Two-Stage Fuzzy Logic Controller for Signalized ... - IEEE Xplore

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 41, NO. 1, JANUARY 2011

Two-Stage Fuzzy Logic Controller for Signalized Intersection Jian Qiao, Naiding Yang, and Jie Gao

Abstract—Traffic efficiency is commonly regarded as the most important target for the control of signalized intersections. However, from the fairness point of view, it can be argued that all vehicles at a signalized intersection should have equal passing opportunities. In this correspondence paper, a two-stage fuzzy logic control model for an isolated signalized intersection has been proposed, where both traffic efficiency and fairness have been considered simultaneously. At the first stage, a green-phase selector has been developed to select the subsequent green phase. At the second stage, a green-time adjustor has been proposed to determine the green time for the selected phase. An offline genetic algorithm (GA) has been developed to optimize the fuzzy rules and membership functions of the two controllers. The simulation results demonstrate that the proposed model outperforms the vehicle-actuated control model and the model proposed by Pappis and Mamdani in 1977 in terms of both traffic efficiency and fairness. The performance of the proposed model can be further improved after its rules and membership functions are optimized by using GA. Index Terms—Fuzzy logic, genetic algorithm (GA), signalized intersection, traffic control.

I. I NTRODUCTION Signalized intersections are the bottleneck of urban road networks. In the past few decades, researchers have proposed various control models with the objective of maximizing the capacities of the intersections [1]. However, the performance of mathematical control models was not always satisfactory owing to the difficulty in precisely describing the complex traffic flow. Since the first fuzzy controller for an isolated intersection (P-Model) [2] was proposed, significant improvements have been made in related studies. First, intersection modeling has been made more realistic. For example, the assumption of singlelane roads was replaced by multilane roads [3], turning traffic was included in the model [4], and two-phase intersections were expanded to multiphase intersections [5]. Second, more factors influencing traffic flow have been considered in the control models. For instance, in the fuzzy models, fuzzy rules could be updated dynamically to adapt to the varying traffic flow [6]. Furthermore, the priority principle for main arteries [7] and the impact of upstream intersections on downstream ones were also considered [8]. Traffic engineers may adjust the signal settings according to their rankings on multiple objectives [9]. Third, it has been realized that phase composing has a significant impact on the capacity of signalized intersections [10], and so does phase sequence [11], [12]. Manuscript received November 1, 2005; revised March 8, 2009; accepted May 24, 2009. Date of publication July 26, 2010; date of current version November 10, 2010. This work was supported in part by the projects RW200725 and 07XE0121 from Northwestern Polytechnical University, and by the project NCET-05-0864 from the Ministry of Education of China. This paper was recommended by Editor W. Pedrycz. J. Qiao is with the Department of Information Management and Information System, School of Management, Northwestern Polytechnical University, Xi’an 710072, China (e-mail: [email protected]; [email protected]). N. Yang is with the Department of Management Science, School of Management, Northwestern Polytechnical University, Xi’an 710072, China (e-mail: [email protected]). J. Gao is with the School of Management, The State Key Laboratory for Manufacturing Systems Engineering, and The Key Laboratory of the Ministry of Education for Process Control and Efficiency Engineering, Xi’an Jiaotong University, Xi’an 710049, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCA.2010.2052606

