A schedule-based Transit Assignment Model addressing the ...

Veröffentlichung / Publication

A schedule-based Transit Assignment Model addressing the Passengers’ Choice among competing Connections

Autoren / Authors: Markus Friedrich PTV AG, Karlsruhe Steffen Wekeck PTV AG, Karlsruhe

Veröffentlicht in / Published in: Friedrich, M., Wekeck, S. (2002): A schedule-based Transit Assignment Model addressing the Passengers’ Choice among competing Connections, Proceedings of Conference „The Schedule-Based approach in Dynamic Transit Modelling: Theory and Applications”, Ischia,. p. 159-173.

Universität Stuttgart Institut für Straßen- und Verkehrswesen Lehrstuhl für Verkehrsplanung und Verkehrsleittechnik www.uni-stuttgart.de/isv/vuv/

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A schedule-based Transit Assignment Model addressing the Passengers’ Choice among competing Connections

Authors: Markus Friedrich, PTV AG Stumpfstr. 1, D-76131 Karlsruhe, Germany Phone ++49-721-9651316, fax ++49-721-9651299, email [email protected] Steffen Wekeck, PTV AG Stumpfstr. 1, D-76131 Karlsruhe, Germany Phone ++49-721-9651339, fax ++49-721-9651299, email [email protected]

Abstract: Transit assignment procedures need to reflect the constraints imposed by line routes and timetables. They require specific search algorithms considering transfers between transit lines with their precise transfer time. The paper will present such an assignment procedure for transit networks using a schedule-based search algorithm. In contrast to existing schedule-based search methods employing a shortest-path algorithm, the described procedure constructs connections using branch & bound techniques. This approach produces better results in cases where slow but cheap or direct connections compete with fast alternatives which are more expensive or require transfers. At the same time it significantly reduces computing time, thus facilitating the use of timetable-based assignment for big networks. A key feature of the presented assignment procedure is the extension of the standard choice model by the connections’ independence to reflect interaction effects between competing alternatives.

Keywords: Transit Assignment, Schedule-based Assignment, Public Transport Modeling

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INTRODUCTION Assignment procedures for transit networks need to model the spatial and temporal structure of traffic supply. Line routes and timetables must be examined in great detail in order to mirror exact transfer times and timetable co-ordination. This type of schedule-based search algorithm can be found in passenger information systems provided by an increasing number of transit operators (e.g. www.bahn.de, www.sbb.ch, www.ns.nl). Here, the search algorithms are applied and optimized for a one-to-one type of problem. Assignment in transport planning, however, needs to examine an entire network and the connections between all traffic zones. This extends the search to a many-to-many problem, which can produce long computing times for big nation-wide networks. To ensure acceptable computing times, assignment models often apply simplified search algorithms based on average values for transfer times (see [4], [9]). These approaches reduce the problem by assuming that the waiting times at boarding or transfer stops depend on the headway of the following transit line. This assumption speeds up computation but fails to consider the coordination of the timetable which is essential for detailed analysis of transfer waiting times or for transit networks with long headways. This paper will present a new schedule-based assignment procedure for transit networks. Considering shortcomings of the widely-used shortest-path techniques, it will be pointed out how the branch & bound strategy employed in the described algorithm manages to produce better assignment results in many cases. Generally, not only the “best” but all sufficiently good connections are found. This is particularly helpful when connections of different characteristics (speed, fare, number of transfers) are at the travellers’ disposal. In addition, the algorithm reduces computation time. Another key property of the presented procedure is the concept of independence which allows a more precise modelling of interaction effects between “similar” connections.

1

EXISTING APPROACHES

Existing approaches for transit assignment can be divided into two categories: 1. Line-based assignment simplifies the search by using average values for the transfer time (see for example [4], [7], [9]). Accordingly, the produced results are aggregates, not precise values. Each transit line is modelled through a sequence of stops, through the running times between the stops and through the headway of the transit line. Transit lines with no fixed-rhythm headway are described by their mean headway. Such procedures do not explicitly calculate transfer times but assume that they depend on the headway. In other words, the co-ordination of the timetable is not considered. Usually, one assumes that the wait times at the boarding stop or at transfer stops are equal to half of the line's headway. Assignment based on lines guarantees acceptable assignment results only for urban areas with a dense network and short headways. 2. Schedule-based assignment considers the timetable of each transit line with its exact departure and arrival times. This type of assignment has become more popular in recent years (see for example [2], [4], [6], [10]). Most of the suggested algorithms apply some type of shortest-path algorithm which determines the optimal connection between two traffic zones for a given de-

