The average control delay (sec/vehicle) has been applied in many pieces of research to estimate the Level of Service (LOS) and then determine the traffic situation of a signalized intersection. As discussed above, the most popular model is Webster’s formula for a minimum average delay as well as optimal cycle length. However, in the case of an over-saturated condition or a high degree of saturation flow rate, the Webster’s formula becomes unreasonable. After comparison and experimental test, Francois Dion et al. [14] and Ahmed Y. Zakariya et al. [15] suggested another method, that should be generated for a wide range estimation, is the average control model for each lane group in Highway Capacity Manual (HCM). The average control delay could be calculated by the rate between total control delay per vehicle and the sum of vehicular traffic.

(1)

Where, AD_I is the average vehicle delay of the complex intersection includes n phases and m lane groups (lg), t_ij,lg is the effective green time for phase i and lane group j, C is the cycle length, X_ij,lg is the v/c ratio (the ratio of traffic flow to the capacity of lane group j in phase i), T is analysis period which equals 15 minutes or 0.25 hour in this case study, k is an incremental delay factor, I is the upstream filtering adjustment factor which equals 1.0 for particular isolated intersection, v_ij is the traffic flow rate of lane group j for phase i.

The queue length at the isolated intersection can be determined by the subtraction of departing vehicles from arriving vehicles at the intersection [16]. While Punyaanek Srisurin et al. [17] and Akçelik, Rahmi [18] suggested that the queuing length should be specified by multiplying the corresponding idle time (corresponding red time) by the rate of arrival of the vehicle at the isolated intersection. Typically, the maximum throughput vehicle in rush hour is extremely important based on the arrival rate of vehicles at the intersection from traffic flow data. Therefore, we suppose the average queue length per cycle calculated by equation (2).

(2)

Where, AQ_I is the average queue length in cycle traffic signal timing at the intersection, AV_ij is arrival vehicle rate in the rush hour of lane group j for phase i, R_ij is the corresponding idle time (equals total effective green time in other phases), Q_ij is the initial queue length of lane group j for phase i.

The information of minimum cycle length could be directly found using a previous study [13]. The lowest suggested value of required cycle length for the phasing diagram and the lane group volumes of an intersection have been given by:

(4)

Where, L is the total lost time for the cycle in seconds at the intersection, is the sum of the critical lane group flow ratios calculated by traffic volume (v) and saturation flow rate (s), X_c is critical v/c ratio for the intersection that is defined by the traffic volume (v) and traffic capacity (c) at the intersection. In the case of designing an intersection operation at its full capacity, the value of X_c=1 should be used. The same idea has been found in a previous study [11] and Ahmeh Y Zakaria et al. [15] recommended the minimum value of cycle length C_min as a duration time that provides enough time to the arriving vehicles to pass through the intersection in the same cycle length by the following equation:

(5)

Furthermore, Ahmeh Y Zakaria et al. [15] mentioned that the maximum value of cycle length C_max depends on the empirical analysis and decision-making experiments. The long cycle length includes a long red time interval leading to a long time delay at the intersection. Normally, the value of 120 sec should be set for the upper limit of cycle length. In some exceptional cases, the value should be considered by cycle lengths in excess of 3 minutes (equals 180 seconds) [13], for example, the intersection in the arterial road, complex urban intersection, or oversaturated traffic flow condition.

(6)

(7)

The appropriate constraints enable the fitness function to escape the local optimum values and approach the global optimal value. Several researchers have attempted to assume the limitation of effective green time based on professional skill, however, with a lack of theoretical basis [2, 4-6, 20], we suggest the use of minimum effective green time in [12] that meets the requirement for safe pedestrian crossings.

(8)

(9)

Where, D is the distance of the crosswalk, S_p is the walking speed of pedestrians, W_E is the width of the crosswalk, N_ped is the number of pedestrians crossing during the green interval.

