An experimental comparison of periodic timetabling models

Computers & Operations Research 40 (2013) 2251–2259

Contents lists available at SciVerse ScienceDirect

Computers & Operations Research journal homepage: www.elsevier.com/locate/caor

An experimental comparison of periodic timetabling models Michael Siebert, Marc Goerigk n Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Germany

art ic l e i nf o

a b s t r a c t

Available online 11 April 2013

In the Periodic Timetabling Problem, vehicle arrivals and departures need to be scheduled over a periodically repeating time horizon. Its relevance and applicability have been demonstrated by several real-world implementations, including the Netherlands railways and the Berlin subway. In this work, we consider the practical impact of two possible problem variations: firstly, how passenger paths are handled, and secondly, how line frequencies are included. In computational experiments on real-world and close-to real-world networks, we can show that passenger travel times can significantly benefit from extended models. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Periodic timetabling Iterative timetabling Integrated passenger routing Event activity networks

1. Introduction The mathematical optimization of timetabling problems has found much interest in the recent literature. In [5], both robust and non-robust problem formulations for finding periodic and nonperiodic schedules are surveyed. In [1], various models for the related problem of track allocation are discussed, where for a given set of train route requests a conflict-free subset of maximum value has to be determined. The authors of [6] consider the problem of finding a timetable for a single track connecting two major stations, where trains can overtake in some of the intermediate stations. They present an integer linear program and solve it using a Lagrange relaxation approach. The authors of [3] consider the non-periodic train timetabling problem, where the objective is to find a set of timetables yielding maximum profit, and model their problem with comparability graphs representing which train paths are compatible, i.e., do not block each other due to capacity constraints. The problem of adding freight trains into an existing timetable of passenger trains is modeled in [4], where a corresponding integer linear program is solved using a Lagrangian heuristic. Both [27] and [2] discuss the problem of finding timetables that are insensitive to disruptions, i.e., robust. While the former proposes a bicriteria approach to balance passenger travel times and robustness, the latter modify the Lagrangian optimization scheme to generate a set of solutions with different trade-off between travel time and robustness in a similar spirit. In this work, we consider the Periodic Event Scheduling Problem (PESP) as introduced in [28], which is used to model periodically

n

Corresponding author. Tel.: +49 6312053878. E-mail addresses: [email protected] (M. Siebert), [email protected] (M. Goerigk). 0305-0548/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cor.2013.04.002

reappearing events due to given time span constraints. It has been successfully applied to the Periodic Timetabling Problem (see, e.g. [22,14,20]), which is usually stated as follows. Given a directed graph EAN ¼ ðE; AÞ (the event-activity network), where we call E the set of events, A the set of activities, as well as a period T∈N, lower and upper duration bounds la ; ua ∈N for each activity a∈A and passenger weights wa ∈R≥0 , a∈A, find a solution x∈NjAj of activity durations minimizing the total passenger travel time ∑a∈A wa xa , such that all duration bounds are fulfilled and for every cycle in EAN, the sum of durations is a multiple of T. It turns out that there are multiple ways to actually build the EAN from what data is available to an operation planner. Specifically, the model as introduced above assumes that (1) passenger paths are known, especially before the actual timetable is known, and (2) line frequencies are implicitly cared for in the set of activities. The first problem has already been considered in recent works (see, e.g. [16,11,25,24,17]), where the authors consider iterative rerouting approaches as well as integrated problems. The problem of different arc frequencies has been considered in [28] and [19]. In [7] a stronger integer programming formulation was introduced. Our contribution. In computational experiments, we show that the quality of the timetabling model crucially depends on how the problems (1) and (2) are dealt with. To the best of our knowledge, these properties have not yet been considered at the same time. We present two ways to handle (2): by frequency as attribute (FA), which is to either ignore frequencies or implicitly care for them and corresponds to the approach from [19], and by frequency as multiplicity (FM) which is to include frequencies by adding additional activities, and derive model properties. Specifically, we

2252

M. Siebert, M. Goerigk / Computers & Operations Research 40 (2013) 2251–2259

discuss that the global optimum of (FM) does not necessarily correspond to an extreme point of the (FA) constraint polyhedron. Furthermore, we consider rerouted timetables that use the combined advantages of both models in order to converge to better solutions than when using only one of these models. In extensive computational experiments using datasets covering the size of the German long-distance rail network, the metro of Athens, and an artificial network, we study the effect of (1) and (2) on passenger travel times. Our data suggests that travel durations can be significantly reduced by our approach. Overview. The remainder of this work is structured as follows: in Section 2, we elaborate on how different line frequencies can be included in the EAN. In Section 3 we consider the iterative retimetabling approach under the light of these findings, introduce variants to perform retimetabling in Section 4 and finally present computational results in Section 5.