Many control models for multiple intersections have also been developed. These include the model for multiple intersections with twoway roads [13], fuzzy-logic- and multiagent-based models [14], [15], changeable-phase-sequence-based model [16], models for both isolated and multiple intersections [17], [18], multiagent-, fuzzy-logic-, and artificial-neural-network-based models [19], [20], model with bus priority [21], etc. The aforementioned studies aim to maximize the traffic efficiency (capacity) of signalized intersections. In spite of the overwhelming importance of traffic efficiency, each driver waiting at a signalized intersection is reluctant to stay for a longer time than others. Hence, the traffic control system should provide equal passing opportunities for all waiting vehicles as much as possible when a higher traffic efficiency can be achieved, which is called “traffic fairness.” However, to our knowledge, this concept has hardly been discussed in the existing literature. In this correspondence paper, a two-stage fuzzy logic control model has been proposed for an isolated signalized intersection, with the objective of maximizing both traffic efficiency and fairness. To further improve its performance, an offline genetic algorithm (GA) has been designed to optimize the rules and membership functions of the model. The performance of the model without optimization (i.e., experimental model, E-Model) and the optimized model (O-Model) has been compared with that of the vehicle-actuated control model (V-Model) [22] and the P-Model via simulation experiments. II. T RAFFIC PARAMETERS AND P ERFORMANCE I NDICES In this corresponding paper, a cross-signalized intersection has been considered. It has been assumed that each direction of the intersection includes three entry lanes: straight-through, left-, and right-turn lanes. There are two loop detectors on each entry lane. One is embedded under the stop line, and the other is embedded at an upstream location with a certain distance away from the stop line. The number of vehicles between the two detectors can be calculated according to the number of passing vehicles counted by the two detectors. In this section, the traffic parameters and the performance indices used in the model and the simulation have been introduced. A. AR The number of vehicles detected by the upstream detector of entry lane i within (t − 1, t] is defined as the arrival rate (AR) of the lane at the end of time t and is denoted by ARit . The sum of ARit of all the entry lanes in the same direction is defined as the AR of this direction. Effective traffic control should be based on the precise forecasting of varying traffic flow. This is a complex problem and is beyond the scope of this correspondence paper. To focus on the traffic control problem, it is assumed that the AR curves of all directions are known. B. NDV At the end of time t, the number of vehicles between the two detectors of entry lane i is defined as the number of detected vehicles (N DV ) of the lane at the moment and is denoted by N DVit . Let Dist be the distance between the two detectors of each entry lane, l be the average length of the vehicles, and δ be the minimum interval between the two adjacent vehicles. Thus, the maximum N DV of an entry lane is

1083-4427/$26.00 © 2010 IEEE

N DVmax = Dist/(l + δ).

(1)


where wit=0 = 0. Therefore

Therefore

N DVit+1 = min max N DVit + ct+1 − git+1 , 0 , N DVmax i

=

0, , min s, N DVit + ct+1 i

for the red phase for the green phase

AW Tit+1 = wit+1 /N DVit+1 .

(2)

and git+1 denote the numbers of where N DVit=0 = 0, and ct+1 i arriving and departing vehicles on entry lane i within (t, t + 1], respectively. Arriving vehicles can be counted by the upstream detector of the lane. In simulation, they are generated randomly according to the AR curve of the lane, and git+1 is counted by the stop-line detector of the lane git+1

179

(3)

where git=0 = 0 and s denotes the saturation flow rate, i.e., the maximum traffic flow volume of an entry lane within one time unit. We define ss , sl , and sr as the saturation flow rates of the straight-through, left-, and right-turn lanes of a direction, respectively. In general, they are different from one another.

(9)

E. TUD Traffic urgency degree (T U D) represents the urgency degree to which the waiting vehicles on an entry lane are eager to pass through the intersection. The T U D of an entry lane is determined based on its W N DV and AW T . It is a nondimensional value. F. EGT Extension of green time (EGT ) is used to adjust the green time of the selected phase. It is determined on the basis of the N DV and AR of the selected entry lane in the selected phase. The control performance indices have been introduced in the subsequent paragraphs of this section. G. NPV

C. WNDV The weighted N DV (W N DV ) of lane i at the end of time t is defined as follows: W N DVit = fit × N DVit

(4)

where fit is the weighted factor. It is defined as fit = ARmax /ARit

(5)

with

(6)

i,t

D. AWT The arriving vehicles are detected when they pass through the upstream detectors. For a vehicle arriving at each simulation step, there is a time interval between the beginning of the step and the detection of the vehicle by the upstream detector. This interval is defined as the denote the summed arriving time arriving time of the vehicle. Let wct+1 i of all the arriving vehicles on lane i within (t, t + 1]. By assuming that these arriving vehicles pass through the upstream detector evenly, we can obtain the following:

ct+1 i

=

j=1

1 1− j− 2

1 ct+1 i

=

ct+1 i . 2

(7)

