Efficiency-Based Mixed Network Design Considering

2018 21st International Conference on Intelligent Transportation Systems (ITSC) Maui, Hawaii, USA, November 4-7, 2018

Efficiency-based Mixed Network Design considering Multi-typed Traffic Demands Yuxin He Department of Systems Engineering and Engineering Management City University of Hong Kong Kowloon, Hong Kong [email protected]

Yang Zhao Centre for Systems Informatics Engineering City University of Hong Kong Kowloon, Hong Kong [email protected]

Jin Qin School of Traffic and Transportation Engineering Central South University Changsha, China [email protected]

Abstract— Transportation network efficiency is a comprehensive reflection of the operation of transportation networks. An effective quantitative evaluation method for the transportation network efficiency is important as it can provide a feedback mechanism of network operation conditions in the process of network design, which gives a theoretical basis for the optimization of urban transportation network. In general, a welldesigned transportation network should be adapted to multityped traffic demands by considering their characteristics after reconstructing. Thus, on the choice of an effective quantitative evaluation method for the transportation network efficiency, this paper proposes a bi-level programming model with the objective of maximizing transportation network efficiency in mixed network design, which has two lower users' equilibrium models corresponding to two kinds of traffic demands. A hybrid Genetic Algorithm (GA) and Frank-Wolfe Algorithm is then developed to solve the proposed problem. Results of the case study show that the network designed by the proposed model a) results in a more rational distribution of traffic flow, b) improves the adaptability of the transportation network and alleviates the traffic congestion, and c) economizes on the use of land, providing a solid foundation for the sustainable development of transportation network. Keywords— traffic modeling, transportation network efficiency, mixed network design, multi-typed traffic demands, bi-level programming, genetic algorithm

I. INTRODUCTION In order to effectively relieve the traffic congestion in the past few decades, thorough studies are conducted on Transportation Network Design Problem (TNDP). Since Morlok proposed a seminal quantitative model of TNDP in 1973, it soon became one of the focal problems in traffic planning research and practical work. Some representative research summarized the TNDP at various stages respectively [1]-[3]. In terms of the nature of the decisions considered, TNDP can be classified into three groups: a) Discrete Network Design Problem (DNDP), which only deals with discrete design decisions such as constructing new roads, adding new lanes, determining the directions of one-way streets, and determining the turning restrictions at intersections; b) Continuous Network Design Problem (CNDP), which is only This research work was partly supported by the Research Grants Council Theme- based Research Scheme (Project No. T32-101/15-R). XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE 978-1-7281-0323-5/18/$31.00 ©2018 IEEE

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Kwok Leung Tsui School of Data Science City University of Hong Kong Kowloon, Hong Kong [email protected]

concerned with continuous design decisions such as expanding the capacity of streets, scheduling traffic lights, and determining tolls for some specific streets, and c) Mixed Network Design Problem (MNDP), which contains a combination of continuous and discrete decisions. MNDP is usually more complicated than DNDP and CNDP, making problems more difficult to model and solve. A combination of continuous and discrete decisions makes MNDP well extended in practice compared with DNDP and CNDP. Thus, this paper focuses on MNDP. Among previous research of transportation network design problem, scholars always set the minimum total travel costs, minimum total investment, minimum network reserve capacity, and/or maximum user surplus as the objective to optimize network design on the premise of meeting the constraints of transportation network. However, this results in a lack of feedback mechanism in the process of decision-making for the comprehensive operation conditions of transportation network. The optimization schemes are thus usually difficult to ensure the overall network performance. For instance, He et al. (2017) demonstrated that the transportation network optimization scheme with the objective of maximizing network efficiency outperformed the optimization scheme with the other objectives in terms of resources saving and balancing traffic flows distribution [4]. Transportation network efficiency as a comprehensive reflection of the operation of transportation networks, is primarily determined by its network structure, the volume and distribution of travel demand, and drivers' routing behavior. A reasonable quantitative evaluation method for the transportation network efficiency can provide a feedback mechanism of network operation condition in the process of transportation network design, which gives a theoretical basis for the optimization of urban transportation network. [5]. Moreover, road network usually carries multi-typed traffic demands. This could result from, for example, daily commuter activities and special events. Ignoring the effects of multi-typed traffic demands in the process of network design will absolutely fail to accommodate the updated type of traffic demands, and it will also cause local road congestion resulting in a poor operation efficiency of road network. Therefore, in order to increase the efficiency of transportation network, the influence of multi-typed traffic demands on user's choice behavior and traffic flow distribution should be considered.

