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Procedia - Social and Behavioral Sciences 96 (2013) 1218 - 1226

13th COTA International Conference of Transportation Professionals (CICTP 2013)

Joint optimization of a rail transit route and bus routes

in a transit corridor

Yang Suna*, Xiaonian Suna, Baoqing Lia, Dehui Gaob

aIntegrated Transport Research Center, China Academy of Transportation Sciences, Beijing, 10029, China bUrban Transport Institute, China Academy of Urban Planning and Design, Beijing, 100037, China

Abstract

Urban transit network is mainly composed of rail transit routes and bus routes in many large cities of China. Previous studies have been made to optimal either rail transit network design or bus network design. This paper was concerned with joint optimization of a rail transit route and bus routes in a transit corridor. Firstly, a method for classifying bus routes under a given rail route transit was proposed. Then, a multi-objective model was developed for designing an integrated rail transit and bus network to maximize rail ridership and minimize total passenger travel time. An algorithm for solving the proposed model based on genetic algorithm was presented. At last, a numerical example was given. The results demonstrate feasible and effective of the proposed model and solution methodology, and show that rail ridership increases and total passenger travel time declines after optimization.

© 2013TheAuthors.Publishedby ElsevierLtd.

Selectionandpeer-review underresponsibilityofChineseOverseasTransportation Association (COTA). Keywords: Transit network design; Rail transit route; Bus route; Integrated network; Optimization

1. Introduction

With the growth of large cities, traffic congestion becomes a serious problem, which affects the life quality of residents. Public transit is an efficient way to solve the traffic congestion problem. Due to longer journey and increasing transit trips, only the bus system of big cities in China can not satisfy the current and future need of transit users. An efficient urban transit system should integrate more than one transit modes, such as bus, rail transit, bicycle transit, bus rapid transit. The rail transit system can provide high level and quality of service. Therefore, numbers of urban rail transit network have been constructed and are been building in many large cities

* Corresponding author. Tel.: +86-13810333246; fax: +86-10-58278827. E-mail address: sunyang_1983@163.com

1877-0428 © 2013 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of Chinese Overseas Transportation Association (COTA). doi:10.1016/j.sbspro.2013.08.139

of China during the last few years. The bus and rail transit becoming the main transit modes, their network layouts are closely related with level of service.

As an important part of public transit planning, the transit route network design has been researched by numerous scholars, such as Ceder and Wilson (1986), Baaj and Mahmassani (1991), Fan and Machemehl (2006), Zhao and Zeng (2008). Their approaches contribute to optimize single transit mode. The integrated transit system, which consists of bus routes and rail transit routes, provides transit service to most of transit users in many large cities of China. Limited studies are available for integrated optimization of rail transit routes and bus routes. Chien and Schonfeld (1998) proposed an optimization model for joint optimization of a rail transit route and its feeder bus system. It was assumed that the bus routes were parallel. Kuan et al. (2006) described the feeder bus network design problem as a hierarchical network design problem (Current et al., 1986). It was pointed out that the primary path represented the rail route and the secondary paths represented the feeder bus routes. In both Chien and Schonfeld (1998) and Kuan (2006), a new transit system which contains no existing bus routes and one existing rail route is assumed, and all types of bus route are feeder bus mode. However, a rail route will be built on one major transit corridor, and there are numbers of bus routes existing before the operation of the new rail transit route. The key to joint optimize of rail transit routes and bus routes is that how to adjust existing related bus routes and design new bus route routes depend on the new rail transit routes.

In order to analyze our problem as far as possible, attention is focused on a transit corridor with a new rail transit route. The integration of one rail transit route design and related bus routes optimization is a complicated problem. For simplicity of analysis, the layout of the rail transit route and the locations of its stations are assumed to given. The objective of this paper is to provide an approach that how to optimize the existing bus routes layouts and add new bus routes under the operation of the rail transit route.

The rest of the paper is organized as follows. A method for bus routes classification under the operation of a rail transit route is presented. Next, a multi-objective mode is proposed. An solution procedure based on genetic algorithm is developed to solve the proposed model. Finally, a numerical example is provided to demonstrate feasible and effective of the proposed model and solution methodology.

