About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 390128, 14 pages
http://dx.doi.org/10.1155/2012/390128
Research Article

Research on Public Transit Network Hierarchy Based on Residential Transit Trip Distance

School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China

Received 9 May 2012; Revised 9 July 2012; Accepted 26 July 2012

Academic Editor: Wuhong Wang

Copyright © 2012 Gao Jian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

To the problem of being lack of transit network hierarchy theory, a research on public transit network hierarchy optimization based on residential transit trip distance is conducted. Firstly, the hierarchy standard of transit network is given, in addition, both simulating electron cloud model and Rayleigh distribution model are used to fit the residential transit trip distance. Secondly, from the view of balance between supply and demand, the hierarchy step of transit network based on residential transit trip distance is proposed. Then, models of transit’s supply turnover and demand turnover are developed. Finally, the method and models are applied into transit network optimization of Baoding, Hebei, China.

1. Introduction

Giving priority to the development of public transportation system plays a pivotal role in alleviating urban traffic congestion. Recently, with the fast expansion of the city size, the public transit lines get much larger and higher density. Many researchers attended to ensure each transit lines’ function by hierarchical public transit network, and then a public transportation service system with clear hierarchy can be established, which can guide residence’s travel behavior scientifically.

The notion of multilevel transit network planning has been widely acknowledged and adopted. Carrese and Gori [1] considered three-route hierarchies; the other studies focus on specific linkages such as feeder routes for a rail network; Bagloee and Ceder [2] divided the transit network into three degrees: mass route, feeder route and local route, and the method to determine hierarchy of a route is studied; Van Goeverden and van Nes [3] describes how the public transport system consists of different network levels; van Nes [4] proposed multilevel network optimization for public transport networks and he [5] also conducted research on multiclass urban transit network design; Salzborn [6] and Knoppers and Muller [7] also optimized the transit network based on hierarchy concept. Jian and Gang [8], Wei et al. [9] proposed the multilevel transit network planning concept during planning the public transit network; Fangqiang [10] conducted transit network optimization based on transit network hierarchy from the view of coupling residential trip with transit network; Kuah and Perl [11] presented a mathematical model for feeder-bus network-design problem. They solved this model by a heuristic method called savings heuristic. Shrivastava and O’Mahony [12] designed feeders for one of heavy rail suburban service stations and coordinated schedules with the aid of genetic algorithm. Guillot [13] and Higgins [14] also conducted research on the bus network of a city was coordinated with the rail network. Chien et al. [15] applied genetic algorithm to design the feeders of a real network and its delay at intersections. Mohaymany and Gholami [16] used multiple modes with various capacities and performances in the feeder network design based on the minimization of user, operator, and social costs. Verma and Dhingra [17, 18] designed feeders of rail transit and presented a synchronized scheduling for rail and its feeders. Shrivastav and Dhingra [19] applied their heuristic feeder route generation algorithm to make a feeder network.

Although many researchers have conducted study related to hierarchy of public transit network, most of them did not propose any quantitative methods to calculate reasonable length for each hierarchical public transit network. In addition, travel behavior is widely studied and many transportation problems are solved based on it [20], so this paper attempts to put forward a hierarchy of public transit network which is applicable to certain urban development patterns and suitable for urban residents who travel by public transit in view of the fact that, at the moment, there is no such hierarchy based on the different demands of ridership and trip distance (resulting in the lack of arteries and local route lines).

2. Analysis on Transit Trip Distance Distribution

2.1. Method of Determining Hierarchy of the Transit Network

A rational public transit network should include different hierarchy types which operate with different standards. This paper categorizes public transit lines into 3 types: (1) mass route, which is the skeleton of the network; (2) feeder route, which operates inside of the district; (3) Local route, which serves as support facility to the mass route.

