The optimization task is to minimize the cost of the slots and the costof the trip construction at the yards Any production schedule canbe represented in D by a set of feasible paths ie one for each ship-ment In the integer programming model the paths of the shipmentsare coupled with the construction of trips at the yards to respect theoperational rules and the shipment positions We will briefly explainthe formulation First we use trivial 01 variables xft to determinewhich trip t is used for slot f The idea of the modeling technique is tointroduce 01 variables yq to control the creation of trips and to force
5 Line Planning 24
the ldquorealrdquo operational routing of the shipments at the classificationyards by means of inequalitiessum
tisinZf
xft minussumqisinQf
yq = 0 forallf isin F
The set Qf can be interpreted as a certain subset of arcs in an auxil-iary graph that represents the construction of trip t in the departureyard of slot f On the hand if some trip t is selected for slot f by set-ting xft = 1 then the construction of that train in the departure yardmust be feasible which is ensured by setting the ldquorightrdquo variables yq toone On the other hand if trip t is not used on slot f all correspondingconfiguration variables yq have to be zero If no degrees of freedomfor selecting slots are given then this model only propagates the op-erational rules at the classification yards In addition an optimizedselection of slots is a strategic question that can be answered by thosemodels using a reasonable set of slots
That example serves only for motivational purposes of a general mod-eling technique that couples and integrates problems appropriately Inaddition it should give the reader some insights in the source of theparticular train slot requirements of a freight railway operator Sincetrain slots defined and used by single wagon freight service operatorsserve as direct input for track allocation problems
5 Line Planning
Once the infrastructure of the passenger transportation system is de-termined lines have to be defined and associated with individual fre-quencies A line is a transportation route between two designated butnot necessarily different terminal stations in the transportation net-work Usually there are some intermediate stops but especially in longdistance passenger railway transportation direct lines ie in Germanycalled Sprinter are used to offer very fast connections between majorcities A train line also includes the specification of the train type ietype of engine number of wagons and its frequency in case of regularperiodic services For example this can be four times an hour duringpeak-hour traffic and two times an hour in off-hour traffic The LinePlanning Problem is to select a set of feasible lines and their frequenciessubject to certain constraints and pursuing given objectives
5 Line Planning 25
5804
5806
5808
581
5812
5814
5816
5818
582
5822
Golmminus(P)minusBahnhof
NeuminusFahrlandHeinrichminusHeineminusWeg
BahnhofminusMedienstadtminusBabelsberg
BahnhofminusParkminusSanssouciPlatzminusderminusEinheit
MagnusminusZellerminusPlatz JohannesminusKeplerminusPlatz
Rathaus
SminusBabelsbergPost
Kirschallee
SminusPotsdamminusHbf
Figure 6 Visualization of line plan for Potsdam
In particular the line plan tries to meet the passenger travel demandand respect existing simplified network capacities and properties Com-mon but obviously contradictory objectives of a line plan are the min-imization of operating costs and the maximization of the service ortravel quality Travel quality or attractiveness of a line plan can bemeasured by the number of direct connections and travel times for pas-sengers But of course the passenger satisfaction of a line plan mainlydepends on the operated and experienced timetable implementing theline plan see Schittenhelm (2009) [186]
Significant work on line planning can be found for example in Bussieck(1997) [49] and Goossens van Hoesel amp Kroon (2006) [101] Laternovel multi-commodity flow models for line planning were proposedby Schobel amp Scholl (2006) [192] and Borndorfer Grotschel amp Pfetsch(2007) [35] Its main features in comparison to existing models arethat the passenger paths can be freely routed and lines are generateddynamically From a general perspective these models are also ldquocou-plingrdquo models The line variables provide ldquocapacitiesrdquo that passengerflow variables utilized for transfers
Properties of this model its complexity and a column-generation al-gorithm for its solution are presented and tested on real-world datafor the city of Potsdam Germany A recent research field is the incor-poration and handling of transfers eg the change-and-go model ofSchobel amp Scholl (2006) [192] However for large scale instances themodel is hardly computational tractable
6 Timetabling 26
Figure 7 Screenshot of visualization tool for public transport networks
Therefore Borndorfer amp Neumann (2010) [29] propose a novel ldquocom-pactrdquo integer programming approach to deal with transfer minimiza-tion for line planning problems even for larger instances Therein theyincorporate penalties for transfers that are induced by ldquoconnectioncapacitiesrdquo and compare a direct connection capacity model with achange-and-go model In Figure 6 a line plan for the city of Potsdamcan be seen each color represents one line
Finally the resulting line plan serves as a direct input for the periodictrain timetabling problem where valid arrival and departure times forthe given lines and frequencies have to be found However the finaldecision of which transport mode a user chooses depends on the avail-able options provided by the public transport network Figure 7 showsthe complete public transport network of the city of Potsdam ie bustram subway and city railway
6 Timetabling
The train timetabling problem has many names - such as train schedul-ing problem train routing problem or sometimes track allocation prob-lem The timetable which is the solution of the train timetabling prob-
6 Timetabling 27
lem is the heart of a public transportation system In the end this isthe offer a railway undertaking presents to the passengers In the caseof a freight train operator the corresponding train slots are the basisto implement and operate the transportation service
It is a main problem of the planning process of railway traffic - simplybecause it asks for the efficient utilization of the railway infrastruc-ture which obviously is a rare good In addition the service qualityof an offered timetable depends directly on the concrete allocation Ina segregated railway system additionally the crucial interconnectionbetween railway undertakings and infrastructure managers has to betaken into consideration
Nevertheless optimization models and techniques are not that widelyused for timetabling in practice in contrast to the subsequent resourceplanning problems ie vehicle and crew scheduling Most timetablesare minor modifications of their predecessors so that basically timeta-bles are historically grown One reason is that a timetable is notonly in Germany a huge political issue Whether a German city willget access to the system of long-distance passenger trains ndash high-speedtrains that are connecting important cities ndash will be decided in elon-gated negotiations between the railway operator DB Fernverkehr thefederal state and the German government ie the Federal Ministryof Transport Building and Urban Development (wwwbmvbsde) Aprominent subject of dispute in the recent years was the rather smallcity Montabaur that got access to the ICE transportation network Inan idealized world network design planning for long-distance passen-ger trains would answer such questions and provide the input for thetimetabling In addition decisions on the service quality of an urbanrapid transit system eg the Berlin S-Bahn will be preassigned andis mainly subsidized Lobbying swayed the decisions more than theresults of quantified analysis
In the following sections we will focus on three different aspects of time-tabling in more detail Section 61 will discuss the ongoing deregulationof the European railway market We give a brief literature review onperiodic and individual trip train timetabling in Section 62 FinallySection 63 will briefly discuss standard railway models of differentscale
6 Timetabling 28
61 European Railway Environment
Railway transportation services require very accurate planning of op-eration in contrast to other modes This is due to the fact that railwayundertakings have to promote their railway transportation services forpassengers far prior to the actual railway operation A published andonly rarely annually changed train timetable allows the customer to userailway transportation services efficiently Moreover uncontrolled rail-way operation is particularly prone to deadlocks Train drivers needto obtain the moving authority for a certain part of the railway in-frastructure from a centrally authorized controlling instance which as-sures a high level of safety An annual initial schedule helps to controlrailway operation since it reduces the vast complexity of real timeoperational planning Nevertheless the liberalization and introductionof competition in the European railway system will break down theseold-established and rigid structures in the near future However incomparison to airline transportation and urban bus transport the rail-way system is very rigid and hardly innovative
Furthermore railway systems consist of very expensive assets In or-der to make best use of these valuable infrastructure and to ensureeconomic operation efficient planning of the railway operation is indis-pensable Mathematical optimization models and algorithmic method-ology can help to automatize and tackle these challenges
In 2009 there were 300 railway undertakings operating in the Germansecondary railway market 60 of them do request railway capacity forpassenger trains From an economic perspective railway undertakingsoffer transportation services on the primary railway market Thus themarket where railway capacity is traded is called secondary railwaymarket
However DB Regio is still the biggest railway undertaking request-ing railway capacity for passenger trains In 2002 Deutsche Bahn AGestablished a ldquoCompetition officerrdquo in order to guarantee the correctimplementation of the European framework for railway capacity allo-cation
Within a competitive railway market the train slot requests submittedby concurrent railway undertakings are more likely to conflict This as-sumption is backed by current statistics of the competition reports ofthe German railway system The number of conflicting trains slot re-quests climbs from 10000 up to 12000 from 2008 to 2009 ie that is an
6 Timetabling 29
impressive increase of 20 In the same period the conflicts reportedby the Trasse Schweiz AG for the allocation process in Switzerlandincrease from 103 to 127
A detailed discussion of the legal environment of the European railwaymarket can be found in Mura (2006) [164] and Klabes (2010) [129]In there all European directives and legal definitions are given as wellas various references to the discussed statistics We will summarizethe most important facts Article 18 of the EU Directive 200114ECcontains all relevant deadlines for the capacity allocation process in theEuropean railway system Of course some flexibility is given to thenational infrastructure managers They can determine these deadlineswithin certain tolerances However they have to publish them so thatthey are available to all licensed railway undertakings to establish afair and open-access market The main regulations are listed in thefollowing
The working train timetable shall be established annually
Infrastructure managers have to declare a specific date and timewhen the shift of one train timetable to the new one takes places
The final date for receipt of annual train slot requests must notbe earlier than 12 months before the new timetable is operated
Not later than 11 month before the new timetable is operatedthe infrastructure managers shall ensure that the internationaltrain slot requests have been allocated provisionally2
Four months after the deadline for submission of the annual trainslot requests by railway undertakings a draft timetable shall beprepared
Furthermore four types of slot request are to be distinguished
long term train slot requests
international train slot requests
annual train slot requests
and ad hoc train slot requests
The planning time horizon which is the time period between the datewhen a train request is submitted and the date when the train pathrequest is included into the working timetable are from 5 up to 15
2The allocation of international train slot requests should be adhered to as faras possible because at least two different national railway infrastructure managersand one railway undertaking are involved
6 Timetabling 30
years in case of long term slot requests This shall insure reliabilityfor the future planning of railway infrastructure managers and railwayundertakings by so called framework agreements International trainslot requests require capacity from at least two different internationalrailway infrastructure providers Annual train path requests have to besubmitted annually to be included into the annual timetable They canbe requested until a deadline that can be determined by the infrastruc-ture manager usually 8 months before the new timetable is operatedDue to the necessary cooperation between the concerned national in-frastructure managers an independent organization RailNetEurope
(wwwrailneteuropecom) was set up International train slot re-quests are directly submitted to RailNetEurope which is responsiblefor the coordination between the involved national infrastructure man-agers
Ad hoc train slot request are as the name already suggest submitted atshort notice In particular this applies to cargo trains which are plannedin a much more flexible way than passenger trains Such train slots arerequested from two weeks to 24 hours in advance In Figure 8 only thebeginning of ad hoc requests concerning the new annual timetable isshown Ad hoc requests for the actual timetable are of course possibleat any time
Most infrastructure managers already plan suitable train slots some-times called system slots in advance without binding them to a specificrailway undertaking In case of ad hoc slot requests or individual slotrequests in the course of the year such anticipated system slots canbe assigned Deciding how much capacity should be reserved a priorifor those ad hoc requests is by no means trivial Of course this isalso done due to the complex planning even for the case of only oneadditional single slot We see a huge potential to support this task byoptimization models and algorithms A reliable track allocation modeland solver could easily analyze the effect of adding another slot with-out the price of time-consuming simulation runs Moreover we willpresent a general approach that guarantees the re-transformation ofthe optimization results into the simulation frameworks
The procedure of capacity allocation is illustrated in Figure 8 Thedeadlines denoted by xminus 11 and y as well as the interaction betweenrailway undertakings (RU) and infrastructure managers (IM) can beseen The first month of operation of the timetable is denoted by xIn addition we highlight the stage where the infrastructure managers
6 Timetabling 31
deadline for internationaltrain slot requests
deadline for annual trainslot requests
draft timetable isestablished
annual timetable is inoperation
x-11
y
y+4
x
time
RU
RU
RU IM
RU
international andlong term requests
annual requests
coordinationphase
ad hoc requests
Figure 8 Timeline for railway capacity allocation in Europe source Klabes(2010) [129]
have to solve track allocation problems Of course the internationallong term and the annual requests can also be planned at the point ofsubmission but conflicts at that time are very rare In the end of theprocess a working (annual) timetable or track allocation is determinedTherefore the names train timetabling and track allocation problemare used for essentially the same problem only the point of view differsOn the one hand railway undertakings are interested in their acceptedslots to offer a suitable timetable for their various purposes On theother hand infrastructure managers are interested in a high and sta-ble utilization of the network by the complete allocation of all railwayundertakings Finally long term international and annual requestsare considered in a draft train timetable at y + 4 Due to the limitedrailway infrastructure capacity the occurrence of conflicts is very likelyespecially in highly frequented parts or bottlenecks However in the
6 Timetabling 32
coordination phase of the railway capacity allocation process all con-flicts have to be resolved This is were optimization can significantlysupport the planning process Even more is required by most Euro-pean directives and laws In Germany sect9 passage 5 of the Regulationfor the use of railway infrastructure see Federal Ministry of Transportamp Housing (2005) [86] states
ldquoThe network provider has to compare the charges to de-cide between equally ranked types of traffic under the termsof passage 4 In case of a conflict between two train slotrequests the one with the higher charge takes or has pri-ority in case conflicts between more than two train slotrequests the allocation or choice with the highest charge intotal takes or has priorityrdquo (translation by the author)
In a first step the infrastructure managers try to resolve the occur-ring conflicts as best as they can In particular slot requests that areinvolved in conflicts are altered Of course when realizing an exact op-timization approach with all ldquodegrees of freedomrdquo it can occur that thebest decision affects also slots that are not directly in conflict beforeIn Figure 9 a trivial situation is shown Each line represents a trainrun on track j from left to right ie the boxes on the sides representthe connecting stations Imagine that the first and the last train (blue)are already scheduled and the other train (red) requested to run onj at the depicted time On the left hand side one can see that onlythe last two trains are in conflict on j ie the crossing of both linessymbolizes a ldquocrashrdquo at that time As a result sticking exactly to therequested times leads to a schedule with maximal two trains Howeveron the right hand side one can see a solution that allows to run alltrains by choosing slightly earlier departure times for the first ones Infact we assume that the slot contracts for the train slots allow for thepropagted departure shift ie we choose an arbitrary safety distanceto avoid crossings
This requires the coordination and cooperation between railway in-frastructure managers and all those railway undertakings whose trainpaths need to be altered Usually at the end of this process a con-flict free draft timetable is determined However in some cases trainslot requests are rejected in the coordination phase It is clear thatthere is some discrimination potential and therefore independent agen-cies are in charge of controlling these procedures eg in Germany
6 Timetabling 33
j
tim
e
j
Figure 9 Simple conflict example and re-solution for track allocation
MacroscopicTimetabling
Periodic
Quadratic semi-assignment
PESP
Tailored methodsMixed IntegerProgramming
Non periodic
HeuristicsMixed IntegerProgramming
Figure 10 Principal methods in the literature for macroscopic timetabling byCaimi (2009) [57]
the Federal Network Agency (Bundesnetzagentur) see httpwww
bundesnetzagenturde
62 Periodic versus Trip Timetabling
Lusby et al (2009) [159] give a recent survey on the track allocationproblem and railway timetabling Nevertheless we want to enlightensome aspects and present a general classification according to solutionmethods used by Liebchen (2006) [148] and Caimi (2009) [57] In Fig-ure 10 the approaches on macroscopic railway timetabling are basicallydivided into two categories periodic and non-periodic scheduling
6 Timetabling 34
621 Periodic Timetabling
Periodic timetables are first and foremost used for passenger trafficEven if there are some works on quadratic semi-assignment modelseg Klemt amp Stemme (1988) [131] most authors consider anothermodel the Periodic Event Scheduling Problem (PESP) It is a powerfuland well-studied model for macroscopic scheduling Serafini amp Ukovich(1989) [199] introduced a general version and Schrijver amp Steenbeck(1994) [194] applied it at first to train scheduling Since that time thePESP has been intensively studied and many extensions and variantswere presented see Odijk (1997) [169] Lindner (2000) [154] Kroonamp Peeters (2003) [141] Kroon Dekker amp Vromans (2004) [142] andLiebchen amp Mohring (2004) [150] The PESP model was successfullyapplied as the core method for the generation of the 2005 timetableof the Berlin underground see Liebchen (2006) [148] and Liebchen(2008) [149] and for the generation of the 2007 railway timetable inthe Netherlands Kroon et al (2009) [140] Furthermore commercialsoftware eg TAKT see Nachtigall amp Opitz (2008) [165] based on thePESP model was developed and entered the market The degrees offreedom for PESP are on a global interacting level between the trainsIt is always assumed that the route or path is already decided ieall headway parameters are calculated under this fixed assumptionas well as the connection times inside the stations Furthermore it isexpected that all trains can be scheduled with respect to their frequen-cies otherwise the complete problem is stated to be infeasible Thisdisadvantage of the model formulation was for a long time negligibledue to sufficient capacity for appropriate scenarios Obviously from anoptimization point of view this has to be revisited and at least feedbackon locals conflicts has to be given which is one particularity of TAKT
Recent research work focuses on the integration of robustness aspectssee Odijk Romeijn amp van Maaren (2006) [170] Kroon et al (2006)[139] Cacchiani et al (2008) [53] Liebchen et al (2009) [152] Liebchenet al (2010) [153] and Fischetti Salvagnin amp Zanette (2009) [91] aswell as integration of flexibilities to improve the interaction betweenmacroscopic and microscopic scheduling see Caimi (2009) [57] andCaimi et al (2007) [59] The contributions of Caimi (2009) [57] aremainly in the area of integrating and improving the interaction be-tween microscopic and macroscopic models for planning passenger traf-fic The idea and goals can be found in Burkolter Herrmann amp Caimi(2005) [48] For example the extension of the PESP to flexible event
6 Timetabling 35
times (FPESP) allows for more degrees of freedom in the subsequentmicroscopic scheduling
The (passenger) timetable itself is the core of all railway activitiesFrom a historical and from a customer point of view national rail-way operators offer almost exclusively periodic timetables for passen-ger traffic On the one hand this is much easier to remember andrecall for passengers and on the other hand the whole process of de-termining a valid timetable becomes much easier ie the planning ofall system-oriented components like infrastructure rolling stock andcrews Furthermore most people expect symmetric transport chains ifthey make a round trip An historical overview is given in Figure which demonstrates the dominance in European subway and railwaysystems today Summarizing a periodic timetable is easy to use easyto understand and easier to operate
However Borndorfer amp Liebchen (2007) [28] showed in a theoreticalwork that periodic timetables can become inefficient compared to triptimetables from an operator point of view Sub-optimality and ineffi-ciency of periodic timetables are accepted and well known Even morespecializations such as synchronized periodic timetables (ITF) are pop-ular in practice and usually used for passenger traffic A synchronizedperiodic timetable is a periodic timetable that additionally providesreasonable transfer times at certain stations
In our rapid growing information society the reasons for periodicitycould become negligible in the future The development in traffic engi-neering of traffic management systems will bring more and more help-ful real-time information to the passengers as well as to the operatorsThe necessity of easy manageable timetables will then cease to applyin the future If an acceptance for non-periodic and fully individual ordemand dependent timetables increases railway operators could offermuch more efficient timetables A trend which can already be observedfor large public events in sports music and so on Deregulation andcompetition will assist this development as well
In a future world of full and real-time available information passengerswill not be insistent that trains have to be scheduled with a fixed cycleperiod More important will be that the timetable covers the demandefficiently and reliably The frequency in peak hours has to be higherbut it will not be mandatory that departure and arrival times will followan exact periodic pattern as long as enough connections are providedThe service quality experienced by the passenger depends more on the
6 Timetabling 36
reliability of the service ie the deviation between expected waitingtimes and real waiting times
Let us discuss timetabling from a passenger traffic perspective The lineplanning determines passenger lines with their frequencies for differentdemand periods ie the lines can be different in peak hours or onweekends The task of timetabling is now to define exact arrival anddeparture times eg in minutes at each station of the lines It is clearthat the requirements and constraints are somehow different to the onesof freight traffic especially in contrast to long-distance railway servicesPassenger trains have in general a fixed stopping pattern with respectto the line definition and of course a tight dwell time interval to fulfillOne the one hand maximum dwell times are needed to offer passengersfast services On the other hand they have to be at least large enoughto allow for transfers ie desired and favorite connections of differentlines at certain major stations For freight railway traffic the situationis different and other aspects mainly affect the service quality egrequired arrival times at certain stations and long possession timesare needed to perform shunting and loading activities The costs for afreight train are much more unpredictable due to the fact that brakingunforeseeable stops and acceleration have a huge effect on the energyconsumption and the total running time
622 Non periodic Timetabling
For networks where freight traffic is predominant and for freight traf-fic in general non-periodic macroscopic timetables are broadly usedAlready in the 1970s Szpigel (1973) [206] studied this problem andproposed a mixed integer programming formulation Later many tech-niques like constraint programming by Silva de Oliveira (2001) [201]Oliveira amp Smith (2001) [171] and Rodriguez (2007) [183] artificial in-telligence approaches by Abril Salido amp Barber (2008) [2] and resourceconstrained formulations by Zhou amp Zhong (2007) [217] were appliedProblem or even case specific heuristic approaches were developed egCai amp Goh (1994) [55] Cai Goh amp Mees (1998) [56] Higgins Kozanamp Ferreira (1997) [115] Dorfman amp Medanic (2004) [76] Ghoseiri Szi-darovszky amp Asgharpour (2004) [98] Semet amp Schoenauer (2005) [198]Lee amp Chen (2009) [146] and Zheng Kin amp Hua (2009) [216] How-ever the most popular and successful solution approaches are integerprogramming based formulations as proposed in the seminal works ofBrannlund et al (1998) [44] and Caprara et al (2006) [63] The most
6 Timetabling 37
important advantage of exact optimization approaches is that in addi-tion to solutions also a guarantee on the solution quality is given Thisallows for precise estimations on optimization potential for the variousplanning challenges
Freight transportation is innately non-periodic ndash a large number ofoperated freight or cargo trains are even not known at the beginningof the timetable planning process Only for some standardized typesof cargo trains slots will be allocated or reserved - later these slotswill be assigned to the real operating trains and an adaption of theschedule has to be done The reason is that the exact weight andlength of a train which is committed only a short period before theday of operation is needed to compute realistic running times Thusthis can lead to some minor changes of the scheduled departure andarrival times of these trains and probably also for other trains due tosafety margins and headway times Modeling the railway safety systemwill be described in detail in Chapter II
One of the earliest publications on the optimization of trip train sched-ules is from Szpigel (1973) [206] The focus of his work is a long singletrack railroad in eastern Brazil which is used by trains to transportiron ore in both directions The line is divided into a number of tracksections with each track section linking two stations In stations ad-ditional tracks are available to allow trains to stop or overtake eachother The main contribution of the author is to identify strong simi-larities between train scheduling problems and the well known job-shopscheduling problem In the train scheduling context trains can be seenas jobs They require the use of several track sections that are the ma-chines to complete their designated route To prevent track sectionsfrom hosting more than one train operation at any given time order-ing constraints are introduced Finally he solves the problem with abranch and bound approach until reaching a feasible meet and passplan Nowadays we would call this method a lazy constraints approachthat ignores the ordering constraints in the linear relaxation and thenbranch if the solution contains trains in conflict However models andtechniques presented in that work for a simple single line are the basisof considering complicated routing situations
Later enumeration based methods were used by Sauder amp Westerman(1983) [185] and Jovanovic amp Harker (1991) [125] to construct feas-ible meet and pass plans based on a MIP approach To the best ofour knowledge the model and algorithm of Jovanovic amp Harker (1991)
6 Timetabling 38
[125] was the first one which leads to a software system that alreadyincludes a simulation modul to work with reasonable times for the trainmovements
Carey amp Lockwood (1995) [65] consider an almost identical network tothat of Szpigel (1973) [206] but propose a different modeling and solu-tion approach The authors present a large MIP formulation similar tothat of Jovanovic amp Harker (1991) [125] Each binary decision variablecontrols the order of a pair of trains on a given track section
Cai amp Goh (1994) [55] propose a simple greedy heuristic for the sameproblem The heuristic considers trains in chronological order and as-sumes that the start time and location are known Later in Cai Gohamp Mees (1998) [56] the authors extend their work to the case that theinitial location of a train is fixed A successful implementation of thealgorithm is reported for an Asian railway company where up to 400trains run per day with as many as 60 trains in the system at any giventime
Brannlund et al (1998) [44] introduce the notion of packing constraintsto restrict the number of trains using any track or block section to atmost one instead of control the order explicitly This work can beseen as the first resource based model approach to the track allocationproblem The authors propose a set packing integer programming for-mulation to solve the problem for a bidirectional single line connecting17 stations in Sweden An acyclic time-space network consisting ofdifferent arc types is use to model each trainrsquos movement Paths in thetime-space network reflect different strategies for the associated trainto complete its itinerary The scheduling horizon is discretized intointervals of one minute each The objective is to maximize the profitsof the scheduled trains with a penalty for unnecessary waiting timesThe author suggests to solve the problem with Lagrangian relaxationtechniques After relaxing all packing constraints the problem decom-poses into n independent subproblems where n is the number of trainsTo construct integral solutions a train priority based heuristic is usedand performs well for the considered instances ie solutions with anoptimality gap of only a few percent are reported A comprehensivesurvey of optimization models for train routing and scheduling up tothe year 1998 is given by Cordeau Toth amp Vigo (1998) [70]
Caprara et al (2001) [61] and Caprara Fischetti amp Toth (2002) [62]further developed the graph theoretical formulation using an event ac-tivity digraph In addition the authors proved that the classical stable
6 Timetabling 39
set problem can be reduced to TTP such that the problem isNP hardIndeed the optimal track allocation problem can be seen as a problemto a find a maximum weight packing with respect to block conflictsof train routes in a time-expanded digraph This framework is fairlygeneral see further articles by Cacchiani Caprara amp Toth (2007) [52]Cacchiani Caprara amp Toth (2010) [54] Fischer et al (2008) [90] andCacchiani (2007) [51] for comprehensive discussions how such a modelcan be used to deal with various kind of technical constraints
Finally Table 2 lists the sizes of the largest instances solved so far bythe various authors The research of Fischer et al (2008) [90] andFischer amp Helmberg (2010) [89] focus primarily on solution techniquesfor relaxations of the problem ie we marked scenarios for which onlyheuristic solutions are reported However a fair comparison is not onlycomplicated by the different scale of the models In particular Lusby(2008) [158] and Klabes (2010) [129] consider microscopic railway mod-els In fact several additional parameters determine the degrees of free-dom and the computational tractability of any TTP instance Here isa short list of the most important ones
routing possibilities within the network
discretization of time
selection of train types
options for running times
time windows of arrival and departure events
complexity of the objective function
and flexibility to let trains stop and wait
623 Conclusion
We conclude with the vision that train schedules will be become moreand more flexible in the near future Information systems and state ofthe art optimization techniques will allow track allocation problems tobe solved for real world application Hence infrastructure managerswill be able to improve the solutions of the coordination phase Morescenarios can be handled and additional cargo requests or ad hoc re-quest will be answered much faster That will lead to a more efficientutilization of the infrastructure Even a completely different handlingand marketing process of ad hoc requests is imaginable to take advan-tage of the new allocation possibilities Furthermore railway opera-tors will be able to react faster on major demand changes in passenger
6 Timetabling 40
reference stations tracks trains
Szpigel (1973) [206] 6 5 10Brannlund et al (1998) [44] 17 16 26Caprara Fischetti amp Toth (2002) [62] 17 16 221
102 101 41Cacchiani Caprara amp Toth (2007) [52] 17 16 221
102 101 41Cacchiani Caprara amp Toth (2010) [54] 65 64 775Fischer et al (2008) [90] 104 193 251Fischer amp Helmberg (2010) [89] 104(445) 193(744) 137Fischer amp Helmberg (2010) [89] 1776 3852 3388
Lusby (2008) [158] (microscopic) asymp 120 524 66Klabes (2010) [129] (microscopic) 2255 2392 32
Chapter IV Section 4 (microscopic) 1154 1831 390Chapter IV Section 4 18 40 390Chapter IV Section 1 37 120 1140
Table 2 Sizes of the solved instances in the literature for the TTP instance
transportation ie the offered timetable will be more flexible Oneprediction for instance is that innovative railway infrastructure man-agers will be able to construct creative solutions and hence will beable allocate ldquomorerdquo train slots As a result railway operators willmore and more rely on ad hoc slots and also become more flexible indesigning their timetables and their operations However we proposethat the railway system needs some time to implement this flexibilityWe rather assume that primarily railway infrastructure managers willuse mathematical optimization models to evaluate more strategic andtactical planning questions concerning track allocations
The highly dynamic aviation environment is the perfect role model ofa free market where the competitors have to satisfy the customersdemands and have to anticipate innovation potential - otherwise thecompetition will squeeze them out of the market The ongoing Euro-pean liberalization of railway traffic will support this process It is notclear that this process can be successfully finished and ldquorealrdquo competi-tion will be introduced ndash however railway transportation has to find itsway to establish efficient offers to compete with the other transporta-tion modes The integration of state of the art mathematical modelingand optimization techniques can immediately support the allocationprocess of railway capacity
6 Timetabling 41
63 Microscopic versus Macroscopic Models
The level of detail of a railway infrastructure or operation model de-pends on the quality and accuracy requirements for generating appro-priate results and of course on the availability and reliability of thedata For long term and strategic planning problems high accuracydata is often not manageable might not exist or can not be providedon time without causing expenditure eg Sewcyk (2004) [200] Inaddition it makes no sense to deal with highly detailed railway mod-els if the question to answer will relate only on some parameters Aprominent example is timetable information where neither the rail-way infrastructure or the rolling stock have to be observed preciselyMoreover formal and legal reasons might prohibit free access to highlydetailed infrastructure data that are classified as essential facilities bysome European railway infrastructure managers These are reasonswhy models of different scale has been established
Microscopic models require high detailed data to produce reliableand high quality results ie for running time calculation and thesimulation of timetables and railway operations
Mesoscopic models are produced if no microscopic data is avail-able standard assumptions are made for missing microscopic el-ements They are used in most eastern European countries thatdo not want to put a lot of effort in generating and maintaininga microscopic database
Macroscopic models embrace coarse and aggregated structuresreal-world applications are vehicle circulation long term trafficplanning strategic infrastructure planning and travel informa-tion systems
Obviously optimization on a microscopic level is still inconceivable dueto the enormous size and granularity of the data Even more it is notnecessary because the decision to run a train or let a train wait can bedone on a macroscopic level that is based on microscopic evaluationsFor example all macroscopic running times are deduced by microscopicsimulation data assuming a standard acceleration and braking behav-ior of the standard train compositions Thus all relevant switchesinclines curves or other velocity impacts are considered implicitly
The literature has suggested a number of top-down approaches egKlemenz amp SSchultz (2007) [130] and Caimi (2009) [57] In a top-downapproach to model railway systems an overview of the entire system is
7 Rolling Stock Planning 42
first formulated specifying but not detailing any ldquorealrdquo sub-systems Atop-down model is often specified with the assistance of ldquoblack boxesrdquoHowever black boxes may fail to elucidate elementary mechanisms torealistically validate the model Solving track allocation problems isonly useful if the railway system is modeled precisely with respect toresource consumptions ie the calculation of running and headwaytimes must be incorporated in detail
The focus of Chapter II will be to develop a novel bottom-up approachfor automatic construction of reliable macroscopic railway models basedon very detailed microscopic ones We will start with a realistic mi-croscopic railway model that indeed might be too large to be solvedin reasonable time to optimality However this model could be sim-plified and aggregated by well defined rules and error estimations ierunning and headway times are incorporated almost exactly This ap-proach turns out to be more reliable and thus more convincing thancontrary top-down approaches that try to integrate more and moredetails in weak and questionable base models
7 Rolling Stock Planning
The goal of the rolling stock planning the vehicle scheduling problemor the aircraft rotation problem is to find a cost minimal assignment ofrolling stock vehicles or aircrafts to the trips stemming from the time-tabling Input for the rolling stock planning are the timetabled tripsand the possible deadhead trips of the vehicles the rolling stock or theaircrafts The timetabled trips are the trips that transport passengersDeadhead trips give the possible concatenation of timetabled trips intorotations The set of timetabled trips and deadhead trips together issimply called trips Each trip has a start- and end-time and a start-and end-location further we need to know the length and the drivingtime of each trip The problem naturally give rise to a rolling stockscheduling graph That is a standard event activity digraph represent-ing space and time In the following we want to discuss the specialproblem of vehicle scheduling (VSP) in urban public transport Thecost of a vehicle schedule is composed of a fixed cost per used vehiclecost per driven distance and cost per time away from a depot of avehicle
8 Crew Scheduling 43
An extensive literature survey of the VSP until 1997 can be found inKokott amp Lobel (1997) [135] Kliewer Mellouli amp Suhl (2006) [132]and Steinzen et al (2010) [203]
The set of available vehicles is called a fleet The maximum number ofvehicles used can be a constraint of the VSP or be part of its resultEach vehicle has a unique vehicle type Typical vehicle types in caseof bus traffic are standard bus double decker or articulated bus Eachvehicle type has a set of characteristics which is relevant for the plan-ning process such as the number of seats an average speed minimummaintenance intervals or maximum length of covered distance with-out refueling Not all vehicle types are able to service all trips Forinstance long buses cannot go around narrow curves double deckersmay not pass low bridges or a larger bus is preferred for trips with highpassenger volume Each vehicle of a fleet is associated with a uniquegarage at a certain location Each garage contains vehicles of varyingtypes in certain quantities We call a vehicle typegarage combinationa depot We may have a maximum number of vehicles of certain typesper garage or in total These numbers are called capacities of the de-pots or vehicle type capacities Obviously similar restrictions are givenin case of planning aircraft rotations or rolling stock rotations
A rotation sometimes also called block is an alternating sequence ofdeadhead and timetabled trips that begins and ends in the same depotRotations can be combined to courses A course is a set of rotationsthat can be driven by a single vehicle We call a set of courses thatcovers all timetabled trips a vehicle schedule
State of the art solution methods for large real-world instances of thevehicle scheduling problem are either based on Lagrangian relaxationheuristics see Kokott amp Lobel (1997) [135] or by heuristic prepro-cessing and solving the resulting problem by standard MIP solvers asproposed by Kliewer Mellouli amp Suhl (2006) [132] Finally Figure 11shows a partial vehicle scheduling graph for a rolling stock scenarioie only the passenger trips are visualized as arcs in a standard week
8 Crew Scheduling
The crew scheduling problem arises not only in railway traffic but alsoin urban public transport and airline transportation From a practicalpoint of view these problems may all differ in their structure needs
8 Crew Scheduling 44
Figure 11 A partial cyclic rolling stock rotation graph visualized in our 3D visu-alization Tool TraVis using a torus to deal with the periodicity
rules and especially their sizes From a theoretical mathematical pointof view they can be formulated as a general model and solved by equiv-alent techniques with a proven optimality gap for almost all practicalrelevant sizes - even for very large scale instances
That is one reason why we will discuss this problem in the followingparagraphs Another one is that the author gathered many valuableexperiences in solving large-scale airline crew scheduling problems inpractice The corresponding mathematical optimization model andsome key constructions are shown in detail Finally the general algo-rithmic solution approach is presented
