Transit route network design-maximizing direct and transfer demand density

Transportation Research Part C 22 (2012) 58–75

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

Transit route network design-maximizing direct and transfer demand density Bin Yu a,⇑, Zhong-Zhen Yang a, Peng-Huan Jin a, Shan-Hua Wu a, Bao-Zhen Yao b a b

Transportation Management College, Dalian Maritime University, Dalian 116026, PR China School of Automotive Engineering, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e

i n f o

Article history: Received 29 December 2009 Received in revised form 2 December 2011 Accepted 5 December 2011

Keywords: Transit network design Demand density Transfer coefficient ACO

a b s t r a c t Transit network design is an important part of urban transportation planning. The purpose of this paper is to build on direct traveler density model and extend it to design transit network considering demand density relating to direct demands and transfers, and lengths of routes. The proposed method aiming to maximize demand density of route under some resource constraints divides transit network design problem into three stages, i.e., skeleton route design, main route design and branch route design, based on the objective functions with different transfer coefficients. An ant colony optimization (ACO) is used to solve the model. The model and algorithm are illustrated with data from Dalian city, China and results show that the approach can improve the solution quality if the transfer coefficient is reasonably set. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Transit network design is an important problem in transportation planning and development, which has been studied for several decades. There have been many worthwhile researches on transit network planning. Dubois et al. (1979) designed transit network by identifying the roads needed for bus routes and choosing the set of bus routes. Then, frequencies of the }m (1981) proposed a designed routes were computed through a model aiming to minimize user waiting time. Hasselstro mathematical programming approach for transit network design by choosing the routes and determining frequencies concurrently. The methodology was capable of implementation and application for realistic network sizes. Ceder and Wilson (1986) summarized the bus network design approaches and presented a new approach and an algorithm to design bus routes based on both passenger and operator interests. Baaj and Mahmassani (1995) argued that a bus network could be generated by optimizing the route and the frequency, simultaneously. Van Nes et al. (1998) presented a transit route design method, in which route or frequency optimization was based on an economic criterion. Gao et al. (2004) presented a bi-level programming model for transit network design problem, which incorporated a transit equilibrium assignment model. Guan et al. (2006) used integer programming to optimize the line layout and assign trips in a given network, simultaneously. The model can be solved by standard branch and bound method. Pattnaik et al. (1998) presented a genetic algorithm (GA)-based optimization method to design transit network. The objective of their optimization model was to minimize the total cost of user and operator. Agrawal and Mathew (2004) presented an optimization model for transit network aiming to minimize the total system cost which is the sum of the operating cost and the generalized travel cost. They used parallel GA to solve the model. Bielli et al. (2002) developed a heuristic based on GA to design transit network. In the heuristic, a multi-criteria analysis was used to estimate the fitness values. Furthermore, they also applied an external envelope of an assignment algorithm when computing the fitness function. Yang et al. (2007) proposed a mathematical model for transit network design aiming to maximize direct traveler density that meant the number of direct travelers carried by per unit length of a route. The direct traveler density ⇑ Corresponding author. E-mail address: minlfi[email protected] (B. Yu). 0968-090X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.trc.2011.12.003

B. Yu et al. / Transportation Research Part C 22 (2012) 58–75

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method was superior to the traditional direct trip method aiming to maximize the number of direct transit trips, since the method considered the route length while it maximized the direct transit demand. However, in metropolitan areas, transit service cannot always provide direct service among all origins and destinations. Although much progress in transit network design has been achieved in the past several decades, the methods that can consider the total demands of the routes including direct trips and transfers in transit network optimization have received relatively little attention. Zhao (2004) proposed a model for large-scale transit network aiming to minimize transfers and optimize the route/network directness. Mauttone and Urquhart (2009) presented a constructive algorithm for transit network design problem, which took the interests of both users and operators into account. In the model a parameter was defined to describe the proportion of the total demand covered by routes indirectly. Schöbel and Scholl (2006) presented integer programming models and suggested a solution approach using Dantzig–Wolfe decomposition for solving the LPrelaxation. The model was aimed to minimize the number of transfers or the transfer time. Borndörfer et al. (2007) proposed a column-generation approach for route planning in which the passenger paths can be routed freely, and the routes were generated dynamically. In the model, transfers between different modes were handled by linking the networks with appropriate transfer edges, and the edges were weighted by estimated transfer times. The purpose of this paper is to build on direct traveler density model proposed by Yang et al. (2007), and extend it to design transit network aiming to maximize transit trip density that is related to the total demands and the lengths of routes under the length, the directness and the demand constraints and so on. To compute transfer demands of the designed routes, incremental assignment method (Ferland et al., 1975), as an approximate equilibrium assignment approach, is used to assign passengers to transit route network. During passenger assignment, each proportion of passenger demand is placed to the path with the shortest travel time. The travel time of the shortest path contains waiting time at stops that is related to frequencies of routes (the waiting time = 1/2  60/frequency), running time and the penalty of transfer. i.e., travel time = running time + waiting time + penalty of transfer. Usually, many big cities in China all have a great population. And passenger transportation in these cities mainly relies on efficient public transportation systems. For large cities, public transit network can be divided into three levels: skeleton routes, main routes and branch routes according to the features of bus routes. Skeleton routes are main transit corridors with large passenger demand in urban public transit network. These routes are usually covered by some fast, large capacity transport modes, such as rail routes or bus rapid transit (BRT). Skeleton routes are mainly in relatively prosperous district, for example, large commercial center, regions with better transportation facilities or central residential areas. Main routes provide transit service between subway stations and nearby residential areas. They usually lay in the major trunk road with a high passenger demand. Compared with skeleton routes, main routes can cover larger areas. Skeleton routes and main routes together become the main forces in urban transit passenger transportation. The last one is branch route network. Branch routes link the road far away from the central city to skeleton routes or main routes. These routes are often covered by village buses, and mainly for transfers. They expand the coverage and accessibility of transit network. Although passenger volume on these routes is relativity small, they can extend within the community, thus improving the convenience of public transportation trips. In this study, transit network design is divided into three periods according to the feature of transit routes. In the first period, skeleton routes are firstly designed based on transit trip origin–destination (OD). Then, main routes are designed based on the transit trip OD left by the designed skeleton routes. Lastly, branch routes are designed to cover the left transit trip OD by the above designed routes as possible. For transit network optimization, e.g., skeleton, main or branch network, it is difficult to be solved through classical optimization techniques (Newell, 1979; Agrawal and Mathew, 2004). Recently many studies have proved that heuristic algorithms are suitable for large-scale transit network optimization problems, such as ant colony algorithm (Poorzahedy and Abulghasemi, 2005; Yang et al., 2007), genetic algorithm (Pattnaik et al., 1998; Chien et al., 2001; Bielli et al., 2002; Chakroborty, 2003; Agrawal and Mathew, 2004) and simulated annealing algorithm (Zhao and Zeng, 2006), etc. Transit network design stated simply, relates to the determination of a set of routes with satisfying some pre-defined objectives. Transit route is designed by deciding the sequence of the stations including the origin terminal, intermediate stations and the destination terminal. Ant colony optimization (ACO) (Dorigo et al., 1996), inspired by the behavior of ants seeking food in the real world, is a probabilistic technique, and it has been used for solving approximately combinatorial optimization problems. The process designing transit route is very similar to the ant-foraging process in the real world, which may be described as follows. Ants start from the nest (i.e., the origin terminal) and search the food (i.e., the destination terminal). During the searching process, ants select the passing nodes (i.e., intermediate stations) according to some rules, and finally reach the food. If the origin terminal is considered as the nest and the destination terminal is considered as the food, transit route design can be described as the process of searching an optimal path from the ‘‘nest’’ to the ‘‘food’’. Thus, ant colony algorithm is used to solve the optimization model in this study. This paper has been organized in the following way: Section 2 is about the optimization model, including the problem formulations and the basic notations of variables; Section 3 describes ant colony optimization for transit network design problem; numerical analysis is carried out in Section 4; and lastly, the conclusions are drawn in Section 5. 2. Optimization modeling In this paper, transit route network optimization consists of three stages. At first, an empty network is built, and then skeleton routes are added in so as to maximize the direct traveler density until some constraints (e.g., route length con-