For traffic control system at the intersection with more than two phases, besides the timing requirements for safe pedestrian crossings, Hao Wang et al. [21] indicated that the reasonable traffic signal timing scheme has to ensure the saturation of another phase which is not big to restrict the traffic jam happening in coordinated traffic control signalized system or traffic network. Based on Hao Wang's model, we supposed the limitation for the other effective green times at the intersection by the following expression:

(10)

According to equation (8), the other minimum effective green times t_ij,min and maximum value of effective green times t_ij,max are constructed.

The relationship between effective green time and cycle length could be found as described in a study [11], by subtracting lost time per cycle from cycle length

(11)

To apply the constraints of allocating green times in GA’s optimal traffic signal timing process, these constraints might be known as linear equalities [13]. The green-time allocations for the phases of the cycle at the intersection are defined by:

(12)

Where, X_ij is v/c ratio for lane group j in phase i, and C is as previously defined.

The fitness function that guided the optimum process has been generated by two objective functions, including average control function and queue length function, equations (1) and (2), respectively. The methodology of this study is to handle the optimal traffic signal timing for a complex intersection with four phases and four approaches. According to the process in section 2.1, real data of traffic pattern demand, and assumed values of some constant indicators discussed above, the first objective function has been generated by five variables, inclu-ding effective green times (t₁, t₂, t₃, t₄) and cycle length (C).

As stated in section 2.1.2., the queuing length could be calculated by equation (2) for each approach by multiplying the rate of the arriving vehicle by the corresponding effective red time for each movement, for example, effective green time t₁ has corresponding idle time is (t₂ +t₃+t₄). Thus, the queue length at the West approach and the East approach is directly defined by the following equation:

(14)

Where, TET and TER are the average arrival rates of Taiwan Blvd's arriving vehicles on the Eastbound (EB) having through-movements and taking right-turns, respectively. TWT and TWR are the average arrival rates of Taiwan Blvd's arriving vehicles on the Westbound (WB) having through-movements and taking right-turns, respectively.

In other movements, the queue lengths have been detected by the same procedure. Hence, the total queue length of Taiwan Blvd- Huichung Rd intersection has been established by equation (2). Notice that we assumed the arrival rate of vehicles is basically stable as well as ignoring the initial queue length of each movement (Q_ij,int= 0) during the entire empirical analysis.

As reported in section 2.2., the constraints of effective green time and cycle length have been presented. While the minimum value of cycle length is calculated by equation (4), the maximum value of cycle length assumed 180 seconds for the complex intersection in an urban area and the total loss time equals 16 seconds per cycle (sum of start-up and clearance delays equal 4 seconds per phase) [12]. The empirical analysis concentrated on the lane group traffic volume for the large-scale and complex intersection by some advantages of significant decreasing queuing length and intersection delay [34].

Furthermore, in agreement with the safe pedestrian crossing issue, two minimum values of effective green time have been defined by the mentioned formulae (8) and (9). The widths of Taiwan Blvd and Huichung Rd are measured by the traffic geometry of this intersection. The distances of the crosswalk are typically designed 4 meters for every large- scale intersection in Taichung city, the number of pedestrians who cross this intersection is collected by video detector data and assumed that the average walking speed of pedestrians is 4 (ft/s) [11]. Other lowest value of effective green times and maximum value of effective green times have been defined by formula (10).

In this manner, the lowest and highest limitation of effective green times and cycle length (t₁, t₂, t₃, t₄, C) have been calculated (35 ≤ t₁ ≤ 88, 11 ≤ t₂ ≤ 131, 44 ≤ t₃ ≤ 119, 05 ≤ t₄ ≤152, 84 ≤ C ≤ 180). Thus, the lower bound (lb) and upper bound (ub) of variable specified as vectors:

(15)

According to formula (11), the linear equality constraint referring to the relationship between effective green time and cycle length has defined:

(16)

Equivalent to:

(17)

Where, total loss time is as previously defined (L= 16 seconds).

