t056 - periodic timetable optimization in public transport

Periodic Timetable Optimization in PublicTransport�

Christian Liebchen

Institut fur Mathematik, Kombinatorische Optimierung und Graphenalgorithmen,Technische Universitat Berlin, [email protected]

Summary.”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 optimizationtechniques that were employed in computing the very first optimized timetable thatwent into daily service: the 2005 timetable of Berlin Underground. This timetablesimproved on both, the passenger travel times and the operating efficiency of thecompany.

The basic graph model, the Periodic Event Scheduling Problem (PESP), isknown for 15 years and it had attracted many research groups. Nevertheless, wereport on significant progress that has been made only recently on issues like solu-tion strategies or modeling capabilities. The latter even includes the integration offurther planning tasks in public transport, such as line planning.

On the theory side, we give a more precise notion of the asymptotical complexityof 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 muchdeeper understanding of cycle bases of graphs, another very hot topic in discretemathematics during the last three years. In 2005, this culminated in both, drawingthe complete map for the seven relevant classes of cycle bases, and the design ofthe fastest algorithms for the Minimum Directed Cycle Basis Problem and for theMinimum 2-Basis Problem.

The book version of this extended abstract is available as reference [8].

1 Timetabling

It is a very important competitive advantage of public transport to be much lessexpensive than a taxi service. This requires many passengers to share the samevehicle. Typically, this is achieved by offering public transport along fixed sets ofroutes, the lines. These serve as input to timetabling.

� This work has been supported by the DFG Research Center Matheon in Berlin.

30 Christian Liebchen

There is a large toolbox of different types of timetables, which we introduce fromthe 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 transfertimes that he faces during his outbound trip are identical to the transfer timesduring 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 competitiveadvantage—at least in long-distance traffic.

Theorem 1 ([7]). There exist example networks showing that each more specializedfamily of timetables causes a nominal loss in quantifiable criteria, such as averagepassenger waiting time.

We are only aware of periodic timetables being able to clearly outweigh theirnominal loss (when comparing with general irregular timetables) by adding benefitin 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 distinguishes

� rush hour service (T = 4 minutes),�

”normal“ service (T = 5 minutes),

� weak traffic service (T = 10 minutes, when retail shops are closed), and� night service (T = 15 minutes, only on weekends).

Computing “the timetable” thus decomposes into computing a periodic timetablefor each period time, and finally glue these together.

2 A Model for Periodic Timetabling

A literature review of different models for periodic scheduling reveals that the mostpromising earlier studies on medium-sized networks are based on the Periodic EventScheduling Problem (Pesp, [18]), see [17, 14, 9, 16]. The vertices in this graphmodel represent events, where an event v ∈ V is either an arrival or a departure ofa directed line in a specific station. A timetable π assigns to each vertex v a pointin time πv ∈ [0,T ) within the period time T . Constraints may then be given in thefollowing form.

T-Periodic Event Scheduling Problem (T-Pesp)

Instance: A directed graph D = (V,A) and vectors �,u ∈ QA.Task: Find a vector π ∈ [0,T )V that fulfills

(πv − πu − �a) mod T ≤ ua − �a (1)

(or πv − πu ∈ [�a,ua]T , for short) for every arc a = (u,v) ∈ A, ordecide that none exists.

Periodic Timetable Optimization in Public Transport 31

In Figure 1 we provide an example instance of T -Pesp, which contains the eventsof two pairs of directed lines and two stations.

Osloer StraßeOl

[1.5,6.5;wa]T

[0.5,1.5;wa]T

[7.0,7.0]T

[1.5,1.5]T

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 orturnarounds of the trains. Besides these most elementary requirements, there havebeen modeled most practical requirements that railway engineers have ([11]). Thiseven includes decisions of line planning and vehicle scheduling, which traditionallywere treated as fully separate planning steps ([1]).

Unfortunately, this modeling power has its price in terms of complexity.

Theorem 2. Let a set of PESP constraints be given. Finding a timetable vector πthat 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 wT (BTπ + Tp)s.t. BTπ + Tp ≤ u

BTπ + Tp ≥ �π ∈ [0,T )V

p ∈ {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 mayreplace the vertex variables π (or node potentials) — which carry time information— with arc variables x (tensions), and/or replace the integer arc variables p withinteger cycle variables z. This way, we end with the following integer program ([14])

min wTxs.t. x ≤ u

x ≥ �Γ Tx− Tz = 0x ∈ QA

z ∈ ZB ,

(3)


where Γ denotes the arc-cycle incidence matrix of an integral cycle basis B of theconstraint graph D = (V,A). There have also been identified lower and upper boundson these integer variables z.

Theorem 3 ([15]). Let C be an oriented circuit and zC the integer variable thatwe associate with it. The following inequalities are valid⎡⎢⎢⎢ 1

T

⎛⎝ ∑a∈C+

�a −∑

a∈C−

ua

⎞⎠⎤⎥⎥⎥ =: zC ≤ zC ≤ zC :=

⎢⎢⎢⎣ 1

T

⎛⎝ ∑a∈C+

ua −∑

a∈C−

�a

⎞⎠⎥⎥⎥⎦ .The following rule-of-thumb could be derived from empirical studies.