2. Classification of bus routes under the given rail transit route

An approach is proposed to define the type of a bus route based on relationship between the bus route and a given rail transit route. If the bus stop is in the coverage area of a rail transit station, it is related with the rail transit station and is defined as the related bus stop. A given bus route, which contains at least one related bus stop, is defined as the related bus route. It is assumed that there are no transfer passengers between the irrelated bus route and the rail transit route. A non-negative integer Ns, which represents the number of related bus stops on a given bus route, indicates whether the bus route is a related bus route. If the value of Ns is more than 0, it means that the bus route is related with the rail transit route and 0 otherwise.

The procedure for bus route classification can be hierarchically structured as in Fig. 1. As shown in Fig. 1, there are four kinds of bus routes. The first, type A, is irrelated with the rail transit route. According to the above assumption, the type A routes have no transfer passengers from and to the rail route and thus does not influence passengers of the rail transit route. It is unnecessary to improve this type route. The second, type B, has origin stop or destination stop in the coverage area of the rail stations and could serve the rail route as a feeder route with less improvement. The third, type C, crosses the rail route and needs less improvement as the same as type B. The fourth, type D, whose alignment overlaps with the rail transit route, competes with the rail transit route and decreases the amount of rail transit passengers. The type D routes influence the quantity of rail transit service and

needs more improvement. An index which evaluates competition performance with rail transit route for type D routes is given in the following.

Fig. 1. Hierarchy of relationship between a rail route and a bus route

For type D, the bus path between one related bus stop and the other related bus stop on a given related bus route is defined as the overlapping path, which competes with rail transit route for passengers. The procedure to evaluate the type D route competition performance with the rail transit route is illustrated with a simple example as shown in Fig.2. There is a rail route A1-A2-A-A4-A5 and a type D route B1-B2-B3-B4-B5-B6-B7-B8. B3, B5, and B7 are the related bus stops. The passenger from origin node B3/A2 to destination node B5/A3 can choose the bus path B3-B4-B5 or the rail transit path A2-A3. The overlapping path is the reason for type D route to compete with the rail route. Therefore, the number of overlapping paths on bus route decides the competition degree of type D route with the rail transit route and is represented by Ps as an index, which evaluates competition performance. To the given rail transit route A1-A2-A-A4-A5, there are three overlapping paths on the bus route in Fig 2: B3-B4-B5, B3-B4-B5-B6-B7, and B5-B6-B7.

In order to evaluate the degree of improvement need for type D, another index is proposed. The index is defined as P = Ps/Pd, where Pd is the number of stop-to-stop path on the type D route. The higher the value of P is, the more improvement the type D route needs. If the value of P is small, the improvement that only changing the layout of the overlapping path is enough. On the other hand, once the value of p is high enough, the type D route may be removed.

Coverage

Rail station Q

I M ■ ~M Rail route - Bus route

Fig. 2. An illustration exampl

3. Mathematical model

In this section, a model is developed to deal with joint optimization of a rail transit route and bus routes. As described above, the service area considered in our study is a transit corridor. The rail transit route layout and its stations locations are given as the input data. Also, there are several bus routes in the rail corridor.

For an effective transit planning, it is important that both view of operator (supply) and user (demand) are considered. On the operator side, the utilization of offered capacity in the integrated transit system is expected as high as possible in order to reduce operation cost. On the user side, travel time is one of the passenger's major concerns and is related to the routes layout, vehicle speed, and route frequency.

Two objectives are considered in the model: (a) maximizing the utilization of rail route capacity and (b) minimizing the total passenger travel time in the transit system. The more numbers of passengers use the rail transit route, the higher degree of utilization the rail transit route gets. Hence, the utilization of rail transit route capacity is estimated by rail passenger ridership. The passenger travel time consists of waiting time, in-vehicle time, and transfer time. The operation is not set out in the route network design and thus waiting time and transfer time could not be decided exactly. Here, it is assumed that the headway of all bus routes is same and is set up by a constant value. Also, the headway of the rail transit route is given as the input data. A constant time is set for each transfer.

The other following assumptions in this study are made as below:

(1)The transit origin destination (OD) matrix is fixed before and after the bus routes improvement.

(2)The speed of bus vehicle is fixed and the traffic congestion in the road is neglected. The travel time on bus route and rail transit route are both constant.

(3)The wait time of the passengers is equal to half of the route headway.

(4)Due to inconveniences caused by transfer, it is assumed the passenger choose the path which needs the minimum transfer number from his/her origin node to his/her destination node. If more than one paths which need

the same transfer number, the passenger will choose the shortest travel time path.