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Fig. 1. An example of transit network.

straint, directness constraint or demand constraint) are overrun. After skeleton route design, main routes are laid into the transit route network according to the maximum traveler density including direct passengers and transfers. Lastly, branch routes are placed into the transit route network including skeleton routes and main routes according to the maximum traveler density. Moreover, when designing a transit network, the ACO algorithm will determine to design skeleton, main or branch routes according to searching process. For example, when there is no route satisfying the constraints (e.g., demand constraint of skeleton route) during designing transit network, this indicates that the skeleton network has finished designing and then the algorithm will begin to construct main routes. Until there is no route can satisfy all the constraints, the transit network design terminates. 2.1. Basic notations An example of transit network is presented in Fig. 1. N is the vertex (station) set. The nodes o and d denote the origin and destination terminals, while the nodes i and j are intermediate stations. The terminals and intermediate stations are the subset of the station set N. lij (the unit is kilometer) denotes road distance between the stops i and j, which consists one or several road sections. rod = the route (that is being constructed) starting from the origin terminal o and ending at the destination terminal d; rmn = the route in the optimized transit network from the origin terminal m to the destination stop n; Dtotal od = total demand density (the unit is person/km) of the constructed route rod; Pdirect = the number of direct travelers (the unit is person) in the constructed route rod; od Ptransfer = the number of transfers (the unit is person) of the constructed route rod; od pij = the number of passengers (the unit is person) between the stops i and j along the constructed route rod; Xod = the station set of the constructed route rod; r xijod = a binary variable that represents whether the trip from the stop i to the stop j in route rod requires no transfer by r r using the constructed route rod. For example, in Fig. 1, xijod ¼ 1, while xi;iod0 ¼ 0 r

xijod ¼



1 if the stops i; j 2 XOD 0 otherwise

ð1Þ

Drijod = a binary variable that represents whether the stop j is the next stop of the stop i in route rod. For example, in Fig. 1, r od Dri;iþ1 ¼ 1 ,while Dijod ¼ 0 1 if the stop j is next to the stop i along the route r od ð2Þ Drijod ¼ 0 otherwise Moreover, the transit route during computation process has a direction that is from the terminal o to the terminal d. As is shown in Fig. 2, if the bus is running from the terminal o to the terminal d, the station j is the next station to the station i, thus Drijod ¼ 1 and Drikod ¼ 0. r r However, Dijod ¼ 0 and Dikod ¼ 1 if the bus is running on the opposite direction.

x = a coefficient relating to the weight of transfers in total demands. Smin, Smax = the minimum and the maximum station spacing between two adjacent stops. In general, large station spacing leads to long walking time of passengers, while short spacing could increase the number of the times of bus start-up and shutdown which results in additional oil consumption and discomfort of passengers. Therefore, when selecting feasible next stations, the stations of the region (Smin < lij < Smax) are regarded as alternative stations. The definition of feasible next station is similar to other researches on transit route optimization. In fact, during the constructing phase of ACO, ants select the next station according to the rule. Moreover, according to Chinese standards and good service level, the station

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Road section Road section

Road section Road section

Road section

Station k

Road section

Terminal o

Terminal d

Road section

Road section

Station i

Station j Route section

Fig. 2. The relation of the stations and route sections.