The time that a vehicle spends between the two detectors of an entry lane is defined as its waiting time. At the end of time t, the mean waiting time of all the waiting vehicles on lane i is defined as average waiting time (AW T ) of the lane at the moment and is denoted by AW Tit . Let wit denote the summed waiting time of all the waiting vehicles on lane i within (t − 1, t]. Then

wit+1

⎧ t w + N DV t − git+1 + wct+1 ⎪ i ⎪ ⎨ i t+1 i t+1 t − gi · AW Tit , N DVi ≥ gi = (8) t+1 t+1 t max N DVi + ci − gi , 0 ⎪ ⎪ ⎩ t+1 t+1 t+1 t · wci /ci

,

N P Vit+1 = N P Vit + git+1

(10)

where N P Vit=0 = 0. The summed N P V of all the entry lanes is defined as the N P V of the intersection. The smaller the N P V , the lower the traffic efficiency. H. AD

t

ARmax = max ARi .

wct+1 i

From the starting time until time t, the number of vehicles that have passed through the intersection via entry lane i is defined as the number of passed vehicles (N P V ) of the lane at the moment and is denoted by N P Vit . Obviously, we have

N DVi < gi

Similar to the arriving vehicles, for a vehicle departing at each simulation step, there is a time interval between the beginning of the step and the detection of the vehicle by the stop-line detector. This denote interval is defined as the departing time of the vehicle. Let wgt+1 i the summed departing time of all the passed vehicles on lane i within (t, t + 1]. Then, we have

git+1

wgt+1 i

=

j j=1

s

=

git+1 git+1 + 1 2s

.

(11)

The sum of the arriving, waiting, and departing times of a passed vehicle is defined as the delay of the vehicle. At the end of time t, the summed delays of all the passed vehicles from lane i is defined as the delay of the lane at the moment and is denoted by dti . Hence

dt+1 = dti + wct+1 + wgt+1 + max N DVit − git+1 , 0 i i i

(12)

where dt=0 = 0. Let T Dt denote the total delay of the intersection i at the end of time t, and let L be the number of entry lanes. Thus, we have T Dt+1 = T Dt +

L

dt+1 i

(13)

i=1

where T Dt=0 = 0. Let C t denote the total number of arriving vehicles within (t − 1, t]. Then C t+1 = C t +

L i=1

ct+1 i

(14)

180


where C t=0 = 0. We define average delay (AD) of the intersection at time t + 1 as ADt+1 = T Dt+1 /C t+1 .

(15)

It is clear that if the AD is smaller, then the traffic efficiency is higher.

I. PSV Let v, a+ , and a− denote the average velocity, acceleration, and deceleration of the vehicles, respectively. The average time for a vehicle to pass through the intersection is α = Dist/v + Gl

(16)

where Gl denotes the loss of green time. The average time for a vehicle to come to a full stop between the two detectors of an entry lane is β = v · (1/a+ − 1/a− ).

t+1

L gi

φ Tijg − Tijc − α, β

(18)

i=1 j=1

where S t=0 = 0, and Tijc and Tijg denote the arriving and departing time points of vehicle j on lane i, respectively. φ(·) is a 0–1 function as follows:

φ Tijg − Tijc − α, β =

1, 0,

Tijg − Tijc − α ≥ β Tijg − Tijc − α < β.

(19)

Hence, percentage of stopped vehicles (P SV ) of the intersection at the end of time t + 1 is P SV t+1 = S t+1 /Gt+1

(20)

where Gt+1 denotes the N P V of the intersection at the end of time t + 1. It can be derived from the following formula: Gt+1 = Gt +

L

git+1

(21)

i=1

where Gt=0 = 0. The smaller the PSV, the higher the traffic efficiency.

