When Periodic Timetables are Suboptimal

Transcription

1 Konrad-Zuse-Zentrum ür Inormationstechnik Berlin akustraße 7 D Berlin-Dahlem Germany RALF BORNDÖRFER CHRISIAN LIEBCHEN When Periodic imetables are Suboptimal his work was unded by the DFG Research Center MAHEON Mathematics or Key echnologies, project B15 Service Design in Public ransport. o appear in Operations Research Proceedings 2007, Springer Verlag. ZIB-Report (October 2007)

2 When Periodic imetables are Suboptimal Ral Borndörer and Christian Liebchen Konrad-Zuse-Zentrum ür Inormationstechnik Berlin, akustr. 7, Berlin-Dahlem, Germany echnische Universität Berlin, Institut ür Mathematik, Str. des 17. Juni 136, Berlin, Germany Summary. he timetable is the essence o the service oered by any provider o public transport (Jonathan yler, CASP 2006). Indeed, the timetable has a major impact on both operating costs and on passenger comort. Most European agglomerations and railways use periodic timetables in which operation repeats in regular intervals. In contrast, many North and South American municipalities use trip timetables in which the vehicle trips are scheduled individually subject to requency constraints. We compare these two strategies with respect to vehicle operation costs. It turns out that or short time horizons, periodic timetabling can be suboptimal; or suiciently long time horizons, however, periodic timetabling can always be done in an optimal way. 1 Introduction he construction o the timetable is perhaps the most important scheduling activity o a railway or public transport company. It has a major impact on operating costs and on passenger comort. he problem has been extensively covered in the operations research literature, see [2] or a recent survey. here are two main timetabling strategies that dier w.r.t. to structural dependencies between individual trips. In a periodic timetable, there is a ixed time interval between two trips; i a single trip is scheduled on a directed line, all other trips o this line are determined. In contrast, in a trip timetable, each trip is scheduled individually, subject to requency constraints. Stipulating appropriate constraints, a trip timetable can be orced to become periodic, or, to put it the other way round, trip timetables eature more degrees o reedom than periodic ones. We investigate in this paper the question whether this reedom can be used to lower operation costs in terms o numbers o vehicles. O course, such improvements (i any) come at the price o diminishing the regularity o the timetable. Supported by the DFG Research Center Matheon (

3 2 Ral Borndörer and Christian Liebchen 2 he imetabling Problem We consider the timetabling problem or a single, bidirectional line between two stations A and B. he line is operated by homogenous vehicles with running times t ab and t ba in directions A B and B A, respectively (these include minimum turnaround times in the respective terminus stations). We want to construct a timetable that covers N time periods o length with a trip requency o vehicles per time period, and such that the minimum headway between two consecutive trips is at least l and at most u. We assume that divides and call / the period time o the timetable (to be constructed). We urther assume l / u and that all mentioned numbers are positive integers except or l, which is supposed to be a non-negative integer. he timetable that we want to construct involves m := N departures at station A, that we denote by U = {u 1,...,u m }, and the same number o departures at station B, that we denote by V = {v 1,...,v m }; let U V be the set o all these departure events. A timetable is a unction t : U V Z that maps departures to times such that the ollowing conditions hold: (i) t(u i ) t(u i+1 ) i = 1,...,m 1 t(v i ) t(v i+1 ) i = 1,...,m 1 (ii) (i 1)/ t(u i ) < ( (i 1)/ + 1) i = 1,...,m 1 (i 1)/ t(v i ) < ( (i 1)/ + 1) i = 1,...,m 1 (iii) l t(u i+1 ) t(u i ) u, i = 1,...,m 1 l t(v i+1 ) t(v i ) u, i = 1,...,m 1. Constraints (i) ensure that the departure times at both stations ascend in time, (ii) guarantees departures in each period at each station, and (iii) enorce a minimum and maximum headway o l and u between two consecutive departures o trips. A timetable is a periodic timetable i condition (iii) is replaced by (iii ) t(u i+1 ) t(u i ) = t(v i+1 ) t(v i ) = /, i = 1,...,m 1, otherwise it is a trip timetable. Note that a timetable can be orced to be periodic by stipulating l = / = u. he timetabling problem is to determine a easible timetable that can be operated with a minium number o vehicles 2. We remark that our problem deinition deliberately omits technical constraints, such as passing sidings, in order to ocus upon purely structural implications. 2 his deinition is made in our context; there are other types o timetabling problems in the literature.