Smin

Smax

i

i current station; feasible stations;

Stations which have been set; infeasible stations;

Fig. 3. The relationship between stop spacing and feasible stop set.

spacing is set to 0.5–0.8 km for the skeleton routes and is set to 0.3–0.5 km for the main routes and branch routes in this study. Assume the station i is the current station, so the stations in the region Smin < lij < Smax are the next feasible stations. Stations in other regions are all infeasible stations for the current location (Fig. 3). The infeasible stations will be in the tabu table of the following ACO method, as well as the stations which have been passing. Lmin, Lmax = the minimum and the maximum length of the laid route. Like station spacing parameters, long routes could increase bus cycle time and probability of breakdown, while short routes lead to more transfers. Here, Lmin = 5 km and Lmax = 15 km; min Disod = the shortest road distance (the unit is kilometer) from the stop o to the stop d; Lod = the length (the unit is kilometer) of the constructed route rod; min Generally, the length should not be extra long or short, i.e., Lmin 6 Lod 6 Lmax . Also, the directness of the route (Disod =Lod ) should be in a rational range. N = the station set on the whole transit network (the nodes in N are based on the stations on the existing transit network); u = the transfer station of the routes rod and rmn; r Liuod = the route length (the unit is kilometer) from the stop i to the stop u along the route rod; Lrujmn = the route length (the unit is kilometer) from the stop u to the stop j along the route rmn; Lentransfer = the transfer distance (the unit is kilometer) from the stop i to the stop j; If assume the stops i and j do not ij belong to the same route, i.e., {i 2 rod and i R rmn and j R rod and j 2 rmn}, the transfer distance between the stops can be written as :

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Lr

Lriuod

mn

uj

Fig. 4. The illustration of the variable Lentransfer . ij

1 2

1

2 1

Fig. 5. An example of road network.

r

Lentransfer ¼ min ðLiuod þ Lrujmn Þ ij

ð3Þ

u¼r od \r mn

r ;r

dijod mn = a binary variable that represents whether the trip from the stop i to the stop j (assume the stops i and j are on the routes rod and rmn, respectively) require transfer. For example, in Fig. 4, the trip from the stop i on the route rod to the stop j r ;r on the route rmn can be achieved by transferring at the station u, then dijod mn ¼ 1.

r ;rmn

dijod

¼

8 > <1

iffrod \ r mn –/g and

> :

otherwise

0



½ði 2 r od and i R r mn Þ and ðj R r od and j 2 rmn Þ[ ½ðj 2 r od and j R r mn Þ and ði R rod and i 2 rmn Þ

 ð4Þ

hij = a binary variable that represents if the transfer distance between a pair stops is excess long. If so the transfer is infeasible.

( hij ¼

min

1 if ðLentransfer =Disij Þ < 3 ij 0

otherwise

ð5Þ

It should be noticed that the value of hij is determined by the actual situation of China. If it is more than three times of the shortest road distance (based on the existing road network), it will discourage passengers to choose the public transportation and turn to other transportation modes. Therefore, the value of hij in the definition is set as 3. Qmin = the minimum number of demand of the laid route (the unit is person). 2.2. Model formulation In this study, the origin or destination terminals in the transit route network are firstly determined based on two rules: (1) the existing terminals of Dalian City; (2) some shopping centers, railway stations and central residential areas, where there was no terminal. Direct demands are the trips between its origin stop and destination stop that require no transfers, while transfers are the trips that cannot get to the destination directly and require using at least two routes. Since transfer is usually annoying and time-consuming for passengers and discourage them using transit mode, the transit trips in this study are assumed to require no more than one transfer, i.e., no transfers or one transfer. The proposed model in our transit route design is developed to maximize the total demand density of the route, which is the number of transit demands divided by the length of the route. And transit demands include direct trips and transfers carried by per unit length of the route. Transit route optimization can be formulated as follows:

max ^ R2R

X P direct þ x  Ptransfer od od Lod r 2R

ð8r od 2 RÞ

ð6Þ

ok

¼ Pdirect od

XX i2N

j2N

r

pij xijod

ð7Þ

B. Yu et al. / Transportation Research Part C 22 (2012) 58–75

Ptransfer ¼ od

XX i2N

Lod ¼

XX i2N

s:t:

r ;r mn

pij dijod

hij

63

ð8Þ

j2N

Drijod lij xrijod

ð9Þ

j2N

8 Lmin 6 Lod 6 Lmax > > > > direct > P þ P transfer P Q min > od od > > > < 0:8 km > lij > 0:5 km for the skeleton route design 0:5 km > lij > 0:3 km for the main route design > > > > > > 0:5 km > lij > 0:3 km for the branch route design > > > : min Disod =Lod < 1:5

^ is the set of all possible sets of routes, and rod is a single route belonging to R. x is used to In Eq. (6), R is a set of routes, R reflect the importance of transfers in the total demands. If x = 0, the method is as the direct demand density method proposed by Yang et al. (2007). With increasing x, transfers account for more importance in the total demands. Usually, it is expected that skeleton routes can carry more direct transit trips while branch routes can feed transfers to skeleton routes or main routes. 3. Ant colony optimization for transit network optimization ACO is an optimization method, which has been successfully applied to some combinatorial optimization problems, e.g. traveling salesman (Dorigo et al., 1996), vehicle routing (Bullnheimer et al., 1999; Bell and McMullen, 2004; Yu et al., 2009; Yu and Yang, 2011), quadratic assignment (Gambardella et al., 1997), and job-shop scheduling (Colorni et al., 1994), etc. But its application to transit network design is relatively new. Besides, for transit network optimization, there also several researches, such as Yang et al. (2007) and Poorzahedy and Abulghasemi (2005). In the real world, ant searches food while laying down pheromone trails. Furthermore, ants’ moves are not random but follow the trails. In the transit route network design of this paper, transit demand density of a route is regarded as ‘‘pheromone’’ of our ACO. Then, ants can search the optimal bus route from the origin terminal to the destination terminal according to the quantity of the ‘‘pheromone’’ on the route sections. The more the ‘‘pheromone’’ is left, the higher the probability will be that this path will be followed by other ants. The process continues until the transit route network can be successfully designed. 3.1. Generation of routes In this study, the domain of the problem is the set of all possible routes. Thus, according to the objective function (Eqs. (6)–(9)), an optimal route can be achieved in the domain under current conditions. Using ACO to implement the above process, each ant can construct its route when it starts from the origin terminal, selects following stops, until reaches the destination terminal. However, a feasible route should also satisfy the length constraint, the directness constraint and the demand (the direct demand or the total demand) constraint. As ants select following stops, they are based on the pheromone strength and the visibility information on the route links from the current stop to the following stops. The pheromone information is the key point for ACO and it usually relates to the optimization objective. Thus, to select the next stop j at the ith stop, the kth ant uses the following probabilistic formula:

yij ðkÞ ¼

8 b a < P sij gij

j R tabuk

:

otherwise

sa gbih

hRtabuk ih

0

ð10Þ

where yij(k) = the probability of choosing to combine stops i and j on the route; sij = the pheromone strength of edge (i, j); gij = the visibility of edge (i, j), which encourages ants to visit the locally optimal path, here gij = pij/lij; a and b = the relative influence of the pheromone trails and the visibility values, respectively; tabuk = the set of the infeasible stops for the kth ant, which includes the stops that have been visited by the ant or violated the station spacing and length constraints. 3.2. Local search After an ant has found a new route by linking the selected stops, the stop set on the route is determined, as well as the passenger demands (including the direct demand and transfers). Then the different sequences among stations in the route will lead to different operating lengths of the constructed route. For the route, the sequence of the stations with the minimum length was optimal. The 2-opt exchange (Bullnheimer et al., 1999; Bell and McMullen, 2004; Yu et al., 2009) is used to

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(a) A transit route before applying 2-opt method

(b) An optimized transit route by 2-opt method Fig. 6. Local optimization of the designed transit route.

see if an overall improvement in the objective function can be attained, which searches the overall stop sequence by testing all possible pairwise exchanges of stop locations in a route. For example, as the road network shows in Fig. 5, a route from an ant is the route 1–2–3–4 (Fig. 6) and its length is 5. It is obvious that the sequence is non-optimal. The 2-opt method through exchanging the stop 2 and the stop 3 can obtain the local optimal route, i.e. the route 1–3-2–4, and its length is 3 (Fig. 6). Since the method does not add or delete stops, it does not change the demand of the route. 3.3. Transit passenger assignment Since the optimization model proposed in this study needs to compute trips of a route, especially transfers, a transit passenger assignment method needs to be developed. Among passenger assignment methods, the equilibrium assignment and the shortest path assignment are the most common methods. Though the equilibrium assignment can improve the solution a lot, it is not suitable for the transit network in practical size, especially for the large scale of networks and nodes in the instance of this paper. On the other hand, the shortest path assignment method can be implemented quickly, but it do not consider the capacity of each route. In fact, however, the capacity of a route is not unlimited. It equals to the product of the capacity of a standard vehicle and the frequency of this route. Each route would have a maximum capacity. To trade off the computational time and quality, the incremental assignment method was used in the paper. And when the OD matrix was divided into proper portions, the result of incremental assignment method was approximate to the one of equilibrium assignment method. The basic idea of incremental assignment method is to divide the OD matrix into several equal portions through some experiments (in this paper, the demand is divided into 50 equal portions), and then assigns each portion to the transit route network according to shortest path method recurrently. After each portion assignment, travel time in transit route network should be updated, i.e., the minimum hyperpath could be changed between the origin and destination. Then, the crowding route section whose passenger demand exceeded the capacity (the capacity is related to the bus frequency of the section and the capacity of a standard vehicle) will be set a high impedance to prevent the following assignments to the section. The process of the transit route network design and frequencies of route updating are shown in Fig. 7. Thus, the steps of this method are as follows: Step 1 Dividing the OD matrix into several equal portions. Step 2 Assigning one portion to the minimum hyperpath. Step 2.1 Selecting a pair of terminals (the origin terminal and the destination terminal) of the current portion. Step 2.2 Searching the minimum hyperpath between the origin and the destination under current conditions. The travel time of the minimum hyperpath contains the running of road sections, the waiting time that is related to the frequency and the penalty of transfer. Step 2.3 Assigning the passenger demand between the pair of terminals to the minimum hyperpath. Step 2.4 Recalculating the travel time of transit route network. Checking passenger demand of each route section whether to satisfy capacity limitation (the capacity limitation of the route section can be computed by the number of the passing vehicles that is related the frequency and the capacity of a standard vehicle). If so, go to Step 2.5, otherwise, the route sections, whose maximum capacities have been reached or exceeded, will be set a very large waiting time to ensure the following assignments not to select the section again. Step 2.5 Repeating Step 2.1–Step 2.4 until the whole passenger demands of the current portion are assigned to the transit network. Step 3 Repeating Steps 2–3 until the whole OD matrix is assigned.

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Adding the optimal route into network; Calculated the frequency of the optimal route Assigning the OD matrix based the frequencies of the current N routes and the frequency of the optimal route; Updating the frequencies of the N+1 routes N+1 routes in current network

N routes in current network

N+2 routes in current network

Adding the optimal route into network; Calculated the frequency of the optimal route Assigning the OD matrix; Updating the frequencies of the current route network Fig. 7. The updating of the frequencies during transit route network design.

Assume that there are two routes (i.e., the routes 1 and 2) through the stops i and j (Fig. 8). The two routes are set with a pre-designed frequency (the frequency would be illustrated later). Then, the distance between the stops i and j along the route 1 will be shorter than the one along the route 2, i.e., LRoute1 < LRoute2 . Thus, in the initial transit passenger assignment, ij ij passengers from the stops i to j will be all assigned to the route section of the route 1. According to the pre-designed frequency, each route would have a maximum capacity limitation of the passenger demands assigned on the route 1. The subsequent passengers could not be assigned to the route 1, and have to select the route 2. In this study, the process is achieved by changing the travel time between the two stops, which includes the running time for buses and the waiting time for passengers and the penalty of transfer. After assigning each portion, the travel time of transit route network would be updated. If the passenger demands on a certain section have reached or exceeded to the maximum capacity of the route section, the waiting time of the section between the stops i and j would be set as a very large impedance to ensure the following assignments not to select the section. Therefore, when the passenger demand of the section from the stops i to j on the route 1 cannot satisfy the capacity limitation, the subsequent passengers will have to choose the route 2. After a route has been laid into the transit route network, the frequency of each route needs to be updated. The frequency of each route is based on the maximum cross-section method according to the assigned passenger demand, i.e., the frequency can be computed by the passenger demand of the maximum cross-section and the capacity of a standard vehicle. Then, the transit route network with new frequencies will be considered as the given network for the further route design. In this study, the frequency of each route cannot be larger than 60 vehicles per hour. To describe the assignment process in this paper, an example of the process of incremental assignment is proposed as follow. Assume that in the current transit network, there is one route, i.e., the route 1. The route 1 is the new optimized route. Assume that the frequency of the route 1 is set to 60 vehicles/hour, i.e., f1 = 60 vehicles/hour. If the capacity of a standard vehicle was 80 passengers, the maximum passenger capacity of a cross-section of the route 1 was 4800 persons. Assumed that before the stop i, there were 4000 passengers who had got on the route 1 and there were