Hierarchy of a route and station spacing has been considered as the criteria of determining the network: (1) All the rail transit lines are regarded as mass routes, (2) public transit lines might qualify as mass routes when they are located along the express way or arterial road and the bus-stop spacing is longer than 800 m, (3) feeder route is the transit line that is located along the arterial road or subarterial road and the bus-stop spacing is between 500 m and 800 m, and (4) the remaining unclassified lines are all counted as local routes. There are still certain points of disagreements on the above criteria. Herein, we set up following provisions to pave the ways for later modeling.(1)One public transit line alone may cover different hierarchy types of route, as shown in Figure 1. In such case, if the trip covers different types, the transfer time is 0.(2)Lines passing the same section of the road are well considered to make sure that they all belong to the same hierarchy type.

390128.fig.001
Figure 1: Sketch of transit network hierarchy types.
2.2. Analysis on Transit Trip Distance Distribution

The simulation of electron cloud model and Rayleigh distribution model are both widely used in the research of trip distance [1618]. The applicability of these two models in the study of transit trip distance has been explored in this paper. Please refer to [17, 18] for more detailed introduction of these two models. Data from twelve different cities (Shenyang, Suzhou, Qinhuangdao, Bengbu, Yinchuan, Wujiang, Changshu, Suzhou, Huaibei, Changde, Chaozhou, and Weifang) have been used to compare the above models.

Cumulative probability distribution function of the electron cloud model and Rayleigh distribution model are: 𝐹(𝑠)=1(2(𝑠/𝑎0)2+2(𝑠/𝑎0)+1)𝑒2(𝑠/𝑎0) and 𝐹(𝑠)1𝑒(0.5𝜆𝑠2) respectively. Table 1 shows the values of parameters involved. The comparative analysis has been conducted in terms of precision in simulating and ability in interpreting:(1)Precision in simulating: values of fitting function 𝑅2 indicate that both models have very high fitting precision. However, after analyzing the mean values and standard deviations of 𝑅2, the electron cloud model turned out to be a better approach.(2)Ability in interpreting: λ derived from the Rayleigh distribution model has no actual meaning, whereas 𝛼0 derived from electron cloud model means the average trip distance. Consequently, the latter performs better.

tab1
Table 1: Parameters and coefficients of determination in simulating electron cloud model and Rayleigh distribution model.

Given those two aspects, electron cloud model is more feasible for the theory stated here.

3. Hierarchy of Transit Network Based on Balance between Supply and Demand

3.1. Concepts of Transit Hierarchy Planning

The optimized allocation of transit network is to study the balance between supply and demand of the public transit system at the macroscopic level. The typical quantitative indicators are the transit’s supply turnover and demand turnover. Attempts have been made to balance the supply and demand through rational categorizing the network.

3.2. Analysis on Demand Turnover of Different Hierarchy Types
3.2.1. Optimal Trip Distance of Public Transit Lines in Different Types of Route

Different public transit lines have varies missions and desired length of passengers’ trips (see Figure 2). When the trip distance is less than 𝑆1, local routes are more favorable. When it is between 𝑆1 and 𝑆2, feeder routes are better options. If the distance is longer than 𝑆2, mass routes have obvious superiority. The problem of how to figure out preferential trip distance can thus be converted to the calculation of critical values 𝑆1 and 𝑆2, which can easily be obtained by equations: 𝑇local=𝑇feeder,𝑇feeder=𝑇mass (𝑇 stands for the minimum trip time). Hence, the following model mainly focused on the trip time.

390128.fig.002
Figure 2: The relationship between transit trip distance and trip time.

(I) Explanation and Hypothesis of the Model
(1)Here, we assume that the layout of the system is rational, and typical square grid with proportional spacing was used, as shown in Figure 3.(2)The transfer of passengers follows a strict order: local route to feeder to mass, or the other way around.(3)Layout of every type of route is arranged in order of hierarchy (from mass to local), one encompasses another, that is, 𝑟3>𝑟2>𝑟1,𝑟3,𝑟2, and 𝑟1 are spaces between mass routes (including express way and arterial road), feeders, and local routes.(4)Average transfer time equals waiting time, exclusive of time spent on walking to the transfer station. 𝑇𝛼𝛽_transfer represents the average transfer time. (𝛼,𝛽=1,2,3 which stands for local, feeder, and mass, resp.).