R and S are the sets of origin nodes and the set of destination nodes respectively. drs is the transit travel demand from origin node r to destination node s. m represents the rail transit route. Let N be the set of bus routes before improvement. Let L be the set of candidate bus routes after improvement. A binary decision variable <pi is given, and its value is 1 if the bus route l is chosen and 0 otherwise. &rsm is a binary variable whose value is 1 if the passenger traveling from r to s will choose the rail transit route and 0 otherwise. &rsl is a binary variable whose value is 1 if the passenger who travels from r to s will choose the bus route l and 0 otherwise.

trsm and trsl are in-vehicle time from r to s along rail transit route and bus route l respectively. The total passenger in-vehicle time T1 is:

T = E YsdrS0rJrSm+^Y?LdrS(Pl0rSltrSl (1)

r^R seS r^R seS leL

The total passenger waiting time T2 can be formulated as:

t2=\YY drS®rsmK+ tZZ Yd^sA

2 ~ / j / j rs rsm m ~ / j / j / j rsrl rsl l

2 reR seS 2 ^eR seS feL

Where hm is the rail transit route headway, and hi is the bus route l headway. The total passenger transfer time T3 can be formulated as:

T3=I (^rsm+2>l0rsl- 1) (3)

^eR seS ltL

Where tc is a constant value for expected transfer time. The multi-objective model is formulated as follows:

maximize = £ £ ^rsm (4)

reR seS

minimize Z2 = T2+ T3 (5)

Vrsm+ZvPrs^ 3 VreR, seS (6)

qmmq^^q^^maxq (7)

Kn<Pl^\(H, L,<Pl )^Kax<Pl (8)

The objective function (4) indicates maximizing rail transit passenger ridership. The objective function (5) is the sum of in-vehicle time, waiting time and transfer time. Constraint (6) ensures that the number of transfer for a trip from node r to node s does not exceed 2 times. Constraint (7) represents the bus route load constraint, where ql, qmin and qmax are bus route l load factor, the minimum load constraint, and the maximum load constraint. Constraint (8) limits the improvement degree for the bus routes. Al is an index that evaluates the degree of bus route l improvement. A should be proposed depend upon local planning guidelines and the transit planners' expertise. Amm and Amax are lower limit and upper limit.

By introducing weighting factors, the two objectives (4) and (5) combine into one. The combined model is given by:

minimize Z3 = —C1Z1 + C2 Z2 (9)

Where C1 and C2 are weighting factors, C1+ C2=1. 4. Solution Procedure based on genetic algorithm

In order to solve our mode which is proposed in previous section, a solution procedure to decide which candidate bus routes are selected in the route design should be given. Once candidate bus routes are generated, the problem becomes combinatorial optimization problem. Genetic Algorithm is effective for combinatorial optimization problem and thus is applied to solve our model.

The decision variables are coded into a binary string. Every digit of the string represents a candidate bus route. The value of digit in code is 1 if this route which the digit represents is chosen and 0 otherwise. The fitness of string is set to equal to function (9).

Candidate bus routes consist of two kinds of routes. One is the type of improved existing bus routes, and the other is the type of new built bus routes. The procedure of generating candidate bus routes based on improving existing bus routes is shown in Fig. 3. The new built bus routes could be generated by k shortest path algorithm and also depend upon the transit planners' knowledge and expertise.

Fig. 3. Procedure of generating candidate bus routes

The procedure of algorithm for solving our model is described as follows:

StepO: Input data, include rail transit route layout, rail transit stations locations, bus routes layout, bus stops locations, and related parameters value;

Stepl: Generate the set of candidate bus routes;

Step2: Initialize population of strings, and compute fitness of population according to function (9); Step3: Crossover operation with type of uniform, and mutation operation with type of random; Step4: Choose operation update old population to generate new population;

Step5: If convergence check is reached, the best string represents the best solution; else, go to step 3. 5. A Numerical example

In this section, a numerical example is provided to demonstrate feasible and effective of the proposed model and solution methodology. The base network contains 16 nodes and 32 links. The OD transit demand matrix is shown in Table 1. For simplicity, the OD transit matrix is symmetric. Fig. 4 shows the test network. There is 1 rail transit route and 4 bus routes, which are represented by sequence node numbers. The rail route is 1-3-7-11-14. The bus routes are bus route no. 1 (1-3-7-11-12-16), bus route no. 2 (4-1-3-2-5-9), bus route no. 3 (6-7-8-12-16), and bus route no.4 (6-10-13-14-15). Bus travel time is shown without bracket next to each link, and rail travel time is shown within bracket next to rail link in Fig. 4.