J. ADD Fair traffic signifies that all vehicles at a signalized intersection have equal passing opportunities, i.e., these vehicles experience equal delays. Hence, traffic fairness can be measured by the differences of the delays and is indicated by average deviation of delay (ADD). At the end of time t, ADD of the intersection is defined as follows: t

P Vi L N t dij − ADt ADD = t

i=1

j=1

III. D ESCRIPTION OF THE M ODEL A. Structure of the Model Investigation on the driving behaviors indicates that it is dangerous to change phase composing dynamically, because this may lead to drivers’ nervous tension and judgment error. In our proposed model, it is assumed that phase composing is predetermined and that phase sequence and signal timing are changeable. The control process is composed of two sequential stages, namely, green-phase selection and green-time assignment, and two fuzzy controllers, namely, greenphase selector (GPS) and green-time adjustor (GTA), have been developed for the two stages, respectively. At the first stage, the T U D of all the entry lanes is calculated using GPS, based on the collected traffic data, and then, the next green phase is selected in light of the T U D. At the second stage, the green time of the selected phase is determined by GTA according to the traffic status of the entry lanes in the selected phase.

(17)

Consider that each vehicle has at most one full stop between the two detectors of an entry lane. The number of full stops for all the passed vehicles at the end of time t + 1 is

S t+1 = S t +

where dtij denotes the delay of passed vehicle j on lane i at the end of time t. The smaller the ADD, the better the traffic fairness.

L i=1

B. GPS GPS is triggered to select the next green phase when the current green phase is over. The control process can be outlined as follows. First, the N DV and AW T of all the entry lanes are calculated based on the collected traffic data. Second, the T U D of each lane is calculated in light of its N DV , AW T , and AR. Finally, the green phase is selected according to the T U D of these lanes. If the entry lanes with the largest T U D belong to the same phase, this phase is then selected as the next green phase. If the lanes belong to more than one phase, then the phase with the largest N DV will be selected. If more than one phase meets the selection criteria, then the next green phase is selected randomly. The inputs of GPS consist of the W N DV and AW T of all the entry lanes, and GPS outputs the T U D of the lanes. According to the definition of W N DV , an entry lane with a small AR can have a large W N DV as long as its AW T is large enough. Hence, this lane can have a large T U D, because T U D is based on both W N DV and AW T rather than merely on AR. As a result, the passing opportunities for the vehicles on this lane are increased. It is obvious that GPS aims to find a balance between traffic efficiency and traffic fairness. In this correspondence paper, the discourse domain of W N DV has been set as [0, 20], and that of AW T has been set as [0, 300], according to engineering experience. As T U D is a nondimensional variable, its discourse domain can be defined arbitrarily and has been set as [0, 60] here. In our simulations, these three discourse domains have been mapped to [0, 32], [0, 320], and [0, 64] correspondingly. The linguistic value sets of W N DV , AW T , and T U D are {Small (S), Medium (M), Large (L)}, {Short (S), Medium (M), Long (L)}, and {Very Low (VL), Low (L), Medium (M), High (H), Very High (VH)}, respectively. The total number of feasible fuzzy rules is 3 × 3 = 9, each of which can be generalized in the form of “if W N DV and AW T then T U D.” The trapezoidal-shaped membership function has been adopted for all the fuzzy subsets. Mamdani’s algorithm and the center-of-gravity method [23] have been applied in fuzzy reasoning and defuzzification, respectively.

C. GTA N P Vit

(22)

After selection of the green phase, GTA will be triggered to adjust the green time for the selected phase. The control process can be


outlined as follows. First, the entry lane with the largest N DV in the green phase is selected. Second, the EGT of the green phase is calculated according to the N DV and AR of the selected lane. If the current green phase is selected again as the next one, then its green time is extended by EGT ; otherwise, its initial green time is set as EGT . The inputs of GTA consist of the N DV and AR of the selected lane, and the output is the EGT of the selected phase. A large N DV indicates more waiting vehicles, and a large AR implies more arriving vehicles. Hence, the selected lane with large N DV and AR may need a long green time. Obviously, traffic efficiency is the main objective of GTA. The discourse domain of N DV is set as [0, 20], and those of AR and EGT are set as [0, 3600] and [0, 60], respectively, according to engineering experience. As in GPS, the discourse domain of N DV is also mapped to [0, 32]. The linguistic value sets of N DV , AR, and EGT are {Small (S), Medium (M), Large (L)}, {Very Small (VS), Small (S), Medium (M), Large (L), Very Large (VL)}, and {Extremely Short (ES), Very Short (VS), Short (S), Medium (M), Long (L), Very Long (VL), Extremely Long (EL)}, respectively. The total number of feasible fuzzy rules is 3 × 5 = 15, and they are all of the same format as in GPS. The trapezoidal-shaped membership function has been adopted for all the fuzzy subsets. Similar to GPS, the same fuzzy reasoning and defuzzification methods have been adopted.