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parture time. For different times of departure, there may be various optimal connections differing in the used vehicle trips (runs) or transfer stops. Assignment based on schedules is the appropriate method when precise values for the transfer time, the service frequency and the vehicle loads are expected. This is especially important in rural areas or rail networks, where headways are long and the co-ordination of the timetable is important for the service quality. A schedule-based search using a shortest-path algorithm unfortunately has two weaknesses: it does not find all relevant connections and requires long computing time. The example displayed in Figure 1 illustrates these weaknesses. The given network consists of a bus line and a train line. Passengers travelling from origin zone A to destination zone X may choose between direct bus connections and faster bus-train connections: Figure 1: Example network A-Village (Origin)

X-City (Destination)

Station

Line network

Train Bus

6:00

B-Village

Bus

6:30

Train Bus

7:00 Train

Timetable

Bus 7:30 Train 8:00

A-Village

Station

B-Village

Origin

Timetable of Bus A-Village 6:10 Station 6:22 B-Village 6:42 X-City 6:55 Connections No Departure 1 6:10 2 6:10 3 6:55 4 7:25 5 7:25

6:55 7:07 6:27 7:40 Arrival 6:55 6:41 7:40 8:10 8:01

7:25 7:37 7:57 8:10

In-Vehicle Time 45 min 28 min 45 min 45 min 28 min

X-City Destination

Timetable of Train Station 6:25 X-City 6:41

Transfer Time 0 min 3 min 0 min 0 min 8 min

Ride Time 45 min 31 min 45 min 45 min 36 min

7:05 7:21

No. of Transfers 0 1 0 0 1

7:45 8:01

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Quality of results: Since the transit supply in some cases offers more than one connection for a given departure time (see example in Figure 1), a shortest-path algorithm requires a definition of the "best” connection. For this purpose, it is common to apply an impedance function which increases the impedance of a connection for each transfer by means of a predefined penalty. A low penalty prefers time-minimal connections, while a high penalty gives priority to connections with lower transfer frequencies. Varying the transfer penalty, however, increases the computation time as each transfer penalty requires an individual search step. Identifying all five connections in the example network of Figure 1 requires six search steps: three search steps with a penalty of 0 minutes (to find connections 2, 3, 5) and three search steps with a penalty of at least 15 minutes per transfer (to find connections 1, 3, 4). The problem of defining an impedance function for the best connection increases with the introduction of fares. In a network with cheap low-speed trains and expensive high-speed trains, it is necessary to vary not only the transfer penalty but also the value of time. In such a network, even a connection which departs earlier and arrives later than another one may be attractive for some passengers if it is cheaper or requires fewer transfers. Computation time: In order to determine all connections by means of a shortest-path algorithm, it is necessary to perform a search for each possible departure time at the origin stop within the examined time interval. For the example network, this requires three search steps starting at 6:10, 6:55 and 7:25. In a real-life network, more than 100 different departure times at a stop are common. This results in long computation times as every departure requires to build one shortest-path tree. Figure 2 illustrates that this type of search routine may be inefficient: 12 different departure times per hour at the origin stop O result in only four connections from O to D. Obviously, not every route search yields a new connection. Considering this phenomenon, it is possible to speed up the search process by limiting the number of departure times. This can be achieved by either randomly selecting a certain percentage of departure times or searching only at predefined times, e.g. every 10 minutes. As an acceptable computing time may only be achieved through a significant reduction of departure times, this approach will usually fail to find all connections. Figure 2: Departure times at an origin stop: 12 departure times per hour at stop O result in only four connections between O and D. dep. 6:03 6:13 6:23 6:33 6:43 6:53 ⋅⋅⋅

Bus: 10 min headway 10 min run time

Train 15 min headway

D

O dep. 6:00 6:10 6:20 6:30 6:40 6:50 ⋅⋅⋅

dep. 6:12 6:27 6:42 6:57 ⋅⋅⋅

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Extensive analysis of the shortcomings of a timetable-based search procedure using a shortestpath algorithm and requirements of European railway companies (German Railways DB AG & DB Regio, Danish Railways DSB) to model railway networks on a national and European level initiated the development of a new search routine for transit assignment, i.e. for a many-to-many search problem. After analysing several possible methods, an approach was chosen which is based on works by Friedrich [3]. As his branch & bound approach, developed in 1994, uses a preprocessing step which requires more computer memory than conventional shortest-path algorithms, the method could only now be adapted for big networks to work on standard PCs.

