Computational Intelligence and Neuroscience

Volume 2015, Article ID 512715, 9 pages

http://dx.doi.org/10.1155/2015/512715

## Golden Ratio Genetic Algorithm Based Approach for Modelling and Analysis of the Capacity Expansion of Urban Road Traffic Network

Key Laboratory of Road and Traffic Engineering, Ministry of Education, Tongji University, 4800 Cao’an Road, Shanghai 201804, China

Received 27 August 2014; Accepted 2 January 2015

Academic Editor: Xiaobei Jiang

Copyright © 2015 Lun Zhang 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

This paper presents the modelling and analysis of the capacity expansion of urban road traffic network (ICURTN). Thebilevel programming model is first employed to model the ICURTN, in which the utility of the entire network is maximized with the optimal utility of travelers’ route choice. Then, an improved hybrid genetic algorithm integrated with golden ratio (HGAGR) is developed to enhance the local search of simple genetic algorithms, and the proposed capacity expansion model is solved by the combination of the HGAGR and the Frank-Wolfe algorithm. Taking the traditional one-way network and bidirectional network as the study case, three numerical calculations are conducted to validate the presented model and algorithm, and the primary influencing factors on extended capacity model are analyzed. The calculation results indicate that capacity expansion of road network is an effective measure to enlarge the capacity of urban road network, especially on the condition of limited construction budget; the average computation time of the HGAGR is 122 seconds, which meets the real-time demand in the evaluation of the road network capacity.

#### 1. Introduction

The growing demand of urban traffic can never be solved by just increasing road facility. Factors like city economics, road structure, and land use will determine the travel mode, travel path, and average travel distance. In addition, in most cities, the distribution of land used has been decided, and the land values promote high-strength development. Moreover, the newly constructed roads will reduce the travel time but also attract traffic flows from other roads, as well as create the new traffic demand. The road network may return to the original congestion level after a period of time [1]. All these lead to the difficulty of extension and transformation of the existing transportation network [2]. Therefore, three problems, how to analyze capacity of road network, how to evaluate traffic supply conditions and road construction level, and how to decide the scale of new construction and reconstruction of existing network capacity, are key for sustainable development of road infrastructures.

In the aspect of the capacity of road network, experts around the world have proposed different methods to define and calculate the capacity of network, such as graph theory method [3], space-time consume method [4], mathematical programming method (including linear programming method and bilevel programming method) [5], and traffic simulation method [6, 7]. As the capacity of road network is not only a physical network problem, but also a dynamic problem which considers people, as well as delay and costs, both of which change with traffic flows. The travelers’ routing choice behavior and traffic state in the network have significant influence on the capacity of road network [8]. In these methods, many scholars have found the great importance of OD pattern on calculating the capacity of road network. Therefore, applying the bilevel mathematical modelling method on describing the traffic capacity of network and developing efficient solution algorithm becomes research focus. Asakura and Kashiwadani proposed the first model about road network capacity balance and the traffic simulation distribution method [9]. Yang et al. combined traffic distribution and assignment model, and they considered the routing choice and destination of travelers, the physical traffic capacity, and environment of each road as the constraint condition of the capacity of road network. An advanced bilevel traffic assignment method was proposed, which considered not only the physical capacity of road network, but also the balance among traffic individuals [10]. The study offers a new method to calculate the road network capacity model.

Despite the promising progress from network topology and network capacity, effective models development and efficient strategies for urban road network capacity remain to be challenged, especially regarding the following issues: network capacity modeling: various network capacities are defined for different design purposes, and these studies analyzed examples of network design problem so as to optimize the road network capacity; model solution: many algorithms have been proposed to calculate the balance model, such as incremental assignment method and Frank-Wolf algorithm, and so forth, but the applications of these algorithms are limited because of too many variables and constraints. So the intelligent optimization algorithms with low complexity are needed to meet the application requirements of large-scale network design.