2. Periodic timetabling models 2.1. The classic model We assume that in a previous step, a line concept has been determined. For a survey on lineplanning models and methods, see, e.g. [26]. Notation 2.1. A public transport network PTN ¼ ðS; TÞ is an undirected, simple graph in which the nodes represent stations, and the edges represent connections between them. A line l is a path in the PTN, and a line concept L ¼ fðl1 ; f 1 Þ; …; ðln ; f n Þg is given by a set of lines with corresponding integer frequencies. The aim of the periodic timetabling problem is to schedule the departure and arrival times of these lines such that passenger travel times are minimized. We therefore further assume that the passenger demand is known, by means of an origin destination matrix OD ¼ ðodss′ Þs;s′∈S , where odss′ denotes the number of passengers who want to travel from s to s′ within the considered time horizon. The PTN is then transformed to an event activity network EAN ¼ ðE; AÞ, where E ¼ E arr ∪E dep represent the arrivals and departures of vehicles, and A ¼ Adrive ∪Await ∪Achange ∪Aheadway represent driving, waiting, changing activities, as well as safety headways between vehicles. We denote by α : E-S the map that assigns to each event in the EAN the station in the PTN where it takes place. For each stop of a line at a station, an arrival and a departure event are introduced. These are connected by a wait activity, that models the time passengers need to get on and off the vehicle. Departure events are then connected to the subsequent arrival event by driving activities that model the time a vehicle needs between two stations. Changing activities model passengers leaving the arrival event of one line to enter the departure event of another. Finally, headway activities can be used between two departure events if their respective lines share infrastructure. The transformation is discussed in more detail in Section 2.2, see also Fig. 1 for an example. As the travel path a passenger chooses would already depend on the timetable itself, a heuristic path in the EAN for every OD-pair odss′ is determined, resulting in passenger weights wa for every activity a∈A. For a given period T, the original periodic timetabling problem can now be formulated as an integer program

in the following way [28]: min

∑ wij ðπ j −π i þ Tzij Þ

ð1Þ

ði;jÞ∈A

lij ≤π j −π i þ Tzij ≤uij

ði; jÞ∈A

ð2Þ

π i ∈N

i∈E

ð3Þ

zij ∈Z

ði; jÞ∈A

ð4Þ

We call the objective function (1) evaluated at ðw; πÞ the (total passenger) traveling time TT w.r.t. w and π. Note that z is uniquely determined by π, if we w.l.o.g. assume ua −la ≤T−1, for all a∈A. A variable π i gives one representative time for an event i∈E. However, as these events repeat periodically, this also means that event i occurs also at the times f…; π i −2T; π i −T; π i ; π i þ T; …g. To model this repetition, variables zij are introduced, called modulo parameters, as they allow the measurement of time differences modulo T. The parameters l and u denote the minimal and maximal allowed activity duration, e.g. the time a vehicle needs to travel from one station to the other, or the time a vehicle is allowed to spend inside a station. By substituting xij ≔π j −π i , this model can be reformulated to a computationally better tractable problem formulation [18]: min

ð5Þ

∑wa xa a

∑ xa − ∑ xa ¼ zc T;

a∈C þ

a∈C −

la ≤xa ≤ua

C∈C

a∈A

ð6Þ ð7Þ

xa ∈∈

a∈A

ð8Þ

zC ∈Z

C∈C

ð9Þ

Here, the variables xa denote the duration of an activity a∈A. Constraints (6) ensure that for every cycle C∈C, these durations sum up to a multiple of the period T. It is shown [28] that not all cycles need to be considered, but only an integral cycles basis, e.g. the fundamental cycles implied by any spanning tree. 2.2. Event activity networks We now consider the transformation from the PTN to the EAN in more detail. Specifically, we identified two different methods of representation: Frequency as Attribute (FA), and Frequency as Multiplicity (FM). Frequency as Attribute. Every line ðl; f Þ∈L becomes a single sequence of drive- and wait activities. Different line frequencies are ignored. Frequency as Multiplicity. Every line ðl; f Þ∈L is transformed to f sequences of drive- and wait activities. Departure events are additionally connected by synchronization activities, whose lower and upper bounds are set to T=f . In Fig. 1, a small example for both EAN representations is given. On top is a PTN consisting of a single line from s1 to s4 with frequency 3, below are both EAN representations with timetables. Note the synchronization activities in the (FM) model that ensure equal time intervals between departures. We will refer to the transformation from (FA) to (FM) representation as periodic rollout or rollout for short. If ðEAN; πÞ is in (FA) state and we obtain ðEAN′; π′Þ by rollout, we say that ðEAN′; π′Þ corresponds to ðEAN; πÞ. Though FM results in larger networks, the additional information might result in timetables that yield better travel times when passengers are rerouted on shortest paths within the network. This trade-off will be analyzed in the following sections. Furthermore, though the line frequencies are lost when using the original

M. Siebert, M. Goerigk / Computers & Operations Research 40 (2013) 2251–2259

2253

Fig. 1. EAN construction and periodic rollout: drive activity bounds are taken from the PTN, for wait and change activities we assume global defaults ([1,3] for wait and change see Fig. 2).

