2006Periodic Timetabl Optimization in Public Transport

Periodic Timetable Optimization in Public TransportChristian LiebchenInstitut f¨u r Mathematik,Kombinatorische Optimierung und Graphenalgorithmen,Technische Universit¨a t Berlin,Germanyliebchen@math.tu-berlin.deSummary.”The timetable is the essence of the service offered by any provider of public transport.“(Jonothan Tyler,CASPT 2006)Despite this observation,in the practice of planning public transportation,only some months ago OR decision support has still been limited to operations planning (vehicle scheduling,duty scheduling,crew rostering).We describe the optimization techniques that were employed in computing the very first optimized timetable that went into daily service:the 2005timetable of Berlin Underground.This timetables improved on both,the passenger travel times and the operating efficiency of the company.The basic graph model,the Periodic Event Scheduling Problem (PESP),is known for 15years and it had attracted many research groups.Nevertheless,we report on significant progress that has been made only recently on issues like solu-tion strategies or modeling capabilities.The latter even includes the integration of further planning tasks in public transport,such as line planning.On the theory side,we give a more precise notion of the asymptotical complexity of the PESP,by providing a MAXSNP-hardness proof as a kind of negative result.On the positive side,the design of more efficient algorithms gave rise to a much deeper understanding of cycle bases of graphs,another very hot topic in discrete mathematics during the last three years.In 2005,this culminated in both,drawing the complete map for the seven relevant classes of cycle bases,and the design of the fastest algorithms for the Minimum Directed Cycle Basis Problem and for the Minimum 2-Basis Problem.The book version of this extended abstract is available as reference [8].1TimetablingIt is a very important competitive advantage of public transport to be much less expensive than a taxi service.This requires many passengers to share the same vehicle.Typically,this is achieved by offering public transport along fixed sets of routes,the lines.These serve as input to timetabling.This work has been supported by the DFG Research Center Matheon in Berlin.30Christian LiebchenThere is a large toolbox of different types of timetables,which we introduce from the most general one to the most specialized one:timetables that are composed of individual trips,periodic timetables,i.e.the headway between any two successive trips of the same line is the same,symmetric periodic timetables,and so-called “Integrated Fixed-Interval Timetables.”Here,a periodic timetable is called symmetric ,if for every passenger the transfer times that he faces during his outbound trip are identical to the transfer times during his return trip,which here is assumed to have the same route.In particu-lar,the periodic timetables of most European national railway companies are in-deed symmetric,because marketing departments consider this being a competitive advantage—at least in long-distance traffic.Theorem 1([7]).There exist example networks showing that each more specialized family of timetables causes a nominal loss in quantifiable criteria,such as average passenger waiting time.We are only aware of periodic timetables being able to clearly outweigh their nominal loss (when comparing with general irregular timetables)by adding benefit in qualitative criteria.Hence,in the remainder we focus on periodic timetables.Typically,the period time T varies over the day.For instance,Berlin Under-ground distinguishesrush hour service (T =4minutes),”normal“service (T =5minutes),weak traffic service (T =10minutes,when retail shops are closed),and night service (T =15minutes,only on weekends).Computing “the timetable”thus decomposes into computing a periodic timetable for each period time,and finally glue these together.2A Model for Periodic TimetablingA literature review of different models for periodic scheduling reveals that the most promising earlier studies on medium-sized networks are based on the Periodic Event Scheduling Problem (Pesp ,[18]),see [17,14,9,16].The vertices in this graph model represent events,where an event v ∈V is either an arrival or a departure of a directed line in a specific station.A timetable πassigns to each vertex v a point in time πv ∈[0,T )within the period time T .Constraints may then be given in the following form.T-Periodic Event Scheduling Problem (T-Pesp)Instance:A