This paper uses bilevel programming to model the capacity expansion of road network, and an improved hybrid genetic algorithm integrated with golden ratio is developed to solve the upper-level model of capacity expansion, while the lower-level user optimized equilibrium model is solved by classic Frank-Wolfe algorithm. The remainders of this paper are as follows. Section 2 defines the research scope and assumptions. Section 3 models capacity expansion of road network. Section 4 illustrates the solution algorithm combining the golden ratio based genetic algorithm and Frank-Wolfe algorithm. Section 5 evaluates the proposed model and algorithm with numerical analysis of a classic network. Section 6 concludes the work.

#### 2. Research Scope and Assumptions

##### 2.1. Research Scope

The generalized concept of capacity expansion of road network is as follows: through improving influencing factors under the condition of certain economic constraints and geographical environment, the maximum number of traffic volume passing the road section in unit time are increased. These improved measures include, but are not limited to, the road conditions of network, the matching degree of OD distribution and road network structure, the road network layout and hierarchy, the service level of road network, and the route choice behavior of traffic individual. This general concept fully considers the various influence factors of network capacity expansion, but it is not realistic to improve the conditions of each influence factor. For example, changing the road network layout and hierarchy cannot be achieved in short-term management and operation, and optimizing the route choice behavior of traffic individual is very difficult to practice due to subjective factors.

Based on discussions above, this paper mainly focuses on improving road conditions of network as a way of expanding capacity of road network. The narrow definition of expanding capacity of road network is that through improving the road conditions of network under the condition of certain economic constraints and geographical environment, the maximum number of traffic volume passing the road section in unit time is increased, and the dimension is pcu/h.

##### 2.2. Assumptions

According to the narrow concept of expanding capacity of road network, what road sections and intersections are taken as the expansion objects must be determined firstly. Following this, the expansion objects of capacity of road network are divided into three categories: take a certain road section or an intersection of the network as the expansion objects; take all road sections and intersections of the network which flow is greater than or equal to the capacity as the expansion objects; take the critical road sections and intersections of the network as the expansion objects. The intersection capacity expansion is mostly improved by traffic organization optimization and traffic designs under the given traffic demand. To do this, the capacity expansion of critical road sections must be first integrated cooperatively. As a foundation of the expansion of intersection capacity and to simplify the combination optimization problem, this paper just selects the set of critical road sections as the expansion objects.

To extend the capacity of road network, urban road network design problem (NDP) is usually divided into following three types [11]: continuous network design problem (CNDP), discrete network design problem (DNDP), and mixed network design problem (MNDP). The CNDP aims to improve the capacity performance of existing road sections by enlarging new lanes. The DNDP aims to extend existing road network by constructing new roads. The MNDP is the combination measures of the CNDP and the DNDP. The corresponding assumptions in this paper are as follows.(1)According to the research scope of the capacity expansion of critical road sections, this paper discusses the type of the CNDP.(2)As the OD demand from those built facilities of a city is relatively stable in a short time, the OD traffic demand in the network is static.

#### 3. Modelling Capacity Expansion of Road Network

The routing choices of travelers are mainly determined by the lowest travel cost. But traffic managers expect to optimize the network performance, such as reducing traffic congestion and maximizing the throughput. Therefore not only travel cost of travelers, but also usage of network capacity should be taken into account in routing choice. Following this, here, the bilevel programming model is used to model the travel objectives of both travelers and traffic managers.

The expanded capacity of road network, defined in upper-level model, is to realize global optimization, and the routing choice behavior, denoting by , is defined in lower-level model. To facilitate the model presentation, the notations used here and after are summarized in Notations section.

*(1) The Upper-Level Model*. Travel time is a comprehensive index used to evaluate the congested and comfortable level of user’s trip. This paper uses the total travel time as the global optimization objective. It is expected that total travel time is reduced with the network capacity expansion. Following this, the upper model of road network capacity is as the following formulas:

Formula (1) minimizes the total travel time in the network. Formula (2) makes sure that the increase of capacity will never exceed the feasible budget. Formula (3) shows the upper limit and lower limit of increasing capacity of road sections. To eliminate the constraints of budget, Lagrange transform is used to simplify the upper model, as shown in formulas (4) and (5). Consider the following:

In formula (4), is the Lagrange multiplier.