81 Airline Crew Scheduling
We refer to Barnhart Belobaba amp Odoni (2003) [20] for an overview onairline optimization in general and on airline crew scheduling Opera-tional cost for crews are a huge cost factor for every aviation companyin the world Complex rule systems by the government as well asby specific labor unions home-base capacities and balancing require-ments to support the subsequent rostering process lead to very largescale combinatorial optimization problems The goal is to find a costminimal set of duties which cover all relevant legs ie the plannedflights of the airline and fulfills all home-base capacities
8 Crew Scheduling 45
We denote the set of relevant legs by T and the set of home-bases thatare locations of available crews by H We partition all possible dutiesor crew pairings as it is called in the airline industry with respect totheir home-bases ie the start and end location of a pairing must bethe same Let P be set of all pairings with P = cuphisinHPh
82 Crew Scheduling Graph
The crew scheduling problem can be described in terms of an acyclicdirected network G = (VA) The nodes of G are induced by the set oftimetabled flights in railway or bus application by the set of timetabledtrips These are tasks t isin T that has to be performed by personnel ina feasible crew schedule Additionally there are nodes s and t whichmark the beginning and the end of pairings called sink and sourcenodes of G Supplementary tasks can also be considered in G such asflight transport also called deadheads or ground transport We willlater discuss how to handle them implicitly a posteriori
The arcs A of G are called links they correspond to possible directconcatenations of tasks within pairings In addition there are artificiallinks that model valid beginnings or endings of pairings An arc (u v) isinA represents the consecutive processing of task v after u by a pairingtherefore local rules with respect to time and location eg minimaltransfer times or ground times can be handled by the constructionof the graph ie by the definition of the arc set However mostof the pairing construction rules concern the complete pairing suchas maximal landings per pairing minimal and maximal flight timeminimal number of meal breaks and many more We denote by R theset of consumption rules and Ur the upper limit An easy example forsuch a graph is given in Figure 12
Each feasible pairing corresponds to a path in G Unfortunately somepaths may violate the construction rules ie assume in example graphshown in Figure 12 a maximal number of landings of at most two thenthe path p = (s AminusB) (AminusBB minusC) (B minusCC minusA) (C minusA t)is infeasible We will come back to details on pairing generation inSection 85 after formulating the crew scheduling problem as an setpartitioning problem
8 Crew Scheduling 46
s t
A-B B-C C-A
A-C C-B B-A
C-B B-A
artificial node
task node
artificial arc
connection arc
time
Figure 12 Crew Scheduling Graph
83 Set Partitioning
We introduce a binary decision variable xp for each pairing p isin P which is 1 if pairing p is chosen or 0 otherwise To each pairing whichis nothing other than a sequence of tasks (and additional elements likedeadheads ground transports meal breaks etc) We denote by cp acost value If we have restrictions on the number of available crews ona home-base h we introduce a so called base constraint and an upperbound κh Obviously this is the most simple case of a base constraintThere are much more complex rules per day and per pairing type oreven balancing requirements which can be handle in reality Althoughthis leads to base constraints we skipped the details on that for simpli-fication We refer to Borndorfer et al (2005) [33] there the definitionof general linear base constraints with arbitrary coefficients is shownin detail to synchronize crews by using base constraints In additionwe report in that paper on the solution of real world instances for crewscheduling with some thousands tasks Moreover our algorithmic ker-nel has been integrated in the planning system NetLineCrew of thesoftware company Lufthansa Systems GmbH In Figure 13 a screenshotof the planning tool NetLineCrew of Lufthansa Systems GmbH canbe seen
8 Crew Scheduling 47
Figure 13 Set of legs (above) and a set of covering pairings (below) show as aGant chart in the planning tool NetLine
(SPP) (i) minsumpisinP
cpxp
(ii)sum
pisinPtisinp
xp = 1 forallt isin T
(iii)sumpisinPh
xp le κh forallh isin H
(iv) xp isin 0 1 forallp isin P
The objective function (i) minimizes the sum of pairing costs Con-straints (ii) ensure that each task t isin T is covered by exactly onepairing p To ensure feasibility we can assume that there is a ldquoslackrdquopairing type with single-leg parings of high cost M
Sometimes it is also possible to relax these to covering constraintsThis allows more than one pairing to contain each task Then in apost-processing step the decision of which crew really processes thetask and which is only using it as a flight transport has to be taken
8 Crew Scheduling 48
But we want to point out that this can only be done if this change doesnot violate the pairing construction rules eg a number of maximalflight transports can not be controlled anymore and may be violatedThat no homebase capacity κh will be exceeded is guaranteed by con-straints (iii) Finally we require that each variable xp is integer to getan implementable crew schedule
84 Branch and Bound
Ignoring the integrality constraints (SPP) (iv) will lead to a well knownlinear programming relaxation which we denote by (MLP) This modelis used to derive a strong lower bound on the optimal value Unfor-tunately the solution of the relaxation can and will probably be frac-tional so that we have to divide the problem into several subproblemsThe construction of the branches has to ensure that the optimal so-lution of (SPP) will be feasible in at least one new subproblem Thelinear relaxation bound of the subproblems can only increase due tothe new domain restrictions A good branching decision is a crucialpoint in solving integer programs ie for (SPP) constraint branchingproposed by Foster amp Ryan (1991) [92] is much more effective thansingle variable branching Another successful branching rule for thesekind of problems is to choose a large subset of variables to fix to onebased on perturbation techniques see Marsten (1994) [160] Wedelin(1995) [211] and Borndorfer Lobel amp Weider (2008) [37] This can beseen as diving heuristic trying to evaluate different parts of the branchand bound tree in a strong branching flavor to detect a so called mainbranch In Chapter III and Section 3 we will highlight this idea inmore detail and utilize it to solve large scale track allocation instances
85 Column Generation
Unfortunately the number of possible pairings p isin P is too large evento write down the model (MLP) Only for a small number of tasks tocover it may be possible to enumerate all pairings However we cansolve this optimization model by using a sophisticated technique calledcolumn generation The idea was first applied to the crew pairing prob-lem by Barnhart et al (1998) [18] and is as simple as effective Letus therefore recapitulate the main steps of the simplex algorithm tosolve linear programs During the simplex algorithm a solution of a
8 Crew Scheduling 49
Initialize (RMLP)
Minimize (RMLP)
Solve Pricing Problem
Variablefound
Update (RMLP)
(MLP) solved
Yes
No
Figure 14 General column generation approach to solve LPs with a large columnset
linear program will only be improved if a non-basic variable with neg-ative reduced cost can be added to the basis (in case of a minimizationproblem) This pricing step can also be done without constructing allvariables or columns explicitly Let us start with an appropriate subsetof variables then the linear relaxation denoted by restricted master(RMLP) is solved to optimality Only a non-considered variable canimprove the current solution of the relaxation - if we can show thatthere is no variable left with negative reduced cost we have proven op-timality for (MLP) without even looking at all variables explicitly Dueto the fact that we add the necessary variables columns of (RMLP)step by step this procedure is called dynamic column generation Thesuccess and efficiency of such an approach is closely related to the com-plexity and capability of solving the pricing step in an implicit manner
Denoting by (π micro) a given dual solution to (RMLP) where π is as-sociated with the partitioning (MLP) (ii) and micro with the (home-)baseconstraints (MLP) (iii) the pricing question arising for the masterproblem (MLP) is
8 Crew Scheduling 50
(PRICE) existh isin H p isin Ph cp = cp minussumtisinp
πt + microh lt 0
We assume that cp =sum
aisinp ca As all pairings end in the non-leg taskt we can define the reduced cost of an arc (u v) isin A wrt (π micro) as
c(uv) =
c(uv) minus πv v isin Tc(uv) + microh v = t
The pricing problem to construct a pairing of homebase h (and type k)of negative reduced cost becomes a constrained shortest path problemin the acyclic digraph G = (VA) (restricted to homebase h and ruleset of type k)
(RCSP) (i) minsumaisinA
caxa
(ii)sum
aisinδout(v)
xa minussum
aisinδin(v)
xa = δst(v) forallv isin V
(iii)sumaisinA
warxa le Ur forallr isin R
(iv) xa isin 0 1 foralla isin A
Here δst(v) = 1 if v = s δst(v) = minus1 if v = t and δst(v) = 0 otherwiseWe solve this problem using a branch-and-bound algorithm similar toBeasley amp Christofides (1989) [22] using lower bounds derived froma Lagrangean relaxation of the resource constraints (RCSP) (iii) seeBorndorfer Grotschel amp Lobel (2003) [32] for more details on the dy-namic program In addition we used ldquoconfigurablerdquo classes of classicallinear resource constraints and cumulative resource constraints withreplenishment arcs We can handle most pairing construction rules di-rectly by multi-label methods Irnich amp Desaulniers (2005) [120] andIrnich et al (2010) [121] gives a recent survey on resource constrainedshortest path problem and how to tackle them in a column generationframework Some rules however are so complex that these techniqueswould become unwieldy or require too much customization For suchcases we used a callback mechanism that is we ignore the rule in ourpricing model construct a pairing and send it to a general rule veri-fication oracle that either accepts or rejects the pairing This can beseen as adding additional resource constraints for infeasible paths in an
8 Crew Scheduling 51
dynamic cutting plane manner Let |P | be length of p and P a set offorbidden paths then
(iii-b)sumaisinp
xa le |P | minus 1 forallp isin P
ensures feasibility of the paths so that a one to one correspondenceto pairings is reached Even if this allows for a general applicationwe want to point out explicitly that such rules slow down the pricingroutine Therefore we recommend to avoid such unstructured rules ifpossible
86 Branch and Price
The optimal solution value of (MLP) is a global bound on the optimalvalue of the model (SPP) If we unfortunately get a fractional solutionvariable xp we must branch and apply a divide and conquer techniqueto ensure integrality This is the state of art and standard technique tosolve mixed integer programs (MIPs) see once again Achterberg (2007)[3] In addition to the standard preprocessing techniques branchingrules node selections heuristics and cutting plane procedures we haveto resolve the LP-relaxation of the subproblems induced by the branch-ing or in other words fixing decisions In contrast to standard or staticMIP solving we have to keep in mind that in our new branches somenon-generated variables are possibly required to solve these subprob-lems to optimality In addition we have to ensure that the branchingdecisions so far are respected Hence we have to enrich the standardpricing of variables with a dynamic procedure that respects the fixingdecisions as well ie the branch on xp = 0
Added together this leads to an exact approach so called branch andprice algorithm to solve large scale MIPs to optimality For practicalinstances this may be too time consuming and even not appropriatebecause getting a feasible good solution in acceptable time is moreimportant in practice than proving optimality Solving the restrictedvariant of the (SPP) via branch and bound only will lead to poor solu-tions Therefore pricing is required in some branch and bound nodesto ldquocompleterdquo the solution and to generate ldquoundesirablerdquo pairings iefrom a cost or dual perspective in the end This real-world requirementcan be achieved by powerful problem adaptive heuristics which onlyperform pricing in several promising nodes of the branch and bound
8 Crew Scheduling 52
tree Hence a global guaranteed bound and optimality gap can stillpersist
87 Crew Composition
A main difference to duty scheduling in public transport or railwaytransport is that for airline crew scheduling complete crews must beconsidered ie each leg has to be covered by at least two pairingsHowever the rules and costs are quite different due to varying con-tracts and responsibilities ie cockpit crews are paid higher than thecabin crews Furthermore the number of required members of the cabincrew can differ from flight to flight This could lead to noteworthy sav-ings but also to inhomogeneous pairings Of course an aviation com-pany wants to have homogeneous pairings to increase the stability ofthe schedule In case of unavoidable disturbances and cancellations aschedule with constant crew compositions seems to be more stable andrecoverable because only this crew is affected from disturbances
To handle this ldquoregularityrdquo requirement we did some preliminary com-putational experiments for an straight forward sequential approachby using the introduced standard model (SPP) see Borndorfer et al(2005) [33] In a first step the major cost component which is the cock-pit crew is minimized After this these pairings were set as ldquodesiredonesrdquo if they are still valid for the other crew part or at least newones are preferred to be as similar as possible to the fixed one of thecockpit In a second step we re-optimize the cabin pairings using model(SPP) with respect to the adapted cost function and cabin rules Thissequential approach produces homogeneous solutions for cockpit andcabin crew very fast Potthoff Huisman amp Desaulniers (2008) [177]successfully used similar ideas and models for re-scheduling of crews atthe operational stage From our point of view an integrated model forcabin and cockpit crew is only required if the cost structure changessignificantly
Chapter II
Railway Modeling
In this chapter we describe techniques to model railway systems withdifferent granularities of the underlying railway infrastructure In aso-called microscopic representation of the railway system almost alltechnical details are considered The analysis of very detailed modelscan lead to more reliable conclusions about the railway system There-fore microscopic models are basically used to evaluate timetables viarailway simulation systems ie to respect the safety system exactlyThe disadvantage of very detailed models is the vast amount of datathat needs to be acquired and processed Even more computationalcapabilities and data management reach their limits
M Soukup wrote in a Swiss newspaper article in the Sonntagszeitungfrom 24082008 about the new planning system NeTS
ldquoSince 21 July 2008 the first 50 SBB schedulers havebeen developing the timetable for 2010 using the new sys-tem By the date of the changeover to the new timetableon 12 December 2009 500 more people will be workingwith NeTS Huge amounts of information are currently be-ing entered into the system For example when the IC828train leaves Zurich at 3pm heading for Bern the timetableschedulers must first take into account around 200 param-eters including the time of day the rolling stock the typeof train the length of the train the length of the route andconflicts when entering and leaving stations Extrapolatedup to cover the whole timetable this means that NeTS pro-cesses around 36 billion pieces of information and needsbetween 500 and 700 gigabytes of storage spacerdquo
To approach this problem macroscopic models are developed that sim-plify and aggregate the railway infrastructure representation Main
54
55
MicroscopicSimulation
Micro-MacroTransformation
MacroscopicOptimization
netcast
aggregate
disaggregate
Figure 1 Idealized closed loop between railway models of different scale for rail-way track allocation
application of macroscopic models are timetable information systemsOne goal of this work is to extend the usage of macroscopic modelsto capacity allocation Therefore we define microscopic railway infras-tructure resources and their macroscopic counterparts The challengeis to specify a reduced and manageable model which sustains the coreof the system at the same time A classification and comparative dis-cussion of railway infrastructure models can be found in Radtke (2008)[180]
The major contribution of this chapter will be the development abottom-up approach to construct a macroscopic model which conservesresource and capacity aspects of the considered microscopic railway sys-tem ie resulting in the tool netcast Such formalized and aggregatedmodels can be tackled by optimization methods especially integer pro-gramming The main concept of this Micro-Macro Transformation isshown in Figure 1
This will be the topic of the next chapter A highlight will be theevaluation of the proposed network simplification and an aggregationmethod on real world data as presented in Borndorfer et al (2010) [42]Furthermore we establish the theoretical background in Schlechte et al(2011) [190] to quantify the quality of the resulting macroscopic modelThe essential task is here to analyze the information loss and to controlthe error caused by the Micro-Macro Transformation
Most that will be presented in this chapter is joint work with RalfBorndorfer Berkan Erol and Elmar Swarat It is based on several dis-cussions with researchers from institutes on railway transport railwayoperations and operations research as well as railway experts fromdifferent railway undertakings and infrastructure providers
56
Let us name some of them here Soren Schultz Christian Weise ThomasGraffagnino Andreas Gille Marc Klemenz Sebastian Klabes RichardLusby Gabrio Caimi Frank Fischer Martin Fuchsberger and HolgerFlier In particular we want to thank Thomas Graffagnino from SBB(Schweizerische Bundesbahnen) who provided us real world data andexplained us a lot of technical issues Martin Balser who points outand contributed to the rounding and discretization aspects and DanielHurlimann and his excellent support to the simulation tool OpenTrack
To establish an optimization process to the allocation of ldquorailway ca-pacityrdquo we first have to define capacity and derive a resource basedmodel for a railway system in an appropriate way Railway capacityhas basically two dimensions a space dimension which are the physicalinfrastructure elements as well as a time dimension that refers to thetrain movements ie occupation or blocking times on the physicalinfrastructure
A major challenge of both dimensions is the granularity the potentialsize and the arbitrary smooth variation of time Figure 2 shows thethe rather small German station Altenbeken in full microscopic detailie with all segments signals switches crossovers etc
Railway efficiency and the capacity of railway networks are importantresearch topics in engineering operations research and mathematicsfor several decades The main challenge is to master the trade-offbetween accuracy and complexity in the planning optimization andsimulation models Radtke (2008) [180] and Gille Klemenz amp Siefer(2010) [100] proposed the use of both microscopic and macroscopicmodels They applied microscopic models for running time calculationsand the accurate simulation of railway operations and macroscopicmodels for long term traffic and strategic infrastructure planning In asimilar vein Schultze (1985) [195] suggested a procedure to insert trainslots according to predefined priorities in a first step and to test thereliability of this timetable in a second step by simulating stochasticdisturbances An alternative approach to determine the capacity of anetwork are analytical methods They aim at expressing the railwayefficiency by appropriate statistics eg the occupancy rate Thereexist two different approaches The first is the handicap theory byPotthoff (1980) [178] it is based on queuing models The second usesprobabilistic models to compute follow-on delays it is mainly basedon the work of Schwanhauszliger (1974) [196] He also introduced theimportant concept of section route nodes to analyze the performance
1 Microscopic Railway Modeling 57
of route nodes or stations Hansen (2010) [109] presents a probabilisticmodel as an alternative to queuing models for a precise estimation ofexpected buffer and running times
The chapter is organized as follows In Section 1 we will recapitulateand describe microscopic aspects of the railway system to establish adefinition of resources and capacity see Landex et al (2008) [145] Inthe literature several approaches work directly on a microscopic levelwith the disadvantage that only instances of small size can be handledsee Delorme Gandibleux amp Rodriguez (2009) [74] Fuchsberger (2007)[94] Klabes (2010) [129] Lusby et al (2009) [159] Zwaneveld et al(1996) [220] Zwaneveld Kroon amp van Hoesel (2001) [221]
Nevertheless on a planning stage it is not possible to consider all thesedetails and also not necessary Hence the main goal for a macroscopicmodel is to evaluate different timetable concepts or infrastructure deci-sions on a coarse granularity Only recently approaches were developedto tackle larger corridor or even network instances In Caimi (2009)[57] a top-down approach is presented and used to handle the completeSwiss network by a priori decomposition of the network into differentzones In contrast to that we present a bottom-up approach to definea macroscopic railway model in Section 2 The introduced transforma-tion from the microscopic to macroscopic view is described in detailanalyzed with respect to the discretization error implemented as a toolcalled netcast and successfully evaluated on real world scenarios egthe Simplon corridor see Erol (2009) [84] On the one hand these mod-els are precise enough to allow for valid allocations with respect toblocking times on the other hand they are simplified and aggregatedto a coarse level which allows for solving large scale optimization in-stances
1 Microscopic Railway Modeling
Railway traffic is a high-grade complex technical system which canbe modeled in every detail This is necessary to ensure that each mi-croscopic infrastructure element ie block segment is occupied by atmost one train at the same time State of the art simulation systemsprovide accurate estimations of running times with respect to such aprecise microscopic model The time period when a train is physicallyusing a block section is called running time Microscopic data is for
1 Microscopic Railway Modeling 58
Figure 2 Detailed view of station Altenbeken provided by DB Netz AG see Al-tenbeken [11]
example incline acceleration driving power power transmission speedlimitations signal positions
In this section we define all needed microscopic elements and data aswell as all macroscopic objects This work was done in a close col-laboration with the SBB who provided data for the scenario of theSimplon corridor see Borndorfer et al (2010) [42] In Figure 5 themicroscopic infrastructure of the Simplon area based on the simula-tion tool OpenTrack see OpenTrack [172] is shown The microscopicnetwork consists of 1154 nodes and 1831 edges
The input for netcast is the microscopic infrastructure network thatis modeled by a graph G = (VE) OpenTrack uses a special graphstructure where the nodes are so called double-vertices that consistof a left and a right part A convention in OpenTrack is that if apath in G enters a node at the left end it has to leave at the right orvice versa This ensures that the direction of the train route is alwaysrespected and no illegal turn around at switches is done on the wayFigure 3 shows an example of a double-vertex graph from OpenTrackMontigel (1994) [163] proposed this concept to describe microscopicrailway networks Figure 4 shows a straightforward transformation ofa double vertex graph to general directed graph
1 Microscopic Railway Modeling 59
Figure 3 Screenshot of the railway topology of a microscopic network in the rail-way simulator OpenTrack Signals can be seen at some nodes as wellas platforms or station labels
Every railway edge e isin E has some attributes like maximum speedor incline A node v isin V is always defined if one or more attributeschange or if there is a switch a station or a signal on this track Everytrack section between two nodes is modeled as an edge
Our transformation approach is based on a potential set of routes inG for standard trains so called train types The set of train types isdenoted by C Let R be the set of all given routes in G In additionwe are given a mapping θ R 7rarr C for all routes to the rather small setof standard train types It is for example possible to have microscopicroutes to ICE trains which differs in their weight or length due tothe composition and to aggregate them in one standard train type forICEs
Figure 4 Idea of the transformation of a double vertex graph to a standard di-graph
1 Microscopic Railway Modeling 60
A microscopic route is a valid path through the microscopic infras-tructure which starts and ends at a node inside a station or at a noderepresenting a parking track In addition it is possible that other nodeson the route are also labeled as stops where the train could potentiallywait
Furthermore these train routes induce in which direction the micro-scopic infrastructure nodes and edges can be used This will directlyinfluence the definition ie the headway parameter of a macroscopicmodel as we will explain later in Section 2 They ought to be reason-able and conservatively grouped with respect to their train class (heav-iest cargo trains slowest interregional or regional passenger trains)Thus only a minimal difference of the running times within a traintype occurs and each associated train route can realize these times byslowing down if necessary For these standard train routes detailedsimulation data has to be evaluated carefully such that reliable run-ning and blocking times in units of δ ie times provided by the microsimulation are given in seconds see Figure 6 Note that several routesof R belong to the same train type For example in case of a heavycargo train that is allowed to stop at some intermediate station ieat one microscopic node S we simulate two routes the first withoutand the second with stopping at S Hence we have different runningtimes and blocking times with respect to the behavior of the train atthe start or end station ie we will use later the term running modefor this Obviously trains which have to break or accelerate have largerrunning times and hence resource consumptions
Example 22 shows the significant differences between the durationsie the running and blocking times related to S Therefore our macro-scopic approach has to cope with that by considering not only traintype but also event dependencies
In Pachl (2002) [173] and Brunger amp Dahlhaus (2008) [46] the laws ofbasic dynamics are applied to describe the dynamics of a train move-ment Basically three groups of forces are considered tractive inertiaand resistance force If all needed parameters are given eg mass ac-celeration and deceleration of the train (directed) incline of the blocksection running times of train movements can be estimated very accu-rately In state of the art railway simulation software eg OpenTrackall relevant parameters are considered in order to provide plausiblevalues see Nash amp Huerlimann (2004) [166]
1 Microscopic Railway Modeling 61
Figure 5 Microscopic network of the Simplon and detailed representation of sta-tion Iselle as given by OpenTrack
In Europe blocking times are used to quantify the infrastructure ca-pacity consumption of train movements The approach is based onthe early works of Happel (1950) [110] and Happel (1959) [111] andthe intuitive concept to associate the use of physical infrastructure re-sources over certain time intervals with trains or train movements seealso Klabes (2010) [129] Pachl (2008) [174] for a comprehensive de-scription of blocking time theory We will now give a brief discussionof blocking times that contributes to a better understanding of ourtransformation algorithm
The origin of the blocking time stairs shown in Figure 6 is the well-known train protection system called train separation in a fixed blockdistance Nowadays these are train control systems that indicate themoving authority to the train drivers and thus ensures safe railwayoperation In this method the railway network is divided into blocksections which are bordered by main signals A block section must notbe occupied by more than one train at a time When a signal allows atrain to enter a block section the section is locked for all other trainsIn this way the entire route between the block starting main signaland the overlap after the subsequent main signal has to be reserved forthe entering train
1 Microscopic Railway Modeling 62
Sta
tion
A
Sta
tion
B
e1 e2 e3 e4 e5 e6
r1
r2
tim
e
ur2e5lr2e5
Figure 6 Blocking time diagrams for three trains on two routes using 6 blocksIn the lower part of the diagram two subsequent trains on route r2 andat the top one train on the opposite directed route r1 are shown
Figure 6 shows that the time interval during which a route r occupiesa track segment consists of the relative reservation duration lre and therelative release duration ure on edge e isin E The relative reservationduration is the sum of the approach time the signal watching timesometimes called reacting time and time needed to set up the routeThe relative release duration is the sum of the release time the clearingtime sometimes called switching time and time needed by the trainbetween the block signal at the beginning of the route and the overlapThe switching time depends significantly on the installed technologysee Klabes (2010) [129] Schwanhauszliger et al (1992) [197] In orderto prevent trains that want to pass a block section from undesider-able stops or brakings the block reservation should be finished beforethe engine driver can see the corresponding distant signal Then thesection stays locked while the train passes the track between the be-ginning of the visual distance to the caution signal and the main signaland thereafter the block section until it has cleared the overlap afterthe next main signal Then the section is released This regime can
1 Microscopic Railway Modeling 63
be improved in block sections that contain con- or diverging tracksbecause in such cases it is often possible to release parts of the sectionbefore the train has passed the overlap after the next main signal
We only want to mention that our approach can be easily adaptedto other simulation tools that provide accurate running and blockingtimes like RailSys or RUT-K We remark that these tools differ in theirdefinition of objects interfaces and some minor interpretations and thatalthough our exposition is based on the simulation tool OpenTrack themain concepts of running and blocking times are the same and thus themethodology is generic
We summarize the microscopic information that we use
an (undirected) infrastructure graph G = (VE)
a set of directed train routes R r = e1 e2 enr with ei isin E
a set of train types C
a mapping θ from routes R to train types C
positive running time dre on edges e isin E for all routes r isin Rmeasured in δ
positive release duration ure on edges e isin E for all routes r isin Rmeasured in δ
positive reservation duration lre on edges e isin E for all routesr isin R measured in δ
orientation of edges is induced by traversing routes (one or bothdirections)
stop possibilities for some nodes vi isin V are induced by traversingroutes
Remark 11 Though we develop our transformation approach for fixedblock railway operation systems the methodology and models could beeasily applied to moving block systems Future systems like ETCS Level3 can already be modeled in simulation tools Arbitrarily small blocksie blocks with lengths converging to zero are considered in simula-tions to emulate the resulting blocking times see also Emery (2008)[82] and Wendler (2009) [214] for an investigation of the influence ofETCS Level 3 on the headway times Simulation tools have to respectall these technical details From an optimization point of view how-ever it is sufficient to consider abstract blocking time stairs regardlessfrom which safety system they result or how they were computed
2 Macroscopic Railway Modeling 64
macronetwork
trainrequests
TTPlibproblem
solver timetable
Figure 7 IO Concept of TTPlib 2008 (focus on macroscopic railway model)
2 Macroscopic Railway Modeling
In this section we present a formal macroscopic railway model Theestablishment of standard models and standard problem libraries havecontributed to the success in problem solving Such libraries exist forthe famous Traveling Salesman Problem see Reinelt (1991) [181] aswell as for general Mixed Integer Programs see Achterberg Koch ampMartin (2006) [4]
We invented a standardization of a macroscopic railway model andintroduced the library TTPlib for the track allocation or timetablingproblem see Erol et al (2008) [85] Figure 7 illustrates the datahandling of a train timetabling problem Section 21 motivates theaggregation idea and recapitulates the standardization of the result-ing macroscopic infrastructure model Section 22 discusses the dis-cretization problem when transferring microscopic models to macro-scopic ones Finally we introduce an algorithm that performs theMicro-Macro-Transformation in Section 23 Furthermore we will showthat the constructed macroscopic model is reliable such that the resultscan be re-transformed and interpreted in a microscopic model and fi-nally operated in ldquorealityrdquo The introduced algorithm constructs froma microscopic railway model a macroscopic model with the followingproperties
macroscopic running times can be realized in microscopic simu-lation
sticking to macroscopic headway-times leads to conflict-free mi-croscopic block occupations
valid macroscopic allocations can be transformed into valid mi-croscopic timetables
2 Macroscopic Railway Modeling 65
21 Macroscopic Formalization
The desired macroscopic network is a directed graph N = (S J) fortrain types C deduced from a microscopic network G = (VE) andtrain routes R On this level our goal is to aggregate (inseparable)block sections (paths in G) to tracks J and station areas (subgraphs ofG) to stations S
The aggregation will be done in a way that depends on the given routesR and the simplification to train types C imposed by the mapping θsuch that the complexity of the macroscopic network depends only onthe complexity of the interactions between the given train routes andnot on the complexity of the network topology which covers all inter-actions between all potential train routes which is much more This isa major advantage over other approaches because the aggregation isdetailed where precision is needed and compressed where it is possible
We will now describe the idea of the construction by means of an ex-ample First all potential departure and arrival nodes at some stationthat are used by the routes R are mapped to one macroscopic sta-tion node Additional macroscopic nodes will be introduced in orderto model interactions between routes due to shared resources Thepotential interactions between train routes in a double-vertex graphare
complete coincidence ie routes have an identical microscopicpath
convergence ie routes cross at a microscopic node (and traverseit in the same direction)
divergence ie routes separate at a microscopic node (and tra-verse until then in the same direction)
or crossing ie routes cross at a microscopic node (and traverseit in the opposite direction)
Note that two routes can correlate in various and numerous ways Letus discuss some of these interactions between train routes at the exam-ple of the infrastructure network shown in Figure 8
Consider first a single standard train that runs from platform A (Wedenote any place where stopping is allowed as a platform) to platformX Then it is enough to consider just one single track from station Ato X in the macroscopic infrastructure Note that this macroscopictrack could correspond to a long path in the microscopic representa-
2 Macroscopic Railway Modeling 66
A
B X
Y
P
Figure 8 Example of macroscopic railway infrastructure
tion Consider now additional standard trains from A to X Possibleinteractions and conflicts between these train routes are the self correla-tion on the directed track from A to X as well as the platform capacityfor standard trains which allows say exactly one train to wait in Aor X Another standard train running from B to X calls for the def-inition of a pseudo-station P at the track junction in order to modelthe train route convergences correctly (Our model distinguishes be-tween regular station nodes where a train can stop and pseudo-stationnodes which are not stop opportunities ie in our model trains arenot allowed to wait at a pseudo-station or to change their directionthere) The pseudo-station P splits the track from A and X into twotracks from A to P and from P to X The second of these tracks isused to model the resource conflict between converging routes of trainsfrom A to X and trains from B to X which is locally restricted to thetrack from P to X (or more precisely from the first blocks to reservecontaining the switch of P) If it is possible to run trains on the samemicroscopic segment in the opposite direction from X to A anotherdirected track has to be defined in the macroscopic network Besidesthe standard self correlation the conflict for opposing routes also hasto be modeled see Figure 6 Diverging or crossing situations betweenopposing train routes can be handled in an analogous way Along thelines of these examples we can exploit aggregation potentials in theinfrastructure by representing several microscopic edges on a route byonly one macroscopic track Of course macroscopic track attributescan also be compressed For example if we assume that the routefrom A to X and the route from B to X are operated by the same traintype we can use a single value for the running time on the track from Pto X
2 Macroscopic Railway Modeling 67
S X
Y
Figure 9 Example of aggregated infrastructure
After constructing the regular stations the pseudo-stations and thetracks between them the network can be further reduced by a secondaggregation step Again consider the situation in Figure 8 Supposeplatforms A and B belong to the same station S If P is a close junctionassociated with S then it may be viable to contract nodes A and Bto one major station node S with a directed platform capacity of twoas shown in Figure 9 Of course by doing so we loose the accuracyof potentially different running times between different platforms ofS and the other stations and we also loose control over the routingthrough or inside S which both can produce small infeasibilities on theoperational level However one can often achieve significant reductionsin network sizes in this way without loosing too much accuracy
This is exactly a decomposition of the TTP for the microscopic networkto a TTP for a macroscopic network with aggregated stations andseveral TPP for the microscopic station areas The next paragraphswill describe the macroscopic elements and attributes in more detail
211 Train Types and Train Type Sets
As a first component the macroscopic model groups trains with similarproperties to a set of train types C as mentioned above The train setsie and so the train types are structured hierarchically by a tree Inthis tree each node corresponds to exactly one train set f isin F sube P(C)which consists of all leaf nodes The leaf nodes represent train setsconsisting of exactly one train type c isin C For each train set allproperties eg running or turn around times of the parent train setare valid analogously restrictions eg station capacities of all parenttrain sets have to be fulfilled as well as the train set specific ones
Figure 10 shows an example tree If a running time for train set 1 ontrack j isin J is defined then this time is also valid for 4 If a stationcapacity at station s isin S is defined for all trains of set 2 then trains of
2 Macroscopic Railway Modeling 68
1
23
45
6 7 8
ABCDEF
BCDEA
BCDE
C D E
Figure 10 Train types and train sets defined as a poset
set 4 to 8 are also captured by the capacity rule On the right side ofFigure 10 the nodes of the tree are interpreted as sets of train types
In a mathematical interpretation these trees are Hasse diagrams vi-sualizing a partially ordered set see Birkhoff (1967) [26] That is abinary relation of the finite set C which is reflexive antisymmetricand transitive In our setting the set F is ordered by inclusion andthe minimal elements of this poset are the elements of the set of traintypes C
212 Stations
The nodes S of the digraph N = (S J) are called stations We distin-guish three types of them
standard-stations (two-sided labeled with 1 and 2) where it ispossible for a train to pass through turn around or wait
dead-end stations (one-sided labeled with 1) where no passingis possible
and pseudo-stations (two-sided labeled with 1 and 2) where noturn around or waiting is possible
Even if in daily operation trains could stop and wait at pseudo-stationsie if a red signal of the security system is shown in front of thisjunction on a planning level stopping there is strictly forbidden due tothe assumed green wave policy
We restrict ourself to standard cases of station capacities such as max-imal number of trains of a certain train set at one time step at a stationMore precisely we use different running modes of trains which will beintroduced in the next section Therefore we can further restrict thenumber of trains that are stopping in or passing through a stationStation capacity constraints can be many other requirements as wellsuch as
2 Macroscopic Railway Modeling 69
maximum capacity per side of station
maximum capacity of station per time interval
maximum capacity of station at a specific time interval
forbidden combinations of (running) modes per train set
forbidden combinations of modes per combinations of train set
or forbidden meetings in stations
The extension of the model is straightforward for these numerous imag-inable special cases and can be easily achieved as we will see laterFinally we list all attributes of station nodes
name and coordinates
type (standard dead-end pseudo) and number of sides
turnaround times dsf for each s isin S and f isin F
station (event) capacities κsf for each s isin S and f isin F
213 Tracks
The set of arcs J of N = (S J) denoted as tracks correspond to severalblock sections of the railroad network For a standard double-way trackbetween station x isin S and y isin S more precisely between two sidesof them there exist two opposite directed arcs (x y) isin J and (y x) isinJ Physical track segments which can be used in both directionscorresponds to two opposite directed arcs of J and build a single waytrack By definition it is not possible to overtake on a track This isonly possible inside stations by using different tracks ie the stationcapacity must allow this More precisely the order of entering trainson each track can not change at the arrival station This assumptionhas an effect on the definition of the network segmentation as well ason the minimal departure headway times see Definition 28
Block section exclusivity on a microscopic stage which we describedin Section 1 transfers to minimal headway times at departure Theminimal abiding difference of the departure times between two con-secutive trains is defined as the minimal departure headway time toensure safety on each track j isin J
Remark 21 Note that it is possible to have more than one track be-tween station x isin S and y isin S Therefore N = (S J) is a multi-graph(allowing parallel arcs) and we should use consistently the notationa isin J instead of (x y) isin J However in cases were we use (x y) isin Jwe indirectly assume that (x y) is unique Furthermore all single way
2 Macroscopic Railway Modeling 70
preceding succeeding minimum headway time
train running train running simulation roundedtype mode type mode result value
in seconds in minutes
