Mathematical Problems in Engineering

Volume 2016, Article ID 1682394, 12 pages

http://dx.doi.org/10.1155/2016/1682394

## Signal Timing Optimization Based on Fuzzy Compromise Programming for Isolated Signalized Intersection

^{1}State Key Laboratory of Automobile Dynamic Simulation, Jilin University, Changchun 130022, China^{2}College of Transportation, Jilin University, Changchun 130022, China^{3}Jilin Province Key Laboratory of Road Traffic, College of Transportation, Jilin University, Changchun 130022, China

Received 8 December 2015; Revised 3 February 2016; Accepted 29 February 2016

Academic Editor: Konstantinos Karamanos

Copyright © 2016 Dexin Yu 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

In order to optimize the signal timing for isolated intersection, a new method based on fuzzy programming approach is proposed in this paper. Considering the whole operation efficiency of the intersection comprehensively, traffic capacity, vehicle cycle delay, cycle stops, and exhaust emission are chosen as optimization goals to establish a multiobjective function first. Then fuzzy compromise programming approach is employed to give different weight coefficients to various optimization objectives for different traffic flow ratios states. And then the multiobjective function is converted to a single objective function. By using genetic algorithm, the optimized signal cycle and effective green time can be obtained. Finally, the performance of the traditional method and new method proposed in this paper is compared and analyzed through VISSIM software. It can be concluded that the signal timing optimized in this paper can effectively reduce vehicle delays and stops, which can improve traffic capacity of the intersection as well.

#### 1. Introduction

Due to increase of vehicle numbers and growing traffic demand, traffic problems are becoming more and more serious. It is well known that the implementation of traffic signal control at intersections is one of the effective methods to relieve the series of traffic problems [1]. Therefore, researching the signal timing optimization methods is of great significance to improve the operation efficiency of the intersections.

Signal timing optimization has been highly concentrated in many years and there have been many researches in this field. When optimizing the signal timing parameters, it is needed to consider various performance indexes comprehensively, such as road capacity, vehicle delays, vehicle stops, and queue length. Early researchers were mostly focused on the optimization of a single index, typically represented by traditional Webster signal timing method of which the optimization objective was to minimize average vehicle delays [1]. For isolated signalized intersections, Ceder and Reshetnik [2] proposed an algorithm to minimize vehicle queues, which was proved efficient. Wang and Yang [3] established a signal timing optimization model to minimize average delays and applied genetic algorithm to search for optimal signal timing parameters. The results showed that the proposed method could obtain lower average delay and stops than using Synchro software. Under oversaturated state, Liu et al. [4] built a dynamic linear programming optimization model with the objective of maximizing traffic capacity at the intersection. Contributing to the deepening of the sustainable development concept, the optimization objective of signal timing control was changed from single objective to multiple objectives gradually. Based on data fusion, Deng et al. [5] proposed a new multiobjective optimization control strategy for urban signalized intersections. Vehicle delay, queue length, and data fusion technology were considered in the strategy, which was proved to be consistent with the traffic flow characteristics in China. Focused on isolated intersection, Jiao and Sun [6] presented a multiobjective signal control model framework based on dynamic O-D estimation. A state-space model was established first. And then dynamic turning movements were estimated. Compared with Webster method, the results were close to actual values. Anderson et al. [7] specially researched the optimization objectives of signal control system systematically, which indicated that signal control strategies should be flexible to adapt to various traffic states. The importance of each objective was supposed to adjust according to diverse traffic states. Schmöcker et al. [8] chose three evaluation indexes as optimization objectives, including average vehicle delays, average queue length, and pedestrian waiting timing. And then a multiobjective optimization function was established. Bellman-Zadeh fuzzy logic was employed to convert the model to single objective function which was finally solved by genetic algorithm. Li et al. [9] optimized the signal timing scheme of oversaturated intersections and the optimization objectives were maximum throughput vehicle numbers and minimum vehicle queue rate. Employing fuzzy logic approach, Murat and Gedizlioglu [10] proposed a multiphased signal control optimization method of isolated intersections.

In summary, the existing researches did not comprehensively consider enough factors. This may lead to the fact that one or two indexes were optimal but others were not optimal. The signal timing for isolated intersections in most current researches was optimized by adding several performance indexes together simply and giving diverse weight values to different indexes subjectively. It was not appropriate because the indexes were often on different dimensions. In addition, the weight of each index could not be adjusted according to traffic flow state. Therefore, it is essential to optimize multiple objectives at the same time and choose a more eclectic scheme to make each objective in a reasonable range.