L ijRoute2

j

i

L ijRoute1

f1

cv

cR

f1

cv

cR

Fig. 8. The process of transit passenger assignment.

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B. Yu et al. / Transportation Research Part C 22 (2012) 58–75

Stop i

Stop j

Frequency of Route 1: f1 =60 Passenger Volume of Route 1: 5000 persons Bus station

Passenger demands

Road network

Route 1

Fig. 9. Result of the first assignment in the current transit network (one route).

1500 passengers who wanted to go from the stop i to stop j by bus. During passenger assignment, the passenger demands between the two stops were divided into three equal portions, i.e., each portion was 500 persons. Since there was only one route through the stops i and j, the route 1 was the minimum hyperpath between two stops. Thus, the first portion demand was assigned to the route 1, as well as the second portion. After that, the passenger volume of the route 1 reached 5000 passengers, which had exceeded the capacity limitation and could not assign any more. Then, the assignment was terminated and the frequency of the route 1 was updated, i.e., f1 = 5000/80 = 62.5 vehicles/hour. Because the frequency of each route could not be large than 60 vehicles/hour in Dalian city, the frequency of the route 1 was also revised to 60 vehicles/hour (Fig. 9). The updated frequency of the route would be used in the further assignment. Then, the second route, i.e., the route 2, was optimized and added into the current transit network. The frequency of the new route (the route 2) was also set to 60 vehicles/hour. Between the stops i and j, there were two paths, i.e., one path was through the route 1 and the other was through the route 2. Assume that the running time between the stops i and j through the route 1 was 15 min and the running time through the route 2 was 20 min. The passenger demand (1500 persons) between the two stops was still divided into three equal portions. Since the waiting time of the two routes were the same, the path through the route 1 was the current minimum path. Thus, the first and the second portions of the passenger demand were assigned to the route 1. Then, the passenger volume of the route 1 exceeded the capacity of the route. The path between the stops i and j of the route 1 was set a high impendence. Thus, the last portion of passenger demand was assigned the route 2. Assume that there was 1900 persons on the route 2 before the stop i. If the passenger load of the cross-sections after the stop j was not more than the one between the stops i and j, the passenger demand of the maximum cross-section of the route 2 was 2400. After the second assignment, the frequencies of two routes were updated (Fig. 10), i.e., f1 = 60 and f2 = 30. The new frequencies of the route 1 and the route 2 would be used in the future OD matrix assignment. 3.4. Update of pheromone information Before updating pheromone information of transit route network, it is necessary to evaluate the searched route. As ant constructs its route, the direct and indirect passenger demands among the stations that the ant has passed through will

Running time: 20 mins

Running time: 15 mins

Stop i

Frequency of Route 1: f1=60 Passenger Volume of Route 1: 5000 persons Bus station

Passenger demands

Stop j

Frequency of Route 2: f 2=30 Passenger Volume of Route 2: 2400 persons Road network

Route 1

Fig. 10. Result of the second assignment in the current transit network (two routes).

Route 2

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be summed. Then, the objective value of the searched route equals to the number of all the direct and indirect demands divided by the route length. According to the objective value of each route, pheromone of the network will be updated. The update of pheromone trails consists of two stages. The first stage is to reduce the amount of pheromone on all edges by simulating the natural evaporation of pheromone from real ants. The second stage is to compute the pheromone increments of the visited edges that relate to the optimization objective and is also the adaptive learning technique of ACO. The standard pheromone updating equation is written as:

snew ¼ q  sold ij ij þ

X

Dskij

q 2 ð0; 1Þ

ð11Þ

k

where snew = the pheromone on the section (i, j) after updating; sold ij ij = the pheromone on the section (i, j) before updating; q = the constant that controls the speed of evaporation; k = the No. of the ant; Dskij = the increased pheromone on section (i, j) of the route found by the ant k. The pheromone increment updating rule uses the ant-circle method (Badr and Fahmy, 2004). The increased pheromone on link (i, j) of the route found by the ant k can be written as:

( k ij

Ds ¼

Dod

e

0

if section ði; jÞ is on the kthroute otherwise

ð12Þ

where e = a constant, which is set as 150 in this study; D = the objective function of the route found by the ant k (‘‘’’ denotes the superscript ‘‘direct’’ or ‘‘total’’). Here, D equals to Pdirect during skeleton network optimization, while D equals to Dtotal od od during main or branch network optimization. 3.5. The process of ACO

BEGIN Initialization; Emptying the optimized transit route network U NUM = 0 //The number of routes in the optimized transit route network U Qmin = 1 //The minimum demand constraints of the designed route Smin=0, Smax=1 //The minimum and the maximum station spacing between two adjacent stops ER = False //A boolean variable that represents there is no route in the domain, which can //meet all the constraints, such as the demand constraints and the route length //constraints Repeat If ER = True //Exist a route in the domain that can meet all the constraints Adding the optimal route among the alternative route set W into the optimized transit route network U NUM++ //when adding an optimal route among the alternative route set W into the //optimized transit route network U Assigning the OD matrix by incremental assignment method Updating the frequency of each route in the current transit route network U based on the maximum cross-section method //The frequencies are used in the future OD //matrix assignment Else If Qmin > 7000 then //Optimizing the skeleton transit route network Qmin = 7000 Smin = 0.5 Smax = 0.8 Emptying the alternative route set W //The alternative route set W is a temporary route set, //among which the optimal route will be added the //optimized network U Else If Qmin > 4000 then //Optimizing the main transit route network Qmin = 4000 Smin = 0.3 Smax = 0.5 Emptying the alternative route set W Else If Qmin > 1000 then //Optimizing the branch transit route network Qmin = 1000 Smin = 0.3 Smax = 0.5 (continued on next page)