PESP formulation on (FA) networks, they can still be included in the EPESP model as introduced in the next section. 2.3. Extended models In this section, two models are described that extend the scope of the PESP: one that includes variable passenger routing, and one that includes line frequencies within (FA) event activity networks. The Origin Destination aware PESP. This model has been proposed, e.g. in [16]. Instead of fixing the passenger paths heuristically, routing and timetabling are integrated in a single model. Let Ap ¼ Adrive ∪Await ∪Achange D A be the set of passenger visible activities, and C be an integral cycle basis. Denote by ðaaε Þa∈A;ε∈E the arc-node-incidence matrix of ðE; AÞ. Let E sdep ,

denote the events that take place at station s∈S, and define E sarr accordingly. We modify the original PESP formulation by introducing variables pas1 s2 that denote if activity a is used by passengers traveling from s1 to s2, and corresponding flow constraints. The Origin Destination aware PESP (ODPESP) is then given as the following quadratic program: ∑ ods1 s2 pas1 s2 xa

∑

ð10Þ

a∈Ap s1 ;s2 ∈S

|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ :wa

∑ xa − ∑ xa ¼ zc T; ∑ s

C∈C;

a∈C −

a∈C þ

∑ aaε pas1 s2 ¼ 1;

s1 ; s2 ∈S

ð11Þ ð12Þ

1 a∈Ap ε∈E dep

∑ aaε pas1 s2 ¼ 0;

Lemma 2.2 (Lower Bound). A (trivial) lower bound for the ODPESP is n

objODPESP ¼

s1 ; s2 ∈S; ε∈E\ðE s1 ∪E s2 Þ

ð13Þ

a∈Ap

s1 ; s2 ∈S

ð14Þ

2 a∈Ap ε∈E arr

la ≤xa ≤ua

zC ∈Z

ð15Þ

a∈A

xa ∈N; pas1 s2 ∈f0; 1g C∈C

a∈Ap

ods1 s2 pas1 s2 la

ð18Þ

We use this lower bound to measure the quality of a timetable solution. Definition 2.3 (Gap). For a given timetable x, the traveling time gap or gap for short is " s1 ;s2 ∈S; # ∑a ¼ ði;jÞ∈Ap ods1 s2 pas1 s2 x gapðxÞ ¼ 100 −1 ; ð19Þ n objODPESP where pas1 s2 are shortest paths w.r.t. x. Note that gapðxÞ might be larger than zero, even if x is an optimal solution to ODPESP. Another property of the gap is that it is independent of whether we use (FA) or (FM), if we construct the EAN from a PTN as depicted in Fig. 1. The Extended PESP. In this model, which has been considered in [28], we do not assume a common period T for all activities, but instead assume that every activity ði; jÞ∈A can imply a different period Tij. Adjusting the PESP model is straightforward: min

∑ ∑ aaε pas1 s2 ¼ −1; s

∑

s1 ;s2 ∈S;

with pas1 s2 being shortest paths w.r.t. la.

E s ≔fe∈E : αðeÞ ¼ sg

min

Constraints (12)–(14) model the passenger flow. Contrary to the original PESP model, the passenger weights wa ¼ ∑s1 ;s2 ∈S ods1 s2 pas1 s2 are therefore considered as variables. Due to its increased complexity (the PESP is known to be NP-hard [28], and the integrated problem is NP-hard, even if modulo parameters are not considered [25,24]), we do not aim at solving ODPESP directly, but instead use it as a quality measure for other PESP models. To do so, we note that a simple lower bound to ODPESP can be achieved by assuming all x are set to their respective lower bounds, and passengers are routed on shortest paths:

a∈A

ð16Þ ð17Þ

∑ wij ðπ j −π i þ T ij zij Þ

ði;jÞ∈A

lij ≤π j −π i þ T ij zij ≤uij π i ∈N

i∈E

zij ∈Z

ði; jÞ∈A

ði; jÞ∈A

The idea of the EPESP is to use the simple (FA) EAN model and to account for line frequencies by allowing passengers to change at any station s between any two lines ðℓ; ℓ′Þ within the shortest time