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CONNECTION SEARCH

2.1

Preprocessing

The objective of the preprocessing step is the generation of connection segments, which describe a part of a journey. These segments are the building blocks of connections in the search process. A connection segment represents either a walk or a transfer-free ride on one transit line. A connection made up of an access walk, a bus ride and an egress walk, for example, consists of three connection segments. From the line route data, an initial step generates the network’s route segments (see Figure 3). A single route segment describes either • a segment of a transit line between a boarding stop and an alighting stop or • a single walk link or a sequence of walk links for access, egress or transfer walks. Figure 3: Route segments and connection segments Stop 1: Terminal s1

s2

s3

Stop 2 s4

s5

Stop 3 s6

Stop 4: Terminal

Route segments si for one direction of a transit line: • Route segment 1 from stop 1 to stop 2 • Route segment 2 from stop 1 to stop 3 • Route segment 3 from stop 1 to stop 4 • Route segment 4 from stop 2 to stop 3 • Route segment 5 from stop 2 to stop 4 • Route segment 6 from stop 3 to stop 4 The number of connection segments for one direction of a transit line is (n − 1) ⋅ n ⋅ k , where 2 n = number of stops, k = number of vehicle trips (runs).

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Walk link sequences are calculated by applying a shortest-path algorithm on those links of the network which are permitted for “transit walks”. A route segment has an initial stop node, a terminal stop node, a length and a running time. Subsequently, connection segments (or connection legs) are calculated from the set of route segments. A connection segment using a transit line is endowed with a departure and an arrival time, whereas connection segments using walk links do not possess specific times, since they are available to the traveler at any time. Figure 4 shows the route and connection segments for the example network of Figure 1, not considering the access and egress walk links, which are assumed to be 0 minutes. Figure 4: Route segments and connection segments for the example network Route Segment No 1

Route Segments From Node To Node A-Village Station

Transit Line Bus 1

2

A-Village

B-Village

Bus 1

3

A-Village

X-City

Bus 1

4

Station

B-Village

Bus 1

5

Station

X-City

Bus 1

6

B-Village

X-City

Bus 1

7

Station

X-City

Train

2.2

Connection Segments Connection Segment No Departure 1 6:10 2 6:55 3 7:25 4 6:10 5 6:55 6 7:25 7 6:10 8 6:55 9 7:25 10 6:22 11 7:07 12 7:37 13 6:22 14 7:07 15 7:37 16 6:42 17 7:27 18 7:57 19 6:25 20 7:05 21 7:45

Arrival 6:22 7:07 7:37 6:42 7:27 7:57 6:55 7:40 8:10 6:42 7:27 7:57 6:55 7:40 8:10 6:55 7:40 8:10 6:41 7:21 8:01

Connection Search

A dynamic, i.e. time-dependent, multi-path algorithm is applied to determine all potential connections. This algorithm builds a connection tree (see Figure 5) which may contain several paths (connections) from the origin to a destination node. Since the tree’s width largely depends on the number of runs of the transit lines, it may be much wider than a usual shortest-path tree. On the other hand, the use of entire connection legs as tree edges simplifies the tree’s structure to a great extent and limits its depth by the maximum number of transfers. A typical value of this userdefined constant is 4 or 5.