GV stop-stop ICE stop-stop 475 8GV stop-stop ICE stop-pass 487 9GV stop-stop ICE pass-stop 466 8GV stop-stop ICE pass-pass 477 8GV stop-pass ICE stop-stop 469 8GV stop-pass ICE stop-pass 474 8GV stop-pass ICE pass-stop 460 8GV stop-pass ICE pass-pass 464 8
GV pass-stop ICE stop-stop 321 6GV pass-stop ICE stop-pass 333 6GV pass-stop ICE pass-stop 312 6GV pass-stop ICE pass-pass 323 6GV pass-pass ICE stop-stop 315 6GV pass-pass ICE stop-pass 320 6GV pass-pass ICE pass-stop 306 6GV pass-pass ICE pass-pass 310 6
Table 1 Technical minimum headway times with respect to running mode
tracks are specified as disjunctive pairs of J so we use j isin J to denotethe counterpart or complement of track j isin J
As we have already motivated in Section 1 the running dynamics arerelevant for the traversal time on a track and the corresponding head-way times
Example 22 We want to clarify that on real numbers from the sce-nario hakafu simple The simple simulation via RailSys of the or-dered pair of a cargo train (GV) and a fast intercity train (ICE) ontrack FOBR to HEBG produces 16 different headway times in secondsTable 1 lists these numbers as well as the rounded values in minutesIt can be observed that depending on the running mode of the trainsthe headway time can differ more than 3 minutes ie the worst casevalue reserves 50 more capacity than the best case Thus a simpleworst case assumption could lead to an underestimation of the potentialcapacity
By this observation it is necessary to distinguish at least between stop-ping and passing trains Otherwise one could not guarantee feasibility
2 Macroscopic Railway Modeling 71
if we would be to optimistic in chosing the headway time or contrary atoo conservative value would lead to underestimation of the real trackcapacity Let MS = dep(arture) arr(ival) pass be the set of pos-sible events or modes at the stations Furthermore we consider thefollowing standard running modes MJ subeMS timesMS for train runs on atrack
stops at departure node and arrival node (1)
stops at departure node and passes at arrival node (2)
passes at departure node and stops at arrival node (3)
and passes at departure node and arrival node (4)
Minimum headway times can be defined for all modes individuallywhich is reasonable see again Example 22 Furthermore the handlingof the events inside a station can be seen in Example 25 Figure 13shows the interpretation of turn around activities inside a station asdashed arcs In pseudo stations only directed passing and in dead-endstations only arrival and departure events have to be considered Bydefinition passing nodes of side 1 represent trains entering at side 1and leaving at side 2 passing nodes of side 2 represent trains enteringat side 2 and leaving at side 1
A detailed definition and way of calculation of these times with re-spect to the microscopic model is topic of Section 23 After listing allattributes of a track j isin J we will present some small examples
start station (tail isin S) and side (isin 1 2)
end station (head isin S) and side (isin 1 2)
type ie single way track or standard
running times djcm isin N0 depending on train type c isin C andmode m isinMJ
minimum headway times hjc1m1c2m2 isin N 0 for departingtrain pairs ie c1 c2 isin Cm1m2 isinMJ
minimum headway times for departing train on j and a departingtrain on the complement track j if single way track (sets andmode)
Example 23 In Figure 11 a macroscopic railway network is shownwith only two standard tracks connecting standard station A via pseudostation P with dead-end station B Running times of mode (1) are il-lustrated as solid lines and the corresponding minimum headway timesare shown as dotted lines for two different train types The correspond-ing running time values and headway matrices are
2 Macroscopic Railway Modeling 72
5 3
3 2
2
2
1
1
2
4
2
3
A P B1 2 1
Figure 11 Macroscopic modeling of running and headways times on tracks
dAP =
(53
) HAP =
(2 24 2
) dPB =
(32
) HPB =
(1 23 1
)
Example 24 A more complex situation is modeled in Figure 12 Wehave a single way track between P1 and P2 which can be used inboth directions On the one hand blue trains are running from A to Ctraversing P1 and P2 On the other hand red trains from D run via P2and P1 to station B In this scenario the two track arcs correspondingto the segment between nodes P1 and P2 are directed opposite and builda single way pair Only one train can pass this section at a time andtherefore headway times for the combination of a train from P1 to P2and a train from P2 to P1 and vice versa are additionally needed
dAP1 =(
5) HAP1 =
(2) dP1P2 =
(3) HP1P2 =
(2)
dP2C =(
3) HP2C =
(2) dDP2 =
(3) HDP2 =
(1)
53
3
32
2
2
2 2
1
11
4
3
A
B
P1 P2
C
D1 2
1 2
1 2
1 2
Figure 12 Macroscopic modeling of a single way track
2 Macroscopic Railway Modeling 73
A B
6
5
5
4
3 3
0
0 7 7
0
0
departure arrivalpassing
Figure 13 Representation as event-activity digraph G = (VN AN )
dP2P1 =(
2) HP2P1 =
(1) dP1B =
(2) HP1B =
(1)
H(P1P2)(P2P1) =(
4) H(P2P1)(P1P2) =
(3)
Example 25 The extension of the network model to different run-ning modes is shown in Figure 13 All potential running modes onthe track from A to B can be seen in Figure 13 For simplificationwe do not show the complete headway relations in that figure but ofcourse all combinations need to be defined to ensure feasibility on thattrack Furthermore the event nodes involved in a turn around activityin station A and B are connected by dashed arcs In a mathematicalmodel we define a turn around as the change from arrival to departurenodes From a railway operations point of view a turn around is onlyperformed if a train enters and leaves the station at the same side ega turn around has a minimum duration of 3 in station A and 7 inB This shows that it is easy to extend the models to handle differentminimum turn around times for each station side individually
All running time definitions on a track induce a headway definition Wecan trivially bound the dimension of the headway matrix of a standardtrack by |(C timesM) times (C timesM)| and 2|(C timesM) times C timesM | for a singleway track respectively Due to the fact that only a relevant subset ofrunning times and therefore also for headways times should be consid-ered at a specific track we suggest to use always sparse representationsof these matrices H Furthermore we introduce useful definitions forheadway matrices
2 Macroscopic Railway Modeling 74
H(km)
k
m
H(k l)
H(lm)k
l
m
H(km)
H(lm)k
l
m
Figure 14 Implausible situation if headway matrix is not transitive
Definition 26 A headway matrix Hj for track j isin J is called transi-tive or triangle-linear if all entries are strictly positive and the triangleinequality is satisfied
forallc1 c2 c3 isin Cm1m2m3 isinMJ
hjc1m1c3m3 le hjc1m1c2m2 + hjc2m2c3m3
Figure 14 motivates why we can assume that headway matrices Hto be transitive in reality We use the simple notation H(k l) for theentry k l that in fact corresponds to a preceding train type succeedingtrain type each with a certain running mode On the left hand atrain of type k is followed by a train of type m with respect to theminimum headway time H(km) In the middle and on the right handan intermediate train of type l is running on that track after k andbefore m It can be seen that if H(km) gt H(k l) + H(lm) thetrack allocation on the left and in the middle are feasible Howeverthe sequence on the right is violating the headway H(km) But itis completely implausible that running trains of type l after k on thistrack and trains of type m after l with respecting minimum headwayscan become infeasible due to violation of the minimum headway timeof k and m The algorithm presented in Section 23 produces headwaymatrices which are transitive simply because of the underlying blockusages In other words if the situation on the right hand is a conflictbetween k and m based on timed resource usage of that track then thesequence k and l or the sequence l and m must already be in conflict
2 Macroscopic Railway Modeling 75
macroscopic element microscopic counterpart
train type c subset of train routes Rstation s unified connected subgraph of Gtrack j (connecting different stations) unified consecutive block sections ie a path in Grunning time on j for c (in ∆) running times on block sections for routes (in δ)headway times on j for pairs c1 c2 (in ∆) blocking time on sections for routes (in δ)
Table 2 Relation between the microscopic and the macroscopic railway model
Definition 27 A headway matrix Hj for track j isin J is called order-safe if all entries are strictly positive and the order is not changing(no passing on tracks)
forallc1 c2 isin Cm1m2 isinMJ hjc1m1c2m2 + dc2m2 le dc1m1 + hjc2m2c3m3
Definition 28 A headway matrix H is called valid if H is transitiveand order-safe
We summarize the macroscopic infrastructure model that we have de-veloped so far as consisting of a network N = (S J) with a set ofrelevant locations S where train events occur and the set of tracks J where trains can run Furthermore we have seen how detailed macro-scopic information for running turn around and headway times for agiven set of train types C and modes M induce a digraph G = (VN AN)with VN sub Stimes1 2timesMS and AN sub VN timesVN By definition all timesare strictly positive integer values with respect to a fixed discretiza-tion eg the times of the instances provided by the TTPlib are inminutes The digraph G = (VN AN) represents all potential eventsand activities in N = (S J) All activities a isin AN have a positiveduration d(a) isin N The restriction to only one train type c isin C isdenoted by G|c Finally Table 2 identifies the macroscopic elementsand their orginal microscopic counterparts with respect to the railwaysafety system and the railway infrastructure resource consumption
22 Time Discretization
Discrete optimization models for timetabling and slot allocation arebased on the use of space-time graphs ie the time is discretizedSimilar as for the topological aggregation there is also a trade-off be-tween model size and accuracy in the temporal dimension This tradoffis controlled by the discretization stepsize The discretized times in themacroscopic model will be based on microscopic simulation data which
2 Macroscopic Railway Modeling 76
is very precise In fact simulation tools provide running and blockingtimes with an accuracy of seconds (or even smaller) Our aim is toaggregate these values in the macroscopic model We propose for thispurpose a conservative approach which means that running and arrivaltimes will never be underestimated in the macroscopic model
Simulation tools provide running and blocking times with an accuracyof seconds (or even smaller) denoted by δ To decrease the problem sizeof real world instances it is essential and a common approach to usea coarse time discretization in the macroscopic model In addition weneed a discrete model to handle decisions wether a train is running andblocking a section or not In our approach the unit of the macroscopictime discretization is based on the microscopic simulation data Let∆ be a fixed parameter to measure all macroscopic time informationeg units of 60 seconds We propose again a conservatively approachwhich means it is not valid to underestimate running ie and thereforearrival times in the macroscopic model In the following we denoteby drj the microscopic running time of route r on track j by drj thediscretized running time and by εrj the cumulative rounding error (inunits of δ) The total rounding error at the end of each route is denotedby εr (in units of δ) A first approach would then be to simply roundup all the times The error estimation of this method is shown inLemma 29
Lemma 29 Let r isin R be a train route in the macroscopic networkN = (S J) with length nr ie that is the number of macroscopic tracksof route r and running times drj measured in δ for each track j isin r If
we simply round up the running times drj for each track to a multipleof ∆ we get a worst-case rounding error of ∆nr minus nr
Proof For each track we have a maximum possible rounding error of∆minus 1 In the worst-case this could occur to all nr tracks of r
The error estimation shows that this rounding procedure results inrather big differences between the macroscopic and the microscopicrunning times From a theoretical point of view we could assume toround up all the times so that we can always argue that the microscopictrain would fit in the macroscopic planned time corridor by just slowingdown Unfortunately this could lead to unnecessary overestimations ofthe running and headway times and thus to inefficient use of capacity
2 Macroscopic Railway Modeling 77
Algorithm 1 Cumulative rounding method for macroscopic run-ning time discretization
Data track j = (s1 s2) = (e1 em) isin J with s1 s2 isin S andei isin E i isin 1 m a train route r isin R with microscopicrunning time drj gt 0 for track j a cumulative rounding errorεrjminus1 and the time discretization ∆ gt 0
Result running time drj and cumulative rounding error εrjbegin
choose k isin N with (k minus 1)∆ lt drj le k∆
if 0 lt (k minus 1) and drj minus (k minus 1)∆ le εrjminus1 then
drj = (k minus 1)∆ round down
εrj = εrjminus1 minus (drj minus (k minus 1)∆) decrease error
elsedrj = k∆ round up
εrj = εrjminus1 + (k∆minus drj) increase error
return pair(drj εrj)
Therefore we use an alternative approach by a sophisticated roundingtechnique The objective is to control the rounding error by only toler-ating a small deviance between the rounded macroscopic running timeand the microscopic one The idea is pretty simple with respect tothe cumulative rounding error it is sometimes allowed to round downbecause enough buffer time was collected on the way In that case weknow that the train can always arrive one time unit earlier at the targetstation of track j Nevertheless we have to make sure that no runningtime is rounded to zero because this would imply no infrastructureusage and can lead to invalid timetables The exact description of theprocedure done at each track is given in Algorithm 1 Let denote byεrjminus1 the absolute cumulative rounding error which cumulates all errorsof r until the previous track j minus 1 on the route At the beginningof a route r the cumulative rounding error clearly equals zero ieεr0 = 0 The macroscopic running times are in fact attributes of a trackj Hence we identify them by drj where d denotes that it is a runningtime attribute and r the related train route
Lemma 210 states that this cumulative rounding technique gives asubstantial better upper bound on the rounding error
Lemma 210 Let Jlowast = j1 jnr with ji = ei1 eim isin J i isin1 n eik isin E be a train route r in the macroscopic network
2 Macroscopic Railway Modeling 78
N = (S J) with microscopic running times drj gt 0 for each track jmeasured in δ gt 0
If ∆ le drj forallj isin J r isin R for the time discretization ∆ the cumulativerounding error εr of the rounding procedure described in Algorithm 1 isalways in the interval [0∆)
Proof The proof is done by induction over the nr tracks of route rConsider the first track j1 on r The start rounding error is denotedby εr0 = 0 It follows that drj1 minus (k minus 1)∆ gt 0 = εr0 Hence Algorithm1 rounds up and we get εrj1 = k∆minus drj1 By definition of k it follows
that 0 le εrj1 lt ∆ since drj1 gt 0
In the induction step we analyze the rounding error of the track jndenoted by εrjn There are two cases
1 Let drn minus (k minus 1)∆ le εrnminus1 Then we round down and set
εrn = εrnminus1 minus (drn minus (k minus 1)∆)
By reason of the fact that ∆ le drj a rounding down to zero couldnot appear By definition of k it clearly follows that
εrn lt εrnminus1 lt ∆
And due to the ldquoIf rdquocondition in the algorithm it is obvious that
εrn = εrnminus1 minus (drn minus (k minus 1)∆) ge 0
2 Consider the other case that is εrnminus1 lt drn minus (k minus 1)∆ Then εrnis set to εrnminus1 + (k∆minus drn) By drn le k∆ it is evident that
0 le εrnminus1 le εrn
At last we have to consider the upper bound It follows that
εrn = εrnminus1 + (k∆minus drn)
lt drn minus (k minus 1)∆ + k∆minus drn= ∆
2 Macroscopic Railway Modeling 79
With the above described rounding technique there is still one problemleft Lemma 210 does not apply for the case when there exists a trackj where drj lt ∆ Then it is not allowed to round down This couldimply a worse upper bound for our rounding procedure as shown inLemma 211
Lemma 211 We consider the same rounding procedure and the sameassumptions as in Lemma 210 except for the case that there is a setB sube 1 nr where for each b isin B drb lt ∆ holds Then the upperbound for the cumulative rounding error εrnr is equal to (|B|+ 1)∆
Proof We again use an induction technique At the beginning we lookat the first track where drb lt ∆ In this case we have (kminus1)∆ = 0 andtherefore k = 1 Due to the prohibition that a macroscopic runningtime equals zero we set εrb = εrbminus1 + (k∆minus drb) It follows that
εrb = εrbminus1 + (k∆minus drb)= εrbminus1 + (∆minus drb)lt ∆ + ∆minus drblt 2∆
Note that as shown in Lemma 210 the rounding error does not growif the running time on the current track is greater than ∆
Next we consider the case that we have yet a number of i tracks witha running time less than ∆ and the i+ 1 track is occurred To simplifynotations the precedent track is denoted by i Then it follows that
εri+1 = εri + (k∆minus dri+1)
= εri + (∆minus dri+1)
lt i∆ + ∆minus dri+1
lt (i+ 1)∆
Figure 15 shows the difference between microscopic and macroscopicrunning time for a fixed value t = 74 at one track with respect todifferent macroscopic time discretizations ∆ Fine discretizations likeless than 15 seconds produce only very small deviations For larger
2 Macroscopic Railway Modeling 80
0 5 10 15 20 25 30 35 40 45 50 55 6050
60
70
80
90
100
110
120
discretization ∆ in seconds
runnin
gti
me
inse
conds
real (microscopic) running timerounded (macroscopic) running time
Figure 15 Transformation of running time on track A rarr B for time discretiza-tions between 1 and 60 seconds
time discretization the error increases significantly except for somepathological cases were t is a multiple of ∆
Figure 16 compares the two rounding methods by illustrating the min-imum average and maximum rounding errors of the macroscopic run-ning times at the end of example routes for all considered train typesthrough the Simplon corridor with respect to time discretizations vary-ing from 0 to 60 seconds The routes have a length of at most tenmacroscopic tracks It is apparent that cumulative rounding dampensthe propagation of discretization errors substantially already for shortroutes
We want to point explicitly that rounding up or down to the nearestinteger number ie in case of 15 to 2 would also limit the propagationof the rounding error on an individual route However this approachcan not guarantee that the block sections can be allocated conflict-freewith respect to the finer discretization δ It is not hard to formulate acounterexample where rounding up and down come adversely togetherand lead to an invalid macroscopic model eg a deadlock on a singleway track Hence there are feasible macroscopic allocations that cannot be re-translated into feasible microscopic ones Therefore resultsof such an approach are questionable and hardly transformable
2 Macroscopic Railway Modeling 81
0 10 20 30 40 50 60
0
100
200
300
400
500
discretization ∆
roundin
ger
ror
inse
conds minimum
averagemaximum
0 10 20 30 40 50 60
0
100
200
300
400
500
discretization ∆
roundin
ger
ror
inse
conds minimum
averagemaximum
Figure 16 Rounding error for different time discretizations between 1 and 60seconds comparison of ceiling vs cumulative rounding
Algorithm 2 Calculation of Minimal Headway Times
Data Track j = (s1 s2) = cupiei isin J with s1 s2 isin S release durationur1ei and reservation duration lr2ei with r1 r2 isin Rc(r1) c(r1) isin C ei isin E i isin 1 m and time discretization∆ gt 0
Result Minimal headway time h(= hjjc(r1)c(r2)) for train typesequence c(r1) c(r2) on track j
beginhlarrinfinfor x = cupiei|ei isin r1 cap r2 do
h = minur1x + lr2x h update timing separation
return d h∆e
Another important aspect for the macroscopic network transformationis the calculation of the headway times Based on the occupation andrelease times in Figure 17 it is possible to define a minimal time differ-ence after which a train can succeed on the same track or can pass itfrom the opposite direction We want to point out explicitly that werestrict ourself wlog to minimal headway times for the combinationof departure trains In reality especially railway engineers often usethe term headway times for all kinds of potentially train event combi-nations for a reference point eg the headway time between arrival oftrain 1 at station A and departure of train 2 at station B is 8 minutes
Algorithm 2 describes the calculation of the minimal headway timefor the cases of two routes r1 and r2 traversing the track in the samedirection We denote the corresponding train types by c1 c2 isin C
2 Macroscopic Railway Modeling 82
In case of crossing routes r1 and r2 on track j = (s1 s2) another head-way time has to be considered By definition each single way track jhas exactly one counterpart j = (s2 s1) isin J which is directed in theopposite direction In addition to the standard headway times relatedto each track j this kind of track needs another headway matrix toensure block feasibility with respect to the opposing direction Letj = (e1 em) be traversed by the directed route r1 Obviously theminimum headway time for a departure of a train on route r2 at stations2 after a departure of a train on route r1 from station s1 is defined as
hjjc(r1)c(r2) =sum
iisin12m
dr1ei + ur1em + lr2em (1)
Note that in this opposing case the relevant block section is always emIn addition to the minimal technical headway time a standard buffer isadded Each network provider such as DB or SBB has a rule of thumbfor this value Nevertheless the special knowledge and the experienceof the planners can locally lead to more accurate numbers
In Figure 17 the macroscopic output after the transformation for thesituation described in Figure 6 can be seen The infrastructure is re-duced from six undirected block segments e1 to e6 to two directed tracksj1 and j2 Furthermore only two macroscopic stations are needed in-stead of seven microscopic nodes On the microscopic scale the trainmovements are given very precisely It is even possible to identify theacceleration cruising and deceleration phases On the macroscopicscale train movements are linearized and only the state of the train atthe start and at the end is controlled ie we restrict ourself to twopossible states stopping and passing In case of passing it is possibleto traverse microscopic elements with different velocities and thus dif-ferent durations for the same train type can occur In order to receivea conservative macroscopic model we choose the calues for the ldquoworstrdquopassing
This is a reasonable compromise between all possible passing stateswhich could be all allowed velocities between zero and a given maxi-mum speed This would unnecessarily increase the needed simulationruns considered route data and train type definitions These aspectscould be varied in a post-processing step after the macroscopic plan-ning However a simple restriction to the ldquoworst caserdquo of traversinga track that is train stops at the start and at the end can lead tounderestimation of the capacity and thus to wrong identification of
2 Macroscopic Railway Modeling 83
j2
j1
r1
r2
tim
e
Sta
tion
A
Sta
tion
BFigure 17 Headway time diagrams for three succeeding trains on one single track
(j1 j2)
bottlenecks as we have seen in Example 22 Therefore the durationsof our macroscopic model depend on train types and events
The blocking times are transfered into minimal headway times betweentrain departures Instead of controlling all blocking times in each blocksegment we simplify the protection system to valid usages of the tracksIn Figure 17 the minimal headway times are illustrated for the giventrain sequence Note that for the third and last train no headwayarea is plotted because no succeeding train is scheduled Of course aforbidden area based on the blocking time stair of that train and apotentially succeeding train has to be considered
23 An Algorithm for theMicrondashMacrondashTransformation
We developed an algorithm that carries out the transformation fromthe microscopic level to the macroscopic level The whole procedure
2 Macroscopic Railway Modeling 84
Algorithm 3 Algorithm for the Micro-Macro-Transformation innetcast
Data microscopic infrastructure graph G = (VE) set of routes R stationsB(r) c(r) isin C r isin R
Result macroscopic network N = (S J) with stations S tracks J and train types Cbegin
ND Stmp = empty foreach r isin R doforeach b isin Br do
create s create standard station
Stmp = Stmp cup s
foreach (r1 r2) isin (RtimesR) dowhile divergence or convergence between r1 and r2 is found do
create p create pseudo station
Stmp = Stmp cup pwhile crossing between r1 and r2 is found do
create p q create pseudo stations
Stmp = Stmp cup p q
AG S = aggregateStations(Stmp)J = (s1 s2) isin S times S| existr isin R with s2 = nextStation(r s1)
TD foreach j isin J doforeach r isin R do
djc(r) = calculateRunningT ime(j r∆)
foreach (r1 r2) isin (RtimesR) dohjjc(r1)c(r2) =maxhjjc(r1)c(r2) calculateHeadwayT ime(j r1 r2∆)
if j is single way thenhjjc(r1)c(r2) =
maxhjjc(r1)c(r2) calculateHeadwayT ime(j j r1 r2∆)
return N = (S J)
is described in Algorithm 3 In the following we will give some addi-tional explanation to the algorithm We skip the details on the differentrunning modes to simplify the notation There are three main stepsmacroscopic network detection (ND) aggregation (AG) and time dis-cretization (TD)
Macroscopic network detection means to construct the macroscopic di-graph N = (S J) induced by R Let B(r) be the set of visited stationsof route r isin R ie locations (microscopic nodes) where the train stopsand is allowed to wait All visited stations are mandatory macroscopicstation nodes Note that after aggregation different microscopic nodescan belong to the same macroscopic station (area) If a conflict be-tween two routes is detected at least one pseudo station is created Aconflict occurs not only in the case of converging or diverging routesbut especially if microscopic elements are used in both directions egif one route crosses another route This detection is simply done by a
2 Macroscopic Railway Modeling 85
pairwise comparison of the train routes So in any case of using thesame track in opposite directions a conflict is detected and two pseudostations are created to isolate the conflicting part In the same wayonly one pseudo station is created if a con- or divergence occurs Theresulting set of stations Stmp can be further aggregated Note thatmicroscopic nodes for each platform (affected by the routes) inside astation are contained in Stmp The routine aggregateStations() in Al-gorithm 3 enforces the imaginable aggregations as informal describedin Section 2 to a station set S Accordingly the station capacitiesare defined in that function as well as the turn around times for theconsidered train types C
After this step the macroscopic network detection with respect to thestations is finished It remains to divide the routes R into sections ieinto tracks with respect to S The subsequent station of node v on thetrain route r is denoted by nextStation(r v) For the creation of thetracks it is important to mention that there could be more than onetrack between two macro stations especially after aggregation stepseg if there are two tracks between two aggregated macroscopic stationsthat could both be used by trains from the same direction So a trackis clearly identified by the starting and stopping microscopic (station)node and in addition to that by the set of microscopic arcs that weremapped to this track
(TD) the calculation of the rounded running and headway times isthe last step of the algorithm On track j we denote the running timeof train route r by drj (= djc(r)) the headway time hjjc(r1)c(r2) for theself correlation case ie when a train on route r2 follows a train withroute r1 and the headway time for the single way case with hjjc(r1)c(r2)The running times are calculated by the cumulative rounding proce-dure calculateRunningT ime() is implemented by Algorithm 1 Thefunction calculateHeadwayT ime() provides the headway times by Al-gorithm 2 and formula 1 For each route the running times and foreach pair of routes the headway times are calculated and (conserva-tively) aggregated according to the assignment of routes to train typesc isin C If there are several routes for the same train type alwaysthe maximum time of the attribute is taken The details on runningmodes have been omitted because it is only another technical questionNevertheless in netcast running and headway times with respect torunning modes are implemented
2 Macroscopic Railway Modeling 86
Figure 18 Constructed aggregated macroscopic network by netcast
In Figure 18 one of the macroscopic networks for the Simplon Tunnelgenerated by Algorithm 3 is shown Finally we summarize the resultingmacroscopic data
(directed) network N = (S J) with stations ie ldquostation areasrdquoS and tracks J
mapping of subpaths of routes to tracks
mapping of microscopic nodes to stations
running time on tracks for all C measured in ∆
headway time on all tracks for all pairs of C measured in ∆
headway time on single way tracks for all pairs of C measured in∆
each micro element e isin E corresponds to at most two (reverselydirected) tracks
each micro element v isin V corresponds to at most one (pseudo)station
Remark 212 The constructed (technical minimal) headway matricesH in netcast are valid ie transitive and order-safe
Remark 213 We developed our transformation tool netcast basedon a given set of routes The idea is to extract the components ofthese routes and map them to train types so that ldquonewrdquo routes can beconstructed Let routes from station A via C to D and from B via Cto E for the same train type be given Figure 19 shows the situationie both train routes stop at station C After the transformation bynetcast the macroscopic model can even handle trains from A to Eand from B to D for that train type via re-combination This allows toreduce the simulation effort to a standard set of patterns and routes
2 Macroscopic Railway Modeling 87
Remark 214 Furthermore netcast aggregates the microscopic in-frastructure network as much as possible based on the set of routestheir overlappings and their stopping pattern In Figure 20 this ishighlighted on several examples On the left the macroscopic network isshown which is produced by netcast if only High Speed Trains (EC)from Brig to Dommodossola and vice versa are considered Due to thefact that no intermediate stopping for these trains is needed the macro-scopic network shrinks to only two stations and two tracks (each perdirection) In the middle the same is done if you consider regionaltrains which stops at some intermediate stations On the right handthe final network for the Simplon with respect to all different types oftrains can be seen Note this is the same network as in Figure 18only visualized in TraVis using the correct geographical coordinates
Remark 215 netcast provides a re-translation of train paths fromthe macroscopic model to the microscopic model That is the macro-scopic path in N = (S J) will be transfered to a microscopic path inG = (VE) Note that in case of station aggregations some degree offreedom in choosing the precise routing inside a station occurs Further-more the departure and arrival times of the macroscopic model whichare given in ∆ are stated more precisely with respect to the originaldurations given in δ
A
B
C D
E
Figure 19 New routing possibilities induced by given routes
(a) only EC (b) only R (c) all train types
Figure 20 Macroscopic network produced by netcast visualize by TraVis
3 Final Remarks and Outlook 88
3 Final Remarks and Outlook
In this chapter we discussed a standard microscopic railway model anda novel macroscopic one that appropriately represents infrastructureresources and thus capacity We introduced a convenient transforma-tion approach which we implemented as the tool netcast The bigadvantage is that the approach is generally applicable to any micro-scopic railway model ie data of a standard microscopic railway sim-ulation tool In addition the reliability and quality of the results isobviously much higher in an integrated system than isolated applica-tions Our Micro-Macro Transformation algorithm detects the macro-scopic network structure by analyzing interactions between standardtrain routes In this way the algorithm can ignore or compress parts ofthe network that are not used by the considered train routes and stillaccount for all route conflicts by constructing suitable pseudo stationsTime is discretized by a cumulative rounding procedure that minimizesthe differences between aggregated and real running times
Furthermore we analyzed the error propagation of rounding procedurescaused by the transformation and the more coarse discretization Thuswe can directly quantify the quality of a macroscopic railway model incomparison to the originated microscopic one The impact of the timediscretization of a railway model can be enormous We will discuss thison several experiments in Chapter IV and Section 4
However with our approach a fixed discretization ∆ can be determinedto construct a macroscopic model with legitimated and reliable resultsThe question which fixed discretization one should choose arises inseveral optimization contexts eg LPP and PESP and is very rarelydiscussed In most cases software systems in operation work with afixed unit ie minutes in most of the related literature The workof Lusby (2008) [158] is exceptional who is using tints of 15 secondsHence it is an interesting field to evaluate discrete models ie notonly railway models with respect to different time scales Furtherdevelopments will be to introduce a dynamic handling of discretizationinstead of a fixed approach to face up to the major challenge directlyldquoinsiderdquo the solver
Chapter III
Railway Track Allocation
In this chapter we introduce the track allocation problem recapitulateseveral appropriate models from the literature and discuss them Amajor contribution will be the development of an extended formula-tion which yields computational advantages especially for real worldinstances We analyze the polyhedral relations of these models andpresent several extensions Finally a sophisticated algorithm for theextended formulation to solve the track allocation problem based oncolumn generation techniques and the approximate bundle method willbe presented
The novel model approach is joint work with Ralf Borndorfer SteffenWeider kindly provided an implementation of the approximate bun-dle method and of the rapid branching heuristic for set partitioningproblems This code was the basis of the adapted versions in TS-OPTwhich has been implemented by the author of this thesis This chaptersummarizes the current state of our research which has already beenpresented at conferences ie Borndorfer amp Schlechte (2007) [30 31]Borndorfer et al (2006) [34] Borndorfer Erol amp Schlechte (2009) [38]Borndorfer Schlechte amp Weider (2010) [43] Schlechte amp Borndorfer(2008) [188] It has already received considerable recognition in re-search on the track allocation problem visible in recently published lit-erature eg Cacchiani (2007) [51] Cacchiani Caprara amp Toth (2007)[52] Cacchiani Caprara amp Toth (2010) [54] Caimi (2009) [57] Fis-cher amp Helmberg (2010) [89] Fischer et al (2008) [90] Klabes (2010)[129] Kontogiannis amp Zaroliagis (2008) [136] Lusby (2008) [158] Lusbyet al (2009) [159]
90
1 The Track Allocation Problem 91
1 The Track Allocation Problem
The track allocation problem also known as the train timetabling prob-lem (TTP) in the literature is the following problem Given is anmacroscopic railway model and a set of train slot requests The (TTP)is to decide which subset of the train requests should be realized andwhat are the exact departure and arrival times of these trains In thiscontext a train slot is a path through the infrastructure network to-gether with exact departure and arrival times for all visiting stationsFurthermore it has to fulfill the requirements of the request specifica-tion However the precise definition will be evolved in this section
Thereby the solution schedule must be a track allocation which isfeasible and optimal ie the solution satisfies all operational macro-scopic infrastructure constraints and maximizes a given objective iea ldquoprofitrdquo function This is a profit-oriented approach persecuted bynetwork provider governor or marketer in the near future eg DBNetze AG [73]Trasse Schweiz AG [207] or ProRail [179]
One could also ask for a ldquocost-minimalrdquo train schedule for given trainsfrom an operator point of view Online dispatching can also be seenas a track allocation problem as minimizing additional waiting timesof the considered trains Obviously the real time dispatching problemhas a different flavor because it needs a different quality of data andshorter solving times but from a mathematical modeling point of viewit is basically the same problem We already discussed the relatedliterature in Chapter I and Section 6
One part of the input of the track allocation problem the macroscopicrailway model was already presented in Chapter II and Section 21The other one the train demand specification will be introduced inSection 11 of this chapter Together they specify an instance of thetrain timetabling or track allocation problem see Figure 1 This spec-ification was developed as a general auction language for railway usagein Borndorfer et al (2006) [34] Furthermore it is used as a stan-dardization for macroscopic train timetabling problems in the problemlibrary TTPlib see Erol et al (2008) [85]
For passenger traffic which is mainly periodic and cross-linked we re-fer to the work on partial periodic service intention see Caimi (2009)[57] In that setting the definition of connections and time dependen-cies between different trains ie meetings of train slots build the core
1 The Track Allocation Problem 92
macronetwork
trainrequests
TTPlibproblem
solver timetable
Figure 1 Concept of TTPlib 2008 (focus on train demand specification and TTP)
of the specification and models For our purpose individual aspectsare most relevant for example the requirements of cargo trains such asdesired arrival times at certain stations or minimum dwelling timesOur specification is also influenced by the work of Schittenhelm (2009)[186] which provides an extensive discussion of quantifiable timetableaspects Nevertheless we will show how to integrate global schedulerequirements like connections or periodic services in our models in Sec-tion 24 Section 12 gives a precise description and construction of aninstance of the TTP by Definition 15
11 Traffic Model ndash Request Set
Consider a basic setting that allows extensive valuation for individualtrain slot requests of the following general form Denote by I the setof given train slot requests Each slot request i isin I specifies a traintype ci isin C a basic profit bi isin Q+ and a list of station stops withat least two elements namely start and final destination On the onehand for each stop mandatory definitions are required
station s isin S
minimum and maximum departure time tdepmin le tdepmax isin N
minimum and maximum arrival time tarrmin le tarrmax isin N
On the other hand additionally optional intentions for each stop canbe specified
optimal departure time tdepopt isin [tdepmin tdepmax] cap N
optimal arrival time tarropt isin [tarrmin tarrmax] cap N
penalties for exceeding times parr+ pdep+ isin Q+ per time unit
penalties for falling below optimal times parrminus pdepminus isin Q+ per timeunit
minimum and maximum dwell time dmin le dmax isin N
1 The Track Allocation Problem 93
Finally it is possible to guide certain attributes of the complete pathby means of
penalty for exceeding of minimum travel time ptravel+ isin Q+ pertime unit
penalty for additional stops pstops+ isin Q+
By source of those parameters mainly the characteristics of individualcargo trains are reflected We deliberately do not consider to specifyrelations between different trains ie this is necessary for passengertrains to keep the TTPlib simple However future challenges will beto incorporate passenger timetable optimization models like PESP inthe specification of the TTPlib
Train slots can be preferred which realize fast connections between ori-gin and destination by choosing ptravel+ larger than zero In Example 11usual penalty functions are given and explained
Analogously it might be useful that slots on which the train has to un-necessarily brake and accelerate again are penalized by pstops+ Energy-saving see Albrecht (2008) [10] is a hot topic in railway engineeringfrom an operational point of view but can also be considered in plan-ning these slots to some extend However we restrict our considerationand input parameters to the list above but of course some other aspectsmight also be interesting eg penalties for exceeding the minimumroute length to prefer direct and short routes
Example 11 Let the function on the left hand in Figure 2 specifythe penalty ε for deviation from the optimal departure time at the firststation of the train slot It can be seen that shifting the departure timewithin the given time window by one time unit earlier than desired ismore punished than departing by one time unit later The function onthe right hand could be useful to control an arrival event No penaltyε is obtained for arriving before the optimal point but exceeding thattime at this stop is critical for the train and hence it is highly penalizedFigure 3 shows a simple profit function w() with respect to a given basicprofit b and both penalizations
Of course the restriction of that framework to two-stepwise-linear func-tions is nonessential The reason for that is to keep the definition ofthe objective function of any train request as simple as possible Thisallows to define a huge range of different goals by just changing someparameters of each train request Nevertheless we want to point out
1 The Track Allocation Problem 94
ε
ttdepmin tdepopt tdepmax
ε
ttarrmin tarropt tarrmax
pdepminus = 1
pdep+ = 05
parrminus = 0
parr+ = 3
Figure 2 Penalty functions for departure(left) and arrival(right) times
tdepmintdepopt
tdepmax
tarrmin tarropt tarrmax
b
b
Figure 3 Profit function w() depending on basic profit and departure and arrivaltimes
explicitly that it would be possible to use much more complex nonlin-ear functions because in the end these function evaluations only leadto different values for the objective coefficients of some arcs Howeverthe framework should not exceed a certain degree of complexity
The goal for developing this framework is to give a train operator thepossibility to specify easily their requirements with only a few param-eters It is an economic ldquobidding languagerdquo that enables train oper-ating companies to express their train slot requests in a satisfactorytractable and flexible way We present possible extensions to deal withcombinatorial restrictions on the train request in a separate Section 24
Finally we want to clarify some easily mistakable terms for stoppedtrains In the request specification we use the term dwell time whichcan either be a turn around activity or pure waiting Due to the factthat this does not make a difference from an operator point of view we
1 The Track Allocation Problem 95
do not distinguish between them However for the consistency of trainpaths we have to handle turn around activities appropriately
12 Time Expanded Train Scheduling Digraph
We expand our macroscopic railway model along a discretized time axisto model timetables in an event activity digraph D = (VA) the socalled train scheduling digraph All durations of G = (VN AN) and alltimes of I are given with respect to a constant discretization ∆ egone minute We construct multiple copies of the infrastructure nodeset VN over a time horizon one node set for each time and for eachtrain request i isin I ie we expand G|ci The arcs AN associated withtrain type ci isin C are also copied connecting nodes in time layers thatfit with the running or turn around times as well as with the eventdefinition In that large scale digraph certain paths are realizations ofrequests ie these graphs can easily have thousands of nodes and arcseven with a discretization of minutes Sometimes we also use the termpath p implements request i By definition a request can be very flexiblewith respect to the route and the event times We denote the set ofimplementing paths for request i isin I by Pi The formal constructionof D = (VA) is as follows