In this paper, a proper quantitative evaluation method [5] for the transportation network efficiency is applied and a bilevel programming model for MNDP with multi-typed traffic demands in terms of optimal network efficiency is established. This is referred to as “one leader with multiple followers model”. “Multiple followers” refers to multiple lower level traffic equilibrium assignment models under different traffic demands; “One leader” refers to an upper-level optimization model to coordinate the overall network design problem. A hybrid Genetic Algorithm and Frank-Wolfe algorithm is developed to solve the model. To the best of our knowledge, this is the first that a bi-level programming model is built for MNDP with multi-typed traffic demands in terms of optimal network efficiency. In addition, the application on a numerical example is conducted for a thorough understanding of the proposed model.

MNDP considering various traffic demands. The upper model is designed for the network structure optimization model while lower models are different traffic equilibrium models corresponding to different traffic demands, which are all subordinate to the unified upper model.

II. A QUANTITATIVE EFFICIENCY EVALUATION METHOD FOR TRANSPORTATION NETWORKS

max  j  j

The MNDP bi-level model with the objective of network efficiency maximization we proposed can be described as follows: A. The Upper Model The upper model is the road transportation network structure optimization design model with the objective of maximizing the weighted sum of different transportation network efficiency values under different types of traffic demands: J

Depending on the characteristics of research methods, the current study of the transportation network efficiency evaluation can be divided into qualitative research and quantitative research[5]. In qualitative research, transportation network efficiency evaluations are mainly based on the multi-index evaluation method[6]. However, there are too many subjective factors of the multi-index method during the evaluation process to ensure the objectivity and rationality of the evaluation results. In quantitative research, the current quantitative efficiency evaluation methods can be applied to urban transportation networks mainly proposed by Latora and Marchiori (2001)[7][9], Nagurney and Qiang (2007)[10]-[12], and Qin et al.(2014)[4]. The later descriptions are denoted by "LM", "NQ" and "QH" methods. He et al. (2017) provided details of the comparative analysis[13]. Through the comparative analysis in validation networks, “QH” method is affected by many factors compared with “LM” and “NQ” methods, which is in line with the real conditions of the transportation network efficiency. Therefore, based on its reasonability, this paper will establish bi-level programming model of the transportation network design problem based on the network efficiency calculated by “QH” method. “QH” method is described as follows:

EQH =

1 nA

xa

t aA

(2)

j =1

(1)

a

s.t.

l aA

a

 I ( ua , ua0 )  B

(3)

e ( u , u )  L a

0 a

(4)

a A

a Rmin 

xaj

ca

a  Rma x , a  A, j = 1, 2,

ua  ua0 , a  A ua  0,1, 2,

,J

(5) (6)

, n, a  A

(7)

The objective function of the upper optimization model (2) comprehensively measures the transportation network efficiency of multiple lower models and highlights the important traffic demands through the way of weighted sum. Each type of demands j can be given a weight value  j according to the frequency of the occurrence and its importance. Here, the sum of  j equals to 1, j = 1, 2,..., J . For the constraints of upper model (3)-(6), the total construction costs are represented by la  I ( ua , ua0 ) and a A

construction should be in the range of certain restrictions B , so the budget constraint is represented as (3). Besides, the function of the land use scale occupied by the construction of the road a is represented by e ( ua , ua0 ) , and L denotes the a A