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Figure 5 outlines the structure of the connection tree. The root of the tree is the centroid node of the origin zone and the only outgoing branch represents the access from the origin to the first transit stop. Figure 5: Structure of the connection tree walking link connection segment dep. arr. connection segment arr. dep.

alighting stop

destination

transfer stop branches

boarding stop

width

walking link origin connection segment dep. arr.

walking link

boarding stop alighting stop

level 0

level 1

level 2

destination

level 3

level 4 depth

Starting from the origin node, a branch & bound strategy is employed. Given a connection segment from the current tree level, all possible successors are considered. Every connection segment s is described by means of the following attributes: • departure time x s and arrival time y s • travel time t s • fare f s Naturally, these three quantities are also defined for connections (i.e. sequences of connection segments). Furthermore, if a connection c is composed of n connection segments, the number of transfers is nc = n − 1 . The resulting search impedance of a connection c is then defined as v c = χ t t c + χ n nc + χ f f c

(χ t ≥ 0, χ n ≥ 0, χ f ≥ 0)

where the coefficients are user-defined parameters. Note that the fare term f c is assumed to be additive here. Naturally, this is a simplification of the real fare model and is only used during the search process for run-time reasons. When loading the generated connections (see section 3), the real fare of each alternative is considerded. Now, let s be the currently processed connection segment from network node a to network node b. Let c* be the new connection between the origin node and b formed by adding s to some connection c arriving at a. Finally, let C b be the set of all known connections to b.

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Connection segment s is inserted into the tree as an successor of c, if and only if the following conditions hold: • Temporal suitability: The connection segment s departs from node a only after the arrival of connection c plus a minimum transfer wait time Ta− , and before a user-defined maximum transfer wait time Ta+ has elapsed: x s − y c ∈ [Ta− ; Ta+ ] • Dominance: There is no known connection c'∈ Cb such that xc ' ≥ xc* and y c ' ≤ y c* and vc ' ≤ vc* and nc ' ≤ nc* . • Tolerance constraints: None of the following rules is violated: v c* ≤ b1 ⋅ min c '∈Cb vc ' + b2 , t c* ≤ d1 ⋅ min c '∈Cb t c ' + d 2 nc* ≤ e1 ⋅ min c '∈Cb nc ' + e2 ,

nc* ≤ N + ,

where all bi , d i and ei are user-specified global tolerance parameters and N + is the userdefined bound for the number of transfers within a connection. • Loops: Transfers within a transit line are only allowed for lines containing a loop, provided that the passenger can save time by boarding an earlier vehicle trip (run) at the intersection node of the route loop. A tree level is traversed completely before connection segments of the next level are considered. The procedure terminates when no successors are found for an entire tree level or the maximum number of transfers is reached. With regard to the accessibility of the algorithm’s outcome, some search information is stored for each of the network nodes. It comprises the shortest journey time, the shortest walking time, the minimum number of transfers and the minimum fare taken over all connections to that node established up to the present. Other values describing the overall characteristics of a connection are passed through all corresponding segments.

3

CONNECTION CHOICE

The remaining connection choice set is denoted by C. Based on the connection attributes mentioned before, a connection’s impedance is calculated for every time interval of traffic demand. The main input to the impedance function is the perceived journey time. This quantity is a linear combination of specific time components such as in-vehicle travel time tcv and transfer wait time tcw plus a transfer penalty: t cp = α v t cv + α w t cw + ... + αnc

(α v ≥ 0, α w ≥ 0,..., α ≥ 0)

(1)

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Since the travel demand may vary within the day, the utility of a connection largely depends on the time interval under consideration. This aspect is modeled through the temporal disutility d cI depending on demand interval I. This temporal disutility models the difference between the riders' desired and the actual departure time of c . Accordingly, d cI is an isotone non-negative function of dist ( xc , I ) , such that d cI = 0 if and only if xc ∈ I . All connections departing within the considered time interval thus have zero temporal disutility. A connection’s impedance (or overall disutility) for time interval I is then composed of its perceived journey time t cp , its temporal disutility d cI and its fare f c : v cI = γ t t cp + γ d d cI + γ f f c

( γ t ≥ 0, γ d ≥ 0, γ f ≥ 0) .

(2)

This impedance is transformed into the connection’s utility by means of some antitone nonnegative function: ucI = g (vcI ) . In the case of a standard Logit model, g (v) = e −βv .