390128.fig.003
Figure 3: Sketch of transit network.

(II) Modeling Procedure
Definition of the parameters involved are as follow:𝑖=1,2,3—local, feeder, mass;𝑇𝑖 = 𝑇𝑖onfoot+𝑇𝑖bybus + 𝑇𝑖waiting—trip time using 𝑖 as the highest type of route;𝑇𝑖onfoot—total walking time from starting point to the station and from the station to destination;𝑉onfoot—walking velocity;𝑇𝑖bybus—time spent on the vehicle;𝑇𝑖waiting—transfer time at transfer station;A—area of the city;𝐿𝑘—length of each type of route; 𝑘=1,2,3—local, feeder, mass;𝜌𝑘 = 𝐿𝑘/A—road network density;𝑟𝑘—space between routes of the same hierarchy type;1/𝑟𝑘=𝜌𝑘/2,𝑟𝑘=2𝐴/𝐿𝑘;it is assumed that 𝑑𝑘 equals 𝑟𝑘/2;𝜂𝑖𝑛—coefficient of determination on transfer; 𝑛—number of transfers;if there is no transfer during the trip, 𝜂𝑖𝑛=0, otherwise, 𝜂𝑖𝑛=𝑇𝛼𝛽_transfer;𝑛𝑖—average transfer time; 𝑉𝑖—velocity of the vehicle on type of route 𝑖;𝑃trip—trip distance.

(1) Minimum Trip Time Using Local Routes as the Highest Type of Route. Figures 4 and 5 demonstrate the shortest path by this means. Span of travelling on foot is (0,2𝑑1] (see Figure 3), to simplify the model, mean value of 𝑑1 is used.

390128.fig.004
Figure 4: The shortest path using local routes as the highest type of route.
390128.fig.005
Figure 5: The sketch map of shortest path using local routes as the highest type of route.

The minimum trip time is 𝑇1=𝑇1onfoot+𝑇1bybus+𝑇1waiting𝑇1onfoot=𝑑1𝑉onfoot𝑇1bybus=𝑃trip𝑑1𝑉1,𝑇1waiting=𝜂1_1+𝜂1_2++𝜂1𝑛=𝜂1.(3.1)

(2) Minimum Trip Time Using Feeder as the Highest Type of Route. Figures 6 and 7 show the shortest path by this means. We assume that in this case, local routes only play a supplementary role, accordingly, the mean value of 𝑑1 is still used as the travelling on foot, because the span of travelling by local routes is (0, 2𝑑2] (see Figure 3). To simplify the model, mean value of 𝑑2 is used.

390128.fig.006
Figure 6: The shortest path using feeder as the highest type of route.
390128.fig.007
Figure 7: The sketch map of shortest path using feeder as the highest type of route.

The minimum trip time is 𝑇2=𝑇2onfoot+𝑇2bybus+𝑇2waiting𝑇2onfoot=𝑑1𝑉onfoot,𝑇2bybus=𝑇1out+𝑇2ontheway+𝑇1back=𝑑2𝑉1+𝑃trip𝑑1𝑑2𝑉2𝑇2waiting=𝜂2_1+𝜂2_2++𝜂2𝑛=𝜂2,(3.2)

where 𝑇1out is the time spent from starting point to feeder via local route. 𝑇2ontheway is time spent on the feeder. 𝑇1back is time spent from feeder to destination via local route. 𝑇1out= 𝑇1back.

(3) Minimum Trip Time Using Mass Routes as the Highest Type of Route. Figures 8 and 9 show the shortest path by this means. Assume that mass route is the major route taken, feeder and local route act as the supplement, can be obtained in the same way, the mean value of 𝑑1 and 𝑑2 are still used as the travelling on foot and traveling by local route, respectively. The span of travelling by feeder route is (0, 2𝑑32𝑑1] (see Figure 3). To simplify the model, mean value of 𝑑3𝑑1 is used.