directed graph D =(V,A )and vectors ,u ∈Q A .Task:Find a vector π∈[0,T )V that fulfills(πv −πu − a )mod T ≤u a − a (1)(or πv −πu ∈[ a ,u a ]T ,for short)for every arc a =(u,v )∈A ,ordecide that none exists.Periodic Timetable Optimization in Public Transport31 In Figure1we provide an example instance of T-Pesp,which contains the eventsof two pairs of directed lines and twostations.Fig.1.A Pesp model for two lines and two stations Here,the straight arcs model either stops(within the black box that repre-sents the station)or trips,and the dotted arcs model either passenger transfers or turnarounds of the trains.Besides these most elementary requirements,there have been modeled most practical requirements that railway engineers have([11]).This even includes decisions of line planning and vehicle scheduling,which traditionally were treated as fully separate planning steps([1]).Unfortunately,this modeling power has its price in terms of complexity. Theorem2.Let a set of PESP constraints be given.Finding a timetable vectorπthat satisfies a maximum number of constraints is MAXSNP-hard.To make T-Pesp accessible for integer programming(IP)techniques,the modulo-operator is resolved by introducing integer variables:min w T(B Tπ+T p)s.t.B Tπ+T p≤uB Tπ+T p≥π∈[0,T)Vp∈{0,1,2}A.(2)Here,the matrix B is the vertex-arc incidence matrix of the constraint graph D= (V,A).But this is not the only way to formulate T-Pesp as an IP.Rather,we may replace the vertex variablesπ(or node potentials)—which carry time information —with arc variables x(tensions),and/or replace the integer arc variables p with integer cycle variables z.This way,we end with the following integer program([14])min w T xs.t.x≤ux≥ΓT x−T z=0x∈Q Az∈Z B,(3)32Christian LiebchenwhereΓdenotes the arc-cycle incidence matrix of an integral cycle basis B of the constraint graph D=(V,A).There have also been identified lower and upper bounds on these integer variables z.Theorem3([15]).Let C be an oriented circuit and z C the integer variable that we associate with it.The following inequalities are valid⎡⎢⎢⎢1T⎛⎝a∈C+a−a∈C−u a⎞⎠⎤⎥⎥⎥=:zC≤z C≤z C:=⎢⎢⎢⎣1T⎛⎝a∈C+u a−a∈C−a⎞⎠⎥⎥⎥⎦.The following rule-of-thumb could be derived from empirical studies.Remark1([4]).The shorter a circuit C∈B with respect to the sum of the spans u a− a of its arcs,the less integer values the corresponding variable z C may take.Moreover,the less values all the integer variables may take,the shorter the solution times for solving this IP.3Integral Cycle BasesIn a directed graph D=(V,A),we consider oriented circuits.These consist of forward arcs and maybe also backward arcs,such that re-orienting the backward arcs yields a directed circuit.The incidence vectorγC∈{−1,0,1}A of an oriented circuit C has a plus(minus)one entry precisely for the forward(backward)arcs of C.Then,the cycle space C(D)can be defined asC(D):=span({γC|C is an oriented circuit of D}).A cycle basisB of C(D)is a set of oriented circuits,which is a basis of C(D). An integral cycle basis allows to combine every oriented circuit of D as an integer linear combination of the basic circuits.Fortunately,in order to decide upon the integrality of a cycle basis,we do not have to check all these linear combinations.Lemma1([4]).LetΓbe the arc-cycle incidence matrix of a cycle basis.For two submatricesΓ1,Γ2with rank(Γ1)=rank(Γ2)=rank(Γ),there holdsdetΓ1=±detΓ2.(4) Definition1(Determinant of a cycle basis,[4]).Let B be a cycle basis and Γ1as in the above lemma.We define the determinant of B asdet B:=|detΓ1|.(5) Theorem4([4]).A cycle basis B is integral,if and only if det B=1.According to Remark1,in the application of periodic timetabling we seek for a minimum integral cycle basis of D.To illustrate the benefit of short integral cycle we provide the following example.Periodic Timetable Optimization in Public Transport33Fig.treeExample1.Consider the sunflower graph SF(3)in Figure2.Assume each arc models a Pesp constraint of the form[7,13]10,i.e.subject to a period time of T=10.According to the initial IP formulation(2),we could deduce that in order to identify an optimum timetable simply by considering every possible timetable vec-torπwe need to check|{0,...,9}||V|=1,000,000vectorsπ.Alternatively,we might check for|{0,1,2}||A|=19,683vectors p to pervade the search space.It will turn out,that these two perspectives have much redundancies within them.In contrast,the valid inequalities of Theorem3reveal that every4-circuit C in D yields an integer variable z C which may only take the three values{−1,0,1}. Even better,a