Among them, A is the set of roads with n A elements, xa and ta are traffic flow and travel costs on road a at User Equilibrium respectively. III. BI-LEVEL PROGRAMMING MODEL OF MIXED NETWORK DESIGN FOR DIFFERENT TRAFFIC DEMANDS BASED ON NETWORK EFFICIENCY

restrictions of land resources. Then the land use scale constraints can be expressed as (4). Moreover, the average vehicle travel speed on road a depends on the value of xa / ca , which is known as the road load or the degree of congestion. Therefore in order to reach a certain service level and utilize road resources effectively, it is necessary to limit the road load. a This paper defines the minimum and maximum load level Rmin a and Rmax allowed by the road a  A , and transportation network service level constraints can be expressed as (5). In addition, we generally do not close existing roads or reduce

TNDP is usually modeled by bi-level programming model [14]-[16]. This paper also establishes the bi-level programming model of "one leader with multiple followers" to describe the

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their road grades, so constraints of the road grade can be set as (6)[17]. B. The Lower Model Lower model is composed of J traffic equilibrium flow models with the consistent structure but different traffic demands distribution. As the model considers J types of traffic demands, there are J equilibrium traffic flow models as the lower models, among which the equilibrium traffic flow model corresponding to demands j ( j = 1, 2,..., J ) is described as follows: xaj

min C j =   t ( xaj , ca ( ua ) ) d x

(8)

aA 0

f

s.t.

k, j rs

= qrsj , s  S , r  R, j = 1, 2,..., J

(9)

kK rs

xaj =    rsa ,k f rsk , j , a  A, j = 1, 2,..., J

(10)

rR sS kK rs

f krs, j  0, a  A, s  S , r  R, k  K rs , j = 1, 2,..., J (11) IV. SOLVING ALGORITHM FOR THE BI-LEVEL PROGRAMMING MODEL OF “ONE LEADER WITH MULTIPLE FOLLOWERS” BASED ON NETWORK EFFICIENCY Genetic Algorithm (GA) is widely used to generate highquality solutions to optimization and search problems by relying on bio-inspired operators such as mutation, crossover and selection [18],[19]. In view of the complexity of our model, GA is used to solve the model proposed earlier. The designed scheme is as follows: 1) Coding: For MNDP, we use one-dimensional integer coding and set the maximum number of candidate edges (roads) of the network l as the length of coding, wherein the number of existing roads is n1 , the number of roads to be built

Decision Variable ua Road grade

1

2

3

4

5

6

7

8

0

2

2

2

4

4

Design speed(km/h)

0

40

60

80 100 100

Capacity(PCU/d)

0

6

4

6

8

100

120

120

120

2) Greedy algorithm ： In order to satisfy constraints of construction costs, land resources, and other conditions, we use the greedy strategy to filter the initial population. The basic steps are as follows: a) Traverse each individual chromosome of the initial population. b) Calculate the unit upgraded costs and unit land occupation of each road separately and sort them according to the order from small to large, and then select the priority of costs or land use randomly, finally give priority to the road with the smaller unit value to upgrade. c) Repeat the Step b) until the total construction funds or land resources have been used up (since the prior rule is randomly generated, each individual in the population is extremely likely not the same, so as to adjust the initial population to satisfy the constraints. 3) Definite the fitness function: Z + Cmin  0  Z + Cmin fi =  Z + Cmin  0  0

(12)

Z is the value of objective, and Cmin is the minimum objective value up to the current generation after evolution.

4) Selection operator: roulette selection operator,

Pi = f i /  f i

(i = 1, 2,..., m)

(13)

i to be

f

i

is the

This paper designs a hybrid Genetic Algorithm (GA) and Frank-Wolfe Algorithm to solve the proposed bi-level programming model. GA is used to solve the upper network structure optimization model and Frank-Wolfe algorithm is used to solve the lower traffic flow distribution model[20]-[21]. The details of the Frank-Wolfe method can be found in Fukushima (1984) [21]. Due to the lower models consisting of two types of equilibrium traffic flow models, thus we call Frank-Wolfe algorithm twice for different traffic demands, and specific algorithm steps are as follows: Algorithm Hybrid Genetic Algorithm and Frank-Wolfe Algorithm

9

1: Initialization. Set the population size n , maximum iteration times Tmax , crossover probability pc , mutation probability pm , the iteration counter T = 0 , and the allowable error  for termination of the iterative process of the lower model.