390128.fig.008
Figure 8: The shortest path using mass route as the highest type of route.
390128.fig.009
Figure 9: The sketch map of shortest path using mass route as the highest type of route.

The minimum trip time is 𝑇3=𝑇3onfoot+𝑇3bybus+𝑇3waiting𝑇3onfoot=𝑑1𝑉onfoot,𝑇3bybus=𝑇1out+𝑇2out+𝑇3ontheway+𝑇2back+𝑇1back=𝑑2𝑉1+𝑑3𝑑1𝑉2+𝑃trip𝑑2+𝑑3𝑉3,𝑇3waiting=𝜂3_1+𝜂3_2++𝜂3𝑛=𝜂3,(3.3) where 𝑇1out is the time spent from starting point to feeder via local route, 𝑇2out is the time spent from local route to mass route via feeder. 𝑇3ontheway is the time spent on the mass route. 𝑇2back is the time spent from mass route to local route via feeder. 𝑇1back is time spent from feeder to destination via local route. 𝑇1out=𝑇1back,𝑇2out=𝑇2back.

(4) Calculation of Optimal Distance on Every Type of Route. Set 𝑇1 is equal to 𝑇2, 𝑇2, equal to 𝑇3, Then: 𝑆1=𝑑1+𝑑2+𝑉2𝑉1𝑉2𝑉1𝜂2𝜂1,𝑆2=𝑑2+𝑑3+𝑉3𝑉2𝑉3𝑉2𝜂3𝜂2.(3.4)

3.2.2. Analysis on Demand Turnover of Different Types of Route

(I) Proportion of Passengers on Each Highest Type of Route
The proportion of passengers who take local route, feeder, and mass route as their highest type of route is as follow.
Local Route: 𝑤1=𝑠10𝑓(𝑠)𝑑𝑠; Feeder Route:𝑤2=𝑠2𝑠1𝑓(𝑠)𝑑𝑠; Mass Route: 𝑤3=𝑠+2𝑓(𝑠)𝑑𝑠.

(II) Average Trip Distance of Passengers on Each Highest Type of Route
The average trip distance of passengers who take local route, feeder, and mass route as their highest type of route is as follow.
Local Route: 𝑆1=𝑠10𝑓(𝑠)𝑠𝑑𝑠/𝑠10𝑓(𝑠)𝑑𝑠; Feeder Route: 𝑆2=𝑠2𝑠1𝑓(𝑠)𝑠𝑑𝑠/𝑠2𝑠1𝑓(𝑠)𝑑𝑠; Mass Route: 𝑆3=𝑠+2𝑓(𝑠)𝑠𝑑𝑠/𝑠+2𝑓(𝑠)𝑑𝑠.

(III) Demand Turnover on Each Type of Route
Referring to previous studies [21, 22] on this issue, calculation method of demand turnover on each type of route is proposed as follows. (1)𝑍𝑖 is defined as the turnover completed by unit passenger using 𝑖 as the highest type of route, 𝑧𝑖𝑗 the component of 𝑍𝑖 completed on route type 𝑗(𝑗𝑖). Thus, 𝑍𝑖=𝑖𝑗=1𝑧𝑖𝑗.(2)𝑧𝑖𝑗 is affected by the choice unit passenger make on each trip, which makes the solution complicated and time-consuming. Therefore, certain simplification has been made. We assume that the turnover on route type inferior to 𝑖 can be seen as the product between the number of passengers who take 𝑖 as the highest type of route and average trip distance made by passengers who take 𝑖1 as the highest type. In this case, 𝑧𝑖𝑗 approximately is equal to 𝑍𝑖 minus the product. The rest such as turnover completed on route type inferior to 𝑖1 can be done in the same manner.