triangle C in D induces afixed variable z C.Thus,the integral cycle basis B that can be derived from the spanning tree F (Fig.2on the right)already reduces the upper bound on the size of the search space to only1·1·1·3=3possible vectors for z.Moreover,considering the minimum cycle basis of D—which for this graph turns out to be integral as it consists of the circuits that bound the fourfinite faces of this plane graph—we end with just one single vector z describing the complete instance.Ideally,we would like to compute a minimum integral cycle basis of D according to the edge weights u a− a.Unfortunately,we are not aware of the asymptotical com-plexity of this combinatorial optimization problem.However,recently there has been achieved much progress on the the asymptotical complexity for the corresponding minimum cycle basis problems for related classes of cycle bases,see[13,2]and ref-erenced therein.We depict these results in Figure3.Notice that any of these classes demands for specific algorithms,because none of these problems coincide([13]).4Summary of Computational ResultsEarlier,in several autonomous computational studies,there had been applied various algorithms to periodic timetabling.We have executed thefirst unified computational study,which covers algorithms as variegated as a Cut-and-Branch Algorithm for Integer Programs,Constraint Programming,and even Genetic Algorithms([12]).To any of these,quite a number of different promising parameter settings was applied. In particular for Integer Programming this amounts to hundreds of different runs onfive different data sets.All the data sets have been provided to us by industrial partners,and they range from long-distance traffic over regional traffic down to undergrounds,comprising between10and40commercial lines each.With respect to both solution quality and independence of parameter settings, both our Genetic Algorithm(GA)and our Cut-and-Branch Algorithm—which is34Christian Liebchendirectedundirected integral weakly strictly TUM 2-basesO (m 3n...)O (m 2n...)(open)O (n )(open)(open)NPC Fig.3.Map of the complexity of the seven variants of the Minimum Cycle Basis Problem for general graphs ([4,13])using CPLEX 9.1—perform considerably well.On the one hand,IP techniques turn out to be extremely sensitive with respect to the choice of several important parameters.On the other hand,in particular for medium-sized instances for which IP techniques still attain an optimum solution,the quality achieved by the GA is somewhat worse.5Improvements for Berlin UndergroundMost important,in a long-term cooperation with Berlin Underground we continu-ously kept on improving our mathematical models of the real world ([10,6]).Finally,in 2004we were able to formulate a mathematical program which covered all the practical requirements that the practitioners have.As a consequence,the optimum solution that was computed by our algorithms convinced Berlin Underground:By December 12,2004,our timetable became the first optimized timetable that went into service—presumably worldwide.This may be compared to the fact that only in operations planning (vehicle scheduling,duty scheduling),Operations Research had already entered the practice.Compared to the former timetable,with our timetable the passengers of Berlin Underground are offered simultaneously improvements in two key criteria,which typically are conflicting:transfer waiting time and dwell time of trains.In more detail,our achievements are:The number of transfers,for which a maximum transfer waiting time of 5minutes can be guaranteed,increases from 95to 103(+8%).The maximum dwell time of any train in the network was reduced from 3.5min-utes to only 2.5minutes (−30%).The timetable could even be operated with one train less.Periodic Timetable Optimization in Public Transport35 The part of the network in which the most significant improvements have been achieved is given in work waiting time charts emerged as a by-product from our cooperation with Berlin Underground([6,7]).Such charts constitute the first visualization of the transfer quality of a timetable.In particular,they made the discussion of pros and cons of different timetables most efficient,as for instance long transfer waiting times(marked in black)along important transfers(marked as bold arcs)become obvious.work waiting time charts for an excerpt of the Berlin subway network—before and after invoking mathematical optimizationThe successful transfer from theory to practice 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