As claimed by formula (12), the cycle length will be allocated by effective green time based on traffic flow data. In this study, we proposed the effective green-times allocation to the high average rate of arriving vehicle approaches, for example, the major movements of vehicles are the phase T1 and phase T3 (through movements and right-turns) at Taiwan Blvd-Huichung Rd intersection. Therefore, the linear equality constraint referring to allocated green times have been generated:

(18)

(19)

Equivalent to:

(20)

(21)

The linear equality constraints could be described by the following equation:

(22)

Where:

(23)

(24)

(25)

The Genetic Algorithm (GA) is stochastic by the operator’s factors (mutation, selection, and crossover) that can make random solutions. Thus, running the GA solution can have different results. The multiple objective GA and suitable constraint conditions take advantage of the limitation values of fitness solutions in Pareto front. Nevertheless, the true Pareto frontier might be found by the appropriate genetic operator and population size. As analyzed in section 2.3, in traffic signal optimization, the crossover operator can be chosen in a range of 0.5 to 0.95, the mutation rate is from 0.001 to 0.5, associated with a population size between 20 and 160 for 60 to 200 generations. Furthermore, the adaptive GA benefits GA's effectiveness and the correlation between mutation probability and crossover probability is not necessary, as mentioned above.

Thus, the crossover operator rate could be adopted from 0.5 to 0.95 to retrieve optimistic solutions and the stable value of the mutation rate (equals 0.001) should be used to have better GA performance. The most widely used population size is 100 in many empirical analyses, as mentioned in Table 1 (reviewing suggested genetic operators). In the era of cutting- edge technology and computers, the computing process is significantly faster. Therefore, the number of generations of more than 200 could be set for the entire process. Generally, we supposed the performed MOGA' procedure as the following options:

1. Set the mutation rate as 0.001 with an adapted mutation type.
2. Establish the population size as 100, generation as 200, and the selection operator as type selection's tournament.
3. Check empirical results by spread indicator (13) with different values of crossover probability from 0.5 to 0.95.

Overall, the subprogram has been written based on the analyzed model by Matlab programming to handle the real traffic flow at Taiwan Blvd- Huichung Rd intersection.

The use of SOGA has been tested by multi-loop calculations (n times) to capture the mean values and Standard Deviations (SD) of x_k factors, including effective green times, cycle times, and values of the fitness function by the followed expression.

(26)

(27)

As noted above, most of the suggested models in the literature reviews typically assumed the constraint functions for hypothetical models with a lack of theoretical bases. Basically, those models followed the formulae from (3) to (11), and not included the green time allocation condition (formula (12) to identify the constraints and limit the values of infeasible solutions. The application of those assumptions has been executed in this section using multi-runs n=30 to obtain the Table 4.

The outcomes in the table above showed that although the values of the queue length and average delay significantly improved and the approximate values of variables T1, T2, T3, T4 are 39s (or 40s), 11s, 44s, 5s, respectively. However, the above values of effective green times indicated an imbalance when allocating cycle time to phase T1 (major arterial road) with main traffic. Particularly, the duration time of phase T1= 39s (or 40s) was less than the duration time of phase T3 = 44s (at the minor road). Nevertheless, the SD values partially reflected the appropriate accuracy of this method after multi-runs. It means that the supposed model was stuck at local optimization values when only using the mentioned constraints following formulae from (3) to (11). Besides, the test results by traffic simulation software showed that the T1 value was unreasonable, resulting in an increase in average delay at the other adjacent intersections in Taiwan Boulevard. Therefore, it is necessary to re-check the constraints to improve the accuracy of those hypothetical models (Table 4).

Another comparison has been made between the use of SOGA and MOGA when utilizing constraints of allocated green times to minimize the time delay, and the queuing length for the main traffic flows in major and minor roads. The results performed that the application of SOGA is likely to be possible with suitable constraints demonstrated by improved values of SD and green time allocation to the major road. However, the outcomes ​​of the MOGA are more convincing in optimizing traffic signals at a complex intersection. In particular, SOGA computation took a longer time by multi-loop calculation to catch the globally optimum values, while MOGA might obtain results via Pareto Front with appropriate constraint functions. The enhanced SD and fitness values (average delay, queue length) demonstrated the use of MOGA to be better than the use of SOGA in the practical analysis at Taiwan Blvd- Huichung Rd intersection (Table 5).