181

Fig. 1. Parameterized presentation of a trapezoid.

Let a0 , b0 , c0 , d0 denote the first initial membership function, which is given based on engineering experience. The adjustment intervals for a, b, c, and d are determined based on a0 , b0 , c0 , and d0 as follows:

a ∈ [al , ar ] = a0 −

b ∈ [bl , br ] = b0 −

b0 − a0 b0 − a0 , a0 + 2 2 b0 − a0 d0 − c0 , c0 + 2 2

b0 − a0 d0 − c0 , c0 + c ∈ [cl , cr ] = b0 − 2 2

(25)

(26)

(27)

IV. M ODEL O PTIMIZATION A LGORITHM

d0 − c0 d0 − c0 , d0 + d ∈ [dl , dr ] = d0 − . 2 2

GA has been theoretically and empirically proven to provide a robust search in complex spaces [24]. An offline GA is designed to optimize the rules and membership functions of GPS and GTA, in which the reciprocal of AD serves as the fitness.

The membership functions of the inputs and output of GPS/GTA can be denoted by the following sets, respectively:

(28)

{axi , bxi , cxi , dxi |i = 1, 2, . . . , l} {ayi , byi , cyi , dyi |i = 1, 2, . . . , m}

A. Fuzzy Rule Encoding Integral encoding is applied to represent the fuzzy rules. The linguistic value set of each fuzzy variable is mapped to a natural number set. For example, {Small, Medium, Large} can be mapped to {1, 2, 3}. Let x and y denote the inputs and z the output of GPS/GTA. Thus x ∈ {i|i = 1, 2, . . . , l}

{azi , bzi , czi , dzi |i = 1, 2, . . . , n} . Real encoding is used to represent the membership functions. A chromosome is composed of (l + m + n) × 4 genes and is represented as follows: ax1 bx1 cx1 dx1 , . . . , axl bxl cxl dxl ay1 by1 cy1 dy1 , . . . , aym bym cym dym az1 bz1 cz1 dz1 , . . . , azn bzn czn dzn .

y ∈ {j|j = 1, 2, . . . , m} z ∈ {k|k = 1, 2, . . . , n} where l, m, and n denote the numbers of elements in the corresponding linguistic value sets. All the fuzzy rules of GPS/GTA can be expressed by a chromosome

As the adjustment interval of b is the same as that of c, b may be larger than c during random generation of the initial population and genetic operations. When such a logic error occurs, it is corrected using the following equations:

b= g1 g2 , . . . , gr−1 gr gr+1 , . . . , gR , gr ∈ {k|k = 1, 2, . . . , n}

(23)

where R is the chromosome length R = l · m.

c=

(24)

The rth fuzzy rule can be inferred as follows: If x = r/m and y = r − m · (x − 1), then z = gr . B. Membership Function Encoding The trapezoidal shape has been adopted for all the membership functions in the model. A trapezoid can be uniquely defined by its four vertices, which are denoted by a, b, c, and d, as shown in Fig. 1.

(29)

min{b, c}, (b + c)/2,

p=0 p=1

(30)

max{b, c}, (b + c)/2,

p=0 p=1

(31)

where p is a random binary number of 0 or 1. C. Generation of Initial Population The first fuzzy rule chromosome was generated manually based on engineering experience, and the rest in the initial population were generated at random. However, the membership function chromosomes in the initial population were created in a different way. First, for each gene, N equidistant points (including two endpoints) were generated in its

182


adjustment interval. N 2 chromosomes were then constructed by using these points as follows: ax1i bx1j cx1(N −j) dx1(N −i) , . . . , axli bxlj cxl(N −j) dxl(N −i) ay1i by1i cy1(N −j) dy1(N −i) , . . . , aymi bymj cym(N −j) dym(N −i) az1i bz1j cz1(N −j) dz1(N −i) , . . . , azni bznj czn(N −j) dzn(N −i) , i, j = 0, 1, . . . , N − 1.