2254

M. Siebert, M. Goerigk / Computers & Operations Research 40 (2013) 2251–2259

in the (FM) model over all changes between ðℓ; ℓ′Þ at s, see Fig. 2 for a visualization. For every activity a∈A that is created by a line of frequency f, we assume a periodicity of T=f . 2.4. Model discussion At this point, we have introduced the following timetabling models: (1) PESP on (FA) and (FM) networks (2) ODPESP on (FA) and (FM) networks (3) EPESP on (FA) networks PESP (FA) and ODPESP (FA) both ignore line frequencies, while PESP (FM), ODPESP (FM) and EPESP (FA) are capable of including them. In the following, we discuss some of the model differences; in particular, we analyze the visibility of passenger connections and describe an example where solutions to (FA) and (FM) networks differ. Visibility of connections. Depending on the model, we obtain different sets of activities A and especially Ap , the sets of passenger visible activities. They determine the ODPESP objective (10) and the (E)PESP objective (5) as well by determining wa, for all a∈A. They also provide the subgraph available for the shortest paths in the retimetabling method we introduce in Section 3. Fig. 2 shows a small example of the connection visibility for (FA)/ (FM) PESP and (FA) EPESP. Two lines ℓ and ℓ′ that cross at a station s are given. Their respective frequencies are 2 and 3, their arrival and departure times π and π′. A passenger who wants to change from ℓ to ℓ′ could therefore use one of 6 possible connections. In this situation, the solid connection between the representative events is the only one that is visible within the (FA) PESP model, while the (FA) EPESP model finds the shortest change duration along the dashed arc. In the (FM) model, all change connections are visible. The Timetabler's Nightmare. We present an example in which the discussed difference of connection visibility has a large impact on the calculated timetable. Fig. 3 shows the PTN under consideration: it consists of four stations fs1 ; s2 ; s3 ; s4 g and an OD matrix in which only one passenger wants to travel from s1 to s4. Two lines are available: A long line that connects s1, s2, s3 and s4 with a frequency of 2, and a short line connecting s2 and s3 once per time period T¼ 10. However, ℓ1 needs at least five time units to get from s2 to s3, while ℓ2 only one. Do the passengers profit from a change? Depending on the timetabling model, the answer might be different. We first consider the optimal solution to the (FA) ODPESP model.

Fig. 3. Example, PTN for the timetabler's nightmare.

The passenger enters line 1 at s1, drives one time unit to s2, waits for another time unit, then uses the same line to proceed to s3, waits, and drives to s4. The total travel time needed is 9 time units, see Fig. 4(a). Changing into line 2 at s2 does not improve the travel time, as the passenger would still need to wait for the slower line 1 at s3. We now consider if this travel time can be improved if we fix the timetable, perform a periodic rollout, and reroute the passenger. The resulting network is shown in Fig. 4(b). As line 1 has a frequency of 2, another instance is generated with an offset of 5 time units. Still, the passenger cannot arrive at s4 in less than 9 time units. With the (FM) model on the other hand, there is the possibility to arrive at s4 in just 5 time units, if the driving time of line 1 from s2 to s3 is increased from 5 to 6. The passenger starts with line 1 at s1, changes to line 2 at s2, then uses the second instance of line 1 to travel on to s4. As this example demonstrates, the (FA) ODPESP optimum can be worse than the (FM) ODPESP optimum. It is a special case of a class of networks discussed in [29], for which the ratio between both optimal values may grow linearly in the period length or (alternatively) the number of stations. There are two reasons why it may be considered as a timetabler's nightmare: firstly, iterative methods to obtain an (FA) ODPESP optimum may worsen the travel time, which actually happens for real networks as documented in Section 5. Secondly, there does not seem to be a chance to losslessly reduce (FM) to (FA), see Remarks 3.4 and 3.5.

3. Retimetabling As the PESP is already challenging to solve for real-world instances, the even larger ODPESP is unlikely directly solvable. However, there is an evident heuristic: Retimetabling, as proposed in [20,11,16,29], which consists of calculating activity weights by routing passengers on shortest paths through the EAN, and solving the resulting timetabling problem iteratively. Definition 3.1 (Retimetabling Step). Solving the PESP for the passenger distribution w with timetable x and rerouting passengers along shortest durations w.r.t. x is called Retimetabling step or ReTim step for short. Definition 3.2 (Retimetabling). A loop of ReTim steps, i.e., solving the PESP and distributing passengers along shortest paths w.r.t. the last timetable yields a sequence of passenger distributions and timetables w0 ; x0 ; w1 ; x1 ; … and is called Retimetabling or ReTim for short. Let TTn be the (total passenger) traveling time w.r.t. xn and the passenger distribution wn. The ReTim limit is defined as limReTim ≔limn-∞ TTn .

Fig. 2. Change activities and their visibility for (FA) PESP, (FA) EPESP and (FM) PESP. In our instances, they all have the same bounds, e.g. ½5; T þ 4 with T ¼ 60 fits here.

Fixing the passenger distribution transforms ODPESP into a PESP, and fixing the durations into an all pairs shortest paths problem. Therefore, ReTim only solves subproblems and if it n converges it holds limReTim ≥objODPESP . In [20] it is shown that

M. Siebert, M. Goerigk / Computers & Operations Research 40 (2013) 2251–2259

2255

Fig. 4. The timetabler's nightmare EAN, derived from Fig. 3: (a) and (c) depict the (FA) resp. (FM) with their optimal ODPESP solutions. Note that after rollout, the solution from (a) cannot be improved by a retimetabling step as depicted in (b). (a) FA, (b) FA + rollout and (c) FM.