We denote the time horizon by T = t0 tmax sube N ie t0 is thefirst time of an event and tmax the last The set of time-nodes associatedto train request i isin I is Vi = (v t) v isin VN t isin T sube VN times Twith VN = S times 1 2 times arr dep passing ie (v t) is the copy ofinfrastructure event node v isin VN of side one or two and station s isin Sat time step t for request i isin I
The next paragraphs will describe four different types of arcs I to IVTwo time-nodes (u τ) and (v σ) are connected by a (running) time-arc((u τ) (v σ)) of train type ci if nodes u and v are connected by an arca isin AN in the infrastructure network G In addition the running timed(a) = dj(a)cim(a) from u to v for a train of type ci must be equal toσ minus τ where j(a) denotes the corresponding track of arc a and m(a)the considered running mode respectively Note that node u can be ofmode dep passing and v of mode arr passing We denote the setof running time-arcs by AI
1 The Track Allocation Problem 96
The second set of potential time expanded arcs are rdquorealldquo turn aroundactivities inside a station Analogously we connect time-nodes (u τ)and (v σ) by a time-arc ((u τ) (v σ)) of train type ci if a turn aroundarc a isin AN in the infrastructure network is defined between this arrivaland departure pair and d(a) = σ minus τ Note that in this case node umust be an arrival and v a departure node on the same side of thestation ie o(u) = o(v)
The third type of arcs is useful to model additional waiting We dis-tinguish between two possibilities
explicit waiting on a turn around arc from arrival to departurenodes
implicit waiting on a waiting time-line between departure nodes
It depends on the considered degree of freedom which waiting policy ismore reasonable For train requests with a restrictive maximum waitingor dwell time at a station ie most passenger trains we suggest explicitwaiting on turn around arcs between arrival nodes and departure nodesThe arrival node (v τ) is then connected with departure node (u σ)if a turn around arc a isin AN with duration d(a) = dsf and ci isin fis defined in the infrastructure network between v isin VN and u isin VNand if dmin(s i) le dsf = σ minus τ le dmax(s i) Hence the duration of awaiting arc respects the given waiting interval for train i in station sand the minimal turn around time dsf Note that in that model thetotal duration of a time expanded turn around arc consists of the timeneeded to perform the turn around1 and a valid waiting expansion
Remark 12 Let m be the number of potential arrival points in timeand n the number of departure points in time then explicit waitingcould lead to at most m middot n turn around arcs
In cases where the length of the waiting interval inside a station couldbecome arbitrary large and is a priori not bounded we use a timelineconcept Timelines are applied to a lot of planning problems wherethe number of potential arcs can become too large to handle themexplicitly see Desrosiers Soumis amp Desrochers (1982) [75] KliewerMellouli amp Suhl (2006) [132] Lamatsch (1992) [143] Weider (2007)[213]
A turn around arc from each arrival node is created to enter the depar-ture timeline on the other station side Thus a minimum waiting time
1For the artificial case of o(u) 6= o(v) the duration dsf might be zero
1 The Track Allocation Problem 97
can be ensured Note that these arcs are the only ones in D = (VA)with a potential duration of zero The departure nodes v isin VN are theconsecutively connected via waiting arcs time by time In particularwaiting at node v is modeled by a time-arc ((v t) (v t+ 1)) of type IV
for all t isin t0 tmax minus 1
Remark 13 Let m be the number of potential arrival points in timeand n the number of departure points in time then implicit waitingcould lead to at most m+ nminus 1 turn around and waiting arcs
In Figure 4 both model approaches are shown The advantage of ex-plicit waiting arcs is that not only minimum but also maximum du-ration can be handled Furthermore it is possible to define arbitraryobjective values and attributes for each arrival and departure pair
In a timeline this information is lost and decomposed The arcs onthe left in Figure 4 are replaced by the tree on the right Each arcis represented by a path in the timeline and vice versa Fortunatelyin our setting the valuation and attributes of an arc are linear in thecomponents of the representing path because of the dependence of timeNevertheless in an implicit waiting representation the control of themaximum waiting time is lost This is compensated by a much smallerrepresentation see 12 and 13 Both representations are available inTS-OPT However default setting is to use the sparse timeline conceptbecause a maximum waiting requirement is rather rare and can furtherbe interpreted as a soft constraint in our instances In the case thata hard maximum waiting is required it is possible to use the explicitmodel for that request However both arc types ie II and IIIare representing waiting with the difference that the first one connectarrival with departure nodes and the second one connect only departurenodes
Finally we define a dummy source node si and sink node ti for eachrequest i isin I The source node si represents the start of request iand is connected via dummy arcs with all valid departure time-nodesv = (s om τ) isin V Node v must be a departure (or passing2) onewith s equal to the start station of i isin I and τ must be inside the givendeparture time window Analogously we connect a valid node v withsink ti if v is an arrival (or passing) node of the final station of i and ifτ is inside the arrival time window
2Passing nodes are allowed at begin or end to handle ldquofly inrdquo or ldquofly outrdquo traffic
1 The Track Allocation Problem 98
t=1
t=2
t=3
t=4
t=5
t=6
t=7
t=8
Figure 4 Explicit and implicit waiting on a timeline inside a station
To simplify the notation we denote the time of time-node v isin V byτ(v) which is the last element of this 4-tupel Analogously we usem(v) s(v) and o(v) as a mapping to access the event m station s andthe side or orientation of o(v) of node v In the same way we denotethe track mode and train type of a running arc a isin AI by j(a)m(a)and c(a) respectively
Due to this construction we can partition the set of arcs Ai with respectto the four following arc types
I running arcs on tracks j isin J
II turn around arcs inside stations s isin S
III waiting arcs inside station s isin S
IV artificial arcs for begin and end of a train request i isin I
Note that by definition s(u) = s(v) for all (u v) isin AII cup AIII ands(u) 6= s(v) for all (u v) isin AIcupAIV with s(si) = s(ti) = empty respectivelyTo make the notation clear we use sometimes the set Ai which is thesubset of all time-arcs related to request i isin I By AI the set of allrunning arcs a isin A are denoted Thus the set of arcs A is a disjunctiveunion middotcupiisinIAi as well as A = AI middotcupAII middotcupAIII middotcupAIV
Furthermore we associate with each arc a isin A an utility or profit valuewa which reflects the objective parameters of the request definitionThe idea is that the profit or utility value wp of a path p isin Pi which
1 The Track Allocation Problem 99
implements request i isin I can be expressed as the sum of all incidentcomponents ie this value is linear with respect to incident arcs
wp =sumaisinp
wa
To avoid unnecessary notational overhead we restrict ourselves to thebasic case of two mandatory stops that is departure at origin andarrival at destination station The special case where a train requestasks for more than two stops can be appropriately reduced to the basiccase stop by stop However to ensure that each intermediate (station)stop is visited in an si minus ti-path several copies of time-nodes have tobe considered
Let vtraveli be the optimal values3 for the duration of the requests i isin Ithat is the difference between arrival time at final station and departuretime at first station of request i isin I Then the objective values wa ofa = (u v) isin Ai are defined as follows
wa =
minusptravel+ (τ(v)minus τ(u)) if a isin AI cupAII cupAIII
bi + vtraveli ptravel+ minus pdepminus (tdepopt minus τ(v)) if a isin AIV u = si τ(v) le tdepopt
bi + vtraveli ptravel+ minus pdep+ (τ(v)minus tdepopt ) if a isin AIV u = si τ(v) ge tdepopt
minusparrminus (tarropt minus τ(v)) if a isin AIV v = ti τ(u) le tarropt
minusparr+ (τ(v)minus tarropt ) if a isin AIV v = ti τ(u) ge tarropt
The result is a space-time network D = (VA) =⋃iisinI(Vi Ai) in which
train slots correspond to directed paths proceeding in time In partic-ular directed paths from si to ti are slot realizations of train requesti isin I
Observation 14 The train scheduling graph D = (VA) is acyclicand therefore there exists a topological order of the nodes4
Obviously we have to perform the time expansion in an efficient man-ner because of the enormous number of potential nodes and arcs Theidea is to identify non-redundant station nodes and track arcs for eachrequest individually in a first step A priori shortest path computations
3These can easily be determined by appropriate shortest path computations withrespect to the duration in G = (VN AN )
4Even if we allow (artificial) turn around inside a station which could havea duration of zero the strong monotony of time on all other arcs especially alloutgoing arcs of departure nodes prevent cycles
1 The Track Allocation Problem 100
Algorithm 4 Construction of D
Data network N = (S J) and requests I (discretized in ∆)Result train scheduling graph D = (VA)init V larr empty Alarr empty foreach i in I do
compute time expansion of Di = (Vi Ai)
compute irreducible digraph Di = (Vi Ai)
compute profit maximizing path in Di = (Vi Ai)
set D =⋃iisinIDi
ie for each train type help to avoid time expansion in unnecessarydirections of the network (VN AN)
After this trivial route preprocessing we only perform the time expan-sion of the remaining network part to reduce the number of consideredtime-arcs and time-nodes Finally invalid sources which are not con-nected to at least one valid sink or invalid sinks which cannot bereached by at least one source are eliminated
Figure 5 shows an example ie in network hakafu simple for a trainrouting graph before preprocessing with 123 potential event nodes and169 activity arcs The corresponding train wants to depart from FSONin time interval [0 5] and arrive at station FCG in time interval [0 15]Depicted are all potential event nodes (station event side time) whichare reachable from the dummy source s in the given time window Afterpreprocessing the graph shrinks to 12 nodes and 13 arcs see Figure 6
Algorithm 4 spans the graph for each individual train request i isin Istop by stop ie from the first station to next specified stop of therequest and produces an irreducible graph representation Di = (Vi Ai)for request i isin I In particular no redundant time nodes or arcs arepresent Furthermore we compute a profit-maximizing path for eachrequest i isin I that is a longest path with respect to weights w in eachacyclic digraph Di The sum of these values is a trivial a priori upperbound of the TTP In Example 16 and in Figure 7 a preprocessednetwork D = (VA) is shown in detail
The space-time network D = (VA) can also be used to make all po-tential conflicts between two or more train slots explicit In fact each
1 The Track Allocation Problem 101
Figure 5 Complete time expanded network for train request
conflict corresponds to timed resource consumption on tracks or insidestations and can be defined by an appropriate subset of time-arcs A
For a potential headway conflict on a track consider two train slots oftype c1 and mode m1 and type c2 and mode m2 departing from thetrack j isin J via arcs a1 isin A and a2 isin A arriving at times t1 and t2respectively wlog let t1 le t2 There is a headway conflict betweenthese slots if t2 lt t1 + hjc1m1c2m2 This conflict can be ruled out bystipulating the constraint that a conflict free set of slots can use only
1 The Track Allocation Problem 102
Figure 6 Irreducible graph for train request
one of the arcs a1 and a2 Doing this for all pairs of conflicting arcsenforces correct minimum headways
For a station capacity conflict consider train slots pi of train typeci isin C i = 1 k entering station s isin S with capacity κsf ci isin fat time t The capacity at time t is exceeded if more than κsf trainsbelonging to that train set are present at this station at time t Notethat we assume that departing trains at time t do not count at time tbecause they are leaving the station at this moment
This conflict can be ruled out in a similar way as before by stipulatingthe constraint that a conflict free set of slots can use at most κsf ofthe following arcs
uv isin AI cup AIV which enters an arrival or a passing node v ofstation s at time t ie s(v) = s and τ(v) = t
1 The Track Allocation Problem 103
uv isin AII cup AIII which starts before time t ie τ(u) lt t andends after time t ieτ(v) gt t
This definition for a general station capacity κsf illustrates the flexi-bility of the model and the possibility to handle more specific stationrestrictions which can easily be modeled by appropriate definitions ofthe restricted subset of A
Cacchiani (2007) [51] construct conflicts sets for consecutive arrivalsconsecutive departures and overtaking situations at certain intermedi-ate stations
Fischer et al (2008) [90] consider for instance station capacities de-pending on the side of the station to control the incoming trains perdirection
This flexibility of the conflict sets is not needed if the network cor-responds exactly to the microscopic infrastructure as in the work ofBrannlund et al (1998) [44] Lusby (2008) [158] and Fuchsberger(2007) [94] However on this scale only small scenarios can be handledand further requirements which are arising non-naturally eg forbid-den meetings of trains are very hard to incorporate
We denote an arbitrary conflict by γ the set of all conflicts by Γ theset of conflict arcs associated with conflict γ by Aγ and the maximumnumber of arcs from Aγ that a conflict-free set of slots can use by κγIf a chosen set of si minus ti paths is conflict-free with respect to Γ wesometimes use the term simultaneously feasible The train timetablingor track allocation problem can then be defined as follows
Definition 15 Given train slot requests I a corresponding digraphD = (VA) a profit value wa for each time-arc a isin A and an ex-plicit definition of conflicts Γ on the time-arcs A the problem to finda conflict-free maximum routing from si to ti is called optimal trackallocation problem In other words we seek for a profit-maximizing setof simultaneously feasible si minus ti paths in D = (VA)
This is a natural and straightforward generalization of the train time-tabling problem described in Brannlund et al (1998) [44] CapraraFischetti amp Toth (2002) [62] and Caprara et al (2007) [64] to the caseof networks There only the case of a single one-way track corridoris considered For convenience we will use the acronym TTP to de-note the optimal track allocation problem It was shown in CapraraFischetti amp Toth (2002) [62] that the TTP is NP-hard being a gen-
1 The Track Allocation Problem 104
requestbasic train stop at time window preferencesvalue type station (tmin topt tmax pminus p+)
blue 10 PTX (1 3 4 1 2)Z (3 5 6 0 1)
red 10 CTX (1 3 3 2 0)Z (5 6 7 2 0)
Table 1 Definition of train request set
eralization of the well-known maximum stable set problem see Gareyamp Johnson (1979) [97]
Example 16 Consider again a tiny network graph consisting of threestations and only two tracks Assume that the infrastructure can be usedby two different train types called blue and red who need one respec-tively two time units to pass the given tracks and each has to respect aminimal headway of one minute on each track For simplification thesetrains can only perform a running mode of type 1 With the introducednotation we have given
stations S = X Y Ztracks J = (X(2) Y (1)) (Y (2) Z(1))train types C = PT CTrunning times djPT1 = 1 djCT1 = 2 forallj isin J and
minimal headway times hjc11jc21 = 1forallj isin J c1 c2 isin C
We consider two train requests Both should start in station X andtarget station Z and are allowed to stop in Y for an arbitrary timeThe first train should start in the time interval [1 4] and arrive in thewindow [3 6] while the second train should depart in [1 3] and arrivein [5 7] As we see we obtain a time horizon of T = 1 7 for the totaltrain routing graph In Table 1 the preferences and valuations of therequests are listed which consists only of a basic value and penalties forscheduled arrival and departure times The graph D = (VA) producedby Algorithm 4 is shown in Figure 7
The given request valuations of Table 1 were transferred to objectiveweights wa of the time-arcs see labels in Figure 7 In this exampleonly the artificial begin and end arcs of the ldquotrain routingrdquo flow havevalues wa different from zero
1 The Track Allocation Problem 105
t=1
t=2
t=3
t=4
t=5
t=6
t=7
s1
s2
t1
t2
X Y Z2 1 2 1
8
10
6
9
108
8
-1
-2
Figure 7 Preprocessed time-expanded digraph D = (VA) of example 16
The optimal track allocation problem is then to find a utility maximizingset of conflict-free si minus ti -flows Here is a complete list of the conflictset Γ
γ1 = ((X 2 dep 1) (Y 1 arr 2)) ((X 2 dep 1) (Y 1 arr 3))γ2 = ((X 2 dep 2) (Y 1 arr 3)) ((X 2 dep 2) (Y 1 arr 4))γ3 = ((X 2 dep 3) (Y 1 arr 4)) ((X 2 dep 3) (Y 1 arr 5))γ4 = ((Y 2 dep 3) (Z 1 arr 4)) ((Y 2 dep 3) (Z 1 arr 5))γ5 = ((Y 2 dep 4) (Z 1 arr 5)) ((Y 2 dep 4) (Z 1 arr 6))γ6 = ((Y 2 dep 5) (Z 1 arr 6)) ((Y 2 dep 5) (Z 1 arr 7))
The best path for the red and blue request has value 10 each but unfor-tunately the simultaneous routing on track (X(2) Y (1)) is invalid withrespect to the headway conditions ie the red and the blue train wantto departing at node X(2) at time 3 To finish the example an optimalsolution realizing a profit value 19 is to schedule the blue train on path
2 Integer Programming Models for Track Allocation 106
p1 = (s1 (X 2 dep 3) (Y 1 arr 5) (Y 2 dep 5) (Z 1 arr 7) t1)
with utility value wp1 = 10 and the red one on path
p2 = (s2 (X 2 dep 2) (Y 1 arr 3) (Y 2 dep 3) (Z 1 arr 4) t2)
with wp2 = 9 respectively
2 Integer Programming Models for Track
Allocation
Section 21 discusses standard integer programming formulations tothe track allocation problem based on the train scheduling graph D =(VA) Furthermore we develop an alternative formulation to take ad-vantage of the structure of the headway conflicts in Section 22 Dueto the very large size of real world problem instances static arc for-mulations are limited To overcome this limitation path versions areoften formulated These are suitable to be solved by sophisticated col-umn generation approaches or approximate bundle methods as we willpresent in Section 3
In Section 23 the models are theoretically compared and analyzedWe will also show that our coupling formulations are extended formu-lations of the original packing formulations Finally we present severalpractical extensions to the problem and models in Section 24
21 Packing Models
As mentioned before operational railway safety restrictions can be han-dled by conflict sets in D = (VA) =
⋃iisinI Di This modeling approach
was introduced by the pioneer works of Brannlund et al (1998) [44]and Caprara et al (2006) [63] on railway track allocation Each con-flict γ isin Γ consists of a subset of arcs Aγ sube A and an upper boundκγ isin Z To formulate the track allocation problem as an integer pro-gram we introduce a zero-one variable xa (ie a variable that is al-lowed to take values 0 and 1 only) for each arc a isin Ai If xa takes
2 Integer Programming Models for Track Allocation 107
a value of 1 in an (APP)prime solution this means that a slot request iassociated with arc a passes through arc a clearly this implies thatslot request i has been assigned On the other hand xa = 0 means thatarc a is not used by a slot associated with slot request i independentlyof whether slot request i is assigned or not Furthermore we are givenwa for each arc a of slot request i in order to account for the overallproceedings or utility of a track allocation Let us finally denote byδin(v) = (u v) isin Ai the set of all arcs entering a time-node v isin ViSimilarly let δout(v) = (v w) isin Ai be the set of arcs leaving time-node v With these definitions and the notation of Section 1 the trackallocation problem can be formulated as the following integer program
(APP)prime
maxsumiisinI
sumaisinAi
waxa (i)
stsum
aisinδout(si)
xa le 1 foralli isin I (ii)sumaisinδin(ti)
xa le 1 foralli isin I (iii)sumaisinδout(v)
xa minussum
aisinδin(v)
xa = 0 forallv isin Vi si ti i isin I (iv)sumaisinAγ
xa le κγ forallγ isin Γ (v)
xa isin 0 1 foralla isin Ai i isin I (vi)
In this model the integrality constraints (vi) state that the arc variablestake only values of 0 and 1 Constraints (ii)ndash(iv) are flow constraints foreach slot request i they guarantee that in any solution of the problemthe arc variables associated with slot request i are set to 1 if and onlyif they lie on a path from the source si to the sink node ti in D =(VA) ie they describe a feasible slot associated with slot request iThey are all set to 0 if no slot is assigned to slot request i Notethat constraints (iii) are redundant because (APP)prime (ii) and (APP)prime
(iv) already define the flow see Ahuja Magnanti amp Orlin (1993) [5]Constraints (v) rule out conflict constraints as described before
The objective function (i) maximizes total network utility by summingall arc utility values wa This integer program can be seen as a ldquodegen-eraterdquo or ldquogeneralizedrdquo multi-commodity-flow problem with additionalarc packing constraints In the sense that even though all train flowsare individual longest path problems in acyclic digraphs Di they areconnected by conflict set Γ and constraints (v) respectively
2 Integer Programming Models for Track Allocation 108
As we already mentioned Caprara et al (2001) [61] and Caprara Fis-chetti amp Toth (2002) [62] defined conflict sets for departures arrivalsand overtakings to ensure operational feasibility Although this formu-lation allows for a very flexible definition of conflicts a disadvantage ofmodel (APP)prime is the ldquohidden structurerdquo the detection and the poten-tially large size of Γ We will examine this issue for the case of headwayconflicts for which constraints (v) are packing constraints ie κγ = 1This can be done as follows We create a conflict graph Λ = (AI E)with node set AI of all running time-arcs As already described inSection 1 for a potential headway conflict on a track j isin J we canconsider two train slots of type c1 and mode m1 and type c2 and modem2 departing from the track j via arcs a1 isin A and a2 isin A arrivingat times t1 and t2 respectively Wlog let t1 le t2 then there is aheadway conflict between these slots if t2 lt t1 + hjc1m1c2m2
Each pair of conflicting arcs a1 and a2 defines an edge (a1 a2) isin E anda corresponding conflict set γ containing both time-arcs and an upperbound κγ = 1 Doing this for all pairs of conflicting arcs enforces correctminimum headways We denote this preliminary model by (APP)primebecause further observations will lead to much stronger formulations
It is clear that these pairwise conflict sets can be enlarged to inclusion-maximal ones which correspond to cliques in Λ In the following we willcollect some basic facts about detection and occurrence of maximumcliques in special graph classes The statements translate directly intoour setting The case of ldquofull block occupationrdquo can be seen as thesimplest one that is the headway time is set to the correspondingrunning time of the train Keep in mind that in this setting headwaysare completely independent from the type of the successor train theydepend only on the departure time The graph Λ becomes an intervalgraph Figure 8 illustrates the construction of Λ and the maximalcliques in that case
Lemma 21 In a block occupation model all maximal conflict sets canbe found in polynomial time since Λ is an interval graph
Proof The cliques in the conflict graph are collections of compact realintervals By Hellyrsquos Theorem see Helly (1923) [113] the intervals ofeach such clique γ isin Γ contains a common point p(γ) and it is easyto see that we can assume p(γ) isin τ(AI) = τ(v) v isin AI It followsthat the conflict graph Λ has O(AI) inclusion maximal cliques whichcan be enumerated in polynomial time In Booth amp Lueker (1976) [27]
2 Integer Programming Models for Track Allocation 109
Figure 8 Example for maximum cliques for block occupation conflicts
and Habib et al (2000) [107] linear time recognition algorithms can befound
Example 22 In Figure 8 the relation between headway conflict setson a track j isin J and the corresponding conflict graph Λ is shown Onthe left hand six trains are shown with the corresponding departure andarrival times In the middle the blocked intervals are projected On theright hand the induced conflict graph Λ can be seen Furthermore wehighlighted all maximal cliques in that small example by shaded areas
Observation 23 The train timetabling problem with full block occu-pation conflicts on a single track is equivalent to finding a maximumindependent set in interval graphs
In general the separation of the maximal clique constraints is not triv-ial This is because the entries5 of the headway matrix H are in generaldifferent for each train type and for each stopping behavior combina-tion
Furthermore realistic minimal headway matrices as presented in Sec-tion 21 are transitive see Definition 26 and in the majority of casesasymmetric Lukac (2004) [157] gives an extensive analysis of the struc-ture of clique constraints arising from triangle-linear and quadrangle-linear matrices and proves that the time window of interest is boundedby twice the maximum headway time However in realistic cases thiscan be quite large Since the number of constraints (APP)prime(v) canbe exponential in the number of arcs Fischer et al (2008) [90] pro-
5In case of full block occupation all entries are equal to the corresponding runningtime
2 Integer Programming Models for Track Allocation 110
pose to use a greedy heuristic to find large violated cliques Note thatconstraints (APP)prime(v) induced by station capacities can be separatedby complete enumeration We denote the arc sets corresponding to allmaximal cliques in Λ by Γmax and receive
(APP)max
sumiisinI
sumaisinAi
waxa (i)
stsum
aisinδout(si)
xa le 1 foralli isin I (ii)sumaisinδin(ti)
xa le 1 foralli isin I (iii)sumaisinδout(v)
xa minussum
aisinδin(v)
xa = 0 forallv isin Vi si ti i isin I (iv)sumaisinAγ
xa le κγ forallγ isin Γmax (v)
xa isin 0 1 foralla isin Ai i isin I (vi)
Note that constraints (APP) (iii) are again redundant The packingmodel can also be formulated with binary decision variables xp foreach path instead of arc variables xa Consequently we define theproceedings of a path p as the sum of its incident arcs
wp =sumaisinp
wa
The resulting version (PPP) reads as follows
(PPP)max
sumiisinI
sumpisinPi
wpxp (i)
stsumpisinPi
xp le 1 foralli isin I (ii)sumpcapAγ 6=empty
xp le κγ forallγ isin Γmax (iii)
xp isin 0 1 forallp isin Pi i isin I (iv)
Constraints (PPP) (ii) ensure that each request is implemented byat most one path Conflict constraints (PPP) (iii) make sure thatno headway or station conflict is violated (PPP) (iv) state that allpath variables xp are zero or one Finally objective (PPP) (i) is tomaximize the profit of the schedule
2 Integer Programming Models for Track Allocation 111
formulation variables non-trivial constraints
(APP) O(A) O(A)(PPP) O(P ) O(V )
(APP)prime O(A) O(A2)(PPP)prime O(P ) O(A2)
Table 2 Sizes of packing formulation for the track allocation problem with blockoccupation
The packing formulations of the optimal track allocation problem withblock occupation conflicts only have the sizes listed in Table 2 Fora set S we write O(S) = O(|S|) Model (PPP)prime is thereby a pathformulation based on pairwise headway conflict sets
We have seen for the block occupation case that the number of maxi-mal conflicting sets can be bounded by the number of nodes and can beefficiently constructed Unfortunately in the general case which aremodels (APP)prime and (PPP)prime it might lead to conflicts sets quadrati-cally in the number of running arcs
22 Coupling Models
We propose in this section an alternative formulation for the optimaltrack allocation problem that guarantees a conflict free routing by al-lowing only feasible route combinations and not by excluding conflict-ing ones as described in Section 21 The formulation is based on theconcept of feasible arc configurations ie sets of arcs on a track withoutheadway conflicts Formally we define a configuration for some trackj = (x y) isin J as a set of arcs q sube Aj = (u v) isin AI s(u)s(v) =(x y)or j((x y)) = j such that
|q cap Aγ| le 1 forallγ isin Γ
Denote by Qj the set of all such configurations for track j isin J and byQ the set of all configurations over all tracks The idea of the extendedmodel is to introduce 01 variables yq for choosing a configuration oneach track and to force a conflict free routing of train paths p isin Pthrough these configurations by means of inequalities
2 Integer Programming Models for Track Allocation 112
sumpisinPaisinp
xp lesum
qisinQaisinq
yq foralla isin AI
In Section 23 we will prove that this is equivalent to the packing con-straints (APP) (v) and (PPP) (iii) in case of headway conflicts Inthe following we will show that these feasible time-arc configurationsor sequences for each track j isin J can be constructed very efficientlyunder several reasonable assumptions
In a first step we introduce a headway conflict equivalence class foreach running arc a isin AI if their resource consumption on a track isequal The reason is that many time-arcs share the same headwayrestrictions ie the next potential departure times are equal evenif other attributes might be different (objective train type requestmode etc)
Definition 24 Two arcs a = (x y) and b = (w z) with a b isin AI areresource equivalent ie a sim b if
j(a) = j(b) (same track)
τ(x) = τ(w) (same departure time)
τ(y) = τ(z) (same arrival time)
and hj(a)c(a)m(a)kl = hj(b)c(b)m(b)klforallk isin C l isin M (same head-way time for any succeeding train type and mode)
Obviously the relation defined by 24 is reflexive symmetric and tran-sitive and thus a equivalence relation In Figure 9 running arcs of tworequests on track (X Y ) can be seen Assume that they fulfill addition-ally the Definition 24 then a hyperarc represents the correspondingequivalence class
Denote by AΨj the set of all equivalence classes on track j isin J and
AΨ =⋃jisinJ A
Ψj of all running arcs AI respectively Due to the headway
definition ie all minimal headway times are strictly positive onlyone arc of each class can be chosen However it does not matter whichone The idea is to define local feasible flows which ensure headwayfeasibility on each track and couple them appropriately with the trainor route flows Even if this trivial observation might complicate thenotation it is a crucial and necessary point to aggregate and strengthenthe models Otherwise this would lead to too many and foremostweaker constraints Instead of directly writing down a correspondingmodel however we propose a version that will model configurations
2 Integer Programming Models for Track Allocation 113
t=1
t=2
t=3
t=4
t=5
t=6
X Y2 1
Figure 9 Example for an equivalence class and a hyperarc
as paths in a certain acyclic routing digraph if the headway matrixis valid The advantages of such a formulation will become clear inthe following The construction extends the already described routingdigraph D = (VA) to a larger digraph as illustrated in Figure 10 Wewill denote the extended digraph by D = (V cup V A cup AΨ cup A)
The construction is as follows Let sj be an artificial source and tj anartificial sink node to define a flow on track j = (x y) Consider therunning arc classes AΨ
j on track j Denote by Lj = u (u v) isin AΨj
and Rj = v (u v) isin AΨj the associated sets of event nodes at the
start and end station of track j Note that all arcs in AΨj go from Lj to
Rj We denote by n(τ1 c1m1 c2m2) isin Z for v = (minus c1m1 τ1) isin Rj
the next possible departure time of a train of type c2 isin C and m2 isinMafter a train c1 isin C has departed with mode m1 isin M at τ1 Now letAj = (v u) v isin Rj u isin Lj be a set of ldquoreturnrdquo arcs that go backin the opposite direction and represent the next potential departure onthat track they connect the end of a running arc on j (or node sj)with all possible follow-on arcs (or node tj) on that
n(τ1 c1m1 c2m2) = τ1 minus djc1m1 + hjc1m1c2m2 (1)
(v u) isin Aj hArr τ(u) ge n(τ1 c1m1 c2m2) (2)
2 Integer Programming Models for Track Allocation 114
AΨjLj Rj sj
tj
sj
tj
Figure 10 Example for the construction of a track digraph
It is easy to see that the configuration routing digraph Dj = (LjcupRjcupsj tj AΨ
j cupAj) is bipartite and acyclic if all minimal headway timesare strictly positive
In Figure 10 the construction is shown on a small set AΨj On the left
the set of arcs (one per equivalence class) of track j isin J and the nodesets Lj and Rj can be seen In the middle the constructed graph Dj
is shown with dashed and dotted auxiliary arcs for the easy case of fullblock occupation
The graph size can be significantly reduced by merging structural nodesand introducing a time-line In the trivial case of full block occupationthe next possible train departure on track j is independent of the pre-ceding and succeeding train type or running mode ie the formula 2simplifies to
n(τ1 c1m1 c2m2) = τ1 minus dc1m1 + hjc1m1c2m2 = τ1
Since n(τ1 c1m1 c2m2) is exactly the arrival time of the consideredrunning train on track j we can merge nodes of set Lj and Rj if theirtimes match Therefore we connect consecutive departure nodes ofLj ie sj with the first one and the last one with tj respectivelyInstead of constructing all possible return arcs each arrival node in Rj
is only connected once with the time-line ie with the next potential
2 Integer Programming Models for Track Allocation 115
departure node Lj (or tj) On the right side of Figure 10 this reducedgraph based on a time-line concept can be seen The precise time-lineconstruction and corresponding mathematical formulas can be foundin Borndorfer amp Schlechte (2007) [30]
Hence sjtj-paths a1 a1 ak ak ak+1 in Dj (without time-lines) andconfigurations a1 ak in Qj are in one-to-one correspondence forthe case of block occupation Let us formally denote this isomorphismby a mapping
middot Qj rarr Pj q 7rarr p j isin Jwhere Pj denotes the set of all sjtj-paths in Dj = (Vj Aj) howeverwe will henceforth identify paths p isin Pj and configurations q isin QjIn the following we will discuss the construction for the general head-way case It is easy to see that the construction rule (2) can again beapplied to ensure consecutive valid headway times However Figure 11gives an example what can happen if Hj is not transitive On the leftthree running arcs on track j and in the middle the constructed trackdigraph D = (VA)j with respect to Hj are shown Note that arc kand l as well as l and m are feasible successor but k and m are notconnected due to non-transitivity of Hj On the right a sjtj-path inDj is highlighted which violates a minimum headway time of trainswhich are not direct successors
Therefore transitivity of H is a necessary condition to allow for anexact construction via Dj Otherwise D(xy) defines only a relaxationof the configuration Qj because there are s(xy)t(xy)-paths which couldviolate non-consecutively headway times
Lemma 25 There is a bijection from all sjtj-paths in Dj to the setof valid configurations Qj on track j isin J if the headway matrix H istransitive
Proof We provide two variants of the proof to facilitate the under-standing Let Dj be the track digraph induced by headway matrixH
1 assume H is transitive then the following map middot is a bijection
middot Pj rarr Qj p = a1 a1 ak ak ak+1 7rarr q = a1 ak j isin J
2 or assume H is not transitive then we can construct a path p isinDj which is not a valid configuration see Figure 11 In that case
2 Integer Programming Models for Track Allocation 116
k
l
m
AΨjLj Rj sj
tj
sj
tj
Figure 11 Example for a path which does not correspond to a valid configurationif the headway times violate the transitivity
no bijection can exist between these spaces of different dimensionwhich is a contradiction
Remark 26 The idea of reducing the huge number of potential returnarcs by a time-line in Dj can be transfered We only have to distin-guish between the basic equivalence classes induced by Definition 24ie independent of the times τ In the worst case these are C timesMdeparture time-lines one for each train type c and running mode mWe do not give a precise formulation for this construction Howeverin our software module TS-OPT a timeline concept that is based on theequivalence classes is implemented
Remark 27 In Section 2 we have introduced an algorithm that pro-vides a macroscopic network with transitive headway matrices on alltracks Technical minimal headway times are naturally transitive forreal world data
Henceforth we have defined all objects to introduce an extended for-mulation of the TTP Variables xa a isin Ai i isin I control again the useof arc a in Di and yb b isin AΨ
j cup Aj j isin J in Dj respectively
2 Integer Programming Models for Track Allocation 117
(ACP)max
sumaisinA
waxa (i)
stsum
aisinδiout(v)
xa minussum
aisinδiin(v)
xa = 0 forall i isin I v isin Visi ti (ii)sumaisinδiout(si)
xa le 1 forall i isin I (iii)sumaisinδiout(v)
ya minussum
aisinδiin(v)
ya = 0 forall j isin J v isin Vjsj tj (iv)sumaisinδiout(sj)
ya le 1 forall j isin J (v)sumaisinb
xa minus yb = 0 forall b isin AΨ (vi)
xa yb isin 0 1 forall a isin A b isin AΨ cupAj (vii)
The objective denoted in (ACP) (i) is to maximize the weight of thetrack allocation Equalities (ii) and (iv) are well-known flow conserva-tion constraints at intermediate nodes for all trains flows i isin I and forall flows on tracks j isin J (iii) and (v) state that at most one flow ietrain and track unit is realized Equalities (vi) link arcs used by trainroutes and track configurations to ensure a conflict-free allocation oneach track individually ie the hyperarcs b isin AΨ are coupled with thearc set AI Finally (vii) states that all variables are binary
Remark 28 Note that conflict constraints induced by station capaci-ties are not considered in that construction In the work of Erol (2009)[84] the configuration idea was also applied to these kind of constraintsActually we prefer a ldquolazyrdquo approach to add them only if needed Eventhough they do not arise naturally In fact only the aggregation oftracks inside and in the area around a station leads to them
Remark 29 Conflict constraints induced by single way usage of twoopposing tracks can be easily considered in that construction as wellThe main difference is the definition of the return arcs which decidewhat a valid successor after each running arc is In that case they canbe adjacent to both stations of the track because the next departure caneither be in the same or in the opposing direction on track j Conse-quently we have departure time-lines on both sides of the track Dueto the properties of headway times for single way tracks the resultinggraph Dj remains acyclic Note that a minimal technical headway timefor the opposing direction must be larger than the running time of thepreceding train see formula 1 in Section 23
2 Integer Programming Models for Track Allocation 118
Pure static approaches and models are handicapped due to memorylimitations The presented digraphs and thus the model formulationcan easily become very large and exceed 8GB of main memory evenfor instances with some hundred trains Explicit numbers are given inChapter IV and Section 1 To overcome these restrictions dynamic ap-proaches to create and solve these models are very efficient and success-ful We already presented the idea of column generation and branchand price in Section 85 To apply these techniques we developed apath based formulation of the (ACP) called (PCP) which will bethe topic of Section 3 The path coupling model (PCP) is formulatedwith binary decision variables xp for each path instead of arc variablesxa and yq for each configuration (ldquopathrdquo) instead of arc variables yb asfollows
(PCP)max
sumpisinP
wpxp (i)
stsumpisinPi
xp le 1 foralli isin I (ii)sumqisinQj
yq le 1 forallj isin J (iii)sumpisinPbisinp
xp minussum
qisinQbisinqyq le 0 forallb isin AΨ (iv)
yq isin 0 1 forallq isin Q (v)
xp isin 0 1 forallp isin P (vi)
The objective denoted in (PCP) (i) is to maximize the weight of thetrack allocation Inequalities (ii) and (iii) are set packing constraintsto ensure that for each request i isin I and each track j isin J at most onepath or configuration is chosen Inequalities (iv) link arcs used by trainroutes and track configurations to ensure a conflict-free allocation oneach track individually We say that b isin AΨ is an element of path pb isin p if there is an arc a isin p with a isin b Finally (v) and (vi) statethat all variables are binary
Let γ isin R|I| π isin R|J | and λ isin R|AΨ| be dual vectors Consider thelinear program arising from (PCP) (i) to (iv) with yq ge 0 q isin Qand xp ge 0 p isin P Because of (PCP) (ii) and (iii) the upper boundconstraints yq le 1 and xp le 1 are redundant and therefore we canignore them for the dualization We get the following dual problem
2 Integer Programming Models for Track Allocation 119
(DLP)min
sumjisinJ
πj +sumiisinIγi (i)
st γi +sum
aisinpb3aλb ge wp forallp isin Piforalli isin I (ii)
πj minussumbisinq
λb ge 0 forallq isin Qj forallj isin J (iii)
γi ge 0 foralli isin I (iv)
πj ge 0 forallj isin J (v)
λb ge 0 forallb isin AΨ (vi)
Furthermore we receive the corresponding pricing problem for the x-variables
(PRICE(x)) exist i isin I p isin Pi sumaisinp
wa minussum
aisinpb3a
λb minus γi gt 0
Remember that each arc a isin AI is exactly coupled with one resource-equivalent hyperarc b isin AΨ denoted by b(a) Solving this pricingproblem is equivalent to answer the question whether there exists arequest i isin I and a path p isin Pi with positive reduced cost Due to thefact that all Di are acyclic this problem decomposes into |I|- longestpath problems with arc lengths la = wa minus λb(a) if a isin AI and la = waotherwise For the y-variables we get
(PRICE(y)) exist j isin J q isin Qj sumbisinq
λb minus πj gt 0
Analogously the pricing problem for the y- variables decomposes into|J |- easy longest path problems one for each acyclic digraph Dj Thepricing of configurations Qj is equivalent to find a shortest sjtj-pathin Dj using arc lengths lb = λb b isin AΨ and 0 otherwise Since Dj isacyclic this is polynomial By the polynomial equivalence of separationand optimization see Grotschel Lovasz amp Schrijver (1988) [104] hereapplied to the (DLP) we obtain
Lemma 210 The linear relaxation of (PCP) can be solved in poly-nomial time
Let us state in this pricing context a simple bound on the LP-valueof the path configuration formulation (PCP) We set b(a) = empty for
2 Integer Programming Models for Track Allocation 120
a isin AI to simplify notation In practical implementations this boundmight be utilize to detect tailing-off effects in a column generationprocedure ie one can stop the column generation with a certain op-timality gap at the root node and start so-called ldquoearly branchingrdquoNamely computing the path lengths maxpisinPi
sumaisinpwa minus
sumaisinpb3a λb
for all i isin I and maxqisinQjsum
bisinq λb for all j isin J yields the followingLP-bound β = β(γ π λ)
Lemma 211 Let γ π λ ge 0 be dual variables6 for (PCP) andvLP(PCP) the optimum objective value of the LP-relaxation of (PCP)Define
ηi = maxpisinPi
sumaisinp
(wa minussum
aisinpb3a
λb)minus γi foralli isin I
θj = maxqisinQj
sumbisinq
λb minus πq forallj isin J
β(γ π λ) =sumiisinI
maxγi + ηi 0+sumjisinJ
maxπj + θj 0
ThenvLP(PCP) le β(γ π λ)
Proof Assuming the pricing problems are solved to optimality wehave
γi + ηi gesumaisinp
(wa minussum
aisinpb3a
λb)rArr γi + ηi +sum
aisinpb3a
λb) ge wp foralli isin
I p isin PI πj + θj ge
sumbisinq
λbrArrπj + θj minussumbisinq
λb ge 0 forallj isin J q isin Qj
(maxγ+η 0maxπ+θ 0 λ) (the maximum taken component-wise) is dual feasible for the LP-relaxation of (PCP)
Remark 212 Note that this is true in general for all column gen-eration approaches where the pricing is solved exactly If the pricingproblem could not be solved to optimality then solving a relaxation ofthe pricing problem can also provide a global bound We analysed thisapproach for the multiple resource constraint shortest path problem byusing enhanced linear relaxations see Schlechte (2003) [187] and Wei-der (2007) [213]
6Note that these will be global infeasible during a column generation
2 Integer Programming Models for Track Allocation 121
23 Polyhedral Analysis