In conventional choice models, the share of the travel demand of time interval I assigned to connection c is proportional to its utility: pcI

=

ucI ucI '

∑ c '∈C

(3)

In principle, this approach is acceptable since it comprises the notion of the passengers’ individual and incomplete perception of connection utilities. The application of such a choice model makes the connection choice a stochastic network loading. Many case studies show, however, that this concept fails to mirror the impact of interactions among similar connections. We introduce an extension of the standard choice model which overcomes the problem by including a measure of a connection’s independence in (3). Analogous problems emerge in the context of drivers’ route choice where highly overlapping paths are wrongly regarded as independent alternatives by the standard choice model. Cascetta et al. introduced a route’s commonality factor to address this topic [1].Closer examination shows that the “independence” needed here should • be included in the choice model irrespective of the given utility calculation function g • range within ]0,1], where 1 is the independence of an isolated connection • assign an independence of

1 k

to each of k identical connections in order to solve the commonly

known red-bus-blue-bus problem • be linear in temporal distance • optionally include distance in perceived journey time and fare, also linearly • be able to model asymmetric pulls between connections of different quality

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In view of these aspects, the independence of a connection c ∈ C is defined as

λ c :=

∑

1 = hc (c' ) 1 +

c '∈C

1

∑ h (c')

,

(4)

c c '∈C , c ' ≠ c

where hc is a non-negative evaluation function, for which hc (c) = 1 and hc (c' ) ≤ 1 ∀c, c'∈ C . hc is to model the pull other connections exert on c . The general rule is: the greater the similarity of two connections, the higher the impact on one another. In our model, hc is defined as +

+

⎛ δ ( c, c ' ) ⎞ ⎛ s 3 | δ 2 ( c, c ' ) | + s 2 | δ 3 ( c, c ' ) | ⎞ ⎟⎟ , ⎟ ⋅ ⎜1 − hc (c' ) := ⎜⎜1 − 1 s1 ⎟⎠ ⎜⎝ s 2 s3 ⎝ ⎠ where δ1 (c, c' ) = 12 | xc − xc ' | + 12 | yc − yc ' |

(5)

is the temporal distance between c and c' ,

δ 2 (c, c' ) := t cp' − t cp the perceived journey time overhead of c' with respect to c and δ3 (c, c' ) := f c ' − f c the corresponding quantity in terms of the fare. Note that all functional relations are modeled linearly.

Obviously, the parameters s k control the range of influence of temporal distance, difference in perceived journey time and difference in fare, respectively: if the temporal distance between two connections c and c’ is larger than s1 , the first factor in (5) vanishes, and hence hc (c' ) = hc ' (c) = 0 . A similar statement holds for the two other attributes. Naturally, the s k must depend on the connecting set C. Case studies have shown that the mean waiting time of a randomly accessing passenger at the origin stop is a good choice for s1 . Furthermore, it makes sense to assume that the impact of one connection on the other need not be symmetric. We let ⎧sk0 if δk (c, c' ) ≥ 0 sk := ⎨ 0 for k = 2,3 . (6) ⎩2sk if δk (c, c' ) < 0

Again, use cases have shown that reasonable values for the underlying constants are s 20 = 13 t cp and s30 = 13 f c . In other words, if the difference between two connections’ perceived journey times or fares exceeds one third of the related arithmetic mean, they are considered substantially distinct and do not have an impact on one another. Using the MNL model enhanced by the new attribute of independence, pcI becomes

pcI

ucI λ c := . ucI λ c

∑ c '∈C

(7)

Consider the following simple example where 120 passengers are faced with several connections in time interval I = [8 : 00, 9 : 00] . Since all connections depart within I, d cI = 0 for all c, mean-

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ing that the travelers are indifferent as to the precise departure time. For simplicity, let γ t = 1 and

γ d = γ f = 0 . We use a standard Logit model with β = 0.1 . The first example (Figure 6) shows three identical connections with a fixed headway. All connections are considered independent, so that λ c = 1 for all c and each connection is assigned the same amount of passengers.

Figure 6: Connection choice example 1 – three connections with a fixed headway c

departs

t cp

λc

pcI not using λ c

pcI using λ c

1 2 4

8:15 8:30 8:45

20 min 20 min 20 min

1.00 1.00 1.00

40 40 40

40 40 40

In the second example (Figure 7) a further connection is added which departs at the same time as connection 2. Here the use of λ c causes the split to assign just as many passengers to connections 2 and 3 as to each one of the connections 1 and 4. In this way, it is taken into account that 2 and 3 are in fact only one connection.