ReTim converges if the PESP is always solved to full optimality. We extend this results to the use of heuristics in the timetabling step. Theorem 3.3 (Retimetabling Convergence). If ∑a∈Ap wna xna ≤∑a∈ Ap wna xn−1 for all n≥1, TTn decreases monotonically and converges. a n−1 with s1 ; s2 ∈S, a∈Ap be a feasible solution with Proof. Let pn−1 as1 s2 , x objective value TTn−1 . Let pnas1 s2 , a∈A, s1 ; s2 ∈S be shortest paths from the next ReTim step with passenger distribution wna ¼ ∑s1 ;s2 ∈S pnas1 s2 , for all a∈Ap , and wn−1 analogous for pn−1 . By definition, it holds that ∑a∈Ap n−1 wna xn−1 ≤∑a∈Ap wn−1 a a xa . A timetabling iteration provides xn. Using our assumption, ReTim is therefore monotonic decreasing and bounded, and thus converges. □

Note that ∑a∈Ap wna xna ≤∑a∈Ap wna xn−1 is fulfilled when PESP is a solved to optimality, but also even for suboptimal solutions if the timetable of the previous iteration is used as a starting solution for the next. We now consider if it is possible to construct an objective function for the (FA) PESP that has the following property: a rollout applied to an optimum yields an optimum of the corresponding rolled out ODPESP instance. If that was the case, we could do without (FM); we would not need to consider the path decision variables anymore and thus could reduce the problem size drastically. Consider again the network from Fig. 4. Here, xd ¼ 5 leads into a local retimetabling optimum, even after performing a periodic rollout. We show that no choice of nonnegative coefficients in a PESP objective function prevents this from being an (FA) optimum. Since there are no modulo parameters involved, our results apply to aperiodic timetabling as well. A general linear objective function is given by ~ c þ γ~ x′c þ C~ ~ d þ βx min αx

ð20Þ

~ γ~ ; C~ ≥0. It holds ~ β; with α; xc þ x′c −1 ¼ xd

ð21Þ

and thus we can rewrite the objective to ~ c þ ð~γ þ αÞx′ ~ c þ ðC~ −αÞ: ~ minðβ~ þ αÞx

ð22Þ

With β ¼ β~ þ α~ and γ ¼ γ~ þ α~ which can still be chosen freely and without constant, there is min βxc þ γx′c :

ð23Þ

If β and γ are both zero the objective function is constant and therefore any feasible solution like xd ¼ 5 is optimal. Let w.l.o.g. γ 4 0. It remains min αxc þ x′c ;

ð24Þ

with α ¼ β=γ. There are four possibilities: α ¼ 0, α ¼ 1, α∈ð0; 1Þ and α 4 1. For α ¼ 1 the objective is equivalent to minimize xd by Eq. (21), for α ¼ 0 it holds x′c ¼ 2, so still xd ¼ 5 would be optimal, the latter two are equivalent trough permutation of xc and xc′ . Therefore, only α 4 1 remains, which yields xc ¼ 1 and since then x′c ¼ xd þ 2 again xd ¼ 5 may take on its lower bound. We can therefore state the following remark. Remark 3.4. In general, no matter how nonnegative coefficients are chosen for a linear (FA) PESP objective function, no choice guarantees that the corresponding (FM) ODPESP solution is optimal, even after rerouting passengers. We now formulate this statement in terms of constraint polytopes. Remark 3.5 (Linear Programming Interpretation). There exist (FA) ODPESP constraint polytopes for which every extreme point has the property that the corresponding rolled out point is suboptimal in the (FM) ODPESP constraint polytope. To see this, consider [20] in which it is shown for the PESP that      π π ∈Q≔conv:hull : lij ≤π j −π i þ Tzij ≤uij ; zij ; π i ∈Z z z is an extreme point of Q if and only if it may be represented by a so called spanning tree structure T ¼ ðT l ; T u Þ⊂A, with T being a tree and xa ¼la resp. xa ¼ ua for all a∈T l resp. a∈T u . Obviously, the only unique optimum in Fig. 4 (modulo adding a constant to all times) is taken on in (c). In the FA model in (a), a spanning tree can consist of ðxd ; xc Þ, ðxd ; x′c Þ or ðxc ; x′c Þ. The latter one is infeasible: neither xc nor x′c may take on their upper bounds. If that happened, the xd constraint cannot be satisfied anymore. Both taking on lower bounds is infeasible for the same reason. The other two possible trees contain xd , which may either be 5 or 7, i.e., lower resp. upper bound. In the case of 5, the change from the first ℓ1;1 to the second instance ℓ1;2 of ℓ1 cannot happen within one period T, since line ℓ2 takes at least 3 time units, but there are only 2 in between. If xd ¼ 7, then 6 time units are needed to get from s1 to s4, which is worse than the global optimum of 5. The statement follows, since adding f0; 1g path decision variables has no influence on a solution being an extreme point or not.