To overcome the drawbacks mentioned above, a new signal timing optimization method for isolated signalized intersections is proposed in this paper based on fuzzy programming. The overall benefits of the intersection are considered in this paper. For establishing a multiobjective function to optimize signal timing parameters, several performance indexes are chosen first, including traffic capacity, total vehicle delays, and vehicle stops in a cycle, as well as exhaust emissions. In order to avoid the subjective factors as much as possible, fuzzy compromise programming is employed to give each optimization objective various weight coefficients. In this paper, weight coefficients can vary with different traffic flow states. When traffic flow is high, the main optimization objectives are to extremely increase the traffic capacity and reduce cycle vehicle delays. On the contrary, when traffic flow is low, the main optimization objectives are to reduce cycle vehicle delays and cycle stops. Then the multiobjective optimization problem is converted to a single objective optimization problem. To obtain the optimized signal cycle and effective green times, genetic algorithm is used to solve the model by MATLAB software. Finally, the signal timing scheme proposed in this paper is compared with other methods, employing the VISSIM simulation software. The results illustrate that the scheme proposed in this paper can effectively reduce vehicle delay and stops, improving the traffic capacity and the whole operation efficiency of the intersection.

The rest of this paper is organized as follows. Section 2 formulates the multiobjective programming model including optimization objectives selection presented in Section 2.1 and multiobjective programming modeling presented in Section 2.2. In Section 3, optimization objective functions are solved by fuzzy compromise programming and genetic algorithm. In Section 4, the model proposed in this paper is verified by choosing a typical intersection in Changchun, China. The results are compared with other traditional methods. Section 5 reveals the conclusions and recommendations for future research.

#### 2. Multi-Objective Programming Model

##### 2.1. Optimization Objectives Selection

It is known that the optimization goals of signal timing are to improve the passing efficiency and reduce vehicle delays and stops at the intersection. Therefore, considering the overall efficiency of the intersection, four traffic performance indexes including traffic capacity, cycle vehicle delay, cycle vehicle stops, and exhaust emission are chosen as the optimization objectives to obtain the optimized signal timing in this paper.

Several assumptions are given to formulate the models. Firstly, the unsaturated intersections are focused on in this paper. The condition when the intersections are oversaturated is not included. Then assume that the distance between the adjacent intersections is long enough and the vehicle arrival of downstream intersection is little affected by the signal setting at the upstream intersection. At the analysis interval, vehicle arrival rate remains stable and obeys a certain traffic arrival distribution, usually Poisson distribution. Although vehicles queue may be formed due to the random arrival in some cycles, the arrival of vehicles keeps balance with vehicle dissipation in the whole analysis period.

###### 2.1.1. Traffic Capacity of the Intersection

Based on existing researches, the intersection capacity can be calculated by adding the capacity of each approach, respectively [11]. Assume that an intersection contains key phases, and then the intersection capacity can be calculated by where denotes the intersection capacity, is the saturation flow for a particular phase , is the number of key phases at the intersection, and is the proportion of effective green time for phase which is equal to the ratio of effective green time and cycle by where denotes the cycle time of the signalized intersection, is the effective green time for phase , and others are the same as above. Then the whole intersection capacity can be obtained as follows:

###### 2.1.2. Cycle Vehicle Delays

In urban road network, most of the vehicle delays are generated at signalized intersections. As a consequence, vehicle delays can reflect the state of the road network to some extent and it is often chosen as signal timing optimization objective. When calculating the vehicle delays at a signalized intersection, Webster model [1] is usually used of which the characteristics are that the model is only suitable for low saturation of traffic flow and assume that the vehicle accumulation is continuous. Therefore, vehicle average delay model [12] is selected as where denotes the traffic flow and denotes the flow ratio of phase , respectively. Substitute formula (2) into formula (4) and the following formula can be obtained. The others are the same as above. Consider

Then the cycle total vehicle delay of the intersection can be calculated as follows:

###### 2.1.3. Cycle Vehicle Stops

Due to the influence of signal light, vehicles often need to stop when approaching the intersection. So vehicle stops are also an important index to measure the traffic state of the intersection. The steady state model [13] proposed by Akçelik and Rouphail is employed to calculate vehicle stops as follows:where and denote the vehicle stop rate and saturation level of phase , respectively. Then the total vehicle stops in a cycle can be obtained as follows:

###### 2.1.4. Vehicle Exhaust Emission

The exhaust pollutants generated in the process of driving have a great impact on environment. For this reason, vehicle exhaust emission is taken as the forth optimization objective. The vehicle exhaust emissions at the intersection consist of two parts. One part is the idle exhaust emissions generated on the import approaches and another part is the exhaust emissions discharged in the intersection area [14, 15]. Hence, vehicle exhaust emission can be formulated as follows: where denotes the emission of pollutant (g/h), is the pollutant emission factor for standard vehicle under idle condition (g/(pcu·h)), is the pollutant emission factor for standard vehicle (g/(pcu·km)), is the average vehicle delay time for phase (s/pcu), and is the length of the approach for phase (km). Usually, the values of and are 5 g/(pcu·h) and 45 g/(pcu·km), respectively. Substitute the average delay into formula (10), and the final emission model could be obtained.

##### 2.2. Multiobjective Programming Modeling

Considering the whole operation efficiency of the intersection, four objective functions are established of which the objectives are maximizing intersection traffic capacity, minimizing vehicle cycle delay, minimizing cycle stops, and minimizing exhaust emissions, denoted as , , , and , respectively.