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Emptying the alternative route set W Else Exit Repeat End If Repeat Selecting an origin terminal and a destination terminal from the stop set Repeat gen = 1 Constructing a route between the origin terminal and the destination terminal according to Eq. (10) Optimizing the route using the local search method Assigning the OD matrix by incremental assignment method based on the frequency of each route in the current transit route network U Evaluating the route and computing the objective value; Updating the pheromone of the network gen = gen + 1 Until (Convergence criteria or the maximum number of generations) Recording the searched route between o and d into the alternative route set W under the current condition Until (Enumerating all the pair of terminals) If there is no route in the domain can meet all the constraints, then ER = False End If Until (The maximum number of generations) End

4. Numerical test The transit demand density model proposed by this study is tested with two sets of data. The first one attempts to illustrate the transit network design process on a simple network; and the second one aims to test the performance of the model and ACO method for designing a real bus network in Dalian city, China. 4.1. Test 1 Test 1 is used to describe the process of optimizing the transit network by the proposed method, i.e., designing skeleton routes, main routes and branch routes in a sequence. In Fig. 11, there is a sample network, which has six nodes and nine links. The integer number on each node represents a stop, and the real numbers on the links between adjacent stops represent the station spacing in kilometers. For example, the number 0.60 means the distance between stations 1 and 3 is 0.60 km. Assume the stop 1 and the stop 2 as the origin terminals, the stop 5 and the stop 6 as the destination terminals and assume

Fig. 11. An example network.

Table 1 Transit demand matrix. Stop

1

2

3

4

5

6

1 2 3 4 5 6

0 22 35 82 66 100

22 0 40 44 95 88

35 40 0 33 67 46

82 44 33 0 74 51

66 95 67 74 0 33

100 88 46 51 33 0

69

B. Yu et al. / Transportation Research Part C 22 (2012) 58–75 Table 2 All possible set for each pair of origin–destination terminals in skeleton route planning. Origin–destination

Candidate route

Coverage of direct transit services

Length

Direct demand

Direct demand density

1M5

1M3M5 1M3M4M5 1M4M5 1M4M3M5

{1 M 3, {1 M 3, {1 M 4, {1 M 4,

3 M 5} 1 M 5, 3 M 4, 3 M 5, 4 M 5} 4 M 5} 1 M 5, 4 M 3, 4 M 5, 3 M 5}

2.36 4.04 2.68 3.80

254 474 308 474

107.63 117.33 114.93 124.74

1M6

1M3M6 1M3M4M6 1M4M6 1M4M3M6

{1 M 3, 1 M 6, 3 M 6} {1 M 3, 1 M 4, 1 M 6, 3 M 4, 3 M 6, 4 M 6} {1 M 4, 1 M 6, 4 M 6} {1 M 4, 1 M 3, 1 M 6, 4 M 3, 4 M 6, 3 M 6}

2.56 3.76 2.40 4.00

324 570 378 570

126.56 151.60 157.50 142.50

2M5

2M3M5 2M3M4M5 2M4M5 2M4M3M5

{2 M 3, {2 M 3, {2 M 4, {2 M 4,

2 M 5, 3 M 5} 2 M 4, 2 M 5, 3 M 4, 3 M 5, 4 M 5} 2 M 5, M 4 M 5} 2 M 3, 2 M 5, 4 M 3, 4 M 5, 3 M 5}

2.66 4.34 2.54 3.66

294 464 280 464

110.53 106.91 110.24 126.78

2M6

2M3M6 2M3M4M6 2M4M6 2M4M3M6

{2 M 3, {2 M 3, {2 M 4, {2 M 4,

2 M 6, 2 M 4, 2 M 6, 2 M 3,

2.86 4.06 2.26 3.86

302 498 288 498

105.59 122.66 127.43 129.02

1 M 5, 1 M 4, 1 M 5, 1 M 3,

3 M 6} 2 M 6, 3 M 4, 3 M 6, 4 M 6} 4 M 6} 2 M 6, 4 M 3, 4 M 6, 3 M 6}

the stop 3 and the stop 4 as the medium stops. The symmetrical transit demand matrix for the six stops is randomly generated, as shown in Table 1. In this simple example, there are only six stops and it is impossible to have many routes in the simple network. Thus, it is assumed that there can just contain a skeleton route and a branch route in the simple network, i.e., there is none of main routes. 4.1.1. Skeleton transit route planning During designing a skeleton transit route, the transfer coefficient is set to zero. The mechanism for skeleton route design may be stated systematically as: First, set the destination terminal; second, select upstream stops and compute demands from the upstream stops to the downstream stops; then repeat until the origin terminal. For the example illustrated in Fig. 11, Table 2 shows the enumeration of all possible sets for each pair of the origin–destination terminals during finding the skeleton transit route regardless of the minimum passenger constraint and the length constraint. From Table 2, the route 1–4–6 (Fig. 12) should firstly be laid to the network. Even though the route does not carry the most passengers, it is the most efficient route in fact. Then, the demand matrix should be revised and the new demand matrix is shown in Table 3.

Fig. 12. Optimal skeleton route with maximum direct demand density.

Table 3 Revised transit demand matrix after adding the skeleton route.

The numbers with shadow are the direct demands carried by the laid skeleton route(s) (1–4–6).