2256

M. Siebert, M. Goerigk / Computers & Operations Research 40 (2013) 2251–2259

4. Solution approaches for retimetabling In Section 3 we described the retimetabling method from a theoretical point of view. This section deals with the actual heuristics to be used in Section 5. We obtain randomized initial timetables by applying permutations to the event indices, using a constraint satisfactory solver [12] to find an initial solution to the resulting PESP. This procedure takes a few seconds per timetable. Taking a randomized initial timetable and performing retimetabling steps until final convergence is achieved will be called a retimetabling run or run for short. For passenger routing, we use randomized shortest paths, i.e., if there are several paths with shortest duration, we pick a random one among them. Since this makes it impossible to ultimately say whether the retimetabling iteration converged or not, we use a heuristic criterion and define convergence as two consecutive TT improvement failures by retimetabling steps. For the timetabling step, we consider two methods: the simple one is to fix the modulo parameters from the previous timetable and to solve the remaining (aperiodic) totally unimodular subproblem. The more powerful, but also more complex method is to use the modulo simplex [21,10]. The latter is able to manipulate the modulo parameters; applied to the same PESP instance, it yields a TT at least as good as the former. We use these two methods in three variations: NoMs No modulo simplex: only use fixed modulo parameters. On convergence, if in (FA) mode, perform a rollout: the so-called convergence rollout; else if in (FM) mode: STOP, final convergence is reached. MsConv Modulo simplex on convergence: like NoMs, but we try to escape local optima by applying the modulo simplex on NoMs convergence and reset the NoMs failure counter to zero. If the modulo simplex already failed to improve TT for two consecutive times, perform a convergence rollout in (FA) or STOP in (FM) instead. MsOnly Only modulo simplex: again like NoMs, but we use the modulo simplex instead of fixed modulo parameters in every step. Note that after a convergence rollout, the run continues in the (FM) event activity network. If we used (FA) EPESP, we switch to PESP changes in (FM).

Table 1 LinTim instance dimensions. Cyclebases w.r.t. fundamental improvement from [13]. Property

Toy

Athens

Rail-small

Rail-large

Stations Edges Lines OD pairs 4 0 Events Activities Best gap found Hard constraints Hard cyclebase width

8 8 5 44 52 182 0 118 20.82

51 52 4 2385 208 234 0.84 200 0.0

250 326 53 48,842 3664 24,670 6.52 10,034 2012.26

319 452 86 77,878 4932 33,446 6.90 10,928 1387.66

and Rail-large. The actual need for ram however was below 200 MB (Java) for the largest instance. We use Cplex 12.1 for solving the aperiodic timetabling problem in NoMs and MsConv. 5.2. Measurement We now discuss how to make the different models comparable. For this purpose we make two main distinctions. Changes: (FA) (E)PESP vs. (FM) PESP. In a (FA) truncated network, in every retimetabling step we evaluate a rollout peek, i.e., we make a backup of our network, perform a periodic rollout, either reroute passengers if we are in a rerouting step or take the passenger distribution from the old rolled out network if in a timetabling step. Former gives us an idea on how well our new timetable performs, the latter on how well it would have performed if passengers kept taking their paths in the rolled out network. When we compare two objective functions, we always measure in the rolled out network, i.e., by using the rollout peek with rerouted passengers if in (FA) state. Passengers: Classic (PESP) vs. Retimetabling (ODPESP). As the classic model embedding we consider (FA) PESP without the benefit of randomization techniques from Section 4, i.e., a single retimetabling step, taking the average over initial timetables and shortest path runs. Further, every PESP heuristic competes with solving the fixed modulo parameters subproblem, which may be considered as the trivial method. Therefore, we make the following definition. Definition 5.1. The classic PESP improvement is defined as " # Δclassic ¼

5. Experiments In this section we experimentally evaluate the impact of the different modeling parameters from Section 2 and solution approaches from Section 4. 5.1. Instances and environment We used the following instances from the LinTim toolbox [9,8]: “Toy” which is a small artificial network, “Athens”, which is a realworld network describing the subway network of Athens, and “Rail-small”, “Rail-large”, which are close-to real-world networks describing the long-distance public transport of Germany. Table 1 gives an overview to the properties of these instances. Especially, the cyclebase width, which is a measure for network complexity and between 5–20 times greater than in [15,23], actually forces us to use heuristics in the timetabling step. Calculations were carried out on a mid 2010 MacBook Pro with 2.4 GHz and 4 GB for the instances Toy and Athens, and a 4 core Intel Xeon with 24 GB main memory for the instances Rail-small

N;M

∑ gapðxði;jÞ;FA;peek; Þ−gapðxði;jÞ;FA;peek; Þ =ðNMÞ; NoMs;1 MsOnly;1

ð25Þ

i;j ¼ 1

where (N,M) are the number of randomized initial timetables resp. shortest paths runs, with (i,j) denoting the resp. run. Further, xði;jÞ;FA;peek; MsOnly;1

resp:

xði;jÞ;FA;peek; NoMs;1

are the timetables obtained after a single retimetabling (FA) step with MsOnly resp. NoMs, rollout peeks with rerouted passengers measured. The classic PESP improvement is depicted as the gray bar in Fig. 6. Our new method, explorative retimetabling we define by taking the best timetable obtained in the whole experiment. Configurations. We parameterize by whether to start with FA, whether to use PESP or EPESP changes and whether to use MsOnly, MsConv or NoMs. We evaluated six of them, four instances, ten initial randomized timetables per instance and three runs per timetable with randomized shortest paths. Results. The best timetable for the Toy instance (gap 0) could only be found by using CPLEX to solve the ODPESP to full optimality. However, with ReTim we got as close as 0.5%.