In this section we show that (PCP) and (ACP) are extended formu-lations of (PPP) and (APP) respectively Furthermore some basicpolyhedral observations are presented using the standard notation anddefinitions that can be found in Ziegler (1995) [219] Starting points arethe LP-relaxations of the configuration formulations and those of thepacking formulations As the LP-relaxations of (APP) and (PPP)and of (ACP) and (PCP) are obviously equivalent via flow decom-position into paths it suffices to compare say (APP) and (ACP)Furthermore we consider models (APP) based on the simple case ofblock occupation conflicts only The case of general headway conflictswould only unnecessary complicate the notation However in case ofstation capacity conflicts a more general definition of ldquoconfigurationsrdquoand hence different models are needed ie see Erol (2009) [84] Letus shortly list the needed sets
A set of all ldquostandardrdquo time-arcs representing train operations
AI set of time-arcs representing track usage
AΨ set of resource equivalence classes representing track usage
Vj set of time-nodes of track digraph induced by track j
Γj subset of conflict set induced by track j
and A = cupjisinJAj set of all ldquoauxiliaryrdquo time-arcs representing theconsecutive succession of arcs on track j
Lemma 213 Let
PLP(APP) = x isin RA (APP) (ii)ndash(v)PLP(ACP) = (x y) isin RAtimesAΨtimesA (ACP) (ii)ndash(vi)
πx RAtimesAΨtimesA rarr RA (x y) 7rarr x
be the polyhedron associated with the LP-relaxations of (APP) and(ACP) respectively and a mapping that produces a projection ontothe coordinates of the train routing variables Then
π(PLP(ACP)) = PLP(APP)
Proof Let Γj = γ isin Γ γ sube Aj j isin J be the set of block conflictcliques associated with track j Consider the polyhedron
2 Integer Programming Models for Track Allocation 122
PLPIP (APP) PLPIP (ACP)
PLPIP (PPP) PLPIP (PCP)
πx (x y) 7rarr x
πx (x y) 7rarr x
Γx x 7rarr (x φ(x))
Γx x 7rarr (x φ(x))
x xa = x(Pa)Λx x 7rarr λ(x) Λx (x y) 7rarr λ(x) λ(y) x xa = x(Pa) y ya = y(Pa)
Figure 12 Relations between the polyhedra of the different models
P = x isin RA (APP) (ii) (iii) (v)P j = x isin RAj
+ sumaisinγ
xa le 1 forallγ isin Γj j isin J
Qj= y isin RAΨj timesAj
+ sum
aisinδ+j (v)
ya =sum
aisinδminusj (v)
ya forallv isin Vjsj tjsumaisinδ+
j (sj)
ya le 1 j isin J
Rj = x isin RAj+ existy isin Qj x le y j isin J
P j is integer because Γj is the family of all maximal cliques of aninterval graph which is perfect Qj is integer because it is the pathpolytope associated with an acyclic digraph finally Rj is integer be-cause it is the anti-dominant of an integer polytope Consider integer
2 Integer Programming Models for Track Allocation 123
points it is easy to see that P j and Rj coincide ie P j = Rj j isin J It follows
PLP(APP) = P cap⋂jisinJ
P j = P cap⋂jisinJ
Rj = π(PLP(ACP))
This immediately implies our main Theorem
Theorem 214 Denote by v(P ) and vLP(P ) the optimal value of prob-lem P and its LP-relaxation respectively with P isin (APP)prime (APP)(PPP) (ACP) (PCP) Then
vLP(APP)prime ge vLP(APP)
vLP(APP) = vLP(PPP) = vLP(ACP) = vLP(PCP)
v(APP)prime = v(APP) = v(PPP) = v(ACP) = v(PCP)
Figure 12 illustrates the transformation between the different modelsThe given projections show that coupling models are extended formu-lations of the original packing ones More details on extended formula-tions and projections of integer programming formulations can be foundin Balas (2005) [16] The idea of extended formulations is shown in Fig-ure 13 On the left hand side the rough structure of the packing for-mulation (PPP) can be seen ie with appropriate binary matrices Aand R On the right hand side the structure of model (PCP) after thetransformation of the packing constraints associated with matrix R isshown Matrix B denotes the auxiliary configuration partitioning partand C and D the necessary coupling part
Lemma 215 PLP (PCP) = x isin RPcupQ (PCP) (ii)ndash(iv) is full-dimensional
Proof To show that PLP(PCP) is full-dimensional we have to con-struct |P|+|Q|+1 affinely independent and feasible points in PLP (PCP)For each path p isin P (q isin Q) we denote the set of arcs incident to p(q) and contained in AΨ by Ap (Aq) The set of all coupling hyper-arcsis again denoted by AΨ
First consider for each p isin P the associated path-configuration inci-dence vector φ(p) isin 0 1P ν(p) isin 0 1Q with k isin P and l isin Qconstructed as follows
2 Integer Programming Models for Track Allocation 124
w
A
R κ
1
w
A
B
0
0
1
1
0C D
Figure 13 Idea of the extended formulation (PCP) for (PPP)
φk(p) =
1 if k = p
0 otherwise(3)
νl(p) =
1 if Al = b sube AΨ b = b(a) foralla isin Ap j(a) = j(l)0 otherwise
(4)
The entries νl(p) ldquoactivaterdquo exactly the minimum configuration ontrack j(l) ldquoconsumedrdquo by path p ie only the arcs b isin AΨ b(a) isin pcapAI
are used in configuration l Request and track packing constraints aretrivially fulfilled because we only chose one path to be at one andbecause at most one configuration is used by path p for each track jThe coupling constraints are fulfilled for all b isin AΨ by the definition ofν(p) since p is a feasible path Thus (φ(p) ν(p)) is obviously containedin PIP (PCP) for all p isin P Next consider for each ldquoconfigurationrdquoq isin Q the qth unit vector (φ(q) ν(q))
We have constructed |P +Q| many vectors which form the matrix
2 Integer Programming Models for Track Allocation 125
(φ(p) φ(q)ν(p) ν(q)
)=
(E|P| 0ν(p) E|Q|
)
where En denotes the n-dimensional identity matrix
These vectors are linearly independent due to the fact that the deter-minant of this lower triangular matrix is obviously 1 Together withthe feasible vector 0 isin R|P+Q| we have constructed |P|+|Q|+1 affinelyindependent points of PLP (PCP) proving our proposition
Lemma 216 Constraint (PCP) (iii) associated with track j isin Jdefines a facets of PIP (PCP) if Qj 6= empty
Proof We have to show that the hyperplaneHj = (φ ν) isin [0 1]|P|+|Q| sumqisinQj yq = 1 contains |P| + |Q| affinely independent points of the
polyhedron PLP (PCP)
First for each p isin P we construct a vector (φ(p) ν(p)) based on thevector (φ(p) ν(p)) as follows If path p contains at least one couplingarc of track j then define vector (φ(p) ν(p)) = (φ(p) ν(p)) and other-wise let (φ(p) ν(p)) = (φ(p) ν(p)) + (0 eqj) where (φ(p) ν(p)) is thevector from formula 3 and 4 and eqj is the qjth unit vector for someconfiguration qj isin Qj
Obviously (φ(p) ν(p)) is feasible and satisfies packing constraints(PCP) (iii) associated with track j with equality
Next for each ldquoconfiguration pathrdquo q isin Qj we define (φ(q) ν(q)) =(0 eq) with eq as the qth unit vector and otherwise (if q isin Q Qj) let(φ(q) ν(q)) be the sum of the (0 eq) and (0 eqj) Hence (φ(q) ν(q))is a feasible point of PIP (PCP) and Hj
Finally we have constructed |P|+|Q|many vectors which are containedin Hj and PIP (PCP) Re-sorting the vectors in an appropriate waywe obtain a lower-triangular matrix such that the last row and columncorresponds to configuration qj then we get
(φ(p) φ(q)ν(p) ν(q)
)=
E|P| 0 0 0middot middot middot E|QQj | 0 0middot middot middot 0 E|Qjqj | 0
middot middot middot 1 0 1
2 Integer Programming Models for Track Allocation 126
Since the determinant of this matrix is one the vectors are linearlyindependent proving that Hj cap PIP (PCP) is a facet
Remark 217 The analysis of the packing constraints (PCP) (ii) andthe coupling constraints (PCP) (iv) remains as an open problem It isnot trivially clear in which cases these constraints are facet defining ornot Even if this is more a theoretical research question we believe thatdeep polyhedral insights can support the algorithmic solution approachHence we hope that in the future these questions might be answered
We want to point out that this is not only a basic theoretical analysisof the model The main motivation was to find out whether there isa structural reason why the coupling models perform better than therecounterparts Even if we can only provide some theoretical answer forthat we believe that this an interesting topic for future research Toanswer the question in which cases coupling constraints are facets mightbe useful in designing and further development of solution algorithms
24 Extensions of the Models
In the last section we analyzed in detail the track allocation problemwith respect to ldquohardrdquo combinatorial constraints In this part we wantto discuss how to handle global combinatorial requirements on the setof train request and rather ldquosoftrdquo constraints on the implicit buffertimes
Manifold reasons cause combinatorial interaction between train slotsOur definitions are based on the bidding language of an auction designintroduced in Borndorfer et al (2006) [34] therefore we use synony-mously bid and train slot request Three potential sources for combi-natorial bids are mentioned tours to support rolling stock planningregular service intentions to allow for attractive offers for the passen-gers and operator neutral connections to establish reliable and fastinterlining connections
Another extension is based on the potential of the extended formu-lation to control the implicit buffer times on each track We exploitthis structural advantage by introducing a robustness measure on theldquoreturnrdquo arcs and developed a straight-forward bi-criteria model inSchlechte amp Borndorfer (2008) [188] This allows for evaluating the
2 Integer Programming Models for Track Allocation 127
trade-off between efficiency ie the utilization of the macroscopic net-work and the stability or robustness ie in terms of the implicit buffertimes of consecutive trains
241 Combinatorial Aspects
A main point in the discussion on railway models is whether it is possi-ble to deal with complex combinatorial technical and economical con-straints in a real-world setting or not We do of course not claimthat we can give a real answer to this question but we want to givean example of a more realistic scenario to indicate that our approachhas potential in this direction To this purpose we discuss a settingthat extends the previous one ie see Section 11 by allowing forcombinatorial AND and XOR requirements
With these extensions it is possible to model most features of the bid-ding language ie the specification of train requests in an auctionenvironment described in Borndorfer et al (2006) [34] Bids for com-plete tours can be expressed as AND connected bids and an optionalstop can be expressed as a XOR connection of requests for slots withand without this stop An AND relation could further be useful to in-dent slots for a frequent service Railway undertakings which can onlyoperate a limited number of train slot could further be interested informulating XOR bids A way how to incorporate general connectionsfor passengers is described in Mura (2006) [164] ie an auxiliary flowis defined that is induced by and coupled with the connective trainslots
Let a combinatorial bid k refer to some subset Ik sube I of bids for singletrain request it may either be an AND or an XOR bid An AND-bidstipulates that either all single slot bids in Ik = i1 i2 imm ge 2must be assigned or none of them A XOR-bid states that at most oneof the bids in the set Ik can be chosen Let IAND denote the set ofAND bids and IXOR the set of XOR bids
The arc based formulations (APP) and (ACP) can be easily extendedby introducing a zero-one variable zi for each train request i that is 1 ifbid i is assigned and 0 else These variables are useful in dealing withcombinatorial bids by the following constraints
2 Integer Programming Models for Track Allocation 128
sumaisinδiout(si)
xa minus zi = 1foralli isin I (5)
zin minus zin+1 = 0foralln isin 1 2 |Ik| minus 1 k isin IAND (6)sumiisinIk
zi le 1forallk isin IXOR (7)
Constraints 5 make sure that zi is only one if train i is scheduledConstraints 6 and 7 enforce combinatorial AND and XOR bids ie anadditional one for each XOR set and |Ik| minus 1-many for each AND setk
242 Robustness Aspects
We exploit the possibility to use the additional variables of the extendedformulations (ACP) and (PCP) to measure robustness in terms of im-plicit available buffer times of a timetable We refrain from supportingthis by recent statistics to punctuality and reliability of any railwaycompany But obviously decision makers are more and more sensitiveto the importance of finding a good compromise between profitable andreliable timetables
Robust optimization that means the incorporation of data uncertain-ties through mathematical models in its original definition as proposedby Soyster (1973) [202] is not applicable to large scale optimizationproblems Moreover these models produce too conservative solutionswhich are resistant against all considered eventualities but far awayfrom implementable in real world Robust optimization however hasbecome a fruitful field recently because more and more optimizationproblems can be solved in adequate time This opens the door to addi-tionally deal with stochastic assumptions instead of only nominal givendata In Ben-Tal amp Nemirovski (1998) [23] and El-Ghaoui Oustry ampLebret (1998) [81] less conservative models were introduced which ad-just the robustness of the solution by some protection level parametersBertsimas amp Sim (2003) [25] survey robust optimization theory andits network flow applications Fischetti Salvagnin amp Zanette (2009)[91] Kroon et al (2006) [139] Liebchen et al (2007) [151] Liebchenet al (2009) [152] apply these robust considerations to the world of
2 Integer Programming Models for Track Allocation 129
railways ie to the periodic railway timetabling They investigatea cyclic version of the timetabling problem modeled as a PeriodicEvent Scheduling Problem and introduce a stochastic methodology ofLight Robustness and Recoverable Robustness For the detailed rout-ing through stations or junctions Caimi Burkolter amp Herrmann (2004)[58] and Delorme Gandibleux amp Rodriguez (2009) [74] proposed ap-proaches to find delay resistant and stable routings The aim of theseconsiderations is to gain more insights into the trade-off between effi-ciency and robustness of solutions and find a practical ldquoprice of robust-nessrdquo
We focus on a pure combinatorial optimization approach which issomehow related to Ehrgott amp Ryan (2002) [79] and Weide Ryan ampEhrgott (2010) [212] broaching the issue of robustness in airline crewscheduling We consider robustness (available buffer times quality ofday-to-day operations) and efficiency (used track kilometers plannedcapacity utilization) to be incomparable entities and consequently fa-vor a bi-criteria optimization approach Later Schobel amp Kratz (2009)[191] applied the same methodology to the problem of periodic railwaytimetabling
We extend models (ACP) and (PCP) to measure robustness whichleads directly to a bi-criteria optimization approach of the problem Todetermine efficient solutions ie the Pareto-frontier of the bi-criteriamodels we used the trivial so-called scalarization and ε-constraint methodMore details on the general theory and solution of multi-criteria opti-mization problems can be found in Ehrgott (2005) [78]
In Schlechte amp Borndorfer (2008) [188] details on a straight-forward col-umn generation approach to solve the scalarized optimization problemcan be found ie we proved that the LP-relaxation of the (PCP) in-cluding an additional ε-constraint remains solvable in polynomial time
However let us explain the incorporation of some ldquorobustnessrdquo on asimple example By rq we denote a robustness value for each config-uration q isin Q We assume that a high robustness value rq meansconfiguration q is robust and a smaller the contrary As a simplifica-tion we expect rq =
sumaisinq ra ie the robustness of a configuration can
be expressed as the sum of the robustness of its incident arcs
Figure 14 illustrates the idea on a single track Considering a trackdigraph Dj induced by three train requests Straight forwardly maxi-mizing the number of scheduled trains in our setting will always lead
2 Integer Programming Models for Track Allocation 130
sj
tj
q1
sj
tj
q2
sj
tj
q3
Figure 14 From fragile q1 and q2 to robust configuration q3
02
46
810
02
46
8100
radicb
2
radicb
0
radicb
2
radicb
Figure 15 Robustness function r of two buffer arcs
to a schedule with profit value 3 but as you can see this can result ina lot of varying schedules In fact all sjtj-paths are solutions eg thethree shown in Figure 14 We are given a desired implicit buffer b isin Nie 5 minutes which we maximally want to hedge against Note thatthese are soft buffer times between train succession Standard buffertime which must be strictly adhered to are already incorporated in theheadway times
Then the following robustness function r R|A| rarr R with
2 Integer Programming Models for Track Allocation 131
r((u v)) =
radicb (u v) isin Aj and t(v)minus t(u) gt bradict(v)minus t(u) (u v) isin Aj and t(v)minus t(u) le b
0 otherwise
will measure the available buffers appropriately Note that only ldquoreturnarcsrdquo contribute to the robustness measure The function r benefitsarcs with duration values close to or above b Moreover this functionbalances the partition of the available implicit buffer times by its con-caveness see Figure 15 Assume b = 2 in our example in Figure 14Then the first configuration q1 has value rq1 = 0 for the second con-figuration rq2 is
radic2 and the third one has rq3 = 2 For the sake of
completeness we set rq to a sufficiently big M for an empty configura-tion q ie we use the b times half the length of the longest path in DjTo find all efficient solutions we propose a straight-forward combinedweighted sum and ε-constraint hybrid method see Ehrgott (2005) [78]Considering model (PCP) this leads to the following objective func-tion with a scalar α isin [0 1]
max α(sumpisinP
wpxp) + (1minus α)(sumqisinQ
rqyq)
As a result we can compile an analysis of the crucial parameters tosupport track allocation decisions as shown in Figure 16 In additionsuch a computational experiment produces a broad spectrum of solu-tions Thus new problem insights are provided and planners have thepossibility to try complete new track allocation concepts
We only present and discuss results for the linear relaxation of model(ACP) In Schlechte amp Borndorfer (2008) [188] the settings and fo-cus of these experiments are explained more precisely On the rightboth objectives depending on α are shown The extreme cases are asexpected For α = 1 only the robustness measure contributes to theobjective and is therefore maximized as much as possible at the cost ofscheduling only some or even no trains For α = 0 the robustness mea-sure does not contribute to the objective and is therefore low while thetotal profit is maximal With decreasing α the total robustness mono-tonically decreases while the total profit increases On the left part ofFigure 16 the Pareto frontier can be seen Note that each computedpair of total robustness and profit constitutes a Pareto optimal pointie is not dominated by any other attainable combination Conversely
3 Branch and Price for Track Allocation 132
420
430
440
450
460
470
480
490
150 200 250 300 350 400 450 500 550
pro
fit
robustness
150
200
250
300
350
400
450
500
550
0 02 04 06 08 1 150
200
250
300
350
400
450
500
550
α
profitrobustness
Figure 16 Pareto front on the left hand and total profit objective (blue left axis)and total robustness objective (green right axis) in dependence on αon the right hand
any Pareto optimal solution of the LP relaxation can be obtained asthe solution for some α isin [0 1] see eg Ehrgott (2005) [78]
3 Branch and Price for Track Allocation
This Section discusses sophisticated algorithmic approaches to solvevery large scale instances of the track allocation problem Standardinteger programming solver such as CPLEX SCIP or GuRoBi can solvestatic model formulations like (APP) and (ACP) up to a certain prob-lem size However to tackle large-scale instances we developed theoptimization module TS-OPT It solves the dynamic model formulation(PCP) by taking advantage of the approximate bundle method and arapid branching heuristic to produce high quality solutions with a mod-erate running time even for very large scale instances The aim of thischapter is to provide a comprehensive understanding of the less thanconventional branch and price approach ie the tailor made methodsin TS-OPT
31 Concept of TS-OPT
Schrijver (1998) [193] and Nemhauser amp Wolsey (1988) [167] providea comprehensive discussion on the general theory of integer program-ming State of the art techniques to solve mixed integer programs ieeven the more general class of constraint integer programs can be foundin the prizewinning thesis Achterberg (2007) [3] The basic method-ology of branch and price was introduced in Barnhart et al (1998)
3 Branch and Price for Track Allocation 133
[18] Details can also be found in Villeneuve et al (2005) [210] In thefollowing sections we apply these technique to the model (PCP)
In Figure 17 the concept of TS-OPT is shown In a first step the problemis constructed This entails reading in all data ie the macroscopicrailway network and the train request set subject to the specificationof the TTPlib constructing the train scheduling graph D = (VA)as proposed in Algorithm 4 and constructing the track digraphs asdiscussed in Section 22
Besides that the main algorithm can be divided in two parts On theone hand the linear programming or Lagrangean relaxation is solvedby a dynamic column generation approach ie using an approximatebundle method or a LP solver to produce dual values The pricing ofvariables are shortest path computations in large acyclic digraphs withrespect to these duals Fischer amp Helmberg (2010) [89] propose a dy-namic graph generation to solve these pricing problems for very largegraphs ie the original objective function has to fulfill the require-ment that an earlier arrival is always beneficial Unfortunately for ourinstances this is not always the case However this seems to be a fruit-ful approach to shrink the problem size of the pricing problems thatcould be extended to arbitrary objective functions The idea is simpleto use only a subset of the nodes and arcs and to define a border-setthat will we adapted with respect to the duals and the solution of theldquorestrictedrdquo pricing problem
On the other hand a branch and price heuristic ie rapid branchingis used to produce high quality integer solutions Instead of an exactbranch and price approach we only evaluate promising branch andbound nodes and perform some partial pricing Furthermore we onlyexplore the branch of variables to 1 because there will be almost noeffect when setting path and configuration variables to 0 The decisionwhich subset is chosen is highly motivated by the solution of the re-laxation ie the best candidate set with respect to a score functiondepending on the bound and the size of the candidate set for a reason-able perturbation of the objective function Section 32 and Section 33will describe the components in more detail
3 Branch and Price for Track Allocation 134
Initialize ProblemConstruct D = (VA)
Solve DualRepresentation
Price Paths andConfigs
Rapid BranchingHeuristic
LP solving
IP solving
Figure 17 Flow chart of algorithmic approach in TS-OPT
32 Solving the Linear Relaxation
In this section we use a slightly different notation with the followingappropriate binary matrices ABC and D
A isin 0 1|I|times|P| is the path-request incidence matrix
B isin 0 1|J |times|Q| is the configuration-track incidence matrix
C isin 0 1|AΨ|times|P| is the hyperarc-path incidence matrix
D isin 0 1|AΨ|times|Q| is the hyperarc-configuration incidence matrix
Without loss of generality we can change packing inequalities (PCP) (ii)and (iii) to partitioning equalities by introducing slack variables cor-responding to empty paths p isin P with profit wp = 0 or empty con-figuration respectively Observe that the upper bounds on x and yin model (PCP) are redundant because A and B are binary and wecan assume that the profit coefficients w are positive ie paths withnegative profit value are redundant
(PCP) max wTx (i)st Ax = 1 (ii)
By = 1 (iii)Cx minus Dy le 0 (iv)
y isin 0 1|P | (v)x isin 0 1|Q| (vi)
A standard technique to solve large scale linear relaxation as those of(PCP) is column generation see Chapter I in Section 85 and Fig-ure 14 We have already seen that the pricing problems are shortestpath problems in acyclic digraphs see Section 22 and Lemma 210
3 Branch and Price for Track Allocation 135
However in TS-OPT we implemented a slightly different approach basedon a Lagrangean relaxation
321 Lagrangean Relaxation
Lagrangean relaxation is a technique to find bounds for an optimiza-tion problem eg upper bounds in case of maximization problemsIn Hiriart-Urruty amp Lemarechal (1993) [116 117] Lemarechal (2001)[147] the basics as well as further details can be found Under certaincircumstances also optimal solutions of the ldquoconvexified relaxationrdquoare provided see Frangioni (2005) [93] Helmberg (2000) [114] Weider(2007) [213]
Two time consuming problems have to be solved repeatedly in anycolumn generation approach First of all an optimal dual solution ofthe restricted problem has to be found ie LPs have to be solvedSecondly we have to find new columns or prove that none exists de-pending on the solutions of the LPs ie dual values by solving thepricing problems
However using Lagrangean relaxation and subgradient methods is of-ten faster and less memory-consuming than LP-methods see Weider(2007) [213] Even if in general this approach only gives bounds andapproximated solutions of the relaxed problem We transfer the largeset of coupling constraints into the objective ie therefore they can beviolated by the solution of the Lagrangean relaxation A Lagrangeanrelaxation with respect to the coupling constraints (iv) and a relaxationof the integrality constraints (v) and (vi) results in the Lagrangeandual
(LD) minλge0
maxAx=1
xisin[01]|P |
(wT minus λTC)x+ maxBy=1
yisin[01]|Q|
(λTD)y
Each solution of (LD) gives a valid upper bound of (PCP) Let usdefine functions and associated arguments by
3 Branch and Price for Track Allocation 136
fP R|AΨ| rarr R λ 7rarr max(wT minus λTC)x Ax = 1 x isin [0 1]|P |
fQ R|AΨ| rarr R λ 7rarr max(λTD)y By = 1 y isin [0 1]|Q|
fPQ = fP + fQ
That are longest path problems in acyclic digraphs with respect to λand
xP (λ) = argmaxxisin[01]|P | fP (λ)
yQ(λ) = argmaxyisin[01]|Q| fQ(λ)
breaking ties arbitrarily With this notation (LD) becomes
(LD) minλge0
fPQ(λ) = minλge0
[fP (λ) + fQ(λ)]
It is well known that the Lagrangean dual of an integer linear programprovides the same bound as a continuous relaxation involving the con-vex hull of all the optimal solutions of the Lagrangean relaxation Thefunctions fP and fQ are convex and piecewise linear Their sum fPQ istherefore a decomposable convex and piecewise linear function fPQis in particular nonsmooth This is precisely the setting for the prox-imal bundle method
322 Bundle Method
The proximal bundle method (PBM) is a method to minimize an un-bounded continuous convex and possibly non-smooth function f Rm 7rarr R The PBM can be used in combination with Lagrangean re-laxation to approximate primal and dual solutions of linear programsA detailed description of the bundle method itself can be found inKiwiel (1990) [127] and of its quadratic subproblem solver in Kiwiel(1995) [128]
In the following we will discuss our straight-forward adaption of thegeneral bundle method We use the PBM to approximate LP-relaxationsof model (PCP) via the Lagrangean problem (LD) defined in Sec-tion 321 The corresponding computational results can be found in
3 Branch and Price for Track Allocation 137
Chapter IV The LP-relaxation of (PCP) is in general too large to besolved by standard solvers such as the barrier algorithm or the dualsimplex because theses LPs consist in general of millions of columnsfor the paths and configurations and several thousands of rows for thecoupling constraints ie even if we already reduce theses constraintsby the definition of AΨ
When applied to (LD) the PBM produces two sequences of iteratesλk microk isin R|AΨ| k = 0 1 The points microk are called stability centers they converge to a solution of (LD) The points λk are trial pointsfunction evaluations (line 5 of Algorithm 5) at the trial points resulteither in a shift of the stability center or in some improved approxi-mation of fPQ
More precisely the PBM computes at each iteration for λk linear ap-proximations
fP (λλk) = fP (λk) + gP (λk)T(λminus λk)fQ(λλk) = fQ(λk) + gQ(λk)T(λminus λk)
fPQ(λλk) = fP (λλk) + fQ(λλk)
of the functions fP fQ and fPQ by determining the function valuesfP (λk) fQ(λk) and the subgradients gP (λk) and gQ(λk) by definitionthese linear approximations underestimate the functions fP and fQie fP (λλk) le fP (λ) and fQ(λλk) le fQ(λ) for all λ Note that fPand fQ are polyhedral such that the subgradients can be derived fromthe arguments y(λk) and x(λk) associated with the multiplier λk as
gP (λk) = minus CxP (λk) = minus suma3pisinPbisinAΨaisin[b]
xP (λkb )
gQ(λk) = DyQ(λk) =sum
b3qisinQbisinAΨ
yQ(λkb )
gPQ(λk) = minus CxP (λk) +DyQ(λk)
This linearization information is collected in so-called bundles
JkP = (λl fP (λl) gP (λl) l = 0 kJkQ = (λl fQ(λl) gQ(λl) l = 0 k
3 Branch and Price for Track Allocation 138
fP Q
λ1 λ2
fP Q
Figure 18 Cutting plane model fPQ of Lagrangean dual fPQ
We will use notations such as λl isin JkP gP (λl) isin JkP etc to expressthat the referenced item is contained in some appropriate tuple in thebundle associated to the path variables of iteration k The PBM usesthe bundles to build piecewise linear approximations
fkP (λ) = maxλlisinJkP
fP (λλl)
fkP (λ) = maxλlisinJkQ
fQ(λλl)
fkPQ = fkP + fkQ
of fPQ see Figure 18 Furthermore a quadratic term is added to thismodel that penalizes large deviations from the current stability centermicrok The direction (line 3) to the next trial point λk+1 is calculated bysolving the quadratic programming problem
(QP kPQ) λk+1 = argmin
λfPQ(λ)minus u
2
∥∥microk minus λ∥∥2
Denote by u a positive weight (step size) that can be adjusted to in-crease accuracy or convergence speed If the approximated functionvalue fkPQ(λk+1) at the new iterate λk+1 is sufficiently close to thefunction value fPQ(microk) the PBM stops microk is the approximate solu-tion Otherwise a descent test (line 8) is performed whether the pre-dicted decrease fPQ(microk) minus fkPQ(λk+1) leads to sufficient real decreasefPQ(microk)minus fPQ(λk+1) In this case the model is judged accurate and aserious step is done ie the stability center is moved to microk+1 = λk+1
3 Branch and Price for Track Allocation 139
Algorithm 5 Proximal Bundle Method (PBM) for (LD) of(PCP)
Data (LD) of (PCP) instance starting point λ0 isin Rn weightsu0m gt 0 optimality tolerance ε ge 0
Result primal xP yQ isin R|P |times|Q| and dual approximation microi isin Rn ofoptimal solutions of the (LD)
1 init k larr 0 JkP larr λk JkQ larr λk and microk = λk
2 repeat until tolerance is reached
3 solve problem (QP kPQ)
find direction
4 compute trial point λk+1 gkP gkQ
5 compute fP (λk+1) gP (λk+1) fQ(λk+1) gQ(λk+1)
6 select
Jk+1P sube JkP cup
(λk+1 fP (λk+1) gP (λk+1)
)(λk+1 fP k(λk+1) gkP
)
7 select
Jk+1Q sube JkQ cup
(λk+1 fQ(λk+1) gQ(λk+1)
)(λk+1 fQ(λk+1) gQ
)
update bundle set
8 if fPQ(microi)minus fPQ(λk+1) le m(fPQ(microk)minus fkPQ(λk+1)) then
9 microk+1 larr microk10 else update stability center
11 microk+1 larr λk+1
12 compute uk+1 k larr k + 1 update stepsize
13 until fkPQ(λk+1)minus fPQ(microk) lt ε(1 +∣∣fPQ(microk)
∣∣)
In the other case we call this iteration a null step ie in which onlythe approximation of the function by the bundles was improved
The bundles are updated (line 6 and 7) by adding the informationcomputed in the current iteration and possibly by dropping someold information More precisely vectors gkP and gkQ are aggregatedsubgradients which will be explained in the next paragraph Finallywe adopt the stepsize Then the next iteration starts see Algorithm 5for a complete pseudo code of the PBM
3 Branch and Price for Track Allocation 140
Besides function and subgradient calculations the main work in thePBM is the solution of the quadratic problem (QP k
PQ) This problemcan also be stated as
(QPkPQ) max vP+ vQ minusu
2
∥∥microk minus λ∥∥2
(i) vP minusfP (λλl) le 0 forallλl isin JkP(ii) vQ minusfQ(λλl) le 0 forallλl isin JkQ
A dualization is in the equivalent formulation
(DQPkPQ) argmax
sumλlisinJkP
αPlfP (microkλ) +sum
λlisinJkQ
αQlfQ(microkλ)
minus 12u
∥∥∥∥∥∥ sumλlisinJkP αPlgP (λ) +sum
λlisinJkQ
αQlgQ(λ)
∥∥∥∥∥∥2
sumλlisinJkP
αPl = 1sumλlisinJkQ
αQl = 1
αP αQ ge 0
Here αP isin [0 1]JkP and αQ isin [0 1]J
kQ are the dual variables associated
with the constraints (QP kPQ) (i) and (ii) respectively Given a solution
(αP αQ of ((DQP kPQ) the vectors
gkP =sum
λlisinJkP
αPgP (λl)
gkQ =sum
λlisinJkQ
αQgQ(λl)
gkPQ = gkP + gkQ
are convex combinations of subgradients they are called aggregatedsubgradients of the functions fP fQ and fPQ respectively It can beshown that they are actually subgradients of the respective functionsat the point λk+1 and moreover that this point can be calculated bymeans of the formula
3 Branch and Price for Track Allocation 141
λk+1 = micro+1
u
sumλlisinJkP
αPgP (λl) +sumλlisinJkQ
αQgQ(λl)
Note that (DQP k
PQ) is again a quadratic program the dimension isequal to the size of the bundles while its codimension is only two Forsolving this problem we use a specialized version of the spectral bundlemethod see Kiwiel (1990) [127] Kiwiel (1995) [128] and BorndorferLobel amp Weider (2008) [37] Finally the PBM (without stopping) isknown to have the following properties
The series (microk) converges to an optimal solution of (LD) ie anoptimal dual solution of the LP-relaxation of (PCP)
The series (xkP (λk) ykQ(λk)) defined as
(xkP (λk) ykQ(λk)) =
sumλlisinJkP
αPx(λl)sumλlisinJkQ
αQy(λl)
converges to an optimal primal solution of the LP-relaxation of(PCP)
Furthermore the primal approximation is useful to guide branchingdecision of the primal heuristic as we will describe in Section 33 Thebundle size controls the convergence speed of the PBM If large bundlesare used less iterations might be needed because of the better approx-imation model however problem (QP k
PQ) becomes more difficult Weuse a simple control schema for the stepsize u similar to Weider (2007)[213] The idea is to increase the stepsize if serious steps are performedif the distance of new trial point and the last one is small In case ofnull steps we gradually decrease the stepsize u
In Chapter IV Section 2 we present results of various experiments withdifferent strategies and parameter settings of our bundle implementa-tion
33 Solving the Primal Problem by Rapid Branch-ing
In this section we describe a heuristic approach based on the branchand price principle to tackle very large scale instances In fact it is
3 Branch and Price for Track Allocation 142
a branch-and-generate (BANG) heuristic ie a branch-and-price al-gorithm with partial branching see Subramanian et al (1994) [204]The heuristic can be classified as a special plunging heuristic with aobjective perturbation branching rule
Wedelin (1995) [211] a similar successful heuristic which perturbs theobjective function of large set-partitioning problems in a dual ascentmethod to find integral solution In Weider (2007) [213] this heuris-tic was invented as rapid branching Therein impressive results forlarge-scale instances of integrated vehicle and duty scheduling prob-lems arising in public transport are presented We will adopt mainideas and transfer them to the (PCP) formulation of the track alloca-tion problem
A simple rounding heuristic is used in Fischer et al (2008) [90] toproduce feasible integral solution of the (PPP) but sometimes fails toproduce high quality solutions In Cacchiani Caprara amp Toth (2007)[52] a greedy heuristic based on near-optimal Lagrangian multiplier wasused to produce solutions of the (PPP) In Section 1 we will see thatsimple greedy approaches or rounding heuristics also fails very oftenfor the (PCP)
Instead of branching on variables Foster amp Ryan (1991) [92] proposedanother branching rule which can be generalized as branching on arcsOne branching decision is to fix an arc to one the other branch toignore the arc completely Lusby (2008) [158] discussed this solutionapproach to a generalization of (PPP) This branching rule resultsnormally in more balanced branch and bound trees Koch Martin ampAchterberg (2004) [134] give a general survey on branching rules forsolving MIPs
The motivation of rapid branching given in Weider (2007) [213] appliesalso in our setting to a large extent
The fixing of single variables (path or configuration) to zerochanges the problem only slightly
The fixing of single arcs to zero changes the problem only slightlyie in general the set of arcs is too large
The fixing of single arcs to one is equivalent to fixing a large setof arcs to zero
The fixing of single variables (path or configuration) to one isequivalent to fix all arcs of the corresponding columns to one
3 Branch and Price for Track Allocation 143
Same observations for large scale LPs that are solved by column gen-eration are mentioned in Lubbecke amp Desrosiers (2005) [156] Thusrapid branching fixes a set of variables at once to one Which somehowreflects our goal to explore only a main branch and to reach fast highquality solutions The idea of the perturbation branching rule is tofind one branch called the main branch that fixes as many variablesas possibles to one to quickly find a solution of (PCP) This is done bysolving a series of LP-relaxations of (PCP) with varying profit func-tions w We perturb the profit function from one iteration to the nextto ldquomake the LP more integerrdquo The profit of variables with large pri-mal values are increased to move them towards an even higher valueor to keep the value at one
The other branches are unimportant unless the main branch turns outto either not include a feasible solution or to include only feasible so-lutions with too low profit Borndorfer Lobel amp Weider (2008) [37]see also the thesis of Weider (2007) [213] proposed also an associatedbacktracking mechanism to correct wrong decisions Our setting is ofobvious similarity and it will turn out that rapid branching can indeedbe successfully applied to solve large-scale track allocation problemEven more we are confident that a generalized variant of rapid branch-ing can be a very effective plunging heuristic in standard MIP solvers
Let l u isin 0 1PtimesQ l le u be vectors of bounds that model fixingsof variables to 0 and 1 Denote by L = j isin P timesQ uj = 0 andU = j isin P timesQ lj = 1 the set of variables fixed to 0 and 1respectively and by
(PCP)(l u) max wTx (i)st Ax = 1 (ii)
By = 1 (iii)Cx minus Dy le 0 (iv)
l le(xy
)le u (v)
the IP derived from (PCP) by such fixings Denote further by N subeP times Q = S some set of variables which have at some point in timealready been generated by a column generation algorithm for the so-lution of (PCP) Let (RPCP) and (RPCP)(l u) be the restrictionsof the respective IPs to the variables in N (we assume that LU sube Nholds at any time when such a program is considered ie variables that
3 Branch and Price for Track Allocation 144
have not yet been generated are not fixed) Finally denote by (MLP)(MLP )(w l u) (RMLP) and (RMLP )(w l u) the LP relaxations ofthe integer programs under consideration (MLP) and (MLP )(w l u)are called master LPs (RMLP ) and (RMLP )(w l u) restricted mas-ter LPs (the objective w is included in the notation for (MLP )(w l u)and (RMLP )(w l u) for reasons that will become clear in the nextparagraphs
Rapid branching tries to compute a solution of (PCP) by means of asearch tree with nodes (PCP)(l u) Starting from the root (PCP) =(PCP)(01) nodes are spawned by additional variable fixes using astrategy that we call perturbation branching The tree is depth-firstsearched ie rapid branching is a plunging heuristic The nodes areanalyzed heuristically using restricted master LPs (RMLP )(w l u)The generation of additional columns and node pruning are guided byso-called target values as in the branch-and-generate method To es-cape unfavorable branches a special backtracking mechanism is usedthat performs a kind of partial binary search on variable fixings Theidea of the method is to try to make rapid progress towards a feasibleinteger solution by fixing large numbers of variables in each iterationrepairing infeasibilities or deteriorations of the objective by regenera-tion of columns if possible and by controlled backtracking otherwise
The idea of perturbation branching is to solve a series of (MLP)s withobjectives wk k = 0 1 2 that are perturbed in such a way that theassociated LP solutions xk are likely to become more and more integralIn this way we hope to construct an almost integer solution at littlecomputational cost The perturbation is done by increasing the utilityof variables with LP values close to one according to the formula
w0j = wj j isin N
wk+1j = wkj + wjαx
2j j isin N k = 0 1 2
The progress of this procedure is measured in terms of the potential orscore function
v(xk) = wTx+ δ|B(xk)|where ε and δ are parameters for measuring near-integrality and therelative importance of near-integrality (we use ε = 01 and δ = 1) andB(xk) = j isin N xkj gt 1 minus ε is the set of variables that are set oralmost set to one ie also called candidate set The perturbation iscontinued as long as the potential function increases if the potentialdoes not increase for some time a spacer step is taken in an attempt
3 Branch and Price for Track Allocation 145
to continue Another reasonable criteria could be that the candidateset does not change On termination the variables in the set B(xk)associated with the highest potential are fixed to one If no variablesat all are fixed we choose a single candidate by strong branching seeApplegate et al (1995) [13] Objective perturbation has also been usedby Wedelin (1995) [211] for the solution of large-scale set partitioningproblems and eg by Eckstein amp Nediak (2007) [77] in the context ofgeneral mixed integer programming
Algorithm 6 Perturbation Branching
Data (RMLP )(w l u) integrality tolerance ε isin [0 05) integralityweight δ gt 0 perturbation factor α gt 0 bonus weight M gt 0spacer step interval ks iteration limit kmax
Result set of variables Blowast that can be fixed to one
1 init k larr 0 w0 larr w Blowast larr empty vlowast larrinfin2 while k lt kmax do maximum number of iterations not
reached
3 compute xk larr argmax(RMLP )(wk l u)
4 set Bk larr j xkj ge 1minus ε lj = 05 set v(xk)larr wTxk + δ|Bk|6 if xk is integer then7 set Blowast larr Bk candidates found
8 break
9 else10 if k equiv 0 mod ks and k gt 0 then11 set jlowast larr argmaxlj=0 x
kj
12 set wkj larrM
13 set Blowast larr Bk cup jlowast spacer step
14 else15 if v(xk) gt vlowast then16 set Blowast larr Bk vlowast larr v(xk) k larr minus1 progress
17 set wk+1j larr wkj + αwj(x
kj )
2 forallj perturb
18 set k larr k + 1
19 if Blowast = empty then20 set Blowast larr jlowast larr strongBranching() strong branching
21 return Blowast
Algorithm 6 gives a pseudocode listing of the complete perturbationbranching procedure The main work is in solving the perturbed re-
3 Branch and Price for Track Allocation 146
Skj+1
S3j+1
S2j+1
S1j+1
Sj
S0j+1
Figure 19 The new solution sets at iteration k source Weider (2007) [213]
duced master LP (line 3) and generating new variables if necessaryFixing candidates are determined (line 4) and the potential is evalu-ated (line 5) If the potential increases (lines 15ndash16) the perturbationis continued (line 17) If no progress was made for ks steps (line 10)the objective is heavily perturbed by a spacer step in an attempt tocontinue (lines 10ndash13) However even this perturbation does not guar-antee that any variable will get a value above 1minus ε if ε lt 12 If thishappens and the iteration limit is reached a single variable is fixed bystrong branching (line 20)
The fixing candidate sets Blowast produced by the perturbation branchingalgorithm are used to set up nodes in the branch-and-generate searchtree by imposing bounds xj = 1 for all j isin Blowast This typically fixesmany variables to one which is what we wanted to achieve Howeversometimes too much is fixed and some of the fixings turn out to bedisadvantageous In such a case we must backtrack We propose to dothis in a binary search manner by successively undoing half of the fixesuntil either the fixings work well or only a single fix is left as shown inFigure 19 This procedure is called binary search branching
Here are the details Let Blowast be a set of potential variable fixes andK = |Blowast| Order the variables in Blowast by some criterion as i1 i2 iKand define sets
Blowastk = i1 ik k = 1 K
Consider search tree nodes defined by fixing
xj = lj = 1 j isin Blowastk k = K dK2e dK4e 2 1