Figure 7: Connection choice example 2 - insertion of a connection with simultaneous departure time c

departs

t cp

λc

pcI not using λ c

pcI using λ c

1 2 3 4

8:15 8:30 8:30 8:45

20 min 20 min 20 min 20 min

1.00 0.50 0.50 1.00

30 30 30 30

40 20 20 40

The third example (Figure 8) shows the impact of an express connection on the distribution. Without λ c , the express connection 3 detracts equally many passengers from all other connections, although its impact on connection 1 may assumed to be weaker than on the neighboring connections 2 and 4. The figure shows that this aspect is considered when λ c is used.

Figure 8: Connection choice example 3 - insertion of an express connection c

departs

t cp

λc

pcI not using λ c

pcI using λ c

1 2 3 4

8:15 8:30 8:40 8:45

20 min 20 min 15 min 20 min

1.00 0.86 0.94 0.86

26 26 42 26

28 24 44 24

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APPLICATION AND OUTLOOK

The presented approach is implemented in the transportation model VISUM [8] and is being used by practitioners since the beginning of 2001. One of the most ambitious users is German Railways who apply the procedure to the entire German train network with about 31,600 vehicle trips (runs) per day. They also intend to analyze bigger networks for all of Europe or a more detailed German network including local transit services.

Figure 9: Indicators of the German Rail application German train network model no. of traffic zones no. of transit stops no. of lines with identical line route and running time no. of vehicle trips (runs) per day mean no. of departure times per traffic zone no. of route segments no. of connection segments Computing time* with shortest-path search • • •

mean computing time for one departure time (= one shortest-path tree) mean computing time for one zone total computing time Computing time* with branch & bound search

531 6,300 3,040 31,600 220 273,000 2,737,000 1 sec 106 sec 15 h 40 min

7 sec • computing time for preprocessing 2,2 sec • mean computing time for one zone 19 min 39 sec • total computing time * Computing time using a x86 Family 6 AMD 1666 MHz computer with 1GB of RAM Current and future developments need to cover the problem of capacity restraints. Capacity restraints can restrict the choice of a connection in two ways: • networks with seat reservation (aircrafts, long distance trains): If one leg of a connection is full, the connection is no longer available. • networks with no seat reservation (local transit): Passengers can not board a vehicle trip, if the vehicle is full. They need to use a different service or wait for a later vehicle trip (run) of the same line which actually results in a new connection. The first type of restraint can be solved through an iterative approach similar to highway assignment, the second type requires a simulation-like loading procedure (similar to [7]). In both cases, either computation time (repetition of connection search) or computer memory use (storage of all connections in RAM) increases.

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REFERENCES

1. Cascetta, E., Russo, F., Vitetta, A.: Stochastic User Equilibrium Assignment with Explicit Path Enumeration: Comparison of Models and Algorithms, Proceedings of the 8th IFAC Symposium on Transportation Systems, ed. M. Papageorgiou and A. Pouliezos, 1997. 2. Florian, M.: Deterministic Time Table Transit Assignment, Paper of EMME/2 Users Group Meeting, Shanghai, 1999 3. Friedrich, M.: Rechnergestütztes Entwurfsverfahren für den ÖPNV im ländlichen Raum (“Computer assisted design of public transport systems in rural areas”), in Schriftenreihe des Lehrstuhls für Verkehrs- und Stadtplanung, Volume 5, Technical University of Munich, 1994. 4. Friedrich, M.: Modelling Public Transport - A European Approach. Preprint CD-ROM of 78th Annual Meeting, Transport Research Board, Washington, 1999. 5. Jansson, K.: VIPS II - A new Computer System for Public Transport Planning, Proccedings of PTRC Summer Annual Meeting, University of Bath, 1987. 6. Nuzzolo, A., Russo, F., Crisalli, U.: A Doubly Dynamic Schedule-based Assignment Model for Transit Networks, Transportation Science 35, 2001. 7. Poon, M.H., Wong, S.C., Tong, C.O.: A dynamic schedule-based Model for congested Transit Networks. Review Paper for Transportation Research Part B, submitted by The University of Hong Kong, Department of Civil Engineering, 2001. 8. PTV AG: VISUM, http://www.english.ptv.de/cgi-bin/produkte/visum.pl, February 2002 9. Spiess, H., Florian, M.: Optimal Strategies: A New Assignment Model For Transit Networks, in Transportation Research, 23B(2), 1989. 10.Tong, C.O., Wong, S.C.: A Schedule-Based Time-Dependent Trip Assignment Model for Transit Networks, Journal of Advanced Transportation, 1999.