M. Siebert, M. Goerigk / Computers & Operations Research 40 (2013) 2251–2259

2257

Fig. 5(c) shows the ReTim run that yielded the best timetable for Rail-small. Surprisingly, it has been obtained by MsConv, i.e., by not using the better heuristic in every step. For Rail-large however, MsOnly indeed yielded the best timetable. Fig. 5(d) shows the best EPESP run for Rail-large. As shown in [29], EPESP in general underestimates the rerouted peek TT. However, after the convergence rollout and implied switch to PESP changes, the TT drops even below the already underestimated value. This behavior may be observed in all other EPESP runs as well in Athens and the Rail networks, but also in some runs using the Toy instance. The EPESP rerouted peek gap is up to 80% worse in average on convergence rollout, while the PESP is no more than 10% worse than on final convergence. This especially means that switching to (FM) changes indeed improves the TT. However, the drawback is that when using (FM) as initial model, the final gap in average is several times worse for Toy and Athens resp. up to 5% for Rail compared to FA+rollout, most likely due to the ReTim iteration getting stuck in a local optimum too quickly. Shortest paths randomization can match the classic improvement, while surprisingly, the initial timetable has the greatest impact: up to an eightfold, which can be seen in Fig. 6(d) by comparing the gray bar with the difference of its lower end with the best timetable (lowest dashed line). Fig. 7 depicts this in a larger context: for the Rail networks, all methods perform similar. However, good final results seem to coincide with good initial timetables.

Fig. 6. ReTim overview. The labels 1X, R, MS, resp. F denote “After first ReTim step”, “Convergence Rollout”, “Before Modulo Simplex” resp. “Final Convergence”. Thin lines represent single timetables, thick black lines averages. Note that timetable quality varies by magnitudes of the classic PESP improvement Δclassic from Section 5.2 (gray bar) if we consider different initial timetables. Especially, explorative retimetabling can do an additional eightfold better in our largest instance (d). (a) Toy, (b) Athens, (c) Rail-small and (d) Rail-large.

Fig. 6 gives an overview on the whole experiment. For the Toy and Athens instance we observe the timetabler's nightmare as predicted in Remark 3.4, i.e., that when using (FA), the TT in the rolled out network increases during the ReTim iteration. For Athens, the situation is particularly bad, as can be seen in Fig. 6 (b): firstly, for the (FA)+MsOnly runs, not only the best overall timetable gets lost, but the average worsens during the ReTim iteration. Secondly, in (FM), the NoMs gap is better than that of MsOnly in average. However, this is not a bug in the Modulo Simplex implementation: when applying it to the NoMs PESP instances with their randomized shortest paths, the worsening disappears, which is experimental evidence that the actual passenger paths are not negligible over the OD pairs.

6. Conclusion

Fig. 5. Single ReTim plots, all initially FA. In (a) and (b) a good resp. the overall best found timetable get lost due to the timetabler's nightmare, (c) shows the best overall run for Rail-small and (d) the best EPESP run for Rail-large. Note that for latter, after rollout, the TT drops below the already underestimated value. (a) Toy, PESP, NoMs, timetable 4, shortest paths run 3, final gap: 3.07, (b) Athens, PESP, MsOnly, timetable 6, shortest paths run 3, final gap: 1.76, (c) Rail-small, PESP, MsConv, timetable 5, shortest paths run 2, final gap: 6.52, (d) Rail-large, EPESP, MsConv, timetable 7, shortest paths run 2, final gap: 7.08.

We theoretically and experimentally evaluated a wide spectrum of modeling aspects and their influence on the average traveling time. Our experimental data suggests that these aspects can have a larger impact on the solution quality than one might expect; in fact, the use of different starting solutions and frequency models can be more significant than the quality of the PESP heuristic applied. Explorative Retimetabling is highly scalable in runtime and memory consumption. It is a mix between evolution (retimetabling, randomized shortest paths) and Monte Carlo (randomized initial timetable). Considering former, it is yet to evaluate if quality

2258

M. Siebert, M. Goerigk / Computers & Operations Research 40 (2013) 2251–2259

Fig. 7. Mean gap over the shortest path randomization runs per initial timetable. Good final timetables seem to coincide with good initial timetables. Except for (a), the spread among the single methods is rather low. (a) Toy, (b) Athens, (c) Rail-small and (d) Rail-large.

may be improved by adding simulated annealing or genetics to the approach. Concerning the latter, initial timetabling randomization may be abstracted into a meta heuristic to solve a wide range of other large-scale problems as well. Further research includes applying the meta heuristic to other problems as well as getting theoretical results making certain assumptions about the solver, the problem structure and the randomness involved. Also, robustness considerations under uncertainty should be taken into account.