These nodes are examined in the above order Namely we first try to fixall variables in BlowastK to one since this raises hopes for maximal progressIf this branch comes out worse than expected it is pruned and we
3 Branch and Price for Track Allocation 147
backtrack to examine BlowasteK2d and so on until possibly Blowast1 is reachedThe resulting search tree is a path with some pruned branches iebinary search branching is a plunging heuristic In our implementationwe order the variables by increasing reduced cost of the restricted rootLP ie we unfix half of the variables of smallest reduced cost Thissorting is inspired by the scoring technique of Caprara Fischetti ampToth (1998) [60] The decision whether a branch is pruned or not isdone by means of a target value as introduced by Subramanian et al(1994) [204] Such a target value is a guess about the development ofthe LP bound if a set of fixes is applied we use a linear function ofthe integer infeasibility If the LP bound stays below the target valuethe branch develops according to our expectations if not the branchldquolooks worse than expectedrdquo and we backtrack
Chapter IV
Case Studies
In the last chapter we report on several computational experimentsSection 1 compares standard models and our novel extended formula-tion In Section 2 we present results of several computational experi-ments to analyze the benefit of the algorithmic ingredients of our novelsolution approach ie the proximal bundle method 22 and the rapidbranching heuristic 23
Section 3 discusses results of an auction based track allocation Theseresults and evaluation have a theoretical and visionary character dueto various questionable assumptions Thus we will also discuss puretheoretical and rather philosophical auction design questions
Finally we present computational results for solving track allocationproblems on real-world scenarios for the Simplon corridor in Section 4The basis for the presented results are the contributions of Chapter IIand Chapter III Furthermore it demonstrates the practical applicabil-ity of optimization for railway track allocation To the best knowledgeof the author and confirmed by several railway practitioners this wasthe first time that on a macroscopic scale automatically produced trackallocations fulfill the requirements of the original microscopic model
1 Model Comparison
TS-OPT is implemented in the programming language C++ It is able togenerate the static formulations (APP)prime and (ACP) as well as to solvemodel (PCP) by the proposed branch and price algorithm in Chap-ter III Section 3 All computations in the following were performedon computers with an Intel Core 2 Extreme CPU X9650 with 3 GHz
148
1 Model Comparison 149
6 MB cache and 8 GB of RAM or an Intel Core i7 870 with 3 GHz8 MB cache and 16 GB of RAM
This choice is motivated as follows (APP)prime is the dominant modelin the literature which we want to benchmark (PCP) and (ACP)are equivalent models that improve (APP)prime (APP)prime and (ACP) areboth arc-based rather easy to implement and very flexible
We did not implement the strong packing model (APP) and also not(PPP) because these models are not robust with respect to changesin the problem structure namely their simplicity depends on the par-ticular clique structure of interval graphs If more complex constraintsare considered these models can become hard to adapt In fact theinstances that we are going to consider involve real world headwaymatrices that give rise to more numerous and more complex cliquestructures as mentioned by Fischer et al (2008) [90] Thus an im-plementation of suitably strong versions of models (APP) and (PPP)would have been much more difficult than an implementation of thebasic versions discussed in Chapter III Section 21
In marked contrast to these models is our configuration model in whichheadway constraints are easy to implement The reason is simple thatthey specify possible follow-on trips on a track which is precisely whata configuration does Formulation (PCP) is in this sense very robustto handle headway conflicts if the corresponding headway matrices aretransitive It is also well suited for column generation to deal with verylarge instances as we will discuss in Section 2
We performed computational experiments with both static modelsOur aim was to gather from these test runs information that wouldallow us to choose a ldquowinnerrdquo ie a model that for the range of theproblem instances we address displays the best computational perfor-mance in practice
The instances for the comparison were solved as follows The rootLP-relaxations of the static models (APP)prime and (ACP) were solvedwith the barrier method of IBM ILOG CPLEX 112 (64 Bit 4 threadsbarrier) see CPLEX 12202 [119] Then the MIP solver of CPLEXwas called for a maximum of at most 1h of running time
1 Model Comparison 150
Figure 1 Infrastructure network (left) and train routing digraph (right) individ-ual train routing digraphs bear different colors
11 Effect of Flexibility
In our experiments we consider the Hanover-Kassel-Fulda area of theGerman long-distance railway network All our instances are basedon the macroscopic infrastructure network that is illustrated in Fig-ure 1 It includes data for 37 stations 120 tracks and 6 different traintypes (ICE IC RE RB S ICG) Our project partner from IVE andSFWBB provided this macroscopic data Because of various possibleturn around and running times for each train type this produces anmacroscopic railway model with 146 nodes 1480 arcs and 4320 head-way constraints ndash infrastructure scenario hakafu simple
Based on the 2002 timetable of Deutsche Bahn AG we constructedseveral scenarios We considered all trains inside that area in a timeinterval of about 480 minutes at a normal weekday from 900 to 1700(or smaller) We varied several objective parameters selected subsetsof the request and generated artificial additional freight traffic seeMura (2006) [164]
All instances related to hakafu simple are freely available at ourbenchmark library TTPlib see Erol et al (2008) [85] From the testruns we have made we have chosen to discuss the results of instancehakafu simple and req 36 ndash a scenario with 285 train requests
Table 1 demonstrates that reasonable track allocation problems canbecome very large even if the consider time windows are limited Themain objective is to maximize the total number of trains in the sched-
1 Model Comparison 151
before preprocessing after preprocessing
τ nodes arcs nodes arcs
0 123239 267080 282 3162 140605 300411 863 10054 155607 331631 2611 35896 169989 361927 4228 63728 186049 395688 6563 10515
10 204423 434499 9310 1572612 224069 476431 12380 2173014 245111 522119 15779 2856916 267989 572185 19838 3667318 291473 625083 24374 4588220 316631 681668 29738 56951
Table 1 Size of the test scenarios req 36
nodes
94relevant
906
redundant
arcs
84relevant
916
redundant
Figure 2 Reduction of graph size by trivial preprocessing for scenarios req 36and τ = 20
ule on a secondary level we slightly penalize deviations from certaindesired departure and arrival times ldquoFlexibilityrdquo to reroute trains iscontrolled by departure and arrival time windows of length at most τ where τ is a parameter To be precise let topt be the optimal arrival (ordeparture) time then we set the minimum arrival (or departure) timetmin to topt minus τ
2and the maximum arrival (or departure) time tmax to
topt+τ2 respectively Hence increasing τ from 0 to 20 minutes in steps
of 2 minutes increases flexibility but also produces larger train routingdigraphs and IPs We used a maximum of 20 minutes because in theallocation process for the annual timetable desired times (in minutes)were varied of at most 5 minutes
After graph preprocessing by algorithm 4 (eliminating arcs and nodeswhich cannot be part of a feasible train route) the resulting 11 in-stances have the sizes listed in Table 1 Figure 2 shows the concretebenefit of the graph preprocessing for the largest instance of that set
1 Model Comparison 152
τ rows cols trains ublowast v(LP) vlowast gap tsum bbnin in s
0 288 316 29 3710 3710 3710 ndash 887 12 962 1005 67 9992 9992 9992 ndash 820 14 3134 3589 121 21905 22292 21905 ndash 890 686 5552 6372 143 23867 24625 23867 ndash 949 5708 9584 10515 161 26077 27999 26077 ndash 1114 569
10 15481 15726 185 30954 32247 30954 ndash 1293 51812 23135 21730 198 33663 34829 33493 051 360985 152129814 33004 28569 220 37597 38726 37394 054 361216 120943116 47245 36673 239 40150 40892 39981 042 361297 77338618 66181 45882 254 43978 45845 43808 039 361358 46267020 93779 56951 257 45657 45845 45176 106 361394 303575
Table 2 Solution statistic for model (APP) and variants of scenario req 36
τ rows cols trains ublowast v(LP) vlowast gap tsum bbnin in s
0 835 920 29 3710 3710 3710 ndash 844 12 2418 2895 67 9992 9992 9992 ndash 841 14 6920 9345 121 21905 21905 21905 ndash 863 16 11129 16329 143 23867 24272 23867 ndash 939 18 17393 27470 161 26077 26925 26077 ndash 1329 280
10 24825 41517 185 30954 31404 30954 ndash 4641 57712 33156 57149 198 33493 34266 33493 ndash 11098 52814 42282 74862 220 37394 38145 37394 ndash 25962 78016 53142 96729 239 39981 40533 39981 ndash 146777 148518 65378 124115 254 43808 45048 43808 ndash 239955 51220 79697 156674 257 45477 45830 45176 067 361853 421
Table 3 Solution statistic for model (ACP) and variants of scenario req 36
Tables 2 and 3 show the results for model (APP) and (ACP) respec-tively The tables list
τ length of the time interval
rows number of rows (constraints) of the integer programmingformulation
cols number of columns (variables) of the integer programmingformulation
trains number of scheduled trains in the solution
ublowast proven upper bound
v(LP ) optimal value of the linear relaxation
vlowast objective function value of (best) integral solution
optimality gap1
bbn number of processed branch and bound nodes
and tsum the total running time of TS-OPT
1The relative gap is defined between the best integer objective bestSol and the
objective of the best node remaining bestNode as |bestNodeminusbestSol|10minus10+|bestSol|
1 Model Comparison 153
instance trains reqs rows cols ublowast v(LP) vlowast gap tsum bbn
in in s
req01 8 8 510 555 7000 7000 7000 ndash 128 1req02 11 11 882 676 8401 8414 8401 ndash 020 1req03 8 8 451 538 6800 6800 6800 ndash 021 1req04 19 19 1287 1197 15053 15083 15053 ndash 039 1req05 15 15 1344 877 10889 10942 10889 ndash 028 4req06 14 14 967 916 11574 11583 11574 ndash 028 8req07 42 46 5812 2949 33609 34349 33609 ndash 521 1754req08 46 55 7140 3312 35793 37244 35793 ndash 809 987req09 62 106 25957 6661 51900 54516 50800 217 360101 203976req10 73 198 76700 12525 66255 66755 61023 857 360157 25673req11 62 288 7453 2304 52600 52667 52600 ndash 116 1
Table 4 Solution statistic of model (APP) for wheel-instances
It turns out that in fact model (APP)prime produces for all instancesa significantly weaker LP-bound (upper bounds v(LP ) and ublowast) thanmodel (ACP) In addition we marked the instances where the LP-bound at the root is equal to the objective value of the optimal integersolutions
With increasing flexibility τ the models become trivially larger Al-though the extended formulation (ACP) produces in most cases thelarger model the produced results are almost always better for thistestset Model (ACP) was able to solve all instances to optimalityexcept for the last one Whereas model (APP) could only solve thefirst six instances during the time limit However the reason was thatthe dual bound could not be significantly improved during branch andbound even if the optimal primal solutions were found We reportedmore results of similar experiments with 146 285 and 570 train requestsin Borndorfer amp Schlechte (2007) [30] where the same effects can beobserved
12 Results for the TTPlib
In addition to the hakafu simple instances the TTPlib contains arti-ficial auction instances provided by our project partners ie AndreasTanner from WIP Figure 3 shows the layout of the infrastructure forthe 11 wheel instances Furthermore station capacities are consid-ered as well as minimum dwell time requirements for several trainssee Chapter II Section 212 and Chapter III Section 11
For each run of TS-OPT a time limit of one hour (3600 seconds) wasused to solve the IPs Table 4 and Table 5 show the results of the staticmodels (APP)prime and (ACP)
1 Model Comparison 154
Figure 3 Artifical network wheel see TTPlib [208]
instance trains reqs rows cols ublowast v(LP) vlowast gap tsum bbn
in in s
req01 8 8 1119 1202 7000 7000 7000 ndash 188 1req02 11 11 1273 1364 8401 8401 8401 ndash 043 1req03 8 8 1104 1175 6800 6800 6800 ndash 027 1req04 19 19 2351 2514 15053 15053 15053 ndash 043 1req05 15 15 1596 1706 10889 10889 10889 ndash 034 1req06 14 14 1816 1945 11574 11574 11574 ndash 030 1req07 42 46 5151 5512 33609 33609 33609 ndash 066 1req08 46 55 5747 6133 35793 35793 35793 ndash 070 1req09 62 106 9854 10553 50800 50800 50800 ndash 129 1req10 73 198 16263 17512 61477 61477 61477 ndash 223 1req11 62 288 6353 4912 52600 52667 52600 ndash 196 1
Table 5 Solution statistic of model (ACP) for wheel-instances
Obviously model (ACP) has more variables than model (APP)prime be-cause of the auxiliary track flows But if the conflict constraints ofthe instance ldquoexploderdquo model (ACP) has significantly less rows than(APP)prime eg in case of instances req 07-req 10
CPLEX was able to solve all 11 instances of model (ACP) to optimalityalready in the root node (in only some seconds) In addition in 10 of11 cases the value of the LP-relaxation equals the optimal value of theinteger problem In contrast (APP)prime was only able to solve 9 problemswithin the time limit For scenario req 09 and req 10 only a gap ofapproximately 2 and 8 were reached after 1 hour Only in twocases the value of the LP-relaxation equals the optimal value of theinteger problem In addition CPLEX needs to solve model (APP)prime asignificant number of branch and bound nodes for 6 instances
1 Model Comparison 155
instance trains reqs rows cols ublowast v(LP) vlowast gap tsum bbn
in in s
req01 198 285 3400 2563 39372 39372 39372 ndash 1186 1req02 266 285 28810 19694 46154 50564 45725 094 361241 1156299req03 273 285 62908 35021 48660 50771 48478 038 361486 367354req04 285 285 349241 97135 51237 51251 51195 008 368718 58421req05 152 194 2216 1764 28800 28800 28800 ndash 899 1req06 204 213 17780 14512 34892 37055 34892 ndash 4062 8207req07 178 184 33607 23450 33141 33623 32782 110 360958 705190req08 199 199 182442 68342 37401 37416 37392 002 363915 147562req09 93 114 1369 1112 16682 16682 16682 ndash 793 1req10 104 109 8147 7699 20288 21818 20288 ndash 556 765req11 97 98 12455 11902 16240 16244 16240 ndash 1067 2116req12 113 113 66011 39167 24533 24537 24533 ndash 3773 906req13 28 28 336 308 5946 5946 5946 ndash 262 1req14 33 33 1879 2544 6953 6953 6953 ndash 241 1req15 31 31 3406 4477 4608 4608 4608 ndash 102 9req16 30 30 9281 9436 9162 9164 9162 ndash 247 352req17 215 285 2417 1929 39330 39723 39330 ndash 1731 1req18 274 285 28827 19638 48291 50494 47459 175 361693 1460850req19 278 285 62994 35116 48832 50788 48831 ndash 362181 555884req20 285 285 346438 99306 51259 51265 51249 002 369988 90655req21 170 209 1676 1382 29692 29692 29692 ndash 687 1req22 206 212 18394 14121 33796 34973 33796 ndash 2243 4057req23 191 199 41456 26132 35850 37812 35849 ndash 362267 892094req24 194 194 184853 68282 37186 37193 37186 ndash 173929 57430req25 98 117 959 822 17556 17556 17556 ndash 678 1req26 117 118 8604 7952 19175 19254 19175 ndash 517 181req27 116 118 16268 13981 18546 19815 18546 ndash 9562 49763req28 102 102 63468 35804 19132 19137 19132 ndash 147304 213086req29 20 20 154 144 4071 4071 4071 ndash 347 1req30 31 31 1439 1835 10006 10006 10006 ndash 090 1req31 363 1062 16844 15620 46440 46475 46440 ndash 6270 182req32 261 1140 106091 44112 20285 21458 20285 ndash 5374 486req33 151 570 34911 22056 10533 10986 10533 ndash 2529 505req36 151 285 5907 5712 24258 25534 24258 ndash 1019 556req37 257 334 201529 82937 36573 37125 36573 ndash 19072 523req38 259 334 201529 82937 39877 40587 39877 ndash 10901 549req39 272 358 245968 73324 50518 52102 50518 ndash 49291 6259req40 272 358 245968 73324 54110 55699 53443 125 363077 4650req41 287 382 106728 56037 41911 42716 41911 ndash 6634 544req42 288 382 106728 56037 44227 45322 44227 ndash 7516 555req43 300 409 247756 87209 53144 55359 53144 ndash 222643 38981req44 300 409 247756 87209 55497 58529 55497 ndash 337826 83002req45 264 344 141976 51079 51490 52311 51490 ndash 95605 1852req46 263 344 141976 51079 46273 47275 46273 ndash 16823 5000req47 25 25 2304 3105 4363 4363 4363 ndash 227 1req48 41 41 11585 13314 7681 7681 7681 ndash 554 1
Table 6 Solution statistic of model (APP) for hakafu simple-instances
We also performed this experiment for the remaining instances of theTTPlib ie 50 instances for network hakafu simple The results ofthe experiment are shown in Table 6 and 7 For four instances CPLEXie req 34req 35 req 49 and req 50 was not able to solve theinteger program within 1 hour for both models For the remaining 46instances model (ACP) reached three times the time limit withoutany solution For another three instances TS-OPT terminates for model(ACP) with a small optimality gap of approximately 1 CPLEX wasable to solve all other instances (40) to proven optimality In additionwe marked 16 instances were the objective values of the LP relaxationfor model (ACP) coincide with optimal integer solution
CPLEX was able to produce solutions for model (APP)prime for all 46instances ie also for instances req 39 req 43 and req 44 withinthe time limit However in 8 cases the runs terminated after an hour
1 Model Comparison 156
instance trains reqs rows cols ublowast v(LP) vlowast gap tsum bbn
in in s
req01 198 285 6880 9315 39372 39372 39372 ndash 1366 1req02 266 285 37487 59637 45725 48348 45725 ndash 4661 518req03 274 285 61607 105243 48527 50599 48527 ndash 241921 13947req04 284 285 153226 302292 51214 51220 50878 066 364478 779req05 152 194 4845 6330 28800 28800 28800 ndash 998 1req06 204 213 28212 44623 34892 36557 34892 ndash 2497 529req07 178 184 41997 70415 32782 33623 32782 ndash 4117 527req08 199 199 111069 216059 37392 37402 37392 ndash 227267 1582req09 93 114 3070 3795 16682 16682 16682 ndash 872 1req10 104 109 15209 22972 20288 20722 20288 ndash 543 49req11 97 98 21656 33715 16240 16655 16240 ndash 1125 622req12 113 113 68197 130143 24533 24535 24533 ndash 8375 482req13 28 28 916 915 5946 5946 5946 ndash 288 1req14 33 33 5061 6613 6953 6953 6953 ndash 246 1req15 31 31 8521 11935 4608 4608 4608 ndash 134 1req16 30 30 16894 26468 9162 9163 9162 ndash 501 99req17 215 285 5361 7318 39330 39492 39330 ndash 1764 1req18 274 285 38118 62658 47459 49579 47459 ndash 2775 104req19 278 285 63662 112602 48831 49803 48831 ndash 16309 531req20 284 285 161313 329062 51255 51257 50918 066 365125 840req21 170 209 3909 5078 29692 29692 29692 ndash 800 1req22 206 212 27657 44998 33796 34690 33796 ndash 2107 351req23 191 199 48054 85452 35849 36819 35849 ndash 8696 542req24 194 194 115319 235679 37186 37191 37186 ndash 349365 6446req25 98 117 2395 3047 17556 17556 17556 ndash 743 1req26 117 118 15725 24117 19175 19175 19175 ndash 565 1req27 116 118 26196 44151 18546 19813 18546 ndash 2391 542req28 102 102 62018 121071 19132 19135 19132 ndash 88222 6357req29 20 20 479 472 4071 4071 4071 ndash 180 1req30 31 31 3797 4822 10006 10006 10006 ndash 128 1req31 368 1062 31754 43710 46440 46440 46440 ndash 6910 30req32 297 1140 80183 126924 20285 20285 20285 ndash 6063 1req33 171 570 42416 67443 10533 10533 10533 ndash 2613 1req36 151 285 11855 16392 24258 24664 24258 ndash 1099 1req37 257 334 130148 265556 36573 36674 36573 ndash 365302 9req38 259 334 130148 265556 39877 40144 39877 ndash 316627 1req39 0 358 114397 226407 50754 50754 ndash infin 364214 1req40 272 358 245968 73324 54107 55699 53443 124 363186 4723req41 287 382 97282 202892 41911 42247 41911 ndash 152714 545req42 288 382 97282 202892 44227 44656 44227 ndash 222827 494req43 0 409 140963 303446 53855 53855 ndash infin 365179 1req44 0 409 140963 303446 56385 56385 ndash infin 365104 1req45 264 344 85629 171420 51490 51625 51490 ndash 74486 149req46 263 344 85629 171420 46273 46585 46273 ndash 125249 223req47 25 25 6163 8272 4363 4363 4363 ndash 268 1req48 41 41 24124 40722 7681 7681 7681 ndash 821 1
Table 7 Solution statistic of model (ACP) for hakafu simple-instances
with an optimality gap of approximately 1 The produced solutionwere already the optimal ones nevertheless (APP)prime was not able toclose the gap within the time limit The other 38 instances were solvedto optimality In 12 cases the objective values of the LP relaxation formodel (APP)prime coincide with optimal integer solution
We increased the time limit to one day and solved again the hardinstances Let us explicitly point out that these computations wouldnot be possible on a standard PC at the beginning of the projectHowever thanks to the 16GB main memory we were able to producethese numbers to verify our novel algorithmic approach which will bediscussed in the next section
Tables 8 and 9 show the results for both models For instances req 34req 35 req 49 and req 50 the LP relaxation of model (APP)prime be-came too large ie CPLEX abort with out of memory The other
1 Model Comparison 157
instance trains reqs rows cols ublowast v(LP) vlowast gap tsum bbn
in in s
req34 0 285 3623973 305366 ndash ndash memout infin 93583 ndashreq35 0 285 7974708 514425 ndash ndash memout infin 110057 ndashreq37 257 334 201529 82937 36573 37125 36573 ndash 40152 523req38 259 334 201529 82937 39877 40587 39877 ndash 25878 549req39 272 358 245968 73324 50518 52102 50518 ndash 86291 6259req40 272 358 245968 73324 53532 55699 53532 ndash 4720367 28766req43 300 409 247756 87209 53144 55359 53144 ndash 524588 38981req44 300 409 247756 87209 55497 58529 55497 ndash 760445 83002req49 0 285 2152600 232204 ndash ndash memout infin 17784 ndashreq50 0 285 7974708 514425 ndash ndash memout infin 28270 ndash
Table 8 Solution statistic of model (APP) for hard hakafu simple-instances
instance trains reqs rows cols ublowast v(LP) vlowast gap tsum bbn
in in s
req34 0 285 384563 873904 51267 51267 ndash infin 8647883 1req35 0 285 587570 1394454 51275 51275 ndash infin 8654046 1req37 257 334 130148 265556 36573 36674 36573 ndash 368360 10req38 259 334 130148 265556 39877 40144 39877 ndash 244881 1req39 272 358 114397 226407 50518 50754 50518 ndash 690122 174req40 272 358 114397 226407 53532 54155 53532 ndash 1768808 645req43 300 409 140963 303446 53144 53855 53144 ndash 1816280 553req44 300 409 140963 303446 55497 56385 55497 ndash 2776454 624req49 0 285 311772 703252 50468 50468 ndash infin 8649350 1req50 0 285 587570 1394454 51275 51275 ndash infin 8674436 1
Table 9 Solution statistic of model (ACP) for hard hakafu simple-instances
instances could be solved to optimality within an hour In contrast tothat CPLEX was able to solve all relaxations of model (ACP) withinone day and produced stronger upper bounds for all hard scenariosHowever CPLEX needed more time producing an optimal integer so-lution for model (ACP) than for model (APP)prime for almost all hardinstances Although CPLEX needs less branch and bound nodes tosolve model (ACP) the time needed per node ie to solve the linearrelaxation was significantly higher than for model (APP)prime
13 Conclusion
We have compared the static model formulation (APP)prime and (ACP)for a huge set and variants of instances which are free available atTTPlib First of all CPLEX was able to solve model (APP)prime and(ACP) for instances of reasonable size to proven optimality ie TS-OPTwas only used to construct the (preprocessed) graphs and models Onlyfor some very large scale instances the larger LP relaxation of the ex-tended formulation had a negative effect on the total running time Wehave observed that even if the extended formulation (ACP) tends inmost cases to larger LP relaxations than (APP)prime the benefit from abetter global upper bound transfers often directly to a higher solutionquality and shorter running times In particular these effects are in-
2 Algorithmic Ingredients for the (PCP) 158
model (APP) (ACP)
req 36-instances
produced best upper bound 2 11 no integrality gap 2 3 optimal solution found 11 11 optimal solution proven 6 10
wheel-instances
produced best upper bound 3 11 no integrality gap 2 10 optimal solution found 10 11 optimal solution proven 9 11
hakafu simple-instances
produced best upper bound 25 50 no integrality gap 12 16 optimal solution found 38 40 optimal solution proven 39 40
Table 10 Comparison of results for differrent models on the TTPlib-instances
tensified if the flexibility of the train requests are high eg if the timewindows of the events are large or if the capacity is rare eg if severaltrains compete for the same track resources
The results of our computational experiments made us conclude thatmodel (ACP) outperforms model (APP)prime Table 10 gives a short sum-mary and lists the number of instances for which the models producedan optimal solution number of instances for which the root upperbound has no integrality gap and the number of instances for whichthe upper bound of the root LP relaxation was better or equal than theone produced by the other model If we would establish a system ofpoint scoring model (ACP) will be most likely the winner on ldquopointsrdquoHence (ACP) is suited best for our particular problem instances andreal world application
2 Algorithmic Ingredients for the (PCP)
In this section we want to analyze our different solution approachesto solve model (PCP) which we all integrated or implemented in our
2 Algorithmic Ingredients for the (PCP) 159
module TS-OPT We start with a comparison of our approach withcomputational results from the literature in Section 21 Section 22discusses experiments and results for the bundle method Finally wepresent computational results of the rapid branching heuristic to solvelarge scale track allocation problems in Section 23
21 Results from the Literature
Let us discuss computational results for a variation of the rather rdquosim-pleldquo wheel instances The reason is that Cacchiani Caprara amp Toth(2010) [54] present results for modified versions of these TTPlib in-stances by excluding station capacities In addition their implementedmodel cannot handle train type specific headway times Hence theyonly considered instances of the TTPlib with one train type ie thewheel instances However let us thank them (and all others) for us-ing our instances in their studies which verifies that the TTPlib pro-vides an useful modular and easily understandable standard formatfor track allocation problems
They used a (PPP) formulation of the problem produced upper boundsby solving the Lagrangian relaxation using standard subgradient opti-mization and column generation and constructed solutions by a greedyheuristic based on Lagrangian profits and some refinement procedureThey were able to solve instances req 1-req 8 to proven optimalitywithin a second For instances req 9-req 11 they could produce al-most optimal solutions ie the produced upper bounds prove a gapwithin 2 of the optimum The time needed to produce solutions forproblem req 9 and req 10 is comparatively high (57 and 602 sec-onds) as well as we already observed for the static model (APP) seeSection 1 and Table 4 However in 5 of 11 cases the presented solutionsare also feasible (and hence optimal) in presence of the station capacityconstraints
Table 11 lists the statistic of our column generation approach using thebundle method and the rapid branching heuristic We want to mentionthat our listed absolute values (bounds and objectives) differ to thepublished ones on TTPlib due to a problem specific scaling inside ofTS-OPT In fact we scale all objective values such that the best path hasprofit of 100 Furthermore we used as a stopping criteria an optimalitygap of 10 It can be seen that we only need a very small numberof branch and bound nodes to produce almost optimal solutions (gap
2 Algorithmic Ingredients for the (PCP) 160
instance trains reqs rows cols ublowast v(LP) vlowast gap tsum bbn
in in s
req01 8 8 235 369 7000 7000 7000 ndash 041 1req02 11 11 253 594 8401 8401 8401 ndash 020 1req03 8 8 237 403 6800 6800 6800 ndash 013 1req04 19 19 474 1149 15067 15067 15067 ndash 030 1req05 15 15 304 972 10892 10892 10889 003 039 3req06 14 14 375 765 11580 11580 11574 005 033 2req07 42 46 919 3587 33722 33722 33716 002 207 3req08 46 55 1014 4296 35944 35944 35936 002 590 3req09 62 106 1422 6173 51128 51128 51120 002 632 3req10 77 198 1879 8645 64468 64468 64461 001 1732 2req11 66 288 1176 3014 55616 55616 55600 003 2903 6
Table 11 Solution statistic of TS-OPT and model (PCP) for wheel-instances
below 005) However the re-scaled upper bounds and solutions areconform to the results presented by Cacchiani Caprara amp Toth (2010)[54] There are minor deviations for the solutions values because arenumerical ones respecting the given tolerances see Table 11
To demonstrate that even such small instances have to be solved viaexact optimization approaches we only run the bundle method to solvethe relaxation and used afterwards a simple greedy heuristic in TS-OPT
to produce a feasible integral solution It can be seen that even forthese simple instances it is not trivial to produce high quality solutionsFor some of the instance the produced solutions have a gap largerthan 15 to the optimum Finally Table 12 compares the (PPP)-results of Cacchiani Caprara amp Toth (2010) [54] the (bundle and)greedy approach and the (bundle and) rapid branching approach tosolve model (PCP) with TS-OPT Already this rather easy subset ofthe TTPlib indicates that our configuration model has computationaladvantages both the static variant (ACP) see Section 1 and Table 4and dynamic version (PCP) In particular if the instance give rise tomany conflicts eg instances req 9 and req 10
bundle and greedy (PPP) TS-OPT
instance vlowast gap tsum vlowast gap tsum vlowast gap tsumin in s in in s in in s
req01 350000 ndash 1 350000 ndash 1 350000 ndash 1req02 360000 167 1 422102 ndash 1 420050 ndash 1req03 340000 ndash 1 340000 ndash 1 340000 ndash 1req04 753350 ndash 1 753329 ndash 1 753350 ndash 1req05 473350 151 1 545678 ndash 1 544450 ndash 1req06 578600 01 1 578724 ndash 1 578700 ndash 1req07 1461550 154 2 1691072 ndash 2 1685800 ndash 2req08 1652800 87 2 1795708 ndash 17 1796800 ndash 6req09 2386900 71 4 2604644 18 57 2604600 ndash 6req10 3039100 61 9 3229996 12 602 3223050 ndash 17req11 2580000 78 16 2780000 06 8 2780000 ndash 29
Table 12 Comparison of results for model (PPP) from Cacchiani Caprara ampToth (2010) [54] for modified wheel-instances
2 Algorithmic Ingredients for the (PCP) 161
22 Bundle Method
We evaluated our algorithmic approaches presented in Chapter IIISection 3 on the benchmark library TTPlib see Erol et al (2008)[85] They are associated with the macroscopic railway network modelhakafu simple already described in Section 1
Figure 4 illustrates the column generation process for solving instancereq 05 with the barrier method of CPLEX For each iteration the cur-rent value of the RMLP is shown as well as the upper bound β(γ π λ)see Lemma 211 The general effects of ldquoheading inrdquo and ldquotailing offrdquocan be observed ie we need many column generation iterations toget an upper bound value of 289 Obviously one could try to improvethe performance or convergence of a standard column generation ap-proach by using stabilization techniques or sophisticated strategies forthe generation of columns see Lubbecke amp Desrosiers (2005) [156]
Figure 5 shows exemplary the progress of the bundle method 5 ieit can be seen that a dual bound of 289 is reported after one secondTogether with Figure 4 it gives an intuition of the progress and con-vergence of the bundle method and the standard column generationapproach for solving instance req 05 The mere fact that the time-scales are significantly different prevent us from plotting both runstogether The reason for the significant smaller solution time is that incase of the bundle method in each iteration only a very small QP and
0 200 400 600
260
280
300
320
340
time in seconds
objectivevalue
objective function of RMLP (reduced cost induced) upper bound
300 400 500 600287
288
289
290
time in seconds
Figure 4 Solving the LP relaxation of model (PCP) with column generation andthe barrier method
2 Algorithmic Ingredients for the (PCP) 162
0 05 1 15
260
280
300
320
340
time in seconds
objectivevalue
upper bound
0 05 1 15
1800
2000
2200
2400
2600
time in seconds
number
columns
0 05 1 15
1800
2000
2200
2400
2600
time in secondsnumber
columns
Figure 5 Solving the LP relaxation of model (PCP) with the bundle method
several shortest path problems are successively solved In case of thecolumn generation approach with the barrier method as well as withthe primal or dual simplex method solving large linear programs andalso solving shortest path problems are alternated
Table 13 compares different solution approaches to solve the linear orLagrangean relaxation of model (PCP) for an arbitrary selection ofrequest scenarios of network hakafu simple On the one hand wesolve the linear relaxation by column generation and by using differentalgorithms to solve the LP relaxation ie the rows ldquodualrdquo containthe results of the dual simplex algorithm ldquobarrierrdquo stands for barrieralgorithm and ldquoprimalrdquo for the primal simplex algorithm On the otherhand the rows ldquobundlerdquo show the results for the bundle method Thesizes ie reqs rows and cols of the finally generated modelsare listed as well as the solution time tsum Column ublowast shows the valueof the upper bound β(γ π λ) induced by the reduced cost during thecolumn generation method see Lemma 211 or the best upper boundproduced by the Lagrangean relaxation Column vlowast(LP ) states thevalue of the produced fractional primal solution We mark this valuein case of the bundle method because the produced fractional vectormight violate the relaxed constraints ie the coupling constraints ofmodel (PCP)
We can observe that the standard column generation approach for solv-ing LPs needs much more columns until the relaxation is solved to op-timality for most of the instances In each iteration a noticeable largerLP is solved The number of column generation iterations (iter) isvery high ie several hundreds if we solve the (MLP) to proven op-
2 Algorithmic Ingredients for the (PCP) 163
solver reqs rows cols ublowast vlowast(LP) tsum iterin s
req 02
primal 285 7914 138450 48806 48241 gt week 761dual 285 7914 147831 48722 48277 gt day 1000barrier 285 7914 145146 48929 48277 gt 4hours 1000
bundle 285 7914 146415 48413 48413 449 1514
req 05
primal 194 1157 36691 28804 28781 454 116dual 194 1157 37087 28800 28800 566 187barrier 194 1157 37448 28820 28800 683 230
bundle 194 1157 2521 28824 28824 2 157
req 17
primal 285 1393 24185 39529 39492 450 130dual 285 1393 25344 39515 39483 616 241barrier 285 1393 25901 39512 39492 842 330
bundle 285 1393 3692 39529 39529 35 234
req 21
primal 209 1032 14932 29692 29683 242 179dual 209 1032 13886 29692 29669 148 86barrier 209 1032 14274 29715 29692 287 184
bundle 209 1032 1991 29728 29728 25 142
req 25
primal 117 645 6058 17556 17556 107 229dual 117 645 5410 17556 17535 29 38barrier 117 645 5433 17556 17556 83 36
bundle 117 645 1268 17573 17573 14 122
Table 13 Statistic for solving the LP relaxation of model (PCP) with columngeneration and the bundle method
timality That is no column with positive reduced cost is left Besidesthe higher memory consumption for the larger LPs we observed a con-vergence problem with the primal and dual simplex as well as with thebarrier method
In contrast the bundle method solves the relaxation (RMLP) in an al-gorithmically integrated and sparse way No ldquorealrdquo column generationis needed because the function evaluation step of algorithm 5 can besolved exactly Only in the direction finding step the generated pathsand configurations are used However the produced solutions of theshortest path problems can be seen as generated columns of the bundlemethod ie these are the columns that we store during the bundle al-gorithm to construct a restricted version of model (PCP) and producean integral solution in the end In addition we keep also the paths and
2 Algorithmic Ingredients for the (PCP) 164
configurations induced by columns that leave the bundle set during thealgorithm
Therefore the generation of columns seems to be more guided andonly a small portion of the paths and configurations compared with theother approaches is needed to solve the relaxation see Figure 5 andTable 13 The very large instance req 02 is one of a few exceptionsfor which the bundle method also needs a comparable high numberof columns similar to the other approaches However the solutiontime is always significantly smaller without losing quality In case ofreq 02 the column generation approach is stopped after a fixed limitof 1000 iterations with a bound even worse than produced by the bundleapproach
For our type of problem ie the Lagrangean dual of model (PCP)the parameter calibration of the the bundle method was rather uncom-plicated and straight-forward Figure 6 compares exemplary the effectof different choices for the size of the bundle (2 5 10 15 20 25) on thesolution of the Lagrangean relaxation of some test instances It can beseen that larger bundles lead in general to a reduction in the numberof iterations to a certain limit However larger bundles also producelarger and more difficult quadratic programs in algorithm 5 such thatthe total solution time and the number of iterations increases after acertain point A default bundle size of 15 seems to be a good choicefor our specific problem instances
Table 14 shows the results of our implementation of the bundle methodon solving the Lagrange relaxation of the the model (PCP) Additionalto the columns we have already introduced in former tables columniter displays the number of iterations of the bundle method to solvethe Lagrangean relaxation see algorithm 5 We denoted the optimalvalue of the Lagrangean dual (LD) by vlowast(LD) After that we per-formed a trivial greedy heuristic to find an integer solution for the
2 5 10 15 20 25
2000
2500
3000
bundle size
req32
iterationstime in seconds
2 5 10 15 20 25
500
1000
1500
2000
2500
bundle size
req31
iterationstime in seconds
2 5 10 15 20 25
500
1000
1500
bundle size
req33
iterationstime in seconds
Figure 6 Testing different bundle sizes
2 Algorithmic Ingredients for the (PCP) 165
instance trains reqs rows cols vlowast(LD) vlowast gap tsum iter
in in s
req01 197 285 1618 4613 39395 39355 010 1822 214req02 207 285 7914 146415 48413 36213 3369 93563 1514req03 224 285 12848 202773 50709 39263 2915 198887 1540req04 208 285 31615 138989 51237 38529 3298 135357 320req05 152 194 1157 2521 28824 28800 008 1135 157req06 175 213 6032 118056 36631 31289 1707 52808 1866req07 158 184 8878 154847 33641 29511 1399 87783 1865req08 155 199 23308 68767 37431 28215 3266 28963 120req09 93 114 746 1392 16699 16682 010 902 119req10 98 109 3303 48171 20748 19411 689 6794 753req11 78 98 4633 15224 16662 13465 2374 1319 65req12 95 113 14856 31302 24539 19435 2626 6458 68req13 28 28 244 321 5946 5946 ndash 339 14req14 32 33 1133 2125 6953 6930 034 286 9req15 30 31 1909 3409 4609 4343 612 158 10req16 29 30 3759 5148 9165 8720 510 264 12req17 211 285 1393 3692 39529 38978 141 2281 234req18 220 285 8218 140206 49605 36090 3745 86104 1520req19 216 285 13576 45652 51272 34127 5024 18073 145req20 194 285 34094 70786 51277 31209 6430 56875 122req21 168 209 1032 1991 29728 29541 063 893 142req22 173 212 6003 20820 35502 26232 3534 3646 92req23 150 199 10370 28203 38118 28011 3608 6284 80req24 140 194 24925 38751 37200 25594 4534 12511 45req25 98 117 645 1268 17573 17556 010 808 122req26 101 118 3460 8396 19334 16203 1932 710 32req27 85 118 5692 11737 19818 12097 6382 1260 23req28 81 102 13612 18026 19138 14919 2828 2507 19req29 20 20 145 189 4071 4071 ndash 226 5req30 30 31 867 1709 10006 9639 380 157 8req31 352 1062 6913 28318 46478 45802 148 34706 828req32 292 1140 16489 28191 20305 19262 542 119645 752req33 171 570 9036 12566 10569 10078 487 15985 459req34 149 285 76842 138994 51275 31090 6492 236624 108req35 137 285 116303 49772 51277 24378 11034 167737 44req36 127 285 2602 28385 24700 17823 3858 9910 684req37 169 334 28694 133626 36710 22161 6565 246274 864req38 167 334 28694 145328 40165 24944 6102 288228 1090req39 142 358 24329 158428 50789 19883 15544 369450 1112req40 140 358 24329 176134 54189 25192 11510 392945 1331req41 144 382 22035 135959 42267 21362 9786 262183 1140req42 134 382 22035 138510 44681 19458 12963 262667 1225req43 151 409 30978 170834 53879 29409 8321 461330 1204req44 154 409 30978 176552 56414 27485 10525 455610 1177req45 163 344 18694 112021 51649 33918 5228 144944 1137req46 151 344 18694 112122 46609 25929 7976 159559 1122req47 24 25 1402 2212 4363 4342 049 297 8req48 39 41 5456 5567 7681 7171 712 648 9req49 139 285 63963 105681 51274 30085 7043 186076 120req50 137 285 116303 49772 51277 24378 11034 167398 44
Table 14 Solution statistic of bundle method and greedy heuristic for model(PCP) for hakafu simple-instances
constructed sub-problems The objective value is denoted by vlowast inTable 14
We could observe that the upper bounds produced by our bundle imple-mentation for model (PCP) have the same quality as the ones obtainedby model (ACP) ie better bounds than model (APP)prime There areonly slight differences because of the numerical tolerances In additionthe bundle approach and model (PCP) is faster than static modelsfor very large scale instances eg req 40 req 49 or req 50 Inaddition solving the static models (ACP) and (APP) for instancesreq 34 req 35 and req 50 is critical from a memory point of viewAt least 16GB of main memory is required to solve the root relaxtion
2 Algorithmic Ingredients for the (PCP) 166
In contrast to that our bundle approach uses only 2 GB of memory tosolve the relaxtion of these instances
However for the produced integer solutions of the greedy heuristic nosolution quality can be guaranteed Obviously there are easy instanceseg req 01req 05req 09req 13 or req 29 where a greedy ap-proach is able to produce an optimal or almost optimal solution Butthere are also many instances for which the greedy solution is far awayfrom optimality eg req 39 req 40 or req 42 have a gap largerthan 100
Finally we conclude that the bundle method is the most efficient ap-proach to produce high quality upper bounds for model (PCP) Itoutperforms standard column generation approaches using the sim-plex or interior point methods ie the total running time is order ofmagnitudes smaller and the quality of the upper bounds is roughlycomparable Furthermore we were able with this approach to pro-duce non-trivial upper bounds much faster than with the static modelvariant (ACP) for very large scale instances
23 Rapid Branching
We tested our implementation of the rapid branching heuristic seeAlgorithm 6 presented in Section 3 of Chapter III on instances from thebenchmark library TTPlib see the macroscopic railway network modelhakafu simple described in Section 11 and some larger request sets
Figure 8 shows an ideal run of our code TS-OPT ie the run of sce-nario req 31 and network hakafu simple On the left hand sidethe objective value of the primal solution the upper bound and theobjective of the fixation evaluated by the rapid branching heuristic isillustrated In the initial LP stage (dark blue) a global upper boundis computed by solving the Lagrangean dual using the bundle methodafter approximately 400 seconds In that scenario one can see theimprovement of the upper bound during the bundle method Further-more in that stage the most important path and configuration variablesare generated On the right hand side of the figure the developmentof the number of generated columns the number of fixed to 1 columnsand the number of integer infeasibilities ie the number of integervariables that still have a fractional value in the primal solution of thecurrent relaxation is shown In the first phase (dark blue) the column
2 Algorithmic Ingredients for the (PCP) 167
0 500 1000 1500 2000 2500
0
200
400
600
800
dual bound
greedy solution
final ip solution
time in seconds
obje
ctiv
e
req31
primal valueupper bound
value of fixation
0 500 1000 1500 2000 2500
0
1
2
3
4
middot104
time in seconds
req31
columnsinteger infeasibilities
fixed to 1
Figure 7 Solving a track allocation problem with TS-OPT dual (LP) and primal(IP) stage
generation process during the bundle method can be seen and that fix-ing a large number of the ldquorightrdquo variables at once (to 1) decreases theinteger infeasibilities significantly but not monotonously In fact therapid branching heuristic produced a solution with 061 and was ableto improve the greedy solution computed directly after the first phasewith a gap of 148