*(2) The Lower-Level Model*. In a fixed demand traffic assignment problem, with the given expanded capacity of links and fixed demand between OD pairs, the lower model is a standard user equilibrium model, which describes the user’s routing choice behavior, as in the following formulas:

Formula (6) describes the assignment problem of Wardrop user equilibrium (UE). Formula (7) meets the need of the conservation of traffic flow. Formula (8) shows the relationship between section and route traffic. Formula (9) describes the nonnegativity of route traffic. The optimal solution meets the user equilibrium condition as in formula (10). When the travel time of route is more than or equal to the minimum travel time, the route traffic is zero. When the travel time of route is equal to the minimum travel time, the route traffic is more than zero; that is to say, this route is occupied:

In formula (10), means the travel time of route in the OD pair numbered . means the minimum travel time of routes between the OD pair numbered .

#### 4. Solution Algorithm

Variables of capacity expansion in upper-level model are necessary to solve lower-level model. Thus, upper-level mathematical model is not a linear optimization model, which is hard to solve by traditional integral equation method as least square method. Genetic algorithm (GA) does not depend on gradient information and experiential knowledge and is able to find global optimum. And, hence, Yin used genetic algorithm to solve network design problem, and introduced bionic mechanisms such as simulated annealing, ants feeding, and particle swam preying to improve the local search ability of simple genetic algorithms [12]. However, the improved genetic algorithm has disadvantages as complicated structure or large calculation, which may cause inefficiency and poor portability. Considering nonlinearity and nonconvexity of bilevel expansion models, this paper introduces the golden ratio to integrate with an improved genetic algorithm to solve upper-level model, and the classic Frank-Wolfe algorithm is used to solve lower-level model.

##### 4.1. Frank-Wolfe Algorithm

Use Frank-Wolfe algorithm to calculate the lower-level model under fixed OD travel demand. Main steps are as follows [13].

*Step 1 (initialization). *Set the iteration number and find a feasible traffic mode .

*Step 2 (update the travel time). *Calculate .

*Step 3 (find direction). *According to , use all-or-none (AON) algorithm to get the auxiliary flow collection .

*Step 4 (displacement distance). *As in formula (11), find a better along the direction of the objective function minimized:

*Step 5. *Update road traffic network flow, which is to calculate new traffic volume in links as in the following formula:

*Step 6 (judge the end condition). *If the algorithm reaches the specific judging criteria (such as maximum iterations), end the Frank-Wolfe algorithm. Otherwise and returns back to Step 1.

##### 4.2. Golden Ratio-Based Heuristic Genetic Algorithm

A golden ratio-based heuristic genetic algorithm (GRGA) has been proposed to yield approximate solutions for expanded capacities of urban road network [14]. The proposed heuristic is able to find the closest solution to the best solution by introducing golden ratio (GR) to enhance the local optimal capability of an improved real-coded genetic algorithm [15].

*(1) Golden Ratio Based Local Search*. When genetic algorithms come to the later evolution process, individuals of the population might trap into local minima especially for the optimization problems with a big problem space and many minima. Hence, it is more possible that the fitness value related to the current individual is lower than the random search, and it can be expected that two individual neighbors, which are different from each other in the topology, are located in the same concave in the searching space. There are potential genes between the two closest individual neighbors which have lower fitness value than these two individuals.

In recent years, the golden ratio has also been applied to optimize timings of traffic signal systems with good results [16]. Here, the golden ratio is introduced to find the genes of the local search position. As in Figure 1, Point and Point are two adjacent individual neighbors, and Point is the potential local position determined by the golden ratio of and ; that is the relationship between the segment and the segment satisfies the golden ratio definition. In addition, to expand the local searching area around the current individual positions during the whole searching process, the local position Point can also be determined by the opposite golden ratio as the opposition concept has been used in evolution optimization algorithm and good performance was obtained. Here, the concept of the opposite golden ratio is to rotate the Point by 180 degrees, and the relationship between the segment and the segment still meets the golden ratio definition.