Not 4th- 3rd- 2nd- 2nd- 1st- 1st- High- High- Highconnected level level level level level level speed speed speed

Lane

80 200 500 800 1500 3000 4000 6000 7000

selected, fi is the fitness of the individual i , and accumulative fitness of all individuals.

DECISION VARIABLES OF ROAD GRADE AND THE CORRESPONDING INDEXES IN CHINA 0

0

Wherein Pi is the probability for an individual

is n2 , and l = n1 + n2 . According to the Road Commission of China Association for Engineering Construction Standardization, the road grades are usually divided into 5 levels: high-speed, 1st-level, 2nd-level, 3rd-level, and 4thlevel. On this occasion, it is possible to discretize MNDP by using the road grade decision variables. Decision variables and its corresponding indexes are as shown in TableⅠ[17]. TABLE I.

Unit construction cost(10000 Yuan/km)

2: Generate the initial population. Generate a l  n matrix with the value of each road’s initial grade as the initial

1500 3000 4000 6000 15000 20000 30000 45000 60000

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travel time function proposed by US Bureau of Public Roads (BPR). Therefore, the travel time on the road a is:

population pop . 3: Call the greedy function, so as to get a new initial population.

   xa    ta = ta ( xa , ca ) = t 1 +      c u ( ) a a     0 a

4: T = T + 1 , calculate the value of each individual in the population, and transfer the individuals which meet the constraints to the lower model to assign traffic flows. Call Frank-Wolfe algorithm twice to calculate the traffic and travel time at equilibrium under two types of traffic demands, and then the value of the objective function can be obtained. Store the best individual of each generation P0 , its corresponding optimal value f 0 , the best individual

Note that for any road traffic load constraint, we set it a a as Rmin = 0 and Rmax = 0.85 . Note that there will not be the case

among all generations P0' and the corresponding optimal

a of Rmin = 0 in the actual traffic operation generally.

 4  xa   la   xa   la  =  1 +     =  1 + 0.15    va   ca ( ua )   va   ca ( ua )   

value f , that is, if f 0  f , then f = f 0 , P = P0 . ' 0

' 0

' 0

' 0

Set the unit matching coefficient  = 100 , the upper limit of land use L = 1000 . The function of land occupation scale e ( ua , ua0 ) can be calculated by:

5: Calculate the fitness value of each individual chromosome fi according to the formula (12).

(

6：Select the individual and calculate the probability of individual being selected Pi according to formula (13). Generate the selection probability Ps at random in the [0,1] interval, if Pi  Ps , select the individual and store it as the best individual.

e ( ua , ua0 ) = 8 ( ua − ua0 ) + 30 L ( ua ) − L ( ua0 )

)

(16)

Wherein L ( ua0 ) represents the number of lanes of road a with the grade ua . In the case study, we simply consider two types of traffic demands in order to understand the characteristics of the problem and solve it easily, namely, J = 2 . Here, 1 =0.5 , and 2 =0.5 . There are 5 OD pairs of the network respectively, corresponding to two types of traffic demands:

7: Carry out the single point crossover operation, and generate a random integer k in the [1, l ] interval, and it is the crossover location of the gene. 8：Carry out the mutation operation with the mutation probability Pm . If the number of iterations T  Tmax , turn to step 9, otherwise T = T + 1 , turn to step 4.

The first type: 1 1 q = 4000, q1,12 = 9600, q12,8 = 8200, q12,12 = 9000,q5,12 = 8000. The second type: 1 1,11

9：End of the iteration. Call the Frank-Wolfe algorithm twice for calculating traffic flow and travel time of each road at equilibrium of the current updated network structure under two types of traffic demands. Output the best individual P0' and the corresponding value of the