Hence, demand turnover on each type of route can be calculated as follows.(1) Turnover completed on the local routes only 𝑍1=𝑧11=𝑊𝑤1𝑠10𝑓(𝑠)𝑠𝑑𝑠,(3.5) where, 𝑊 is the total number of trip time.(2) Turnover completed on the feeder routes only 𝑍2=𝑊𝑤2𝑠2𝑠1𝑓(𝑠)𝑠𝑑𝑠=𝑧21+𝑧22𝑧21=𝑆1𝑊𝑤2,𝑧22=𝑍2𝑧21=𝑊𝑤2𝑠2𝑠1𝑓(𝑠)𝑠𝑑𝑠𝑆1.(3.6)(3) Turnover completed on the mass routes only 𝑍3=𝑊𝑤3𝑠+2𝑓(𝑠)𝑠𝑑𝑠=𝑧31+𝑧32+𝑧33𝑧31=𝑆1𝑊𝑤3,𝑧31+𝑧32=𝑆2𝑊𝑤3𝑧32=𝑊𝑤3𝑆2𝑆1,𝑧33=𝑍3𝑧31+𝑧32=𝑊𝑤3𝑠+2𝑓(𝑠)𝑠𝑑𝑠𝑆2.(3.7)(4) Demand turnover on each type of route: demand turnover on local route, feeder; and mass route is represented as 𝑍𝑍1, 𝑍𝑍2, and 𝑍𝑍3 and total turnover 𝑍total. The relationships between 𝑍total, 𝑍𝑍𝑖, 𝑍1, and 𝑧𝑖𝑗 are shown in Table 2.

tab2
Table 2: The relationship among different classes (types) of turnover.
3.3. Analysis on Supply Turnover of Different Hierarchy Types

Supply capacity of each type of route demands departure frequency, operating hours, type of vehicle, and load factor. Assume that supply turnover is 𝐺𝑍𝑖.

Departure Frequency: the average frequency on each type of route 𝑓𝑖=60/𝜇𝑖 is used, where 𝜇𝑖 is the average departure interval (min).

Type of Vehicle: passenger flow varies on different types of route. Therefore, different types of vehicle are equipped accordingly. It is assumed that rated passenger load of vehicles on each type of route is 𝐸𝑖.

Load Factor: load factor 𝜙𝑖 is an important factor to indicate the comfort of the vehicle.

Then, supply turnover of each type of route is 𝐺𝑍𝑖=𝐸𝑖𝑇𝑖run𝑓𝑖𝜑𝑖𝐺𝐿𝑖,(3.8) where 𝑇𝑖_run is the operating hours in one day, 𝐺𝐿𝑖 is the length of each type or route.

3.4. Analysis on Balance between Demand and Supply on Each Type of Route

Based on the analysis in Sections 3.2 and 3.3, here, we assume that supply turnover equal demand turnover, then,   𝐺𝑍𝑖=𝑍𝑍𝐼 (see Table 2), the desired length of each type of route is. 𝐺𝐿𝑖=1𝐸𝑖𝜑𝑖𝑇𝑖_run𝑍𝑍𝑖.(3.9)

4. A Case Study: Baoding

Located in Heibei Province, Baoding has 100 km2 lands for construction and the population had reached 1.06 million by 2009. The length of express way, mass route, feeder, and local route are 82.6 km, 116.3 km, 74.1 km, and 222.7 km, respectively. The total length is 495.7 km. The application’s process and results are as follows.

(1) Values of Parameters Involved
Based on analysis of the public transit survey of Baoding, the values of parameters involved (the detail explanation of the parameters can be found in Section 3.3) are calculated and the results are as follows:(1)𝐸=[𝐸1,𝐸2,𝐸3]=[72,98,98];(2)𝜇=𝜇1,𝜇2,𝜇3=[10,8,6];(3)𝜂=[𝜂1,𝜂2,𝜂3]=[5,9,12];(4)𝑉=[𝑉1,𝑉2,𝑉3]=[15,20,25];(5)𝜑1=𝜑2=𝜑3=0.9; (6)𝑇1_run=𝑇2_run=𝑇3_run=16.