Remark 1 ([4]). The shorter a circuit C ∈ B with respect to the sum of thespans ua − �a of its arcs, the less integer values the corresponding variable zC maytake. Moreover, the less values all the integer variables may take, the shorter thesolution times for solving this IP. ��

3 Integral Cycle Bases

In a directed graph D = (V,A), we consider oriented circuits. These consist offorward arcs and maybe also backward arcs, such that re-orienting the backwardarcs yields a directed circuit. The incidence vector γC ∈ {−1,0,1}A of an orientedcircuit C has a plus (minus) one entry precisely for the forward (backward) arcsof C. Then, the cycle space C(D) can be defined as

C(D) := span({γC |C is an oriented circuit of D}).

A cycle basis B 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 integerlinear combination of the basic circuits.

Fortunately, in order to decide upon the integrality of a cycle basis, we do nothave to check all these linear combinations.

Lemma 1 ([4]). Let Γ be the arc-cycle incidence matrix of a cycle basis. For twosubmatrices Γ1,Γ2 with rank(Γ1) = rank(Γ2) = rank(Γ ), there holds

detΓ1 = ±detΓ2. (4)

Definition 1 (Determinant of a cycle basis, [4]). Let B be a cycle basis andΓ1 as in the above lemma. We define the determinant of B as

detB := |detΓ1|. (5)

Theorem 4 ([4]). A cycle basis B is integral, if and only if detB = 1.

According to Remark 1, in the application of periodic timetabling we seek for aminimum integral cycle basis of D. To illustrate the benefit of short integral cyclewe provide the following example.


Fig. 2. The sunflower graph SF(3), and a spanning tree

Example 1. Consider the sunflower graph SF(3) in Figure 2. Assume each arc modelsa 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 toidentify an optimum timetable simply by considering every possible timetable vec-tor π we need to check |{0, . . . ,9}||V | = 1,000,000 vectors π. Alternatively, we mightcheck for |{0,1,2}||A| = 19,683 vectors p to pervade the search space. It will turnout, that these two perspectives have much redundancies within them.

In contrast, the valid inequalities of Theorem 3 reveal that every 4-circuit Cin D yields an integer variable zC which may only take the three values {−1,0,1}.Even better, a triangle C in D induces a fixed variable zC .

Thus, the integral cycle basis B that can be derived from the spanning tree F(Fig. 2 on the right) already reduces the upper bound on the size of the search spaceto only 1 · 1 · 1 · 3 = 3 possible vectors for z. Moreover, considering the minimumcycle basis of D — which for this graph turns out to be integral as it consists of thecircuits that bound the four finite faces of this plane graph — we end with just onesingle vector z describing the complete instance. ��

Ideally, we would like to compute a minimum integral cycle basis of D accordingto the edge weights ua−�a. Unfortunately, we are not aware of the asymptotical com-plexity of this combinatorial optimization problem. However, recently there has beenachieved much progress on the the asymptotical complexity for the correspondingminimum cycle basis problems for related classes of cycle bases, see [13, 2] and ref-erenced therein. We depict these results in Figure 3. Notice that any of these classesdemands for specific algorithms, because none of these problems coincide ([13]).

4 Summary of Computational Results

Earlier, in several autonomous computational studies, there had been applied variousalgorithms to periodic timetabling. We have executed the first unified computationalstudy, which covers algorithms as variegated as a Cut-and-Branch Algorithm forInteger Programs, Constraint Programming, and even Genetic Algorithms ([12]). Toany of these, quite a number of different promising parameter settings was applied.In particular for Integer Programming this amounts to hundreds of different runson five different data sets. All the data sets have been provided to us by industrialpartners, and they range from long-distance traffic over regional traffic down toundergrounds, comprising between 10 and 40 commercial 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 is

34 Christian LiebchenP

directedundirectedintegralweaklystrictly TUM

2-bases

O(m3n . . . )O(m2n . . . )(open)O(n) (open)(open)NPC

Fig. 3. Map of the complexity of the seven variants of the Minimum Cycle BasisProblem for general graphs ([4, 13])

using CPLEX� 9.1—perform considerably well. On the one hand, IP techniquesturn out to be extremely sensitive with respect to the choice of several importantparameters. On the other hand, in particular for medium-sized instances for whichIP techniques still attain an optimum solution, the quality achieved by the GA issomewhat worse.

5 Improvements for Berlin Underground

Most 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 2004 we were able to formulate a mathematical program which covered all thepractical requirements that the practitioners have. As a consequence, the optimumsolution that was computed by our algorithms convinced Berlin Underground: ByDecember 12, 2004, our timetable became the first optimized timetable that wentinto service—presumably worldwide. This may be compared to the fact that onlyin operations planning (vehicle scheduling, duty scheduling), Operations Researchhad already entered the practice.

Compared to the former timetable, with our timetable the passengers of BerlinUnderground are offered simultaneously improvements in two key criteria, whichtypically are conflicting: transfer waiting time and dwell time of trains. In moredetail, our achievements are:

� The number of transfers, for which a maximum transfer waiting time of 5 minutescan be guaranteed, increases from 95 to 103 (+8%).