Table 1. OD transit demand matrix

O\D 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 - 6 30 6 11 4 25 10 13 7 36 9 2 28 25 17

2 6 - 9 2 8 6 11 14 5 8 15 11 8 13 21 3

3 30 9 - 7 5 7 10 6 10 2 23 3 12 15 31 2

4 6 2 7 - 7 3 1 4 6 1 2 6 8 11 2 7

5 11 8 5 7 - 7 2 2 10 7 5 4 8 3 2 9

6 4 6 7 3 7 - 10 13 3 6 9 9 11 2 5 20

7 25 11 10 1 2 10 - 5 2 1 20 4 9 15 7 8

8 10 14 6 4 2 13 5 - 10 5 2 14 4 15 5 3

9 13 5 10 6 10 3 2 10 - - 3 3 4 1 7 2

10 7 8 2 1 7 6 1 5 - - 40 3 7 4 1 6

11 36 15 23 2 5 9 20 2 3 40 - 4 15 20 18 6

12 9 11 3 6 4 9 4 14 3 3 4 - 6 11 13 9

13 2 8 12 8 8 11 9 4 4 7 15 6 - 3 10 7

14 28 13 15 11 3 2 15 15 1 4 20 11 3 - 5 9

15 25 21 31 2 2 5 7 5 7 1 18 13 10 5 - 5

16 17 3 2 6 9 20 8 3 2 6 6 9 7 9 5 -

Fig. 4. Base network for test

The other input parameters in the example include: for headway, hm = 3 and hl = 6; for transfer time, tc = 1; for weighting factors, C1 = 0.7 and C2 = 0.3. For simplicity, the constraint (7) is neglected. To a related bus route l, index Al in our example is formulated as:

if 4 = 0 if 4 = 1

Where ODl1 is the number of OD served by improvement route l, which is still served by the initial route of l. ODl2 is the number of OD served on the initial route l. For Al constraint, Amin = 30% and Amax = 100%.

The optimization results for bus routes are summarized in Table 2. A bus route 10-11 is new built. Table 3 gives rail transit ridership and total passenger travel time before and after optimization.

Table 2. Optimization results for bus routes

initial bus route

optimized bus route

1-3-7-11-12-16 4-1-3-2-5-9 6-7-8-12-16 6-10-13-14-15

removed

4-2-5-9

6-7-8-12-16

13-14-15

new route 10-11

40% 100% 30%

Table 3. Optimization results for the model objectives

Before optimization

After optimization

Rail ridership

Total passenger waiting time Total passenger in-vehicle time Total passenger transfer time Total passenger travel time

From Table 3, we can make the following observations: (1) the rail transit ridership increase from 1364 to 1620 after bus routes optimization. This is expected because the rail transit route is the mainline for serving passengers. (2) Total waiting time decreases from 4263 to 3879, and total in-vehicle time decreases from 35624 to 33964. The reason is that after optimization, more passengers choose rail transit route, and high speed and high frequency of the rail transit route reduce passenger in-vehicle time and waiting time. Though total passenger transfer time increases from 1418 to 1722 by the integrated transit system, the total passenger travel time declines. The optimization is beneficial to both operator and user.

6. Conclusion

Bus and rail transit are the two main transit modes in many large cities of China. An integrated bus and rail transit system will increase the efficiency of both modes and attractive to users. The key to design integrated bus and rail transit system is that how to improve bus routes to fit with the rail transit route. In this study, attention has focused on a single corridor service area with a given rail transit route. The method that bus routes classification based on the given rail transit route is presented. A multi-objective model is proposed, in which the objective is to maximize rail transit ridership and minimize total passenger transit travel time. In order to solve the proposed model, a solution algorithm based on genetic algorithm is proposed. A numerical example is given to demonstrate feasible and effective of the proposed model and solution methodology. The results show that rail transit ridership increases and total passenger travel time declines after optimization. The proposed model and solution methodology are demonstrated feasible and effective.

References

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