(2) Important Outcomes
Trip distance is simulated with electron cloud model; the probability density function is 𝑓(𝑠)=(4𝑠2/(5.4)3)𝑒2𝑠/5.4, with 𝑅2=0.995. The model’s precision in simulating is high enough to be applied.
There exists a dynamic balance between supply of the transit network and demand of passengers. Dynamic balance coefficient is assumed as 𝜀(𝜀𝑖=𝐺𝑍𝑖/𝑍𝑍𝑖). If 𝜀0.9, supply is inadequate, if 0.9<𝜀0.95 or 1.05<𝜀1.10, supply just matches demand, if 0.95<𝜀1.05, supply matches demand perfectly, if 𝜀>1.10, supply is sufficient.
Table 3 shows the demand (obtained by calculation) and supply (actual data) undervaries situations. Comparative analysis indicates that the public transit network in Baoding has the following problems.(1)Total length is relatively short. Currently, the actual supply length of Baoding’s transit line is 1734 km, in order to reach the level 𝜀=0.9 (the supply length should be 3069 km), nearly 1300 km length of transit line should be added. The main reason why the total length is so short is because the lack of local routes whose main function is to expand service range of the transit network and to make walking distance as short as possible. The actual supply length of local route is only 399 km, which is far from the demand of level (supply length of local route should be 1493 km), therefore, local routes should be relatively longer and have higher densities.(2)The length of mass routes is relatively long. Currently, the actual supply length of mass routes is 690 km, which is much longer (nearly 340 km longer) than the requirement of level 𝜀=0.9 (the supply length can be 347 km). Large-scale distribution centers and functional areas are connected by mass routes which require high-speed transport. However, currently the size of Baoding city is at moderate level, and trip distance of residents is generally short, the length of existing mass routes seems a bit redundant.
To solve these problems, the idea of hierarchy planning is proposed as follows.(1)Construction of local routes should be strengthened in order to shorten the distance between bus station and origin or destination, consequently shorten the walking distance which facilitates bus travel. (2)Increase or decrease the grade of the transit routes to achieve rational route configuration. For example, alter mass route to feeder route or feeder to local route.(3)According to the above two ideas, a specific measure is put out as an example to meet the demand of level 𝜀=0.9: (a) decrease the some mass routes’ grade, thus, there will 350 km mass route will be changed into feeder routes, the length of feeder routes will be about 1000 km; (b) add another 1000 km of local routes. By the above two ways, the hierarch of Baoding’s public transit will be much reasonable and level 𝜀=0.9 can be reached.

tab3
Table 3: Road length of each transit hierarchy in baoding.

5. Conclusion

A hierarchy planning toward public transit network is developed based on the distribution of passengers’ trip distance. Main achievements are concluded as follows. (1) Trip distance is simulated with electron cloud model and Rayleigh distribution model, comparative analysis shows that the former has better precision in simulating and ability in interpreting. (2) A model for optimal trip distance of each hierarchy type of routes is proposed based on features of passengers in the public transit system; (3) A method of macroscopic calculation on hierarchy planning is developed, which is based on turnover balance between supply and demand. The above achievements have enriched the theory of hierarchy configuration of public transit network, and provide a feasible approach to transit network planning.

Acknowledgment

The authors would like to thank for financial support by the national science and technology support projects, under the Contract no. 2009BAG12A10-9.