References [1] Borndörfer R, Schlechte T. Models for railway track allocation. Technical Report, Zuse-Institut Berlin (ZIB), ZR-07-02; 2002. [2] Cacchiani V, Caprara A, Fischetti M. A lagrangian heuristic for robustness, with an application to train timetabling. Transportation Science 2012;46(1):124–33. [3] Cacchiani V, Caprara A, Toth P. Non-cyclic train timetabling and comparability graphs. Operations Research Letters 2010;38(3):179–84. [4] Cacchiani V, Caprara A, Toth P. Scheduling extra freight trains on railway networks. Transportation Research Part B 2010;44B(2):215–31. [5] Cacchiani V, Toth P. Nominal and robust train timetabling problems. European Journal of Operational Research 2012;219(3):727–37. [6] Caprara A, Fischetti M, Toth P. Modeling and solving the train timetabling problem. Operations Research 2002;50(5):851–61. [7] Galli L, Stiller S. Strong formulations for the multi-module pesp and a quadratic algorithm for graphical diophantine equation systems. In: Proceedings of the 18th annual European conference on algorithms, ESA'10. SpringerVerlag; 2010, p. 338–49. [8] Goerigk M, Schachtebeck M, Schöbel A. LinTim—Integrated Optimization in Public Transportation. Homepage. See 〈http://lintim.math.uni-goettingen.de/〉.

[9] Goerigk M, Schachtebeck M, Schöbel A. Dependencies between line planning, timetabling and delay management: Experiments with the lintim toolbox. Technical Report, Preprint-Reihe, Institut für Numerische und Angewandte Mathematik, Georg-August Universität Göttingen; 2012. [10] Goerigk M, Schöbel A. Improving the modulo simplex algorithm for large-scale periodic timetabling. Computers and Operations Research 2013;40(5):1363–70. [11] Kinder M. Models for periodic timetabling. Diploma thesis, Institut für Mathematik, Technische Universität Berlin; May 2008. [12] Lecoutre C, Tabary S. Abscon 112 toward more robustness. In: 3rd international constraint solver competition held with CP'08 (CSC'08); September 2008, p. 41–8. [13] Liebchen C. Finding short integral cycle bases for cyclic timetabling. In: Proceedings of the ESA, LNCS 2832. Springer; 2003. p. 715–26. [14] Liebchen C, Möhring R. The modeling power of the periodic event scheduling problem: railway timetables – and beyond. In: algorithmic methods for railway optimization, Lecture Notes on Computer Science, vol. 4359. Springer; 2007, p. 3–40. [15] Liebchen C, Peeters L. Integral cycle bases for cyclic timetabling. Discrete Optimization. 2009;6(1):98–109. [16] Luebbe J. Passagierrouting und Taktfahrplanoptimierung. Diploma thesis. Institut für Mathematik, Technische Universität Berlin; September 2009. [17] Michaelis M, Schöbel A. Integrating line planning, timetabling, and vehicle scheduling: a customer-oriented approach. Public Transport 2009;1(3):211–32. [18] K. Nachtigall, A branch and cut approach for periodic network programming. Technical Report 29, Hildesheimer Informatik-Berichte, Institut für Mathematik, Hildesheim; 1994. [19] Nachtigall K. Periodic network optimization with different arc frequencies. Discrete Applied Mathematics 1996;69(1–2):1–17. [20] Nachtigall K. Periodic network optimization and fixed interval timetables. Dissertation, Institut für Flugführung, Deutsches Zentrum für Luft- und Raumfahrt Braunschweig; April 1998. [21] Nachtigall K, Opitz J. Solving periodic timetable optimisation problems by modulo simplex calculations. In: Fischetti Matteo, Widmayer Peter, editors. ATMOS 2008. Germany: Dagstuhl; 2008.

M. Siebert, M. Goerigk / Computers & Operations Research 40 (2013) 2251–2259

[22] Odijk MA. A constraint generation algorithm for the construction of periodic railway timetables. Transportation Research Part B: Methodological 1996;30 (6):455–64. [23] Peeters LWP. Cyclic railway timetable optimization. TRAIL thesis series, Erasmus Research Institute of Management; 2003. [24] Schmidt M. Integrating routing decisions in network problems. PhD thesis, Universität Göttingen; 2012. [25] Schmidt M, Schöbel A. The complexity of integrating routing decisions in public transportation models. In: Erlebach T, Lübbecke M, editors. Proceedings of the 10th workshop on algorithmic approaches for transportation modelling, optimization, and systems, OpenAccess series in informatics (OASIcs), vol. 14.. Germany: Dagstuhl; 2010. p. 156–69 Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.

2259

[26] Schöbel A. Line planning in public transportation: models and methods. OR Spectrum 2011:1–20. [27] Schöbel A, Kratz A. A bicriteria approach for robust timetabling, vol. 5868 of LNCS. Springer; 2009. [28] Serafini P, Ukovich W. A mathematical model for periodic scheduling problems. SIAM Journal on Discrete Mathematics 1989;2(4):550–81. [29] M. Siebert. Integration of routing and timetabling in public transportation. Institut für Numerische und Angewandte Mathematik, Georg August Universität Göttingen. See 〈http://num.math.uni-goettingen.de/picap/pdf/E672. pdf〉, April 2011. Diploma thesis.