� The maximum dwell time of any train in the network was reduced from 3.5 min-utes to only 2.5 minutes (−30%).

� The timetable could even be operated with one train less.


The part of the network in which the most significant improvements have beenachieved is given in Figure 4. Network waiting time charts emerged as a by-productfrom our cooperation with Berlin Underground ([6, 7]). Such charts constitute thefirst visualization of the transfer quality of a timetable. In particular, they madethe discussion of pros and cons of different timetables most efficient, as for instancelong transfer waiting times (marked in black) along important transfers (marked asbold arcs) become obvious.

04.5

02.0

Sn

Be

Fp

08.5 06.0

Sn

Be

Fp

Fig. 4. Network waiting time charts for an excerpt of the Berlin subway network—before and after invoking mathematical optimization

The successful transfer from theory to practice has even been reflected by articlesand interviews in nationwide newspapers and radio transmissions:

� Berliner Zeitung, November 9, 2005, in German ([3])http://www.berlinonline.de/berliner-zeitung/archiv/.bin/dump.fcgi/2005/1109/wissenschaft/0002/index.html

� Deutschlandfunk, December 9, 2005, 16:35h, in Germanhttp://www.dradio.de/dlf/sendungen/forschak/446751/

References

1. Michael R. Bussieck, Thomas Winter, and Uwe Zimmermann. Discreteoptimization in public rail transport. Mathematical Programming B,79:415–444, 1997.

2. Ramesh Hariharan, Telikepalli Kavitha, and Kurt Mehlhorn. A FasterDeterministic Algorithm for Minimum Cycle Bases in Directed Graphs. InMichele Bugliesi et al., editors, ICALP, volume 4051 of Lecture Notes inComputer Science, pages 250–261. Springer, 2006.

3. Reinhard Huschke. Schneller Umsteigen. Berliner Zeitung, 61(262):12,2005. Wednesday, November 9, 2005, In German.

4. Christian Liebchen. Finding short integral cycle bases for cyclictimetabling. In Giuseppe Di Battista and Uri Zwick, editors, ESA, volume2832 of Lecture Notes in Computer Science, pages 715–726. Springer, 2003.

5. Christian Liebchen. A cut-based heuristic to produce almost feasible pe-riodic railway timetables. In Sotiris E. Nikoletseas, editor, WEA, volume3503 of Lecture Notes in Computer Science, pages 354–366. Springer, 2005.


6. Christian Liebchen. Der Berliner U-Bahn Fahrplan 2005 – Realisierungeines mathematisch optimierten Angebotskonzeptes. In HEUREKA ’05:Optimierung in Transport und Verkehr, Tagungsbericht, number 002/81.FGSV Verlag, 2005. In German.

7. Christian Liebchen. Fahrplanoptimierung im Personenverkehr—Muss esimmer ITF sein? Eisenbahntechnische Rundschau, 54(11):689–702, 2005.In German.

8. Christian Liebchen. Periodic Timetable Optimization in Public Transport.dissertation.de, 2006. PhD thesis.

9. Thomas Lindner. Train Schedule Optimization in Public Rail Transport.Ph.D. thesis, Technische Universitat Braunschweig, 2000.

10. Christian Liebchen and Rolf H. Mohring. A case study in periodictimetabling. Electr. Notes in Theoretical Computer Science, 66(6), 2002.

11. Christian Liebchen and Rolf H. Mohring. The modeling power of the peri-odic event scheduling problem: Railway timetables – and beyond. Preprint020/2004, TU Berlin, Mathematical Institute, 2004. To appear in SpringerLNCS Volume Algorithmic Methods for Railway Optimization.

12. Christian Liebchen, Mark Proksch, and Frank H. Wagner. Performanceof algorithms for periodic timetable optimization. To appear in SpringerLNEMS PProceedings of the Ninth International Workshop on Computer-Aided Scheduling of Public Transport (CASPT). To appear.

13. Christian Liebchen and Romeo Rizzi. Cycles bases of graphs. TechnicalReport 2005-018, TU Berlin, Mathematical Institute, 2005. accepted forpublication in Discrete Applied Mathematics.

14. Karl Nachtigall. Periodic Network Optimization and Fixed IntervalTimetables. Habilitation thesis, Universitat Hildesheim, 1998.

15. Michiel A. Odijk. A constraint generation algorithm for the constructionof periodic railway timetables. Transp. Res. B, 30(6):455–464, 1996.

16. Leon W.P. Peeters. Cyclic Railway Timetable Optimization. Ph.D. thesis,Erasmus Universiteit Rotterdam, 2003.

17. Alexander Schrijver and Adri G. Steenbeek. Dienstregelingontwikkelingvoor Railned. Rapport CADANS 1.0, Centrum voor Wiskunde en Infor-matica, December 1994. In Dutch.

18. Paolo Serafini and Walter Ukovich. A mathematical model for pe-riodic scheduling problems. SIAM Journal on Discrete Mathematics,2(4):550–581, 1989.

t056 - periodic timetable optimization in public transport

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