2 2 2 2 2 q11,1 = 5600, q12,1 = 7000, q8,1 = 9000, q12,2 = 6400, q12,5 = 9600. The transportation network of numerical example is shown in Fig. 1. Solid arrows indicate the existing roads and dashed arrows indicate the roads to be built. The numbers out of brackets are identifiers of roads. Numbers in brackets represent the length la and initial road grade ua0 of the road a ,

objective f 0' . V. CASE STUDY The construction cost in TableⅠactually is the investment in new road construction per unit length (km), as mentioned earlier. The construction costs of upgrading the road are related to road status before and after construction. Therefore, road construction cost function can be defined as I a ( ua , ua0 ) , which

namely (la , ua0 ) . 1

1(80, 4)

2

2(60, 3)

19(130, 0)

18(100, 0) 5(50, 3)

4(40, 2)

3

3(90, 3)

4

20(120, 0)

6(80, 5)

7(60, 3)

is set as:

I a ( ua , ua0 ) = I a ( ua ) − I a ( ua0 )

(15)

(14)

5

8(60, 4)

6

21(140, 0)

This paper sets the investment upper limit as M=10000 million according to TableⅠ, after determining the grade ua of the road a , the free flow speed va = va ( ua ) (km / h) on the road a can be determined, then the free flow time on the road a is ta0 = la / va .We set the parameters  = 0.15,  = 4 of the

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9

15(60, 4)

7

22(150, 0)

12(60, 4)

11(70, 3)

9(100, 5)

10

Fig. 1. Numerical example network.

8

23(90, 0)

13(55, 4)

16(80, 5)

10(50, 5)

11

14(40, 4)

17(50, 5)

12

The designed algorithm to solve the model is run by Matlab R2013a programming on the platform of Windows 10. In comprehensive consideration of calculation time and solution quality, the population size of GA is determined as 100, the maximum number of iterations is 200, crossover probability

TABLE II.

and mutation probability are 0.9 and 0.01, respectively. In order to analyze the characteristics of “one leader with the multiple followers model”, the results are compared with those before optimization and models considering single type of traffic demands only (see Table Ⅱ):

OPTIMIZATION RESULTS BEFORE OPTIMIZATION AND AFTER OPTIMIZATION OF THREE MODELS After optimization

Before optimization Optimization scheme

One leader with multiple followers model

The model only considering the first type of traffic demands

The model only considering the second type of traffic demands

[4,3,3,2,3,5,3,4,5,5,3,

[4,4,3,2,3,5,3,6,6,5,3,

[4,3,4,2,4,6,3,4,5,5,3,

[4,4,4,4,4,6,3,4,5,5,3,

4,4,4,4,5,5,0,0,0,0,0,0]

5,4,5,5,5,5,0,0,0,0,2,0]

4,4,4,4,6,6,0,0,0,0,1,0]

4,5,4,4,5,6,0,0,0,0,1,0]

Cost of construction

\

884000w

738000w

659000w

Transportation network efficiency

4597.4

5248.8

4631.5

5174.5

Total system impedance[1]

3.1145*105

2.1301*105

2.2800*105

2.0174*105

Land occupation scale

\

624

816

984

Travel time between each OD pair

(2,8)

8.72

5.1743

-40.66%

5.1337

-41.13%

7.73

-11.35%

(1,11)

9.6532

5.9806

-38.05%

6.2413

-35.34%

8.2563

-14.47%

(1,12)

10.291

6.5556

-36.30%

6.7901

-34.02%

8.2961

-19.38%

(2,12)

9.4424

5.595

-40.75%

5.7907

-38.67%

7.4435

-21.17%

(5,12)

3.8054

2.7868

-26.77%

3.4918

-8.24%

3.8054

0

(8,1)

9.816

7.7974

-20.56%

8.0845

-17.64%

4.8976

-50.11%

(11,1)

9.8326

7.6669

-22.03%

8.1107

-17.51%

4.8617

-50.55%

(12,1)

10.407

8.2081

-21.13%

8.6371

-17.01%

5.3942

-48.17%

(12,2)

6.6772

4.1355

-38.07%

4.2917

-35.73%

3.0087

-54.94%

(12,5)