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Table 4 All possible set for each pair of origin–destination terminals in branch route planning. Origin– destination

Candidate route

Coverage of direct transit services

Length

Direct demand

Direct demand density

Coverage of transfers

Transfer demand

Demand density

1M5

1M3M5

{1 M 3, 1 M 5, 3 M 5}

2.36

254

107.63

272

222.88

1M3M4M5

{1 M 3, 1 M 4, 1 M 5, 3 M 4, 3 M 5, 4 M 5} {1 M 4, 1 M 5, 4 M 5} {1 M 4, 1 M 3, 1 M 5, 4 M 3, 4 M 5, 3 M 5}

4.04

372

92.08

(3, 5) M (4, 6) = {3 M 4, 3 M 6, 4 M 5, 5 M 6} (3, 5) M (6) = {3 M 6, 5 M 6}

154

130.20

2.68 3.80

206 372

76.87 97.89

(5) M (6) = {5 M 6} (3, 5) M (6) = {3 M 6, 5 M 6}

66 154

101.49 138.42

{1 M 3, 1 M 6, 3 M 6} {1 M 3, 1 M 4, 1 M 6, 3 M 4, 3 M 6, 4 M 6} {1 M 4, 1 M 6, 4 M 6} {1 M 4, 1 M 3, 1 M 6, 4 M 3, 4 M 6, 3 M 6}

2.56 3.76

148 192

57.81 51.06

(3) M (4) = {3 M 4} -

44 -

75.00 51.06

2.40 4.00

0 192

0.00 48.00

-

-

0.00 48.00

2M3M5 2M3M4M5

{2 M 3, 2 M 5, 3 M 5} {2 M 3, 2 M 4, 2 M 5, 3 M 4, 3 M 5, 4 M 5}

2.66 4.34

294 464

110.53 106.91

526

110.53 228.11

2M4M5

{2 M 4, 2 M 5,<– > 4 M 5} {2 M 4, 2 M 3, 2 M 5, 4 M 3, 4 M 5, 3 M 5}

2.54

280

110.24

396

266.14

3.66

464

126.78

(2, 3, 5) M (4, 6) = {1 M 2, 1 M 3, 1 M 5, 2M 6, 3M 6, 5M 6} (2, 5)M (1, 6) = {1 M 2, 1 M 5, 2 M 6, 5 M 6} (2, 3, 5)M (1, 6) = {1 M 2, 1 M 3, 1 M 5, 2M 6, 3M 6, 5M 6}

526

270.49

2M3M6

{2 M 3, 2 M 6, 3 M 6}

2.86

302

105.59

200

175.52

2M3M4M6

{2 M 3, 2 M 4, 2 M 6, 3 M 4, 3 M 6, 4 M 6} {2 M 4, 2 M 6, 4 M 6} {2 M 4, 2 M 3, 2 M 6, 4 M 3, 4 M 6, 3 M 6}

4.06

398

98.03

(2, 3) M (1, 4) = {1 M 2, 1 M 3, 2 M 4, 3 M 4} (2, 3) M (1) = {1 M 2, 1 M 3}

104

123.65

2.26 3.86

188 398

83.19 103.11

(2) M (1) = {1 M 2} (2, 3) M (1) = {1 M 2, 1 M 3}

44 104

102.65 130.05

1M4M5 1M4M3M5 1M6

1M3M6 1M3M4M6 1M4M6 1M4M3M6

2M5

2M4M3M5

2M6

2M4M6 2M4M3M6

Fig. 13. Optimal branch route with maximum demand density.

4.1.2. Branch transit route planning An important role for branch transit services is to implement convenient transfer besides direct service. Therefore, the branch transit route design should consider direct demand and transfers, simultaneously. Assume there is only one skeleton route (1–4–6) in the above example and the transfer coefficient is set to 1, i.e., x = 1. Like the last stage, after designing the skeleton transit service (Fig. 12), Table 4 shows the enumeration of all possible sets for each pair of the origin–destination terminals during finding branch transit routes. The route with the maximum demand density is the route 2–4–3–5 (Fig. 13). After the route is laid, the remained demand matrix is shown in Table 5 and it can be observed that all passengers can be served by the designed transit network.

4.2. Test 2 The all-enumeration method can obtain the optimal transit network under the small city with a fewer routes, while for a large-scale transit network the model is difficult to solve using the same method within acceptable time. To examine the model and the efficiency of the algorithm, data in Dalian city are used for the numerical test 2. There are totally 89 bus routes

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Table 5 Revised transit demand matrix after adding the branch route.

The numbers with shadow are the direct demands carried by the laid skeleton route(s) (1–4–6). The italic numbers are the direct demands carried by the laid branch route(s) (2–4–3–5). The bold numbers are the transfers carried by the laid branch route(s) (2–4–3–5).

and 1500 bus stops, which extends 1130 km. Passenger stop OD matrix at peak hours is obtained from our former research (Yang et al., 2007). The proposed method in this study consider transfer and direct transit trips and is more complicated than the model presented by Yang et al. (2007). Therefore, route searching is a more time consuming work. Here, the platform, which covers one or several close stops, is induced to reduce the number of stops and to simplify network structure and transfer path search. Then, the stop-based demand matrix is also changed into platform-based demand matrix. After simplification, there are 1007 transit platforms. It is reasonable that the transit network design is divided into several stages including the skeleton network design, the main network design and the branch network design. Therefore, we set different minimum number of travelers of the laid route during each planning stage, i.e., Q min ¼ 7000 during skeleton route searching, Qmin = 4000 during main route searching, and Qmin = 1000 during branch route searching. The parameters in the ACO were a (a = 2) and b (b = 1) (Yang et al., 2007). The method was carried out with Microsoft Visual C++.Net 2003 on windows XP platform environment. 4.2.1. Model calibration To determine the transfer coefficient of each transit network design level, several groups of transfer coefficients are tested, which include fxskeleton ¼ 0; xmain ¼ 0; xbranch ¼ 0g; fxskeleton ¼ 0; xmain ¼ 0:2; xbranch ¼ 0:5g; fxskeleton ¼ 0; xmain ¼ 0:5; xbranch ¼ 1g; fxskeleton ¼ 0; xmain ¼ 0:8; xbranch ¼ 1:2g and fxskeleton ¼ 0; xmain ¼ 1; xbranch ¼ 2g. Based on the different parameters, several transit networks are designed. In the five designed transit networks, there are 77, 73, 71, 67 and 61 routes, respectively. Then, the comparison results of no transfer proportion (direct demands/total service demands), service coverage proportion (total serviced demands/total transit demands), average route demand density and average route directness of each network are shown in Figs. 14 and 15, respectively. Besides, the demand density and route directness are averaged over the routes of the optimized transit network. The first parameter combination is, in fact, direct demand density method and its no transfer proportion is the highest among the five parameter combinations. In the transit network designed by the direct demand density method, there are the most routes compared with the other networks. So the average route directness of the network is the largest. However, average route demand density of the direct demand density -based network is the smallest. This indicates that the direct demand density method is only suitable for large passenger corridor planning, rather than the branch route design. Moreover, compared with the other four optimized transit route networks, the service coverage proportion of the direct demand density-based network is also the smallest. In general, low service coverage proportion