The basic input parameters of the models are the saturation flow , traffic flow , the flow ratio , and the length of the approach for phase (km). And the output variables of the models are cycle time of the signalized intersection and the effective green time for phase . Consider

Some constraints of the objective functions are formulated as follows which consider the actual traffic environment:where the above constrains limit signal cycle length, effective green time, and saturation ratio of the intersection, respectively. The first constrain limits the signal cycle size in which and , where denotes the number of key phase sequences. The second constrain limits the effective green time size where and denote the minimum and maximum effective green time, respectively. The values of them can be determined according to the actual traffic environment [12]. The saturation size which is equal to the ratio of traffic flow and traffic capacity is limited by the third constraint. and denote the minimum saturation size and the maximum saturation size separately of which the values are often equal to 0.7 and 0.95 in order to avoid the waste of green time and too short green time [12].

#### 3. Optimization Objective Functions Solving

Because solving multiobjective optimization problem is complex, so the combination method is often employed to solve this series of problems. By giving each optimization objective different weight coefficients, the multiobjective optimization problem can be transformed into single objective optimization problem. The dimensions of each objective are often different. Therefore, it is not reasonable to add multiple optimization objectives together simply. Then fuzzy compromise programming approach is brought in this paper.

##### 3.1. Fuzzy Compromise Programming

Fuzzy compromise programming approach takes various objectives into consideration synthetically by marginally evaluating individual objectives and then globally evaluating all objectives. In the global subjective evaluation, the decision-maker’s preferences are reflected and various objectives are taken into account [16]. The multiobjective signal timing optimization function is converted to single objective function. According to different traffic state, the optimized signal cycle length and effective green time can be obtained.

The basic idea of fuzzy compromise programming method is to achieve overall optimum by calculating the distance between the feasible solution set and the ideal solution set and determining the feasible solution which is closest to the ideal solution as the final solution [17]. The objective functions in different dimensions can be changed into a dimensionless function through conversion. The values are limited between 0 and 1. Different weight coefficients are given to different optimization objectives. The solving steps are as follows [18].

*Step 1. *Convert the objective functions to minimum forms under the limits of constraints. Calculate the minimum values and maximum values, respectively, first. For each particular objective in the multiobjective programming problem , the upper and lower bounds values are assigned which are denoted by and . In fact, for objective , and represent the highest acceptable level and the aspired level of the achievement, respectively. Then the ideal value vector is constituted by the lower bounds values of all objective functions, denoted as . Similarly, the opposite ideal value vector is constituted by the upper bounds values of objective functions, denoted as :

*Step 2. *Calculate the membership function of each optimization objective function. As it is known, there are several membership functions including linear, hyperbolic, and piecewise-linear [16]. For practicability and simplicity, the linear membership function [18] is adopted in this paper. With the upper and lower bounds values, marginal evaluation can be obtained for each objective . A mapping illustrates what degree the decision makes th objective close to the aspired solution, which is used to denote the membership function of objective . The values of are between 0 and 1, representing the degree of compatibility between the solved value and the aspired value. The marginal evaluation mapping can be obtained by the following formula:

*Step 3. *Calculate the weight coefficient of objective which is denoted as . The membership functions which are more important are constituted as a group . Equivalence class is built by using the relationship . Choose one element from each equivalence class to form a collection which is denoted as . The values of the relationship function are determined by the evaluation function :where the coefficients , , , , , and are all real numbers between 0 and 1 and the formulas and exist.

*Step 4. *Define matrix according to the evaluation function . And establish a relation matrix and denote it as :

*Step 5. *Then the weight coefficient in which belongs to can be obtained as follows: where denotes the boundary value of matrix .

*Step 6. *Convert the multiobjective function to a single objective function: where denotes the distance index of which the value can be different. According to the actual environment, the values are usually 1, 2, and .

##### 3.2. Genetic Algorithm

Many methods have been developed for traffic signal timing optimization. Conventional methods include integer programming and hill climbing. With the requirements of computing, intelligent algorithm is developed, such as genetic algorithm (GA). Using GA to solve the fuzzy goal programming has been researched by Pal and Gupta [19]. The algorithm steps of solving the model are presented as follows.

*Step 1 (representation and initialization). * denotes the binary coded representation of a chromosome in a population as . Define pop-size as the population size and chromosomes are randomly initialized.

*Step 2 (fitness function). *Because the domain of Rosenbrock function is negative, the fitness function can be equal to the negative function. Namely, take the objective function () as individual fitness in this paper.

*Step 3 (take selection, crossover, and mutation operations). *Classical roulette-wheel scheme is used for selecting in the genetic search process [5]. Choose proportional selection operator for selection operation, single point operator for crossover operation, and basic mutation operator for mutation operation.

#### 4. Model Verification

A typical intersection in Changchun city is selected to verify the signal timing optimization method proposed in this paper. The intersection has four approaches and the geometrical characteristic is shown in Figure 1. There are four phases at this intersection. The first phase is used to release the straight traffic flow in the eastern and western approaches. The second phase is used to release the left traffic flow in the eastern and western approaches. Similarly, the third and the fourth are used to release the straight and left traffic flow in the southern and northern approaches, respectively.