4.0367

2.7732

-31.30%

3.8412

-4.84%

3.7017

-8.29%

The comprehensive analysis of the results shows that compared with the initial network, travel time between each OD pair in the network after optimization of “one leader with multiple followers model” has been significantly reduced, while travel time between each OD pair after optimization of the models only considering single type of traffic demands also decreases but not as much as that of “one leader with multiple followers model”. It indicates that “one leader with multiple followers model” lays emphasis on relieving traffic congestion of each road hand in hand instead of making one wax at the expense of the other waning. Additionally, when traffic demands and other conditions are constant, for the network optimized through the model only considering one type of traffic demands, the construction cost is less but land resources occupation is relatively larger than that of the network optimized through “one leader with multiple followers model”, which is the comprehensive optimization of transportation network, so that can economize on the use of land, providing a reliable basis for the sustainable development of transportation network. Through calculation of road traffic load before and after optimization under different types of traffic demands, the number of roads with traffic load greater than 1 before and

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after optimization can be counted. In addition, in order to understand whether the distribution of each road’s traffic flow in the optimized network is balanced or not, we also calculate the standard deviation of traffic flows of non-zero flow roads in each optimized network (see Table Ⅲ). TABLE III. NUMBER OF ROADS WITH TRAFFIC LOAD GREATER THAN 1 AND STANDARD DEVIATION OF TRAFFIC FLOW OF NON-ZERO FLOW ROADS IN EACH OPTIMIZED NETWORK Before optimization

The number of roads with traffic load>1 Standard deviation of nonzero traffic flow

One leader with multiple followers

After optimization Only considering the first type of demands First Second type type OD OD

Only considering the second type of demands First Second type type OD OD

First type OD

Second type OD

First type OD

Second type OD

11

11

5

5

8

11

9

10

0.838

0.753

0.279

0.402

0.479

0.628

0.420

0.491

According to Table Ⅲ, it is found that traffic load of most roads is in decline after the optimization of “one leader with multiple followers model” based on transportation network efficiency. Due to comprehensively considering different types of traffic demands when upgrading existing roads and constructing a new road (6,11) in the network, the capability of the new roads and upgraded roads are enhanced. Therefore, traffic load of most roads is significantly reduced after optimization, and the supersaturation of the road network can be alleviated greatly. Besides, we can also note that the standard deviation of traffic flows of non - zero flow roads in the network optimized through “one leader with multiple followers model” is the smallest among all cases, indicating that the traffic of each road distributed more evenly in the network after optimization of “one leader with multiple followers model”. However, there are a large number of saturated or supersaturated roads in the network and traffic distributed unevenly before optimization and optimized through the model only considering single type traffic demands, which can neither help to relieve the congestion nor lead to a more rational distribution of traffic in global road network system. Therefore, it is noted that the network optimized through “one leader with multiple followers model” based on transportation network efficiency can better adapt to multityped traffic demands, improve the operation efficiency of the road network, achieve a rational distribution of traffic, economize on the use of land, and reduce the degree of traffic congestion when network undertakes different types of traffic demands.

demands. In addition, the optimized network could economize on the use of land. REFERENCES [1]

[2]

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[5]

[6] [7]

[8] [9]

[10] [11]

[12]

VI. CONCLUSIONS On the choice of an effective quantitative evaluation method for the transportation network efficiency, this paper established a bi-level programming model of mixed network design based on transportation network efficiency. Besides, considering the road network after reconstructing should be adapted to multi-typed traffic demands, this paper proposed a “one leader with multiple followers model”, which had two lower users' equilibrium models corresponding to two kinds of traffic demands, following an upper-level optimization model to coordinate the overall network design. A hybrid Genetic Algorithm and Frank-Wolfe Algorithm for solving the model was developed. Results of case study showed that through the optimization of “one leader with multiple followers model”, the travel time was greatly reduced and the obvious optimization effect could be obtained. Additionally, the traffic load of most roads in the optimized network was significantly reduced. In these respects, it played an important role in relieving the traffic congestion in the network. Moreover, as “one leader with multiple followers model” proposed in this paper considered multi-typed traffic demands, the optimized network could better adapt to different traffic demands. It can comprehensively improve the operation efficiency of the road network, achieve a more rational distribution of traffic and reduce the degree of traffic congestion when the network undertook different traffic

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