Figure 8 shows another run of our code TS-OPT ie scenario req 48of network hakafu simple On the left hand side the objective valueof the primal solution the upper bound and the objective of the fix-ation evaluated by the rapid branching heuristic is plotted again Inthe initial LP stage (dark blue) a global upper bound is computed bysolving the Lagrangean dual using the bundle method after approxi-mately 15 seconds In that scenario the upper bound is only slightlybelow the trivial upper bound ie the sum of all maximum profits Inthe succeeding IP stage (light blue) an integer solution is constructedby the greedy heuristic and improved by the rapid branching heuristicIt can be seen that the final integer solution has virtually the sameobjective value as the LP relaxation and the method is able to closethe gap between greedy solution and the proven upper bound On theright hand side of the figure one can see that indeed often large num-bers of variables are fixed to one and several backtracks are performedthroughout the course of the rapid branching heuristic until the finalsolution was found In addition we plotted the development of the in-teger infeasibilities ie the number of integer variables that still havea fractional value
2 Algorithmic Ingredients for the (PCP) 168
0 20 40 60 80 100
0
20
40
60
80
100
120
140
dual boundgreedy solution
best ip solution
time in seconds
obje
ctiv
e
req48
primal valueupper bound
value of fixation
0 20 40 60 80 100
0
200
400
600
800
1000
time in seconds
req48
integer infeasibilitiesfixed to 1
Figure 8 Solving track allocation problem req 48 with TS-OPT
Tables 15 16 and 17 show results for solving the test instances byour code TS-OPT in order to calibrate our method Furthermore weset a limit on the number of backtrack for rapid branching of 5 Thetables list the number of scheduled trains in the best solution foundthe number of requested train the size of the model in terms of num-ber of rows and columns the upper bound produced by the bundlemethod the solution value of rapid branching heuristic the optimalitygap the total running time in CPU seconds and the number of (rapid)branching nodes The computations in Table 15 have been performedwith an aggressive choice of the rapid branching integrality toleranceof ε = 04 Table 17 shows the results for a cautious choice of ε = 01and Table 17 for the default choice of ε = 025 It can be seen that theaggressive choice tends to be faster because more variables are fixed atonce to explore fewer rapid branching nodes but the solution qualityis lower However there are a few exceptions eg instance req 07explores less nodes and terminates with a better solution Choosinga very moderate setting leads to larger computation times and moreevaluation of rapid branching nodes with the adavantage that the so-lution quality is in general higher In addition one can see that therapid branching heuristic sometimes fails to produce solutions eg forinstance req 11 with aggressive or moderate settings However withchoosing ε = 025 high quality solutions for large-scale track allocationproblems involving hundreds of train requests can be computed
The benefit of the our algorithmic approach can be seen for very largescale instances In Table 18 we list the results for instances with morethan 500 requests through the network hakafu simple In additionthese instances have much more coupling rows than the instances of
2 Algorithmic Ingredients for the (PCP) 169
instance trains reqs rows cols v(LP) vlowast gap tsum bbn
in in s
req06 198 213 6032 118056 36631 33064 1079 2703556 26req07 171 184 8878 154847 33641 29791 1292 993344 21req08 160 199 23308 68767 37431 27178 3773 4171871 23req11 0 98 4633 15224 16662 27178 - 201758 29req12 98 113 14856 31302 24539 19317 2703 327355 22req17 216 285 1393 3692 39529 39276 064 3781 15req18 253 285 8218 140206 49605 41528 1945 2757764 39req31 360 1062 6913 28318 46478 46197 061 267549 13req32 257 1140 16489 28191 20305 20244 030 262838 21req33 138 570 9036 12566 10569 10533 034 56039 8
Table 15 Solution statistic of rapid branching with aggressive settings
instance trains reqs rows cols v(LP) vlowast gap tsum bbn
in in s
req06 0 213 6032 118056 36631 - infin 3020458 49req07 172 184 8878 154847 33641 28958 1617 1091631 39req08 166 199 23308 68767 37431 28001 3368 3377172 24req11 0 98 4633 15224 16662 17617 - 338586 28req12 0 113 14856 31302 24539 17617 3929 239354 34req17 216 285 1393 3692 39529 39276 064 4687 5req18 254 285 8218 140206 49605 37453 3245 5172837 42req31 359 1062 6913 28318 46478 45337 252 302606 8req32 257 1140 16489 28191 20305 20240 032 273525 13req33 138 570 9036 12566 10569 10533 034 120950 23
Table 16 Solution statistic of rapid branching with moderate settings
instance trains reqs rows cols v(LP) vlowast gap tsum bbn
in in s
req06 201 213 6032 118056 36631 32777 1176 1612486 29req07 172 184 8878 154847 33641 28275 1898 3398551 31req08 168 199 23308 68767 37431 28779 3006 2971657 27req11 88 98 4633 15224 16662 14522 1474 70381 16req12 96 113 14856 31302 24539 18213 3473 271282 22req17 216 285 1393 3692 39529 39276 064 4887 9req18 253 285 8218 140206 49605 38521 2878 3556213 42req31 357 1062 6913 28318 46478 45639 184 302438 9req32 256 1140 16489 28191 20305 19959 173 283852 24req33 138 570 9036 12566 10569 10566 002 65347 9
Table 17 Solution statistic of rapid branching with default settings
instance trains reqs rows cols v(LP) vlowast gap tsum bbn
in in s
req 506 218 506 30213 282463 27455 26679 291 7018690 2188req 567 247 567 30595 259003 36947 36058 246 6357324 1875req 813 215 813 32287 225482 44145 41858 546 3762705 157req 875 239 875 36206 248922 39510 36822 730 4612819 228req 906 235 906 35155 265837 44116 40906 785 5123458 471
Table 18 Solution statistic of TS-OPT for model (PCP) for very large instances
the TTPlib The associated graphs and static models are too big andcannot be solved on machines with 16GB main memory Using defaultsettings of rapid branching in TS-OPT and a limit maximum backtracksof 100 leads to the shown results This demonstrates that rapid branch-ing is a powerful heuristic to solve large scale track allocation problemsand is able to produce high quality solution with a small optimalitygap
3 Auction Results 170
24 Conclusion
We showed that the bundle method and the rapid branching heuristicis a competitive approach to tackle large scale (PCP) formulationsthat are originating from railway track allocation problems Further-more this illustrates that this solution approach has potential to befurther generalized for solving large scale mixed integer programs Inparticular if the model formulation allows for a strong Lagrangean re-laxation the bundle method has a lot of advantages in comparison tostandard LP solvers eg running time and total memory consump-tion Moreover our novel approach produced much faster high qualityprimal solutions and global upper bounds for several unsolved largescale track allocation instances of the TTPlib
3 Auction Results
We consider in this section the results of a theoretical design of anauction-based allocation mechanism for railway slots in order to estab-lish a fair and non-discriminatory access to a railway network In thissetting railway undertakings (RU) compete for the use of a shared rail-way infrastructure by placing bids for trains that they intend to runThe main motivation and argumentation of that idea can be found inBorndorfer et al (2006) [34] The trains consume infrastructure ca-pacity such as track segments between and inside stations over certaintime intervals and they can exclude each other due to safety and otheroperational constraints even if they would not meet physically as wealready define in detail in Chapter II The auctioneer ie an infras-tructure manager chooses from the bids a feasible subset namely atimetable that maximizes the auction proceeds Such a mechanism isdesirable from an economic point of view because it can be argued thatit leads to the most efficient use of a limited resource However it isclear that this vision can only become reality if the railway industry ac-cepts sophisticated and modern technologies to support their planningand operational challenges Figure 9 shows a general auction mecha-nism that has to be stated more precisely eg definition of roundsactivity rules definition and rules on bids and many more Startingpoint is always the submission of initial bids by the participants In thenext step the winner determination problem is solved until the prede-
3 Auction Results 171
Participants (RU) Auctioneer (IM)
Submit initial bids
Solve winnerdetermination problem
Modify bids
Publish finalallocation and prices
Figure 9 Auction procedure in general
fined conditions on termination are fulfilled eg the maximal numberof rounds is reached or there was no activity of the participants
In the final stage the winner ie the allocation of goods to biddersand the corresponding prices are determined and published A cen-tral question in mechanism design is whether there exists mechanismsensuring efficient allocation ie auctions that ensure that resourceswind up in the hands of those who value them most In other wordsan auction game is efficient if in equilibrium the winner are the buyerswith the highest valuation The precise concept of equilibrium with re-spect to well-defined terminology of bids and valuations can be foundin Milgrom (2004) [162]
In other industries well defined and implementable auction variantsare an established mechanism to allocate scarce goods eg energymarket telecommunication frequencies airport slots and ticketing ofmajor events However the technical complexity and size of the rail-way resources act as a barrier to establish an auction based capacityallocation procedure The winner determination problem of a railwayauction is then to solve the track allocation problem discussed in Chap-ter III Obviously this procedure has to be defined and controlled byan independent agency ie the Federal Network Agency in case ofGermany
In the following sections we will define and discuss different auctiondesigns Some more from theoretical others from a computational andpractically implementable point of view
3 Auction Results 172
31 The Vickrey Track Auction
Vickrey (1961) [209] argued in his seminal paper for the importance ofincentive compatibility in auction design and he showed that a secondprice auction has this property as well as efficiency In a second priceauction the bidder who submitted the highest bid is awarded the objectbeing sold and pays a price equal to the second highest amount bid
William Vickrey was awarded the Nobel Memorial Prize in Economicstogether with James Mirrlees for their research on the economic theoryof incentives under asymmetric information He and independentlyClarke (1971) [68] and Groves (1973) [106] also proposed a sealed-bid auction that generalizes the simple Vickrey auction for a singleitem to the multi-item case the so-called Vickrey-Clarke-Groves (VCG)mechanism which is also incentive compatible Incentive compatibilityis a concept originally proposed by Hurwicz (1972) [118] to describeany set of rules or procedures for which individuals find it in their ownbest interest to behave non-strategically in particular truthfully Thisis important in a variety of contexts such as creating the mechanismfor electing representatives or for deciding who receives benefits withina welfare state Moreover the field of mechanism design is a rather newand fruitful mathematical research area
This classical result pertains to a combinatorial auction in which bidsare placed for bundles of items and two bundles can be allocated ifand only if they do not contain the same item This is however notsufficient for a railway track auction in which more general constraintson the compatibility of slots arise eg from minimum headway con-straints Whatever these constraints may be a second price auctioncan of course also be conducted in such a setting However it is apriori not clear if such an auction is incentive compatible
In Borndorfer Mura amp Schlechte (2009) [40] we formally defined sucha Vickrey Track Auction (VTA) and showed that this is indeed thecase by straight-forward modification of the original proof The proofof Mura (2006) [164] does not depend on the concrete structure ofthe TTP ie it generalizes to combinatorial Vickrey auctions witharbitrary combinatorial winner determination problems For exampleit follows that a VTA with additional constraints on the number of slotsthat can be allocated to a bidder is also incentive compatible becausethis rule can be dealt with by adding constraints to the specific winnerdetermination problem
3 Auction Results 173
Even if the VTA is only a one-shot auction ie only one round isperformed the definition of the prices causes the solution of severalwinner determination problem ie all winner determination problemswith each of the winners excluded Erdogan (2009) [83] focuses onthe computational tractability of this algorithmic mechanism designby extending a branch and bound approach to a branch and remem-ber algorithm that exploit several information of the original winnerdetermination problem ie usage of still valid cuts and solutions aswarmstart information for the MIP solving For artificial auction sce-narios based on the instances of the TTPlib he reported an accelerationratio of two for the Vickrey payment computations ie as well as forthe measured geometric mean of the total number of branch and boundnodes and simplex iterations needed
Indeed this shows that the VTA has theoretically all desired propertiesand even the computation of the payments may be reasonably practi-cable with great efforts Nevertheless it is really challenging to estab-lish such an auction design in reality due to the complex and hardlytransparent price determination process in particular for combinato-rial auctions with a lot of participants Furthermore it is known thatthe ldquogeneralizedrdquo Vickrey auction suffers from several severe practicaldrawbacks see Ausubel amp Milgrom (2005) [14]
It does not allow for price discovery that is discovery of themarket price if the buyers are unsure of their own valuations
It is vulnerable to collusion by losing bidders
It is vulnerable to shill bidding with respect to the buyers
It does not necessarily maximize seller revenues seller revenuesmay even be zero in VCG auctions
The sellerrsquos revenues are non-monotonic with regard to the setsof bidders and offers
In these auctions several criteria besides incentive compatibility meritthe attention of a practical mechanism designer Revenues are an obvi-ous one Auctions are commonly run by an expert auctioneer on behalfof the actual seller and any failure to select a core allocation with re-spect to reported values implies that there is a group of bidders whohave offered to pay more in total than the winning bidders yet whoseoffer has been rejected Imagine trying to explain such an outcome tothe actual seller or in a government sponsored auction to a skepti-cal public Monotonicity of revenues with respect to participation isanother important property of auction mechanisms because its failure
3 Auction Results 174
could allow a seller to increase sales revenues by disqualifying biddersafter the bids are received Another important desideratum is that abidder should not profit by entering and playing as multiple biddersrather than as a single one
32 A Linear Proxy Auction
Designing an auction for the usage of railway infrastructure resourcesis nothing novel Brewer amp Plott (1996) [45] suggest a model wherefeasibility of a train schedule is based on the binary exclusion prop-erty which says that a schedule of trains is feasible if any two trainsare conflict-free Parkes amp Ungar (2001) [175] present an auction-basedtrack allocation mechanism for the case that single-track double-trackand yard segments have to be concatenated to form a single line Theysuggest a hybrid mechanism that combines elements of the simultane-ous and the combinatorial auction formats However these approachesare mainly driven by economic questions and assume almost trivialrailway track allocation models and artificial data sets
In that section we will present results of a more practically imple-mentable iterative auction design with linear prices ie the LinearizedProxy Auction (LPA) We will briefly discuss the main focus of thatwork The precise auction design can be found in Schlechte amp Tanner(2010) [189] It generalizes the Ausubel Milgrom Proxy Auction pre-sented by Ausubel amp Milgrom (2002) [15] Indeed no efficiency can beensured but at least the resulting allocation lies in the core An indi-vidually rational outcome is in the core of an auction game if and onlyif there is no group of bidders who would strictly prefer an alternativedeal that is also strictly better for seller Consequently an auctionmechanism that delivers core allocations has the advantage that thereis no individual or group that would want either to renege after theauction is run in favor of some allocation that is feasible for it and theany non-core agreement made before the auction risks being unwoundafterwards
Our generalized variant (LPA) leads to the possibility of prices ly-ing above the bidder-optimal core frontier in contrast to the gen-eral Ausubel Milgrom Proxy auction Some examples are discussedin Schlechte amp Tanner (2010) [189] However main advantage of thedesign is the use of dual prices ie the dual solution of the LP re-laxation of model (ACP) to enforce activity in the iterative auction
3 Auction Results 175
to decrease the number of auction rounds without loosing too muchefficiency
Table 19 lists the results of an auction simulation for real world de-mand data of the railway network hakafu simple The statistic basisof that data and the explicit auction rules eg minimum incrementstarting time of a bid etc can also be found in Schlechte amp Tanner(2010) [189] Furthermore we scaled the profit values of the bidderswith a constant scaling factor α to analyse the sensitivity of our auc-tioning approach
profit auction rounds
α trivial dual efficiency trivial dual speedup
08 2983 2932 0983 1765 1361 2510 3658 3597 0984 1943 1411 2715 4941 4843 0980 2006 154 2320 6144 5967 0971 2153 172 2025 7272 7065 0972 2177 1823 1640 9720 9374 0964 2296 1984 1460 12233 11879 0971 2312 1959 15
Table 19 Incremental auction with and without dual prices profit and numberof rounds until termination
Table 19 compares two versions of the LPA auction The first versionof the LPA denoted as trivial does not know any minimum price rulefor newly introduced slots so bidders start bidding for slots from pricezero The second version of LPA uses the dual-based minimum pricerule and is therefore labeled with dual We compare the results inefficiency and convergence rate The second and the third column ofTable 19 show the outcome for both LPA versions one can see thatthe minimum price rule does not essentially affect efficiency in the nextcolumn However the last columns demonstrate that the number ofrounds is significantly lower with the dual minimum price rule Weobserve that using dual prices as minimum prices may speedup theauction while the efficiency loss is moderate for our test cases
33 Conclusion
We presented and discussed several aspects of different theoretical auc-tioning procedures for the use of railway infrastructure resources We
4 The Simplon Corridor 176
want to point out explicitly that because of the character of the ex-periments and several assumptions on the auction setting most of ourcontributions are theoretic ones Our experience from discussions withseveral European railway infrastructure managers is that ldquorealrdquo auc-tioning is a visionary idea that is hardly imaginable and implementablein the near future However the iterative resolution of resource con-flicts in the coordination phase see again Figure 8 can obviously beexchanged by more efficient procedures using an automatic track allo-cation tool embedded in an appropriate auction design Still a lot ofdecision makers have to be convinced until the railway industry willagree on such an procedure An adequate auction design with specifiedrules for ldquorailway capacityrdquo as for instance in the telecommunicationmarket for frequencies see Brunner et al (2007) [47] and Ausubel ampMilgrom (2002) [15] has to be defined and supported by the majorityof railway actors
4 The Simplon Corridor
In this section we present the results of the developed models and algo-rithms of Chapter II and III for a real world application ie the Sim-plon corridor in Switzerland The scenarios are extensively describedfrom Section 41 to Section 43 Finally Section 45 provides a capac-ity analysis of the Simplon tunnel using our optimization frameworkfor railway track allocation
41 Railway Network
There are only two north-south railway connections through the Alpsin Switzerland namely the Gotthard corridor and the Lotschberg-Simplon corridor The Simplon connects Switzerland and Italy andis therefore of strategic importance for the international railway freighttraffic It has a length of approximately 45 km and 12 stations Thismay sound like a rather small network at first glance but the rout-ing possibilities at the terminals Brig and Domodossola the routingpossibilities in the intermediate stations Iselle and Varzo and a ratherunusual slalom routing for certain types of cargo trains lead to verycomplex planning situations An OpenTrack network data export forthe part from Brig (BR) in Switzerland to Domodossola (DO) in Italy
4 The Simplon Corridor 177
Figure 10 Micro graph representation of Simplon and detailed representation ofstation Iselle given by OpenTrack
was provided by the SBB Schweizerische Bundesbahnen The micro-scopic network consists of 1154 nodes and 1831 arcs including 223 sig-nals see Figure 10 Even if this network consists of only 12 stationsand has a length of approximately 45 km it is an important corridorin the European railway network According to geographical condi-tions there are only two north-south railway corridors in Switzerlandthe Gotthard corridor and the Lotschberg-Simplon corridor This is inconflict with the fact that Switzerland is an very important country forthe traffic transit between central und southern Europe To that effectthere is a huge and increasing demand on slots through this corridorThe Simplon tunnel is in fact a bottleneck in the European railwaynetwork
This data was macrotized in two steps The first step is resort tostandardized train driving dynamics that lead to the definition of ahandful of train types these are used to compute standardized drivingand headway times This allows to amalgamate larger parts of themicroscopic infrastructure network to a macroscopic network in thesecond step The following subsections describe this process for theSimplon application
4 The Simplon Corridor 178
42 Train Types
The decision which train types to consider is a crucial point becausea more detailed consideration of driving dynamics allows the construc-tion of tighter schedules For a capacity analysis however a modellingstrategy is appropriate that captures the main characteristics but ab-stracts from minor special characteristics of individual trains We usesix different types two for passenger trains and four for freight trains
The different but invariable stopping patterns of regional trains (R)and intercity trains (EC) and their very different driving dynamics(due to the different engines used) result in considerable differencesin running and headway times for such trains They are thereforeconsidered as two train types We do however ignore different traincompositions ie in length and in the number of wagons Hence Rand EC are the two types of passenger trains that we consider
Freight trains come in four different types GV Auto are specialtrain services that transport passengers and their automobiles fromBrig (BR) to the next station after the Simplon tunnel which is Iselle(IS) There these trains cross all other tracks to reach an isolated rampBecause of these unique routing requirements at Iselle we considerthem as belonging to an individual freight train type on their own
GV RoLa and GV SIM are train types that transport freight vehicles(GV RoLa) and containers (GV SIM) They have a larger height andwidth than standard freight trains and they can use only one of thetracks in the tunnel between Iselle and Preglia This results in a so-called ldquoslalom routerdquo that these trains have to take from Brig In Isellethey have to change to the right track2 until Preglia ie it is possibleto change again to the standard side in the intermediate station Varzoto let other trains pass Furthermore the running times of these trainstypes especially for the direction from Brig to Domodossola differsignificantly namely a GV RoLa needs about 7 minutes more than aGV SIM They also use different routes in the area of DomodossolaThus separate train types GV RoLa and GV SIM are introducedFinally GV MTO are standard freight trains which use the standardtracks in the Iselle-Preglia tunnel
SBB was interested in running additional freight trains through theSimplon such that we concentrated on freight traffic We assume in
2In Switzerland trains are usually running on the left side
4 The Simplon Corridor 179
0 4 8 12 16 20 240
4
8
12
16
20
time slot
tr
ains
EC R GV Auto
Figure 11 Given distribution of passenger or fixed traffic in the Simplon corridorfor both directions
particular that the passenger trains are given and cannot be changedHence the slots for passenger trains R and EC from Brig to Domo-dossola and vice versa are fixed In addition the GV Auto trainswhich are not operated all day are also fixed All these trains musthowever be considered with respect to their influence on the remainingtraffic ie with respect to their headways and with respect to stationcapacities Figure 11 shows the passenger train distribution across theday
43 Network Aggregation
The train types introduced in Section 42 can run on 28 different routesthrough G = (VE) The routes differ in their stopping pattern and invarious ways to pass through Varzo These routes are the basis of theaggregation of the microscopic network They partition the networkinto segments on which driving and headway times can be computedindividually In other words if a train route runs on a track segmentand no other routes cross one can compute the parameters that arerelevant for a slot allocation on this segment beforehand and compressthe segment
Clearly the routes meet at the stations such that the macroscopic net-work must necessarily contain a node for each of the twelve stationsSome more macroscopic pseudo nodes are needed to model all trainroute interactions correctly ie divergences convergences and cross-
4 The Simplon Corridor 180
ings Applying the netcast Micro-Macro Transformation algorithmdescribed in Chapter II and in Schlechte et al (2011) [190] producesa macroscopic network N = (S J) with 55 nodes and 87 tracks 32 ofthese nodes are pseudo stations Most of them are located directly inthe front area of stations The other 23 nodes are possible start endor waiting nodes along the corridor
This automatically constructed network was further aggregated in asecond step by applying some reductions that are not yet genericlyimplemented in netcast We kept only those pseudo stations thathandle crossing conflicts namely for GV Auto on the route fromBrig to Iselle and those for a detailed modeling of the station VarzoThe reason for this detailed treatment of Varzo is that the routingthrough this station is crucial for the capacity of the whole corridorIn Varzo the over-width freight trains can pass each other such that alocking of the entire area between Iselle and Preglia can be avoided forGV SIM and GV RoLa trains from the other direction when one ofthem runs through the tunnel All other potential pseudo nodes wereaggregated to the closest station node in a conservative manner iethe headway times for the incident tracks had to be slightly overesti-mated In addition some nodes that represent different platforms atthe same station were aggregated After these modifications the net-work consists of 18 stations and 40 tracks For comparison we alsoconsider a ldquotraditionalrdquo macroscopic network that is solely based onstation nodes clearly a conservative model based on such an aggrega-tion will employ oversized buffers and therefore waste capacity Let uslist the macroscopic networks that we constructed by netcast on thebasis of microscopic OpenTrack data
network with station area aggregation (18 stations and 40 tracks)simplon big
network with full station aggregation (12 stations and 28 tracks)simplon small
After some experiments with these networks the expertise of SBBabout the operational conditions in the Simplon corridor led to theintroduction an additional technical blocking time for combinations ofGV RoLa trains with any other trains in the front area of Domod-ossola The headway times of cargo trains were set to a fixed valueof some minutes instead of the simulation values in order to guaranteecertain departure and arrival distances in the marshaling yard of BrigWe further improved the macroscopic model by adding buffer times for
4 The Simplon Corridor 181
type direction freight trains
name train requests passenger freight BR-DO DO-BR GV RoLa GV SIM GV MTO
4h-tp-as-d 41 15 26 23 18 4 9 134h-tp-as-n 36 8 28 20 16 7 10 114h-tp-s-d 42 15 27 23 19 4 8 154h-f20-s 38 14 24 22 16 6 12 64h-f15-s 46 14 32 26 20 8 16 84h-f12-s 54 14 40 30 24 10 20 104h-f10-s 62 14 48 34 28 12 24 124h-f75-s 78 14 64 42 36 16 32 1624h-tp-as 390 63 327 203 187 69 108 15024h-tp-s-n 219 63 156 110 109 48 54 5424h-tp-s 297 63 234 149 148 60 78 9624h-f24-s 183 63 120 92 91 30 60 3024h-f20-s 207 63 144 104 103 36 72 3624h-f15-s 255 63 192 128 127 48 96 4824h-f12-s 303 63 240 152 151 60 120 6024h-f10-s 351 63 288 176 175 72 144 72
Table 20 Statistics of demand scenarios for the Simplon case study
standard headways and headways for the opposite direction In thisway two more macroscopic networks were generated with netcast
with station area aggregation (18 stations and 40 tracks) andtechnical times simplon tech
with station area aggregation (18 stations and 40 tracks) andtechnical and buffer times simplon buf
44 Demand
In order to evaluate and analyze the Micro-Macro Transformation in-troduced in Chapter II and the optimization models discussed in Chap-ter III we considered various train request scenarios The capacity ofthe Simplon corridor is estimated by saturating it with freight trainsthat are selected from fictional request sets To this purpose we haveconstructed 16 train request sets listed in Table 20 The first eightrequest sets cover a four hour time horizon (prefix ldquo4hrdquo in the requestset name) either from 8am to 12am (suffix ldquodrdquo for day) or from 0am to4am (suffix ldquonrdquo for night) The other request sets are used to calculatea timetable for an entire day (24h)3
Three of the 4h request sets are called ldquotestplanrdquo (tp) which meansthat they are used to evaluate the correctness of the Micro-MacroTransformation on the basis of a microscopically feasible timetable thathas been generated manually by the authors The same applies to thethree ldquotestplanrdquo request sets that cover the whole day Some of the test
3The ldquonrdquo in the second 24h request is a reminder that freight trains drive morefrequently at night
4 The Simplon Corridor 182
Brig-Domodossola Domodossola-Brig
∆ (sec) running headway running headway
1 1778 272 1794 2516 297 46 299 42
12 158 23 149 2130 60 10 60 960 30 5 30 5
300 6 1 6 1
Table 21 Running and headway times for EC with respect to ∆
request sets eg 24h-tp-as have the disadvantage that the requestsare not symmetrically distributed with respect to both directions Wetherefore distinguish between asymmetric (as) and symmetric (s) re-quest sets which do not have this drawback
We also remark that almost all ldquotprdquo request sets do not match thetrain type distribution that is desired by SBB Namely traffic demandin practice takes the form that every second request is a GV SIMwhile the others are GV RoLa and GV MTO in equal parts To ap-proximate this characteristic we generated some more requests usinga uniform distribution according to the desired train demand patternThe resulting request sets are named with the infix ldquofxrdquo where x de-notes the period time of the freight trains We remark that we are awareof the fact that in practice traffic demand is not uniformly distributedhowever for want of better data and for the purpose of demonstrat-ing the principal viability of our model in an analysis of a theoreticalcapacity of the corridor we deem this data good enough
Observation 41 We will briefly discuss the impact of discretizationon the real world data of the Simplon The best usage from a simplecapacity point of view without considering realistic traffic assumptionsis trivially to use only the fastest train as much as possible For thegiven Simplon corridor this is an EC train with times for both directionslisted in Table 21 We denote by d the rounded running time withrespect to ∆ and by h the technical minimal rounded headway timerespectively
Even this trivial consideration of the corridor as a network of only twostations and two tracks documents the sensitivity of the macroscopicmodel with respect to the chosen discretization ∆ Assuming a coarseunit of 5 minutes it is only possible to run 12
(= 3600
300
)trains in each
direction per hour Only when ∆ is smaller than 12 or 6 seconds a
4 The Simplon Corridor 183
maximum capacity of 13 or 14(= 3600
42middot6
)trains per direction and per
hour is theoretically available
45 Capacity Analysis based on Optimization
We provide in this section a capacity analysis of the Simplon corridorusing our micro-macro aggregation approach The goal of this study isto saturate the residual capacity of the corridor by running a maximumnumber of fictitious freight trains (GV MTO GV SIM GV RoLa)between the passenger trains (remember the passenger trains are givenas fixed)
We remark that there are a lot of side-constraints for such additionaltrains that we do not consider Requirements such as desired arrival ordeparture time windows at certain stations dwell time requirementsthe balance of train traffic in opposite directions and other constraintsare ignored partly because of lack of data partly because there is nopoint for such constraints in an analysis of a theoretical capacity max-imum These considerations are also the reason for using the followingsimple objective function
a basis value for each scheduled train depending on type anddirection
a penalty for deviations from optimal arrival and departure times
and very small penalties for travel time increases or avoidablestops
We constructed the macroscopic scenarios associated with all requestsets and with all four macroscopic networks namely simplon smallsimplon big simplon tech and simplon buf Furthermore wevaried the time discretization of the model using step sizes of 6 1030 and 60 seconds The resulting macroscopic track allocation prob-lems were solved using the integer programming based track allocationoptimizer TS-OPT presented in Chapter III the solutions were disag-gregated using netcast and verified by OpenTrack For each run ofTS-OPT a time limit of one day (86400 seconds) was used
Table 22 lists exemplary solution statistics for all request scenarios andnetwork simplon big using a discretization of 10s The tables gives
number of trains (trains)
number of columns of the integer program (cols)
4 The Simplon Corridor 184
instance trains cols rows v(LP) ublowast vlowast gap tLP tIP
4h-tp-as-d 35 70476 30432 14935 14727 14727 ndash 000 18684h-tp-as-n 27 35859 17136 15121 14639 14639 ndash 003 14604h-tp-s 36 106201 45873 9077 7057 7057 ndash 2328 2054044h-f20-s 30 173929 69531 15252 14597 14597 ndash 5423 2397834h-f15-s 34 110920 46870 15176 13690 13690 ndash 1882 1440074h-f12-s 36 211745 84107 18957 18636 18636 ndash 10778 12508984h-f10-s 37 235430 93501 20609 20033 20033 ndash 15358 12124924h-f75-s 37 135746 56968 7926 7215 7215 ndash 3797 118561124h-tp-as 203 462769 196238 103594 98477 98477 ndash 10273 635887724h-tp-s-n 154 284038 117208 79462 76063 76063 ndash 4045 16094224h-tp-s 174 403017 167548 88897 84330 84330 ndash 7602 273918724h-f24-s 143 444199 178162 72229 69712 69712 ndash 9260 44547624h-f20-s 156 471759 195167 79131 75249 75249 ndash 9370 37792524h-f15-s 174 660642 250673 91922 88543 86184 274 23506 864004024h-f12-s 179 662236 259676 98546 95876 95876 ndash 21354 794973724h-f10-s 193 791285 312943 109047 106970 104108 275 42675 8640071
Table 22 IP-Solution analysis of network simplon big with time discretizationof 10s and a time limit of 24h
number of rows of the integer program (rows)
optimal value of the linear relaxation (v(LP))
(best) proven upper bound (ublowast)
(best) objective function value of integral solution (vlowast)
optimality gap in percent
time needed to solve the linear relaxation (tLP )
and the total running time of TS-OPT
A first important result is that TS-OPT is indeed able to compute afeasible ie conflict free slot allocation for all instances within one dayFigure 12 shows an example of a resulting train diagram with a validblock occupation for request set 24h-tp-as network simplon buf anda discretization of 30s The tractability of these instances is to do thenetwork aggregation algorithm of netcast presented in Chapter IIwhich produces reasonably sized macroscopic networks that give rise toreasonably sized track allocation problems There is no instance whereTS-OPT needs more than 600 MB of main memory and TS-OPT cantherefore compute feasible solutions for almost all problems This giveevidence that our micro-macro aggregation approach and our extendedformulation works very well
Not every instance could be solved to proven optimality for each net-work and time setting But the 4h-requests never took more than threeand a half hours to be solved to optimality and even for the really com-plex uniformly distributed daily scenarios feasible solutions with smalloptimality gaps could be computed Moreover the instance with themaximum number of train requests (24h-tp-as with 390 train requests)could be solved to optimality for each network and all time discretiza-tions of 30 seconds and more Table 22 shows that such an instanceproduces a timetable with 203 trains which means that 140 freight
4 The Simplon Corridor 185
Figure 12 Traffic diagram in OpenTrack with block occupation for request 24h-tp-as
train slots out of the requested potential 327 train slots are routed inthe optimal schedule This establishes a theoretical capacity of theSimplon corridor of more than 200 trains per day Adding technicaland buffer times in network simplon buf it is still possible to sched-ule 170 trains This number is almost identical to the saturation inthe timetable that is currently in operation and can be taken as anindication of both the accuracy of the model as well as the quality ofthe current timetable We can also observe that not every request setproduces a saturated timetable that runs between 160 and 200 trainsper day This highlights the fact that the demand ie the number ofrequested trains of different types and the degrees of freedom in routingthem have a crucial effect on the capacity of a corridor
We also analyzed the effects of different time discretizations Table 23and 24 give an overview on the sizes of the resulting track allocationproblems for two test instances We distinguish two different discretiza-tion parameters namely we denote by dep steps the step size for traindeparture events and by wait steps the step size for train dwell activi-
4 The Simplon Corridor 186
24h-tp-as
discretization (sec) 6 10 30 30 60
dep steps (sec) 30 50 150 30 60wait steps (sec) 60 100 300 60 60cols 504314 318303 114934 370150 178974rows 222096 142723 53311 170525 81961t(lp) (sec) 13567 4888 1777 5413 15167t(ip) (sec) 7277455 1240919 11034 8168302 241120size of IP (MB) 50 30 10 36 18trains 196 187 166 188 180
Table 23 Solution data of instance 24h-tp-as with respect to the chosen timediscretization for simplon small
24h-f15-s
discretization (sec) 6 10 30 30 60
dep steps (sec) 30 50 150 30 60wait steps (sec) 60 100 300 60 60cols 649494 375694 115293 392146 172462rows 234529 146044 49458 163388 74200t(lp) (sec) 19036 6459 283 4744 10350t(ip) (sec) 292376 263962 3483 826571 104348size of IP (MB) 64 36 10 38 16trains 176 163 143 155 145
Table 24 Solution data of instance 24h-f15-s with respect to the chosen timediscretization for simplon small
ties respectively As expected problem sizes normally4 decrease withcoarser time discretizations and the same holds for the running timesAnyway TS-OPT can solve even instances with more than 500000 vari-ables
An exception to the rule ndash coarser time discretization implies a decreasein problem size ndash can be observed by comparing the 30s and the 60sinstance This irregularity originates from a different parameter settingwith respect to possible departure and waiting times see Table 23 Inthe first 30s discretization scenario a train can only depart at times thatare multiples of 150 seconds see definition of dep steps and the waitingtimes must be a multiple of five minutes see definition of wait steps
4There is no general relation between problem size and solution time as one cansee by a comparison of the 6s-discretization runs
4 The Simplon Corridor 187
simplon small
simplon big
simplon tech
simplon buf
0 100 180trains
Figure 13 Comparison of scheduled trains for different networks (simplon ) forinstance 24h-tp-as in a 60s discretisation
24h-tp-as 24h-f15-s
scheduled requested scheduled requested
GV RoLa 30 69 21 48GV SIM 41 108 51 96GV MTO 69 150 39 48
all freight trains 140 327 111 192
Table 25 Distribution of freight trains for the requests 24h-tp-as and 24h-f15-sby using network simplon big and a rounding to 10 seconds
That is a rather rough model with a limited degree of freedom Wetherefore changed the parameters for the 60s runs such that the timesteps are narrower and more similar to the 6s case We also did 30sruns with departure and waiting times similar to the 6s cases suchthat the influence of those two parameters could be analyzed It turnsout that there is not only a connection between time discretizationand the number of scheduled trains but there is also an often evenstronger connection between departure and waiting time steps and thenumber of scheduled trains We therefore also must pay attention tothese parameters We finally remark that the combinatorial complexityandor the computational tractability of a particular track allocationinstance can not be reliably predicted or estimated by looking at simplescenario statistics
Another important point is the influence of network aggregation on thenumber of scheduled trains As shown in Figure 13 a more detailednetwork model leads to a major increase in the number of scheduledtrains But by introducing specific headway times we again loose about8 of the trains and an additional 6 by also considering buffer times
4 The Simplon Corridor 188
Up to now we only considered the total number of scheduled trains asa measure for the corridor capacity But it is also important to keepthe structure of the computed timetable in mind Figure 14 shows thetrain type distribution of the three freight train types for two requestsThis little example is representative for the general observation that thetrain type distribution associated with uniformly distributed requests ismuch closer to the desired distribution see Figure 14 than that of therequests based on a testplan timetable The latter timetables feature ahigher fraction of GV MTO requests than desired in fact these trainsdo not run on a slalom route in the corridor and are therefore easier toschedule The higher percentage of GV SIM and GV RoLa trains inthe uniformly distributed request sets often leads to bigger problemsthan that resulting from the testplan request sets see Table 23 andTable 24
Another observation is that the majority of timetables schedules moretrains from Domodossola to Brig than vice versa This is not surprisingas the models due not contain any symmetry constraints We didhowever try to achieve some balance by manipulating the objectivefunction Introduce such global constraints could be an interestingaspect of future work
desired distribution
25
GV MTO
25
GV RoLa
50
GV SIM
24h-tp-as 24h-f15-s
4929
GV MTO
2143
GV RoLa
2928
GV SIM
3513
GV MTO
1892
GV RoLa
4595
GV SIM
Figure 14 Distribution of freight trains for the requests 24h-tp-as and 24h-f15-sby using network simplon big and a rounding to 10 seconds
4 The Simplon Corridor 189
46 Conclusion