4 3 Periodic vs. rip imetables When Periodic imetables are Suboptimal 3 Lemma 1. Consider a public transport line between stations A and B with running times t ab and t ba which include the minimum turnaround times in the respective terminus stations, such that t ab + t ba is an integer multiple o. hen, operating this line or a time duration o at least N > t ab + t ba requires at least Z := t ab + t ba (1) vehicles in an arbitrary timetable. Proo. At least vehicles have to be scheduled in each o the irst Z/ time periods until the irst vehicle can be reused or a second trip in the same direction. Lemma 2. Consider a public transport line between stations A and B with running times t ab and t ba (again including minimum turnaround times). Operating this line at shorter running times t ab t ab and t ba t ba does not increase the number o vehicles that are required or operation in the respectively best arbitrary timetables or this line. Proo. For the optimal timetables or running times t ab and t ba there exist timetables with running times t ab and t ba that can be operated at the same number o vehicles. In act, add a turnaround waiting time at terminus station B o t ab t ab, and a waiting time o t ba t ba at station A. [t ab, t ab + ε] A B [t ba, t ba + ε] Fig. 1. he so-called Pesp-graph (see, e.g., [2]) or the situation o the periodic vehicle circulation as it is considered in Prop. 1. Proposition 1. Consider a public transport line between stations A and B with a period time and running times t ab and t ba (again including minimum turnaround times in the terminus stations) such that t ab + t ba < N /. hen, any periodic timetable requires at least Z 0 := t ab + t ba vehicles or operation. Moreover, any periodic timetable or this line can be operated with at most Z vehicles. Indeed, there exist periodic timetables that can be operated with Z 0 vehicles. (2)

5 4 Ral Borndörer and Christian Liebchen Proo. Using the cycle inequalities due to Odijk [4], it had been observed by Nachtigall [3] that there exists some appropriate ε > 0 such that the ollowing general bounds on the number Z o vehicles are valid or all periodic timetables, and tight or some timetables: Z 0 := t ab + t ba tab + Z ε + t ba + ε Z (3) Lemma 3. Consider a public transport line between stations A and B with running times t ab and t ba which once more include the minimum turnaround times in the respective terminus stations. hen, operating this line or at least a time duration o N > t ab + t ba requires at least tab + t ba Z 1 := (4) vehicles in an arbitrary timetable. heorem 1. For each public transportation line with running times t ab and t ba (including minimum turnaround times), there exists a number N 0 N such that operating the line or a time duration o at least N 0 requires at least Z 0 = t ab + t ba vehicles or operation. In other words, or suiciently long time horizons, the minimum number o vehicles needed to operate a trip timetable is equal to the minimum number o vehicles needed to operate a periodic timetable. Proo. In a time duration o N, or the two directions o the line together there must be scheduled at least 2N trips. In turn, one vehicle can cover no 2N more than o these trips. t ab +t ba 2N 0 Now, choose N 0 such that t ab +t ba becomes integer. hen, the number o required vehicles is bounded rom below by (5) 2N 0 2N 0 t ab +t ba = (t ab + t ba ) (6) = t ab + t ba. (7) Since we must only consider integer quantities o vehicles, (5) ollows.

6 When Periodic imetables are Suboptimal 5 4 Example Let the balancing intervalequal = 60 minutes, and let the number o required trips within this interval be = 3. We consider a parameterized oneway running time t ab = t ba = 60 c, which includes the minimum turnaround times, or c {1, 2,..., 10}. A 01:00 B 02:00 03:00 04:00 05:00 06:00 07:00 Fig. 2. A timetable or t ab = t ba = 52, that uses only ive vehicles. In the irst ive hours, there are three departures rom each terminus station. But in the sixth hour, there are only two departures rom B, which makes this timetable ineasible or N = 6. he lines on the right indicate the vehicle that covers two trips in the corresponding hour. First, observe that whenever c < 10, then the number o vehicles that are required in the best periodic timetables equals t ab + t ba 120 2c 120 2c 60 c Z 0 = = 60 = = = 6. (8) Second, it can be veriied that in any trip timetable, we need at least ive vehicles or operating this line over at least three hours. Last, recall rom

7 6 Ral Borndörer and Christian Liebchen the proo o heorem 1 that ater at least 100 2c hours we can be sure to also require six vehicles when scheduling each trip individually. Yet, in this example we will show that, or certain running times, as early as ater at least six hours we are sure to need the sixth vehicle also in any trip timetable. he irst simple observation is that within each hour, when considering both directions o the line together, there must be six trips in the schedule. Hence, in order to need only ive vehicles, there must be one vehicle within each hour that covers two o these six trips. O course, these two trips must be in opposite direction. Now comes the key observation: I some ixed vehicle covers two trips in hour X, then it cannot cover two trips in any o the hours {X + 1,...,X + c 2}. As a consequence, i the one-way running time was tab = 52, and thus c = 8, then no vehicle can cover two trips in two o the hours {1,...,6}, because o X + c 2 = = 6. Hence, in the sixth hour o operation, the latest, a sixth vehicle has to be put into operation, c. Fig Passenger Waiting imes One can also analyze the dierences between periodic and trip timetables with respect to passenger comort, i.e., waiting times. An analysis o this type can be ound in [1]. here, an example o a transportation network is given, in which passengers spend strictly less waiting time at transers, i a trip timetable is used instead o when a periodic timetable is used. Reerences 1. Christian Liebchen. Fahrplanoptimierung im Personenverkehr Muss es immer IF sein? Eisenbahntechnische Rundschau, 54(11): , In German. 2. Christian Liebchen. Periodic imetable Optimization in Public ransport. Ph.D.thesis, echnische Universität Berlin, dissertation.de Verlag im Internet. 3. Karl Nachtigall. Periodic Network Optimization and Fixed Interval imetables. Habilitation thesis, Universität Hildesheim, Michiel A. Odijk. A constraint generation algorithm or the construction o periodic railway timetables. ransportation Research B, 30(6): , 1996.