Fig. 2.

Phase comprising the cross-signalized intersection.

(32)

Second, s chromosomes were generated randomly to get N 2 + s candidates. Finally, the fitness of these candidates was evaluated via simulation, and s satisfactory individuals were selected as the initial population.

D. Crossover and Mutation The crossover and mutation operators for the fuzzy rule chromosomes are similar to that of simple GA [24]. A max–min-arithmeticalalgorithm-based [25] crossover has been used for the membership function chromosomes. Suppose that the two chromosomes selected for crossover in the tth generation are as follows:

Cvt = ctv1 , . . . , ctvk , . . . , ctvK

Fig. 3. Assumed AR curves of the intersection’s four directions. These directions are north (N), south (S), east (E), and west (W). TABLE I F UZZY RULES OF GPS AND GTA

t Cw = ctw1 , . . . , ctwk , . . . , ctwK . t+1 are generated through the following The offspring Cvt+1 and Cw crossover operation:

ct+1 vk =

ct+1 wk =

⎧ t ac + (1 − a)ctvk , 0 ≤ pv ≤ 1/4 ⎪ ⎨ wk t t acvk + (1 − a)cwk , min {ctwk , ctvk } , ⎪ ⎩ max {ctwk , ctvk } ,

1/4 < pv ≤ 1/2 1/2 < pv ≤ 3/4 3/4 < pv ≤ 1

acvk + (1 − a)cwk ,

1/4 < pw ≤ 1/2 1/2 < pw ≤ 3/4 3/4 < pw ≤ 1

E. Optimization Strategy (33)

⎧ t ac + (1 − a)ctvk , 0 ≤ pw ≤ 1/4 ⎪ ⎨ wk t t t , ctvk } , ⎪ ⎩ min {cwk t t

max {cwk , cvk } ,

(34) V. O PTIMIZATION AND S IMULATION

where a is a constant (0 < a < 1), and pv and pw are two random numbers within the range of [0, 1]. A nonuniform mutation algorithm [24] is applied for the mutation operation of the membership function chromosomes. For a chromosome of the tth generation Cvt , a random position cvk is selected for mutation. Let [clvk , crvk ] be the adjustment interval of cvk . The offspring Cvt+1 is generated by replacing cvk with cvk

cvk =

r cvk + Δ (t, ), cvk − cvk cvk + Δ t, cvk − clvk ,

b=0 b=1

(35)

where b is a random binary number of 0 or 1. Function Δ(·) is defined as follows:

Δ(t, y) = y 1 − r(1−t/T )

h

The optimization of the two controllers is independent of each other. The fuzzy rules and membership functions are optimized in an asynchronous and iterative way. The fuzzy rules are improved by fixing the membership functions first, and then, the membership functions are optimized based on the improved fuzzy rules.

(36)

where r is a random number in the range of [0, 1], T is the maximum number of generations, and h is a given constant. The value of Δ(t, y) is in the range of [0, y] and approaches 0 when the value of t increases. Hence, the mutation operator performs a wider search when t is small, and a local search as t increases.

A simulation program was developed using Visual C++, and a series of synchronous simulations was conducted to compare the performance of different control models.

A. Simulation Parameters The simulations were conducted on a cross-signalized intersection instance. The phase composing of the intersection is shown in Fig. 2. The assumed AR curves are shown in Fig. 3. The unit of time is seconds, and the time step is 1 s. The simulation time is 120 min (7200 s). The distance between the two detectors of each entry lane is 140 m, the average vehicle length is 5.5 m, and the minimum interval of the two adjacent vehicles is 1.5 m. Thus, the maximum detectable queue of each entry lane consists of 20 vehicles. The lower and upper limits of the green time are 15 and 60 s, respectively. The loss of green time and the interval of amber light are both 2 s. The average velocity, acceleration, and deceleration are 36 m/s, 2 m/s2 , and −2 m/s2 , respectively. It is assumed that the vehicles arrive on each entry lane according to the Poisson process. The left- and right-turn rates of each direction are 25%. The saturation flow rates of the straight-through, left-, and right-turn lanes of each direction are 0.6, 0.5, and 0.6 veh/s, respectively.