100

Proportion (%)

No transfer proportion

Service coverage proportion

85 70 55 40

Fig. 14. comparison of no transfer proportion and service coverage proportion in different parameter sets.

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Fig. 15. Comparison of average demand density and average directness in different parameter sets.

185

Demand density (person/km)

Demand density of ACO

Demand density of GA

175

165

155

145

1

2

3

4

5 6 Calculation times

7

8

9

10

Fig. 16. Comparison of solution quality and computation time of each calculation.

would greatly reduce the service level of transit route network, and thus reducing the attractiveness of public transport to passengers. So with respect to transfers, the service level of the optimized transit route network can be improved. In addition, the transit networks with the fourth and fifth parameter combinations can cover large scale passenger demands while their no transfer proportions are low. It can be attributed to their importance of transfers. But the average route directness of each network is very low. This can induce that these optimized routes are meandering to get more demands including direct demands and transfers. Furthermore, the combination fxskeleton ¼ 0; xmain ¼ 0:5; xbranch ¼ 1g has a high average route demand density similar to the fourth and fifth, but the average route directness is the highest among the three combinations. In fact, from Fig. 15, the average directness of the fourth and fifth networks is close to 0.67 and obviously lower than the others. Therefore, the combination fxskeleton ¼ 0; xmain ¼ 0:5; xbranch ¼ 1g is adopted in our transit network design. 4.2.2. Solution algorithm validation To test the validation of the proposed algorithm, a two-stage genetic algorithm is introduced for comparison. In the genetic algorithm, the first stage is a conventional binary code, which indicates the station set on a certain route. In the code, each number represents a stop, where 1 indicates that the stop belongs to the route, and 0 indicates that the stop does not belong to the route. In the second stage, local search is used and the best sequence of the stops can be found. In the genetic algorithm, single point crossover and mutation is used, which is the simplest crossover and mutation method. Then, with the same parameters, we continue experimenting two algorithms 10 times, and Fig. 16 depicts the results of the calculation. As is shown in Fig. 16, the average demand density of 10 times calculations by the algorithm proposed in our paper is 172.23. And it’s much higher than the one calculated by genetic algorithm, which turns to be 163.85. Then, we test

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the stability of the proposed algorithm. It can be seen that the method has an average deviation of 1.74%, which shows a good stability of our proposed algorithm. In addition, the average computation time of the ACO method is about 632 min, while the average time of genetic algorithm is 1016 min (the maximum running time in two algorithms is set as 1200 min). Considering that there is no real-time request in transit network design, however, the time can be regarded as an acceptable time.

Fig. 17. Optimal transit network with best transfer coefficient.

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Fig. 17 (continued)

Fig. 18. Comparison of demand proportion and no transfer proportion of different routes.

4.2.3. Results With the best combination fxskeleton ¼ 0; xmain ¼ 0:5; xbranch ¼ 1g, a transit network can be designed. And the skeleton transit routes, the main transit routes and the branch transit routes in the designed transit network are shown in Fig. 17. During skeleton routes searching, six routes are designed. These skeleton routes go from the residential place to the center or traffic terminals in Dalian city, which are the points generating or abstracting large trips. Compared with the skeleton transit routes, since minimum demand constraints are low, the numbers of the optimized main routes and branch routes are more. The main transit network includes 22 routes, while there are 43 routes in the branch transit network. The service coverage proportions of three networks, i.e., the skeleton, main and branch networks, are 20.5%, 35.2% and 27.6%, respectively. One can see that the service coverage of the main and branch route networks are larger than the one of the skeleton route network. Especially, the branch routes could get to the communities and the routes are usually very winding. Although the passenger volume is not large, the branch routes have a better accessibility. Fig. 18 shows the demand proportions and no transfer proportions carried by three networks in the existing network and optimized network. It is obvious that the main routes carry most passengers in both existing network and optimized network. In optimized network, the no transfer proportion of the skeleton routes, main routes and branch routes turns to be decreased, while there is little change for the no transfer proportion in the existing network. It suggests that the route hier-

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archy in the existing network is not clear. The skeleton routes, in particular, cannot exploit its high-speed and large capacity advantage to the full due to the small passenger demands. In the whole optimized transit network, although there are just six skeleton routes, they can still carry about 24% demands. Furthermore, the high no transfer proportions show the feature of these skeleton routes. Moreover, nearly half of the total demands carried by the branch routes consist of transfers, in accordance with the transfer coefficient. Overall, the method proposed in our article can improve the layout of transit network effectively. 5. Conclusions Transit network design is an important part of urban transportation planning. There have been many researches for transit network planning problem. However, in these researches, transfers have received relatively little attention. This is mainly because that transit network design problem is still difficult to be solved by classical optimization techniques, even if not considering transfers. This paper presented a transit network design model aiming to maximize transit demand density that considered direct trips and transfers. Data from Dalian city was used to test the method and results showed that the optimized transit route network can be improved with respect to transfers. Further study will consider other aspects of the resulting transit network such as the cost of the operators so as to enhance the performance of the proposed methods. Moreover, as the incremental assignment method was used as an approximate equilibrium assignment method, future study will try some equilibrium assignment methods to improve the solutions. 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