References

  1. S. Carrese and S. Gori, “An Urban bus network design procedure,” Applied Optimization, vol. 64, pp. 177–196, 2002.
  2. S. A. Bagloee and A. Ceder, “Transit-network design methodology for actual-size road networks,” Transportation Research Part B, vol. 45, pp. 1787–1804, 2011.
  3. C. D. van Goeverden and R. van Nes, “Hierarchy in public transport networks: the case of Amsterdam,” in Proceedings of the 11th World Conference on Transport Research, 2007.
  4. R. Van Nes, “Multilevel network optimization for public transport networks,” Transportation Research Record, vol. 1799, pp. 50–57, 2002. View at Scopus
  5. R. Van Nes, “Multiuser-class urban transit network design,” Transportation Research Record, vol. 1835, pp. 25–33, 2003. View at Scopus
  6. F. J. M. Salzborn, “Scheduling bus systems with interchanges,” Transportation Science, vol. 14, no. 3, pp. 211–231, 1980. View at Scopus
  7. P. Knoppers and T. Muller, “Optimized transfer opportunities in public transport,” Transportation Science, vol. 29, no. 1, pp. 101–105, 1995. View at Scopus
  8. L. jian and H. Gang, “Level planning method of bus-route network and its application,” Urban Transport of China, vol. 2, no. 4, pp. 34–37, 2004.
  9. W. Wei, Y. Xinmiao, and C. Xuewu, Urban Trnasit Planning Method and Management Techonology, Science press, Beijing, China, 2002.
  10. L. Fangqiang, The Application of Coupling Models between Inhabitant Trip Distribution and Public Transport Network, Southeast University, Nanjing, China, 2010.
  11. G. K. Kuah and J. Perl, “The feeder-bus network-design problem,” Journal of the Operational Research Society, vol. 40, no. 8, pp. 751–767, 1989.
  12. P. Shrivastava and M. O'Mahony, “A model for development of optimized feeder routes and coordinated schedules-A genetic algorithms approach,” Transport Policy, vol. 13, no. 5, pp. 413–425, 2006. View at Publisher · View at Google Scholar · View at Scopus
  13. E. Guillot, “Bus transit interface with light rail transit in Western Canada,” Transportation Research Part A, vol. 18, no. 3, pp. 231–241, 1984. View at Scopus
  14. T. J. Higgins, “Coordinating buses and rapid rail in the San Francisco Bay Area: the case of Bay Area rapid transit,” Transportation, vol. 10, no. 4, pp. 357–371, 1981. View at Publisher · View at Google Scholar · View at Scopus
  15. S. I. Chien, L. N. Spasovic, S. S. Elefsiniotis, and R. S. Chhonkar, “Evaluation of feeder bus systems with probabilistic time-varying demands and nonadditive time costs,” Transportation Research Record, no. 1760, pp. 47–55, 2001. View at Scopus
  16. A. S. Mohaymany and A. Gholami, “Multimodal feeder network design problem: ant colony optimization approach,” Journal of Transportation Engineering, vol. 136, no. 4, pp. 323–331, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. A. Verma and S. L. Dhingra, “Feeder bus routes generation within integrated mass transit planning framework,” Journal of Transportation Engineering, vol. 131, no. 11, pp. 822–834, 2005. View at Publisher · View at Google Scholar · View at Scopus
  18. A. Verma and S. L. Dhingra, “Developing integrated schedules for urban rail and feeder bus operation,” Journal of Urban Planning and Development, vol. 132, no. 3, pp. 138–146, 2006. View at Publisher · View at Google Scholar · View at Scopus
  19. P. Shrivastav and S. L. Dhingra, “Development of feeder routes for Suburban railway stations using heuristic approach,” Journal of Transportation Engineering, vol. 127, no. 4, pp. 334–341, 2001. View at Publisher · View at Google Scholar · View at Scopus
  20. W. Wang, W. Zhang, H. Guo, H. Bubb, and K. Ikeuchi, “A safety-based behavioural approaching model with various driving characteristics,” Transportation Research Part C-Emerging Technologies, vol. 19, no. 6, pp. 1202–1214, 2011.
  21. S. Fei, Research on Grade Proportion and Layout Method of Urban Road, Southeast University, Nanjing, China, 2006.
  22. Z. Zhuping, Road Network Gradation Optimization Model According To Traffic Demand, Southeast University, Nanjing, China, 2009.