183

Fig. 4. Membership functions of GPS. The vertical axes indicate the membership degree (MD). (a) Unoptimized functions. (b) Optimized functions.

Fig. 6. Tendency of the N DV and AW T of all the entry lanes under the control of (a) the V-Model, (b) the P-Model, (c) the E-Model, and (d) the O-Model. Fig. 5. Membership functions of GTA. The vertical axes indicate the membership degree (MD). (a) Unoptimized functions. (b) Optimized functions.

B. Optimization Parameters and Results The genetic parameters for the two controllers are set as the same according to the experiments. The population size and the number of maximum generations are both 50. The crossover and mutation probability are 0.85 and 0.005. The constants a and h are 0.35 and 0.5, and N is set as five. The unoptimized and optimized fuzzy rule chromosomes for GPS and GTA are listed in Table I. The unoptimized and optimized membership functions for GPS and GTA are shown in Figs. 4 and 5, respectively.

lanes. It is shown that when the AR of all the directions is small, the N DV of the straight-through lanes never increases to the upper limit (20 vehicles), although sometimes, it may be larger than those of the other lanes. As the AR of all the directions increases gradually, the N DV of the straight-through lanes increases correspondingly. Durative oversaturation finally emerges, as shown in Fig. 6(a) and (b), and discontinuous oversaturation emerges, as shown in Fig. 6(c); however, oversaturation seldom emerges in Fig. 6(d). It is shown that the AW T and fluctuation of all the lanes are the smallest [see Fig. 6(d)], followed by Fig. 6(a) and (c). In Fig. 6(b), the performance of all the lanes is very satisfactory in the first half of the simulation, but that of the straight-through lanes worsens rapidly in the second half. It has been shown that both the traffic efficiency and fairness of the E-Model are better than those of the V-Model and the P-Model and that the performance of the O-Model is the best.

C. Simulation Process Fig. 6 shows the tendency of N DV and AW T for all the entry lanes of the intersection during the simulation process under the control of the V-Model, P-Model, E-Model, and O-Model. As the traffic volume of the straight-through lanes is much larger than that of the left- and right-turn lanes, the N DV and AW T of the straight-through lanes, such as lanes 2, 5, 8, and 11, are larger than those of the other

D. Simulation Results Fig. 7 shows the tendency of the AD and ADD of the intersection during the simulation process under the control of the V-Model, the P-Model, the E-Model, and the O-Model. The performances of the P-Model and the E-Model are similar in the first half of the simulation,

184


ACKNOWLEDGMENT The authors would like to thank the referees and the Associate Editor for their valuable comments that led to a significant improvement of this paper. R EFERENCES

Fig. 7. AD and ADD curves of the intersection. (a) AD curves. (b) ADD curves. TABLE II F INAL S IMULATION R ESULTS

but that of the P-Model worsens rapidly with the increase in traffic volume. The performance of the V-Model is always poor, unless traffic volume is very small, and that of the O-Model is always the best. This figure further confirms that the E-Model outperforms the V-Model and the P-Model with regard to both traffic efficiency and fairness and that the O-Model is the best. The final simulation results of AD, P SV , ADD, and N P V are listed in Table II. It has been shown that all performance indices are improved by the E-Model when compared with the V-Model and the P-Model, and they are further improved by the O-Model. VI. C ONCLUSION Human beings are an important part of urban road traffic systems. Hence, the control system of signalized intersections should take both traffic efficiency and traffic fairness into consideration. To do this, a two-stage fuzzy logic control model for isolated signalized intersections has been proposed. An offline GA has been designed to optimize the fuzzy rules and membership functions of the model in order to improve its performance. The simulation results show that the proposed model outperforms the V-Model and the P-Model with regard to both traffic efficiency and traffic fairness. Its performance could be further improved after the fuzzy rules and membership functions are optimized by using GA. Therefore, the model is effective and can be regarded as a suggested scheme for the control of signalized intersections.

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