Webster’s formula is a well-known method to optimize the traffic control system for the timing plan at the intersection by the following equation.

(28)

Where, C_opt is the optimization value of cycle length, L is as previously defined, ∑Y_ci is the total critical lane flow ratio.

Then, the effective green time (T_i) allocation for each phase calculated by the below equation.

(29)

The empirical result Table 6 showed that the optimal cycle length calculated by Webster’s formula (C_opt= 153 seconds) is less than the MOGA’s optimistic cycle length (C=162 seconds).

In spite of this, the STSTP demonstrated better effectiveness of effective green time allocation. Precisely, the Webster method just only considered the critical group traffic flow, ignored the time allocation for pedestrian safety and coordinated traffic network issue that suggested by Hao Wang's model [21]. The T4’s duration (T4= 4 seconds) is less than the minimum value of effective green time (T4 = 5 seconds) of the Northbound’s left turns vehicles (NB-L) based on the traffic network coordination problem defined by formula (10). Furthermore, the allocated T3’s duration of the Webster model of through movements on the Southbound (SB-T) is too long (T3 = 55 seconds) for a minor road as Huichung Rd.

In addition, the Webster model normally used the critical lane traffic flow to detect the traffic signal timing optimization result, while this study utilized the critical traffic flow for lane group as a basic unit to define the optimal traffic-timing plan. The use of traffic flow of the lane group at the intersection demonstrated efficiencies in various research, for instance, Chen Xiaoming et al. [35] suggested using lane group as the basic unit to analyze the signalized intersection's reliable capacity and/or DingXin Cheng et al. [36] regenerated the Webster's optimization cycle length based on lane group traffic flow and the average delay in highway capacity manual 2000 (HCM 2000).

The traffic signal timing plan for the intersection is defined by the following formula:

(30)

Where, G_i is the actual green time for phase i, Y_i, Ar_i and T_Li are the yellow time, all red time, and total loss time for phase i, respectively.

The current traffic signal timing plan at Taiwan Blvd - Huichung Rd intersection has been designed in Fig. (6). According to the above equation, the revised signal timing plan has been defined by Fig. (9) assuming that the amber time, all-red time, and turning diagram followed the previous signal timing design.

The compared traffic simulation models have been generated by the SUMO software and the Synchro software for both the Existing Traffic Simulation Model (ETSM) and the Suggested Traffic Simulation Model (STSM) from traffic geometry and current traffic patterns demand.

The outcomes of SUMO software clearly described the traffic situation for isolated intersection by various per-formance indexes. Furthermore, the Synchro software could investigate the effects of the suggested traffic signal timing plan on adjacent intersections of Taiwan Blvd- Huichung Rd intersection in a generated traffic network of Taichung city, Taiwan.

The comparison between the ETSM and the STSM during rush hour from 7 am to 8 am showed that the total max jam length of Taiwan Blvd-Huichung Rd intersection reduced 1611 meters. Additionally, the observed total time loss was 3272.8 seconds shorter between the ETSM and the STSM. The above results were recorded by the lane-area detector (E2) tool in SUMO software.

Fig. (9). Suggested traffic signal timing plan.

Fig. (10). Traffic simulation model of Taiwan Blvd- Huichung Rd intersection by the Synchro software.

The exported results of Synchro software for the STSM showed that the adjacent intersection on the Southbound (the North approach) has change LOS (Level Of Service) from LOS C to LOS B by changing T3's duration that provided much time than the previous model for arriving vehicle passed through the Taiwan Blvd- Huichung Rd intersection. The LOS of the other adjacent intersections is still stable (Fig. 10).