To the best knowledge of the author and confirmed by several rail-way practitioners this was the first time that automatically producedtrack allocations (on a macroscopic scale) fulfill the requirements ofthe original microscopic model Furthermore we strongly believe thatour models and algorithmic solution approaches are already able tosupport the mid-term and long-term planning of track allocations iethe creation of the annual time table Finally we want to completethe thesis with an excerpt from the project conclusions of our industrypartners from SBB
rdquoThe produced timetables from this project are qualita-tively better than all previous results of other projects Forthe first time it was possible to simulate an algorithmic gen-erated timetable in the simulation tool OpenTrack withoutconflicts We would expect a benefit (by introducingsuch a tool) on a strategic middle-term and long-term levelBecause we estimate that we could decrease the planningtime needed for freight train allocation from 2-3 weeks toonly one week In addition much more scenario variationscould be considered and results could be produced muchfasterldquo (translation by the author)
References 190
References
[1] E J W Abbink M Fischetti L G Kroon G Timmer ampM J C M Vromans Reinventing crew scheduling at netherlandsrailways Interfaces 35(5)393ndash401 2005 Cited on page 15
[2] M Abril M A Salido amp F Barber Distributed search in rail-way scheduling problems Eng Appl Artif Intell 21(5)744ndash755 2008ISSN 0952-1976 Cited on page 36
[3] T Achterberg Constraint Integer Programming PhD thesis Tech-nische Universitat Berlin 2007 Cited on pages 9 51 132
[4] T Achterberg T Koch amp A Martin MIPLIB 2003 Opera-tions Research Letters 34(4)1ndash12 2006 URL httpwwwzibde
PublicationsabstractsZR-05-28 ZIB-Report 05-28 Cited onpage 64
[5] R K Ahuja T L Magnanti amp J B Orlin Network FlowsTheory Algorithms and Applications Prentice-Hall Inc EnglewoodCliffs New Jersey 1993 Cited on page 107
[6] R K Ahuja K C Jha amp J Liu Solving real-life rail-road blocking problems INTERFACES 37(5)404ndash419 2007URL httpinterfacesjournalinformsorgcgicontent
abstract375404 Cited on pages 18 19
[7] R K Ahuja R H Mohring amp C D Zaroliagis (Eds) Ro-bust and Online Large-Scale Optimization Models and Techniques forTransportation Systems vol 5868 of Lecture Notes in Computer Sci-ence Springer 2009 ISBN 978-3-642-05464-8 Cited on pages 199202 205
[8] S Albers Online algorithms a survey Math Program 97(1-2)3ndash26 2003 Cited on page 10
[9] S Albers amp S Leonardi On-line algorithms ACM ComputSurv 31(3es)4 1999 Cited on page 10
[10] T Albrecht Railway Timetable and Traffic chap Energy-EfficientTrain Operation pp 83ndash105 Eurailpress DVV Media 2008 Citedon page 93
[11] Altenbeken Altenbeken 2009 URL httpwwwdbdesitebahn
degeschaefteinfrastruktur__schienenetznetzzugang
dokumenteBahnhofSNBHHA__NBSpdf In German available athttpwwwdbdesitebahndegeschaefteinfrastruktur_
_schienenetznetzzugangdokumenteBahnhofSNBHHA__NBS
pdf Cited on pages xxv 58
[12] L Anderegg S Eidenbenz M Gantenbein C Stamm D STaylor B Weber amp P Widmayer Train routing algorithms
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Concepts design choises and practical considerations In R E Lad-ner (Ed) ALENEX pp 106ndash118 SIAM 2003 ISBN 0-89871-542-3Cited on page 14
[13] D Applegate R Bixby V Chvatal amp W Cook Findingcuts in the TSP (a preliminary report) Technical report Center forDiscrete Mathematics and Theoretical Computer Science (DIMACS)March 1995 DIMACS Technical Report 95-05 Cited on page 145
[14] L M Ausubel amp P Milgrom Ascending proxy auc-tions Levinersquos bibliography UCLA Department of Economics2005 URL httpeconpapersrepecorgRePEcclalevrem
122247000000000785 Cited on page 173
[15] L M Ausubel amp P R Milgrom Ascending auctions with packagebidding Frontiers of Theoretical Economics 1(1)1ndash42 2002 Cited onpages 174 176
[16] E Balas Projection lifting and extended formulation in integer andcombinatorial optimization Annals OR 140(1)125ndash161 2005 Citedon page 123
[17] C Barnhart amp G Laporte Handbooks in Operations Research ampManagement Science Transportation North-Holland 2007 Cited onpages 9 10
[18] C Barnhart E L Johnson G L Nemhauser M W PSavelsbergh amp P H Vance Branch-and-price Column gener-ation for solving huge integer programs Oper Res 46(3)316ndash3291998 ISSN 0030-364X Cited on pages 48 133
[19] C Barnhart H Jin amp P H Vance Railroad blocking A networkdesign application Oper Res 48(4)603ndash614 2000 ISSN 0030-364XCited on pages 18 19
[20] C Barnhart P Belobaba amp A R Odoni Applications of opera-tions research in the air transport industry Transportation Science 37(4)368ndash391 2003 Cited on page 44
[21] N Barnier P Brisset T Rivire amp T R Ere Slot allocationwith constraint programming Models and results In In Proc of theFourth International Air Traffic Management RampD Seminar ATM2001 Cited on page 10
[22] J E Beasley amp N Christofides An algorithm for the resourceconstrained shortest path problem Networks 19379ndash394 1989 Citedon page 50
[23] A Ben-Tal amp A Nemirovski Robust convex optimization Math-ematics of Operations Research 23(4)769ndash805 1998 Cited on page128
[24] L Bengtsson R Galia T Gustafsson C Hjorring ampN Kohl Railway crew pairing optimization In F Geraets
References 192
L Kroon A Schoebel D Wagner amp C Zaroliagiis (Eds)Algorithmic Methods for Railway Optimization LNCS pp 126ndash144Springer-Verlag 2007 Cited on page 15
[25] D Bertsimas amp M Sim Robust discrete optimization and networkflows Mathematical Programming 9849ndash71 2003 Cited on page128
[26] G Birkhoff Lattice theory vol 25 American Mathematical Soci-ety Providence RI 3 edition 1967 Cited on page 68
[27] K S Booth amp G S Lueker Testing for the consecutive onesproperty interval graphs and graph planarity using pq-tree algorithmsJ Comput Syst Sci 13(3)335ndash379 1976 Cited on page 108
[28] R Borndorfer amp C Liebchen When periodic timetables aresuboptimal In OR pp 449ndash454 2007 Cited on page 35
[29] R Borndorfer amp M Neumann Models for line planning withtransfers ZIB Report 10-11 ZIB Takustr 7 14195 Berlin 2010Cited on page 26
[30] R Borndorfer amp T Schlechte Models for railway track alloca-tion In C Liebchen R K Ahuja amp J A Mesa (Eds) ATMOS2007 - 7th Workshop on Algorithmic Approaches for TransportationModeling Optimization and Systems vol 07001 of Dagstuhl SeminarProceedings Internationales Begegnungs- und Forschungszentrum furInformatik (IBFI) Schloss Dagstuhl Germany 2007 Cited on pagesii 90 115 153
[31] R Borndorfer amp T Schlechte Solving railway track allocationproblems In J Kalcsics amp S Nickel (Eds) OR pp 117ndash122Springer 2007 ISBN 978-3-540-77902-5 Cited on pages ii 90
[32] R Borndorfer M Grotschel amp A Lobel Duty schedulingin public transit In W Jager amp H-J Krebs (Eds) MATHE-MATICS ndash Key Technology for the Future pp 653ndash674 Springer Ver-lag Berlin 2003 URL httpwwwzibdePaperWebabstracts
ZR-01-02 ZIB Report 01-02 Cited on page 50
[33] R Borndorfer U Schelten T Schlechte amp S Weider Acolumn generation approach to airline crew scheduling In OR pp343ndash348 2005 Cited on pages ii 46 52
[34] R Borndorfer M Grotschel S Lukac K MituschT Schlechte S Schultz amp A Tanner An auctioning approachto railway slot allocation Competition and Regulation in NetworkIndustries 1(2)163ndash196 2006 URL httpwwwzibdePaperWeb
abstractsZR-05-45 ZIB Report 05-45 Cited on pages ii 90 91126 127 170
[35] R Borndorfer M Grotschel amp M E Pfetsch A column-generation approach to line planning in public transport Transporta-
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tion Science 41(1)123ndash132 2007 ISSN 1526-5447 Cited on pages 1025
[36] R Borndorfer M Grotschel amp U Jaeger Planungsprob-leme im offentlichen Verkehr In M Grotschel K Lucas ampV Mehrmann (Eds) PRODUKTIONSFAKTOR MATHEMATIKndash Wie Mathematik Technik und Wirtschaft bewegt acatech DISKU-TIERT pp 127ndash153 acatech ndash Deutsche Akademie der Technikwis-senschaften und Springer 2008 ISBN 978-3-540-89434-6 URLhttpopuskobvdezibvolltexte20081103 ZIB Report 08-20 Cited on page 10
[37] R Borndorfer A Lobel amp S Weider A bundle method forintegrated multi-depot vehicle and duty scheduling in public transit InM Hickman P Mirchandani amp S Voszlig (Eds) Computer-aidedSystems in Public Transport (CASPT 2004) vol 600 of Lecture Notesin Economics and Mathematical Systems pp 3ndash24 Springer-Verlag2008 Cited on pages x xiv 48 141 143
[38] R Borndorfer B Erol amp T Schlechte Optimizationof macroscopic train schedules via TS-OPT In I HansenE Wendler U Weidmann M Luthi J Rodriguez S Ricciamp L Kroon (Eds) Proceedings of the 3rd International Seminaron Railway Operations Modelling and Analysis - Engineering and Op-timisation Approaches Zurich Switzerland 2009 Cited on pages ii90
[39] R Borndorfer M Grotschel amp U Jaeger Planning problemsin public transit ZIB Report 09-13 ZIB Takustr 7 14195 Berlin2009 URL httpopuskobvdezibvolltexte20091174 Toappear in English translations of acatech book Cited on page 10
[40] R Borndorfer A Mura amp T Schlechte Vickrey auctions forrailway tracks In B Fleischmann K H Borgwardt R Kleinamp A Tuma (Eds) Operations Research Proceedings 2008 pp551ndash556 Springer-Verlag 2009 URL httpopuskobvdezib
volltexte20081122 ZIB Report 08-34 Cited on pages ii 172
[41] R Borndorfer I Dovica I Nowak amp T Schickinger Robusttail assignment Technical Report ZIB Report 10-08 Zuse-InstitutBerlin Takustr 7 14195 Berlin 2010 URL httpopuskobvde
zibvolltexte20101231 Cited on page 6
[42] R Borndorfer B Erol T Graffagnino T Schlechte ampE Swarat Optimizing the simplon railway corridor ZIB Report10-24 ZIB Takustr 7 14195 Berlin 2010 submitted to Annals ofOperations Research 4112010 Cited on pages ii 55 58
[43] R Borndorfer T Schlechte amp S Weider Railway trackallocation by rapid branching In T Erlebach amp M Lubbecke(Eds) Proceedings of the 10th Workshop on Algorithmic Approaches
References 194
for Transportation Modelling Optimization and Systems vol 14 ofOpenAccess Series in Informatics (OASIcs) pp 13ndash23 Dagstuhl Ger-many 2010 Schloss DagstuhlndashLeibniz-Zentrum gr Informatik ISBN978-3-939897-20-0 doi httpdxdoiorg104230OASIcsATMOS201013 URL httpdropsdagstuhldeopusvolltexte2010
2746 Cited on pages ii 90
[44] U Brannlund P Lindberg A Nou amp J-E Nilsson Railwaytimetabling using langangian relaxation Transportation Science 32(4)358ndash369 1998 Cited on pages 36 38 40 103 106
[45] P J Brewer amp C R Plott A binary conflict ascending price(BICAP) mechanism for the decentralized allocation of the right to userailroad tracks International Journal of Industrial Organization 14(6)857ndash886 1996 Cited on page 174
[46] O Brunger amp E Dahlhaus Running time estimation chap 4pp 58ndash82 Railway Timetable amp Traffic Eurailpress 2008 Cited onpage 60
[47] C Brunner J Goersee C Holt amp J Ledyard An ex-perimental test of combinatorial fcc spectrum auctions Technicalreport California Institute of Technology Pasadena 2007 URLhttpwwwhsscaltechedu~jkgfcc_smrpbpdf Cited on page176
[48] D Burkolter T Herrmann amp G Caimi Generating dense rail-way schedules In A Jaszkiewicz M Kaczmarek J Zak ampM Kubiak (Eds) Advanced OR and AI Methods in Transporta-tion pp 290ndash297 Publishing House of Poznan University of Technol-ogy 2005 URL httpeuro2005csputpoznanpleprochtmlCited on page 34
[49] M R Bussieck Optimal lines in public rail transport PhD thesisTU Braunschweig 1997 Cited on pages 10 25
[50] M R Bussieck T Winter amp U T Zimmermann Discreteoptimization in public rail transport Mathematical Programming 79B(1ndash3)415ndash444 1997 Cited on pages xxiii 9 10
[51] V Cacchiani Models and Algorithms for Combinatorial Optimiza-tion Problems arising in Railway Applications PhD thesis DEISBologna 2007 Cited on pages 39 90 103
[52] V Cacchiani A Caprara amp P Toth A column generationapproach to traintimetabling on a corridor 4OR 2007 To appearCited on pages 39 40 90 142
[53] V Cacchiani A Caprara L Galli L G Kroon ampG Maroti Recoverable robustness for railway rolling stock plan-ning In ATMOS 2008 Cited on page 34
[54] V Cacchiani A Caprara amp P Toth Scheduling extra freighttrains on railway networks Transportation Research Part B Method-
References 195
ological 44(2)215ndash231 2010 URL httpeconpapersrepecorg
RePEceeetransbv44y2010i2p215-231 Cited on pagesxxiv 39 40 90 159 160
[55] X Cai amp C J Goh A fast heuristic for the train scheduling problemComput Oper Res 21(5)499ndash510 1994 ISSN 0305-0548 Cited onpages 36 38
[56] X Cai C J Goh amp A Mees Greedy heuristics for rapid schedul-ing of trains on a single track IIE Transactions 30(5)481 ndash 493 1998URL httpwwwspringerlinkcomcontentv9t27h636427t066Cited on pages 36 38
[57] G Caimi Algorithmic decision support for train scheduling in a largeand highly utilised railway network PhD thesis ETH Zurich 2009Cited on pages xxv 2 18 33 34 41 57 90 91
[58] G Caimi D Burkolter amp T Herrmann Finding delay-toleranttrain routings through stations In OR pp 136ndash143 2004 Cited onpage 129
[59] G C Caimi M Fuchsberger M Laumanns amp K Schupbach09 periodic railway timetabling with event flexibility In C LiebchenR K Ahuja amp J A Mesa (Eds) ATMOS 2007 - 7th Workshop onAlgorithmic Approaches for Transportation Modeling Optimizationand Systems Dagstuhl Germany 2007 Internationales Begegnungs-und Forschungszentrum fur Informatik (IBFI) Schloss Dagstuhl Ger-many ISBN 978-3-939897-04-0 URL httpdropsdagstuhlde
opusvolltexte20071173 Cited on page 34
[60] A Caprara M Fischetti amp P Toth Algorithms for the setcovering problem Annals of Operations Research 982000 1998 Citedon page 147
[61] A Caprara M Fischetti P L Guida M Monaci G Saccoamp P Toth Solution of real-world train timetabling problems InHICSS 34 IEEE Computer Society Press 2001 Cited on pages 38108
[62] A Caprara M Fischetti amp P Toth Modeling and solving thetrain timetabling problem Operations Research 50(5)851ndash861 2002Cited on pages 38 40 103 108
[63] A Caprara M Monaci P Toth amp P L Guida A lagrangianheuristic algorithm for a real-world train timetabling problem DiscreteAppl Math 154(5)738ndash753 2006 ISSN 0166-218X Cited on pages36 106
[64] A Caprara L Kroon M Monaci M Peeters amp P TothPassenger railway optimization In C Barnhart amp G Laporte(Eds) Handbooks in Operations Research and Management Sciencevol 14 chap 3 pp 129ndash187 Elsevier 2007 Cited on pages 12 103
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[65] M Carey amp D Lockwood A model algorithms and strategy fortrain pathing The Journal of the Operational Research Society 461995 Cited on page 38
[66] L Castelli P Pellegrini amp R Pesenti Airport slot allocationin europe economic efficiency and fairness Working Papers 197Department of Applied Mathematics University of Venice 2010 URLhttpeconpapersrepecorgRePEcvnmwpaper197 Cited onpage 10
[67] A Charnes amp M Miller A model for the optimal programmingof railway freight train movements Management Science 3(1)74ndash921956 Cited on pages xxv 2 5 6
[68] E H Clarke Multipart pricing of public goods Public Choice 219ndash33 1971 Cited on page 172
[69] J Clausen A Larsen J Larsen amp N J Rezanova Disrup-tion management in the airline industry-concepts models and meth-ods Comput Oper Res 37(5)809ndash821 2010 ISSN 0305-0548 Citedon page 15
[70] J-F Cordeau P Toth amp D Vigo A Survey of Optimiza-tion Models for Train Routing and Scheduling TRANSPORTATIONSCIENCE 32(4)380ndash404 1998 URL httptranscijournal
informsorgcgicontentabstract324380 Cited on page 38
[71] F Corman R M Goverde amp A DrsquoAriano Rescheduling DenseTrain Traffic over Complex Station Interlocking Areas pp 369ndash386Springer-Verlag Berlin Heidelberg 2009 ISBN 978-3-642-05464-8doi httpdxdoiorg101007978-3-642-05465-5 16 Cited on page15
[72] A DrsquoAriano F Corman D Pacciarelli amp M Pranzo Re-ordering and local rerouting strategies to manage train traffic in realtime Transportation Science 42(4)405ndash419 2008 ISSN 1526-5447Cited on page 15
[73] DB Netze AG DB Netze AG-Homepage 2010 URL httpwww
dbnetzecom httpwwwdbnetzecom Cited on page 91
[74] X Delorme X Gandibleux amp J Rodriguez Stability eval-uation of a railway timetable at station level European Journal ofOperational Research 195(3)780ndash790 2009 Cited on pages 57 129
[75] J Desrosiers F Soumis amp M Desrochers Routes sur un reseauespace-temps Technical Report 236 Centre de recherche sur les trans-ports Universite de Montreal 1982 Cited on page 96
[76] M J Dorfman amp J Medanic Scheduling trains on a railway net-work using a discrete event model of railway traffic TransportationResearch Part B Methodological 38(1)81 ndash 98 2004 ISSN 0191-2615 URL httpwwwsciencedirectcomsciencearticle
References 197
B6V99-484SFYN-22e474b988e5fca3c08b20c1cf991a960b Citedon page 36
[77] J Eckstein amp M Nediak Pivot cut and dive a heuristic for 0-1mixed integer programming J Heuristics 13(5)471ndash503 2007 Citedon page 145
[78] M Ehrgott Multicriteria Optimization Springer Verlag Berlin 2edition 2005 Cited on pages 129 131 132
[79] M Ehrgott amp D Ryan Constructing robust crew schedules with bi-criteria optimization Journal of Multi-Criteria Decision Analysis 11139ndash150 2002 Cited on page 129
[80] S Eidenbenz A Pagourtzis amp P Widmayer Flexible trainrostering In T Ibaraki N Katoh amp H Ono (Eds) ISAAC vol2906 of Lecture Notes in Computer Science pp 615ndash624 Springer2003 ISBN 3-540-20695-7 Cited on page 14
[81] L El-Ghaoui F Oustry amp H Lebret Robust solutions to un-certain semidefinite programs SIAM J Optim 933ndash52 1998 Citedon page 128
[82] D Emery Enhanced ETCS Level 3 train control system InA Tomii J Allan E Arias C Brebbia C GoodmanA Rumsey amp G Sciutto (Eds) Computers in Railways XI WITPress 2008 Cited on page 63
[83] C Erdogan Computing prices for track allocations Diploma thesisTU Berlin 2009 Cited on page 173
[84] B Erol Models for the train timetabling problem Diplomathesis TU Berlin 2009 URL httpwwwzibdeOptimization
ProjectsTrafficLogisticTrassenBthesis_erolpdf Cited onpages 57 117 121
[85] B Erol M Klemenz T Schlechte S Schultz amp A TannerTTPlib 2008 - A library for train timetabling problems In A TomiiJ Allan E Arias C Brebbia C Goodman A Rumsey ampG Sciutto (Eds) Computers in Railways XI WIT Press 2008URL httpopuskobvdezibvolltexte20081102 Cited onpages ii 64 91 150 161
[86] B Federal Ministry of Transport amp Housing Regula-tion for the use of railway infrastructure 2005 URL httpwww
gesetze-im-internetdeeibv_2005BJNR156610005html Citedon page 32
[87] Federal Transport Infrastructure Planning ProjectGroup Federal transport infrastructure plan 2003 2003URL httpwwwbmvbsdeAnlageoriginal_17121
Federal-Transport-Infrastructure-Plan-2003pdf Citedon pages xxv 1 2
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[88] P-J Fioole L G Kroon G Maroti amp A Schrijver Arolling stock circulation model for combining and splitting of passengertrains European Journal of Operational Research 174(2)1281ndash12972006 Cited on page 14
[89] F Fischer amp C Helmberg Dynamic graph generation and dy-namic rolling horizon techniques in large scale train timetabling InT Erlebach amp M Lubbecke (Eds) Proceedings of the 10thWorkshop on Algorithmic Approaches for Transportation ModellingOptimization and Systems vol 14 of OpenAccess Series in In-formatics (OASIcs) pp 45ndash60 Dagstuhl Germany 2010 SchlossDagstuhlndashLeibniz-Zentrum fuer Informatik ISBN 978-3-939897-20-0 doi httpdxdoiorg104230OASIcsATMOS201045 URLhttpdropsdagstuhldeopusvolltexte20102749 Cited onpages 39 40 90 133
[90] F Fischer C Helmberg J Janszligen amp B Krostitz To-wards solving very large scale train timetabling problems by lagrangianrelaxation In M Fischetti amp P Widmayer (Eds) ATMOS2008 - 8th Workshop on Algorithmic Approaches for Transporta-tion Modeling Optimization and Systems Dagstuhl Germany 2008Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik Germany URLhttpdropsdagstuhldeopusvolltexte20081585 Cited onpages 39 40 90 103 109 142 149
[91] M Fischetti D Salvagnin amp A Zanette Fast approaches toimprove the robustness of a railway timetable Transportation Sci-ence 43(3)321ndash335 2009 ISSN 1526-5447 Cited on pages 34 128
[92] B A Foster amp D M Ryan An integer programming approachto scheduling In Computer Aided Scheduling of Public TransportSpringer Verlag Berlin 1991 Cited on pages 48 142
[93] A Frangioni About lagrangian methods in integer optimization An-nals of Operations Research 139163ndash193 2005 ISSN 0254-5330 URLhttpdxdoiorg101007s10479-005-3447-9 101007s10479-005-3447-9 Cited on page 135
[94] M Fuchsberger Solving the train scheduling problem in a mainstation area via a resource constrained space-time integer multi-commodity flow Masterrsquos thesis Institut for Operations ResearchETH Zurich 2007 Cited on pages 57 103
[95] A Fugenschuh H Homfeld A Huck A Martin amp Z YuanScheduling Locomotives and Car Transfers in Freight TransportTransportation Science 42(4)1 ndash 14 2008 Cited on page 19
[96] A Fugenschuh H Homfeld amp H Schulldorf Single car rout-ing in rail freight transport In C Barnhart U Clausen U Lau-ther amp R Mohring (Eds) Dagstuhl Seminar Proceedings 09261
References 199
Schloss Dagstuhl ndash Leibniz-Zentrum fr Informatik Deutschland 2009Cited on page 19
[97] M Garey amp D Johnson Computers and Intractability A Guide tothe Theory of NP-Completeness WH Freeman and Company NewYork 1979 Cited on page 104
[98] K Ghoseiri F Szidarovszky amp M J Asgharpour A multi-objective train scheduling model and solution Transportation Re-search Part B Methodological 38(10)927 ndash 952 2004 ISSN 0191-2615 URL httpwwwsciencedirectcomsciencearticle
B6V99-4C0053J-12e37583200d0d67abec74538df41f1909 Citedon page 36
[99] GIlgmann The essence of railways GIlgmann 2007 Cited onpage 8
[100] A Gille M Klemenz amp T Siefer Applying multiscaling analysisto detect capacity resources in railway networks chap A 7 pp 73ndash82Timetable Planning and Information Quality WIT Press 2010 Citedon page 56
[101] J-W Goossens S P M van Hoesel amp L G Kroon Onsolving multi-type railway line planning problems European Journalof Operational Research 168(2)403ndash424 2006 Cited on page 25
[102] M F Gorman Statistical estimation of railroad congestion delayTransportation Research Part E 45(3)446ndash456 2009 Cited on page4
[103] M Gronkvist The Tail Assignment Problem PhD thesis ChalmersUniversity of Technology and Goteborg University 2005 Cited onpages 10 14
[104] M Grotschel L Lovasz amp A Schrijver Geometric Algorithmsand Combinatorial Optimization vol 2 of Algorithms and Combina-torics Springer 1988 ISBN 3-540-13624-X 0-387-13624-X (US)Cited on pages 9 119
[105] M Grotschel S O Krumke amp J Rambau Online Optimizationof Large Scale Systems Springer Sept 2001 ISBN 3-540-42459-8Cited on page 10
[106] T Groves Incentives in Teams Econometrica 41617ndash631 1973Cited on page 172
[107] M Habib R M McConnell C Paul amp L Viennot Lex-bfsand partition refinement with applications to transitive orientationinterval graph recognition and consecutive ones testing Theor Com-put Sci 234(1-2)59ndash84 2000 Cited on page 109
[108] T Hanne amp R Dornberger Optimization problems in airlineand railway planning - a comparative survey In Proceedings of theThe Third International Workshop on Advanced Computational Intel-ligence 2010 Cited on page 10
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[109] I Hansen State-of-the-art of railway operations research chap A 4pp 35ndash47 Timetable Planning and Information Quality WIT Press2010 Cited on page 57
[110] O Happel Ein Verfahren zur Bestimmung der Leistungsfahigkeitder Bahnhofe PhD thesis RWTH Aachen 1950 Cited on page 61
[111] O Happel Sperrzeiten als Grundlage fur die FahrplankonstruktionEisenbahntechnische Rundschau (ETR) pp 79ndash90 1959 Cited onpage 61
[112] S Harrod Modeling network transition constraints with hypergraphsTransportation Science 10293ndash310 2010 Cited on page 4
[113] E Helly Uber Mengen konvexer Korper mit gemeinschaftlichenPunkten Jahresber Deutsch Math Verein 32175ndash176 1923 Citedon page 108
[114] C Helmberg Semidefinite programming for combinatorial optimiza-tion Technical report Zuse Institute Berlin October 2000 also ha-bilitation thesis Cited on page 135
[115] A Higgins E Kozan amp L Ferreira Heuristic techniques forsingle line train scheduling Journal of Heuristics 3(1)43ndash62 1997ISSN 1381-1231 Cited on page 36
[116] J-B Hiriart-Urruty amp C Lemarechal Convex Analysis andMinimization Algorithms I vol 305 of A Series of ComprehensiveStudies in Mathematics Springer-Verlag 1993 Cited on page 135
[117] J-B Hiriart-Urruty amp C Lemarechal Convex Analysis andMinimization Algorithms II vol 306 of A Series of ComprehensiveStudies in Mathematics Springer-Verlag 1993 Cited on page 135
[118] L Hurwicz On informationally decentralized systems In C BMcGuire amp R Radner (Eds) Decision and Organization A Vol-ume in Honor of Jacob Marschak North-Holland Amsterdam 1972Cited on page 172
[119] CPLEX 12202 IBM ILOG CPLEX Optimization StudioIBM 2011 URL httpwww-01ibmcomsoftwareintegration
optimizationcplex-optimizer Cited on page 149
[120] S Irnich amp G Desaulniers Shortest Path Problems with ResourceConstraints chap 2 pp 33ndash65 GERAD 25th Anniversary SeriesSpringer 2005 Cited on page 50
[121] S Irnich G Desaulniers J Desrosiers amp A Hadjar Path-reduced costs for eliminating arcs in routing and scheduling IN-FORMS Journal on Computing 22(2)297ndash313 2010 Cited on page50
[122] J Jespersen-Groth D Potthoff J Clausen D HuismanL Kroon G Maroti amp M Nielsen Disruption management inpassenger railway transportation Econometric Institute Report EI
References 201
2007-05 Erasmus University Rotterdam Econometric Institute Jan2007 URL httpideasrepecorgpdgreureir1765008527
html Cited on page 15
[123] J Jespersen-Groth D Potthoff J Clausen D HuismanL G Kroon G Maroti amp M N Nielsen Disruption man-agement in passenger railway transportation In Ahuja Mohring ampZaroliagis (2009) [7] pp 399ndash421 ISBN 978-3-642-05464-8 Cited onpage 11
[124] K C Jha R K Ahuja amp G Sahin New approaches for solvingthe block-to-train assignment problem Networks 51(1)48ndash62 2008ISSN 0028-3045 Cited on pages 18 19
[125] D Jovanovic amp P T Harker Tactical scheduling of rail oper-ations Transportation Science 2546ndash64 1991 Cited on pages 3738
[126] D Kim amp C Barnhart Transportation service network designModels and algorithms In N H M Wilson (Ed) Proc of the Sev-enth International Workshop on Computer-Aided Scheduling of PublicTransport (CASPT) Boston USA 1997 vol 471 of Lecture Notes inEconomics and Mathematical Systems pp 259ndash283 Springer-VerlagBerlin Heidelberg 1997 Cited on page 18
[127] K C Kiwiel Proximal bundle methods Mathematical Program-ming 46(123)105ndash122 1990 Cited on pages 136 141
[128] K C Kiwiel Approximation in proximal bundle methods and de-composition of convex programs Journal of Optimization Theory andapplications 84(3)529ndash548 1995 Cited on pages 136 141
[129] S G Klabes Algorithmic railway capacity allocation in a compet-itive European railway market PhD thesis RWTH Aachen 2010Cited on pages xxv 11 13 29 31 39 40 57 61 62 90
[130] M Klemenz amp SSchultz Modelling aspects of a railway slot allo-cation In 2nd International Seminar on Railway Operations Modellingand Analysis 2007 Cited on page 41
[131] W Klemt amp W Stemme Schedule synchronization for public tran-sit networks In Computer-Aided Transit Scheduling pp 327ndash335Springer-Verlag New York 1988 Cited on page 34
[132] N Kliewer T Mellouli amp L Suhl A time-space net-work based exact optimization model for multi-depot bus schedul-ing European Journal of Operational Research 175(3)1616ndash1627December 2006 URL httpideasrepecorgaeeeejores
v175y2006i3p1616-1627html Cited on pages 43 96
[133] V Klima amp A Kavicka Simulation support for railway infrastruc-ture design and planning processes In In Computers in Railways VIIpp 447ndash456 WIT Press 2000 Cited on page 18
References 202
[134] T Koch A Martin amp T Achterberg Branching rules revisitedOperations Research Letters 3342ndash54 2004 Cited on page 142
[135] A Kokott amp A Lobel Experiments with a dantzig-wolfe decom-position for multiple-depot vehicle scheduling problems Technical Re-port ZIB Report 97-16 Zuse-Institut Berlin Takustr 7 14195 Berlin1997 URL httpwwwzibdePaperWebabstractsSC-97-16Cited on page 43
[136] S C Kontogiannis amp C D Zaroliagis Robust line planningunder unknown incentives and elasticity of frequencies In ATMOS2008 Cited on page 90
[137] C Kopper Zu lange zu groszlig zu teuer 2010 URL httpwww
zeitde201042Bahn-Neubaustrecken Cited on page 17
[138] C Krauchi amp U Stockli Mehr Zug fr die Schweiz Die Bahn-2000-Story (More train for Switzerland The Rail 2000-Story) ZurichAS-Verlag 2004 Cited on page 18
[139] L Kroon R Dekker G Maroti M Retel Helmrich amp M JVromans Stochastic improvement of cyclic railway timetables SSRNeLibrary 2006 Cited on pages 34 128
[140] L Kroon D Huisman E Abbink P-J Fioole M FischettiG Maroti A Schrijver A Steenbeek amp R Ybema The newdutch timetable The or revolution Interfaces 39(1)6ndash17 2009 ISSN0092-2102 Cited on pages 2 17 34
[141] L G Kroon amp L W P Peeters A variable trip time modelfor cyclic railway timetabling Transportation Science 37(2)198ndash212May 2003 Cited on page 34
[142] L G Kroon R Dekker amp M J C M Vromans Cyclic railwaytimetabling A stochastic optimization approach In F GeraetsL G Kroon A Schobel D Wagner amp C D Zaroliagis(Eds) ATMOS vol 4359 of Lecture Notes in Computer Science pp41ndash66 Springer 2004 ISBN 978-3-540-74245-6 Cited on page 34
[143] A Lamatsch An approach to vehicle scheduling with depot capac-ity constraints In M Desrochers amp J-M Rousseau (Eds)Computer-Aided Transit Scheduling Lecture Notes in Economics andMathematical Systems Springer Verlag 1992 Cited on page 96
[144] S Lan J-P Clarke amp C Barnhart Planning for robust airlineoperations Optimizing aircraft routings and flight departure times tominimize passenger disruptions Transportation Science 40(1)15ndash282006 Cited on page 6
[145] A Landex B Schittenhelm A Kaas amp J Schneider-Tilli Capacity measurement with the UIC 406 capacity methodIn A Tomii J Allan E Arias C Brebbia C GoodmanA Rumsey amp G Sciutto (Eds) Computers in Railways XI WITPress 2008 Cited on page 57
References 203
[146] Y Lee amp C-Y Chen A heuristic for the train pathingand timetabling problem Transportation Research Part BMethodological 43(8-9)837 ndash 851 2009 ISSN 0191-2615URL httpwwwsciencedirectcomsciencearticle
B6V99-4VXT0P3-121cce3f2565ca4b86cb04a608124b7c36 Citedon page 36
[147] C Lemarechal Lagrangian relaxation In Computational Combi-natorial Optimization pp 112ndash156 2001 Cited on page 135
[148] C Liebchen Periodic Timetable Optimization in Public TransportPhD thesis Technische Universitat Berlin 2006 Cited on pages 1013 33 34
[149] C Liebchen The first optimized railway timetable in practice Trans-portation Science 42(4)420ndash435 2008 Cited on pages 2 34
[150] C Liebchen amp R H Mohring The modeling power of the peri-odic event scheduling problem Railway timetables - and beyond InATMOS pp 3ndash40 2004 Cited on page 34
[151] C Liebchen M Schachtebeck A Schobel S Stiller ampA Prigge Computing delay resistant railway timetables Technicalreport ARRIVAL Project October 2007 Cited on page 128
[152] C Liebchen M E Lubbecke R H Mohring amp S StillerThe concept of recoverable robustness linear programming recoveryand railway applications In Ahuja Mohring amp Zaroliagis (2009) [7]pp 1ndash27 ISBN 978-3-642-05464-8 Cited on pages 34 128
[153] C Liebchen M Schachtebeck A Schobel S Stiller ampA Prigge Computing delay resistant railway timetables ComputOper Res 37(5)857ndash868 2010 ISSN 0305-0548 Cited on page 34
[154] T Lindner Train schedule optimization in public rail transport PhDthesis TU Braunschweig 2000 Cited on page 34
[155] A Lobel Optimal Vehicle Scheduling in Public TransitShaker Verlag Aachen 1997 URL httpwwwshakerde
Online-GesamtkatalogDetailsidcISBN=3-8265-3504-9 PhDthesis Technische Universitat Berlin Cited on page 14
[156] M Lubbecke amp J Desrosiers Selected topics in column genera-tion Oper Res 53(6)1007ndash1023 2005 Cited on pages 143 161
[157] S G Lukac Holes antiholes and maximal cliques in a railway modelfor a single track Technical Report ZIB Report 04-18 Zuse-InstitutBerlin Takustr 7 14195 Berlin 2004 URL httpwwwzibde
PaperWebabstractsZR-04-18 Cited on page 109
[158] R Lusby Optimization Methods for Routing Trains Through RailwayJunctions PhD thesis The University of Auckland 2008 Cited onpages 39 40 88 90 103 142
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[160] R Marsten Crew planning at delta airlines Talk at the 15th IntSymp Math Prog 1994 Cited on page 48
[161] D Middelkoop amp M Bouwman Train network simulator forsupport of network wide planning of infrastructure and timetables InIn Computers in Railways VII pp 267ndash276 WIT Press 2000 Citedon page 18
[162] P Milgrom Putting Auction Theory to Work Cambridge Univer-sity Press 2004 URL httpeconpapersrepecorgRePEccup
cbooks9780521536721 Cited on page 171
[163] M Montigel Modellierung und Gewahrleistung von Abhangigkeitenin Eisenbahnsicherungsanlagen PhD thesis ETH Zurich 1994 Citedon page 58
[164] A Mura Trassenauktionen im schienenverkehr Diploma thesisTechnische Universitat Berlin 2006 URL httpwwwzibde
OptimizationProjectsTrafficTrassenBdiplom_murapsgzCited on pages 29 127 150 172
[165] K Nachtigall amp J Opitz Solving periodic timetable optimisationproblems by modulo simplex calculations In ATMOS 2008 Cited onpage 34
[166] A Nash amp D Huerlimann Railroad simulation using OpenTrackIn Computer Aided Design Manufacture and Operation in the Railwayand other Advanced Transit Systems 2004 Cited on page 60
[167] G L Nemhauser amp L A Wolsey Integer and CombinatorialOptimization Wiley-Interscience Series in Discrete Mathematics andOptimization John Wiley amp Sons New York 1988 Cited on pages9 132
[168] F Niekerk amp H Voogd mpact assessment for infrastructure plan-ning some dutch dilemmas Environmental Impact Assessment Re-view 1921ndash36 1999 Cited on page 18
[169] M A Odijk Railway Timetable Generation PhD thesis TU Delft1997 Cited on page 34
[170] M A Odijk H E Romeijn amp H van Maaren Generation ofclasses of robust periodic railway timetables Computers amp OR 332283ndash2299 2006 Cited on page 34
[171] E Oliveira amp B M Smith A combined constraint-based searchmethod for single-track railway scheduling problem In P Brazdilamp A Jorge (Eds) EPIA vol 2258 of Lecture Notes in ComputerScience pp 371ndash378 Springer 2001 ISBN 3-540-43030-X Cited onpage 36
References 205
[172] OpenTrack OpenTrack 2010 URL httpwwwopentrackch httpwwwopentrackch Cited on page 58
[173] J Pachl Systemtechnik des Schienenverkehrs Teubner VerlagStuttgart-Leipzig-Wiesbaden 3 edition 2002 Cited on page 60
[174] J Pachl Timetable design principles chap 2 pp 9ndash42 RailwayTimetable amp Traffic Eurailpress 2008 Cited on page 61
[175] D C Parkes amp L H Ungar An auction-based method for de-centralized train scheduling In Proc 5th International Conference onAutonomous Agents (AGENTS-01) pp 43ndash50 2001 Cited on page174
[176] M Peeters amp L G Kroon Circulation of railway rolling stocka branch-and-price approach Computers amp OR 35(2)538ndash556 2008Cited on page 14
[177] D Potthoff D Huisman amp G Desaulniers Column generationwith dynamic duty selection for railway crew rescheduling Econo-metric Institute Report EI 2008-28 Erasmus University RotterdamEconometric Institute Dec 2008 URL httpideasrepecorg
pdgreureir1765014423html Cited on pages 11 15 52
[178] G Potthoff Verkehrsstromungslehre Band 1- Die Zugfolge aufStrecken und in Bahnhofen 3 Auflage transpress Verlag Berlin 1980Cited on page 56
[179] ProRail ProRail 2010 URL httpwwwprorailnlhttpwwwprorailnl Cited on page 91
[180] A Radtke Infrastructure modelling chap 3 pp 43ndash57 RailwayTimetable amp Traffic Eurailpress 2008 Cited on pages 55 56
[181] G Reinelt Tsplib - a traveling salesman problem library ORSAJournal on Computing 3376ndash384 1991 Cited on page 64
[182] N J Rezanova amp D M Ryan The train driver recovery problem-a set partitioning based model and solution method Comput OperRes 37(5)845ndash856 2010 ISSN 0305-0548 Cited on pages 11 15
[183] J H Rodriguez A constraint programming model for real-time trainscheduling at junctions Transportation Research Part B Method-ological 41(2)231ndash245 2007 URL httpeconpapersrepecorg
RePEceeetransbv41y2007i2p231-245 Cited on page 36
[184] A Romein J Trip amp J de Vries The multi-scalar complexity ofinfrastructure planning evidence from the dutch-flemish eurocorridorJournal of Transport Geography 3(11)205ndash213 2003 Cited on page18
[185] R Sauder amp W Westerman Computer aided train dispatchingDecision support through optimization Interfaces 13(6)24ndash37 1983Cited on page 37
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[186] B Schittenhelm Quantitative methods to evaluate timetable attrac-tiveness In I Hansen E Wendler U Weidmann M LuthiJ Rodriguez S Ricci amp L Kroon (Eds) Proceedings of the 3rdInternational Seminar on Railway Operations Modelling and Analy-sis - Engineering and Optimisation Approaches Zurich Switzerland2009 Cited on pages 25 92
[187] T Schlechte Das Resource-Constraint-Shortest-Path-Problem undseine Anwendung in der OPNV-Dienstplanung Masterrsquos thesis Tech-nische Universitat Berlin 2003 Cited on page 120
[188] T Schlechte amp R Borndorfer Balancing efficiency and ro-bustness - a bi-criteria optimization approach to railway track alloca-tion In M Ehrgott B Naujoks T Stewart amp J Wallenius(Eds) MCDM for Sustainable Energy and Transportation SystemsLecture Notes in Economics and Mathematical Systems 2008 URLhttpopuskobvdezibvolltexte20081105 Cited on pagesii 90 126 129 131
[189] T Schlechte amp A Tanner Railway capacity auctions with dualprices In Selected Proceedings of the 12th World Conference onTransport Research 2010 ISBN 978-989-96986-1-1 URL http
opuskobvdezibvolltexte20101233 submitted to SpecialIssue of Research in Transportation Economics 2422011 Cited onpages ii 12 174 175
[190] T Schlechte R Borndorfer B Erol T Graffagnino ampE Swarat Aggregation methods for railway networks In I HansenE Wendler S Ricci D Pacciarelli G Longo amp J Ro-driguez (Eds) Proceedings of 4th International Seminar on Rail-way Operations Modelling and Analysis (IAROR) vol 4 2011 Citedon pages ii 55 180
[191] A Schobel amp A Kratz A bicriteria approach for robust time-tabling In Ahuja Mohring amp Zaroliagis (2009) [7] pp 119ndash144 ISBN978-3-642-05464-8 Cited on page 129
[192] A Schobel amp S Scholl Line planning with minimal travelingtime In L G Kroon amp R H Mohring (Eds) 5th Work-shop on Algorithmic Methods and Models for Optimization of Rail-ways Dagstuhl Germany 2006 Internationales Begegnungs- undForschungszentrum fur Informatik (IBFI) Schloss Dagstuhl Ger-many ISBN 978-3-939897-00-2 URL httpdropsdagstuhlde
opusvolltexte2006660 Cited on page 25
[193] A Schrijver Theory of Linear and Integer Programming Inter-science series in discrete mathematics and optimization Wiley 1998Cited on page 132
[194] A Schrijver amp A Steenbeck Dienstregelingontwikkeling voorrailned (timetable construction for railned Technical report Center
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for Mathematics and Computer Science 1994 Cited on page 34
[195] K Schultze Modell fur die asynchrone Simulation des Betriebes inTeilen des Eisenbahnnetzes PhD thesis RWTH Aachen 1985 Citedon page 56
[196] W Schwanhauszliger Die Bemessung der Pufferzeiten imFahrplangefuge der Eisenbahn PhD thesis RWTH Aachen 1974Cited on page 56
[197] W Schwanhauszliger I Gast K Schultze amp O Brunger Pro-grammfamilie SLS Benutzerhandbuch Technical report DeutscheBundesbahn 1992 Cited on page 62
[198] Y Semet amp M Schoenauer An efficient memetic permutation-based evolutionary algorithm for real-world train timetabling InCongress on Evolutionary Computation pp 2752ndash2759 IEEE 2005ISBN 0-7803-9363-5 Cited on page 36
[199] P Serafini amp W Ukovich A mathematical for periodic schedulingproblems SIAM J Discret Math 2(4)550ndash581 1989 ISSN 0895-4801 Cited on page 34
[200] B Sewcyk Makroskopische Abbildung des Eisenbahnbetriebs in Mod-ellen zur langfristigen Infrastrukturplanung PhD thesis Leibniz UHannover 2004 Cited on page 41
[201] E Silva de Oliveira Solving single-track railway scheduling prob-lem using constraint programming PhD thesis University of LeedsSchool of Computing 2001 Cited on page 36
[202] A Soyster Convex programming with set-inclusive constraints andapplications to inexact linear programming Oper Res 211154ndash11571973 Cited on page 128
[203] I Steinzen V Gintner L Suhl amp N Kliewer A time-spacenetwork approach for the integrated vehicle-and crew-scheduling prob-lem with multiple depots Transportation Science 44(3)367ndash382 2010ISSN 1526-5447 Cited on page 43
[204] R Subramanian R Sheff J Quillinan D Wiper ampR Marsten Coldstart Fleet assignment at delta air lines In-terfaces 24(1)104ndash120 1994 Cited on pages 142 147
[205] L Suhl V Duck amp N Kliewer Increasing stability of crewschedules in airlines In C Barnhart U Clausen U Lau-ther amp R H Mohring (Eds) Models and Algorithms for Op-timization in Logistics number 09261 in Dagstuhl Seminar Proceed-ings Dagstuhl Germany 2009 Schloss Dagstuhl - Leibniz-Zentrumfuer Informatik Germany URL httpdropsdagstuhldeopus
volltexte20092178 Cited on page 6
[206] B Szpigel Optimal train scheduling on a single track railway InProceedings of IFORS Conference on Operational Researchrsquo72 num-ber 72 in 6 pp 343ndash352 1973 Cited on pages 36 37 38 40
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[207] Trasse Schweiz AG Business report 2009 2009 URL httpwww
trassech httpwwwtrassech Cited on pages 13 91
[208] TTPlib TTPlib-Homepage 2008 URL httpttplibzibdehttpttplibzibde Cited on pages xxvii 3 154
[209] W Vickrey Counterspeculation auctions and competitive sealedtenders The Journal of Finance 16(1)8ndash37 1961 URL httpwww
jstororgstable2977633 Cited on page 172
[210] D Villeneuve J Desrosiers M E Lubbecke amp F SoumisOn compact formulations for integer programs solved by column gen-eration Annals OR 139(1)375ndash388 2005 Cited on page 133
[211] D Wedelin An algorithm for a large scale 0-1 integer programmingwith application to airline crew scheduling Annals of Operations Re-search 57283ndash301 1995 Cited on pages 48 142 145
[212] O Weide D Ryan amp M Ehrgott An iterative approach to robustand integrated aircraft routing and crew scheduling Comput OperRes 37(5)833ndash844 2010 ISSN 0305-0548 Cited on page 129
[213] S Weider Integration of Vehicle and Duty Scheduling in PublicTransport PhD thesis TU Berlin 2007 Cited on pages x xiv xxvii3 10 16 96 120 135 141 142 143 146
[214] E Wendler Influence of ETCS on the capacity of lines In Com-pendium on ERTMS Compendium on ERTMS European Rail TrafficManagement System Eurailpress 2009 Cited on page 63
[215] T White amp A Krug (Eds) Managing Railroad TransportationVTD Rail Publishing 2005 ISBN 0-9719915-3-7 Cited on page 4
[216] J W Zheng H T Kin amp M B Hua A study of heuristic ap-proach on station track allocation in mainline railways InternationalConference on Natural Computation 4575ndash579 2009 Cited on page36
[217] X Zhou amp M Zhong Single-track train timetabling with guar-anteed optimality Branch-and-bound algorithms with enhanced lowerbounds Transportation Research Part B Methodological 41(3)320ndash341 March 2007 URL httpideasrepecorgaeeetransb
v41y2007i3p320-341html Cited on page 36
[218] E Zhu T G Crainic amp M Gendreau Integrated service networkdesign in rail freight transportation Research Report CIRRELT-2009-45 CIRRELT Montreal Canada 2009 Cited on page 18
[219] G M Ziegler Lectures on Polytopes Springer 1995 Cited onpage 121
[220] P J Zwaneveld L G Kroon H E Romeijn M Sa-lomon S Dauzere-Peres S P M Van Hoesel amp H WAmbergen Routing Trains Through Railway Stations Model For-mulation and Algorithms Transportation Science 30(3)181ndash194
References 209
1996 URL httptranscijournalinformsorgcgicontent
abstract303181 Cited on page 57
[221] P J Zwaneveld L G Kroon amp S P M van HoeselRouting trains through a railway station based on a node pack-ing model European Journal of Operational Research 128(1)14ndash33 January 2001 URL httpideasrepecorgaeeeejores
v128y2001i1p14-33html Cited on page 57
Lebenslauf
Thomas Schlechte
geboren am 10031979 in Halle an der Saale
1985 bis 1986 Besuch der Grundschule in Halle
1986 bis 1991 Besuch der Grundschule in Berlin
1991 bis 1998 Besuch des Descartes Gymnasiums in Berlin
1998 bis 2004 Studium der Mathematik an der Technischen Uni-versitat Berlin
Seit 2004 Wissenschaftlicher Mitarbeiter am Zuse InstituteBerlin (ZIB)
- Table of Contents
- List of Tables
- List of Figures
- I Planning in Railway Transportation
-
- 1 Introduction
- 2 Planning Process
-
- 21 Strategic Planning
- 22 Tactical Planning
- 23 Operational Planning
-
- 3 Network Design
- 4 Freight Service Network Design
-
- 41 Single Wagon Freight Transportation
- 42 An Integrated Coupling Approach
-
