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Abstract

A dynamic multi-objective genetic algorithm based on partial least squares prediction model (DNSGA-II-PLS) is presented in this paper to solve the mix traffic flow multi-objective timing optimization problem with time-varying traffic demand. Take motor vehicle delay, non-motor vehicle delay, and pedestrian delay as objectives to solve the problem. Make comparison with three improved dynamic multi-objective genetic algorithms based on prediction strategy: dynamic multi-objective evolutionary algorithm based on simple prediction (DNSGA-II-PREM), autonomous regression dynamic multi-objective evolutionary algorithm (DNSGA-II-AR), and mutation-based dynamic multi-objective evolutionary algorithm (DNSGA-II-MUT) under four kinds of test functions. The results show that compared with the other three algorithms, the DNSGA-II-PLS algorithm proposed in this paper shows better performance in convergence and distribution, and this algorithm has less complexity. Finally, taking the intersections of Taiping North Road and Zhujiang Road in Nanjing as the research object, the performance of the algorithm is tested under the simulation environment. The results show that compared with the DNSGA-II–AR, which has best performance among three compared algorithms, the proposed DNSGA-II-PLS algorithm can effectively reduce the delay of motor vehicles by 6.7%, non-motor vehicles by –2.8%, and pedestrian waiting time by 20.5%.

Keywords

Dynamic multi-objective optimization mixed traffic flow least squares prediction single intersection time-varying traffic demand

Introduction

In China, the mixed traffic flow formed by motor vehicles, non-motor vehicles, and pedestrians is one of the main problems leading to serious traffic jam at the intersection. In addition, various kinds of complicated and unexpected factors under mixed traffic flow environment may cause serious interference to drivers and affect their driving behaviors. This may easily lead to errors of driver’s judgments and operation and result in traffic accidents.¹ Therefore, to solve the traffic congestion problem effectively in the cities of China, there is great practical significance to establish an optimal control method for traffic signal timing plan which is suitable for real-time control in China.

At present, among the researches on the optimization of mixed traffic flow signal timing around the world, one method is to consider a single optimization performance objective. For example, Liu et al.² constructed the optimal model of dynamic linear programing under the oversaturated condition with the objective of maximizing the capacity of intersections. Based on the arriving principle of motor vehicles flow between adjacent intersections, Wey³ chooses the total vehicle delay of the flow in the control period as the goal to establish the control model and applies the improved simplex method to solve the problem. Another approach is to synthetically consider a number of performance indicators, and weighting sum method is used to form a general goal. Its essence is to transform the multi-objective optimization problems to single-objective optimization problems to solve. For example, on the basis of the delay model proposed by Akçelik R and Rouphail NM,⁴ the delay model proposed by Wu⁵ chooses three indicators as objective function, which is to maximize traffic capacity of signalized intersection, minimize average vehicle delay, and minimize the number of stopping at intersections, and the stability of traffic capacity is taken as the second objective function. Finally, it is converted into a total objective function by means of linear weighting; Zhang et al.⁶ divide the objective function into two layers: the first one selects three performance indicators: average delay, parking number, and traffic capacity to optimize the traffic efficiency at the intersection, which is defined as the objective function of efficiency control. The second layer selects the standard deviation of vehicle delay to improve the stability of signal control, which is defined as the objective function of robust control. And then weighted sum of two objective functions is weighted by the weight coefficient variation method to construct the objective function. Although it is easier to solve the problem by transforming multi-objective to single-objective, it still belongs to the single-objective optimization model. From the optimization theory point of view, it cannot achieve the optimal or the most satisfactory multiple control objectives at the same time, and cannot analyze the relevance, validity among control objectives, and the sensitivity of control objectives against control parameters.⁷

In addition, in the existing researches, the signal timing scheme is rarely designed from the perspective of dynamic model and dynamic algorithm to adapt to the dynamic and changing traffic demands. Ma et al.⁸ comprehensively analyze the different traffic signal control performance indicators, taking into account the time efficiency of motor vehicles, pedestrian time efficiency, and environmental benefits, to build multi-objective signal timing optimization model, using multi-objective continuous ant colony optimization algorithm to get the optimization solution of timing plan. The results show that the proposed model can provide a number of different timing characteristics of the timing period to meet the needs of different traffic conditions. Li⁹ establishes a signal timing optimization model at intersections with the consideration of both pedestrian and vehicle benefits and uses the model to analyze the change rules of vehicle capacity, pedestrian, and vehicle delays under different mixed traffic conditions. Besides, it studies the standards of pedestrian signal phase setting, which provides a theoretical basis for reasonably establishing the signal timing model of intersections under various traffic conditions. Xiao et al.¹⁰ aim at the mixed traffic flow at the intersection, propose a hybrid traffic flow optimization control method based on high-dimensional multi-objective evolutionary algorithm, which provides five optimal control indicators to enhance the intersections signal control efficiency. Chen et al.¹¹ aim at optimizing the signal timing plan of a single intersection under mixed traffic conditions, constructs a multi-objective signal timing model, and designs a multi-objective hyper-volume evaluation algorithm to obtain a candidate timing scheme set, finally, using the minimum subjective and objective information deviation method to determine the best optimal timing scheme. Although all these studies have made a certain contribution, they all use static multi-objective optimization algorithms. Compared with dynamic multi-objective optimization algorithms, static multi-objective optimization algorithms are difficult to capture real-time circumstance under the dynamic traffic demand changing environment and respond promptly. Considering the fact that most of the researches on signal timing plan in mixed traffic flow are mostly focused on fixed-time control, and less consideration is given to the real-time changes of traffic flow behavior, in response to this problem, this paper presents a real-time optimization control algorithm for signal timing plan, through real-time monitoring of traffic flow changes, and using dynamic multi-objective optimization algorithm for optimizing intersection timing plan.

In solving multi-objective optimization problems, the non-dominated sorting algorithm (NSGA-II) proposed by Deb et al.¹² has been widely used due to the fact that it can reduce the complexity of non-inferior sequencing genetic algorithms and has the advantages of fast running speed and good convergence of solution sets and other advantages. In order to improve the passing efficiency of the single intersection and solve the serious congestion problem on the upstream intersection which caused by the queue vehicles overflowed from the oversaturated intersection, An et al.¹³ uses NSGA-II algorithm to optimize the timing plan for single oversaturated intersections by solving the multi-objective optimization model. Compared with the timing scheme obtained by the traditional F-B method (the theoretical method invented by British scholar F Webster-B Cobber), the multi-objective optimization model can be used to reduce the queuing length and delay time of oversaturated intersection and to reduce the overflow rate. Li¹⁴ proposes a traffic signal control optimization algorithm based on NSGA-II in order to improve the efficiency of traffic signal control at oversaturated intersections, which shows good performance.

Based on the multi-objective evolutionary algorithm NSGA-II, this paper constructs the multi-objective dynamic model of single-intersection under mixed traffic environment with the goal of vehicle delay, non-motor vehicle delay, and traffic capacity at intersections. Aiming at presenting dynamic multi-objective evolutionary algorithm which cannot adapt to environment well and cannot convergence quickly, dynamic multi-objective evolutionary algorithm based on partial least squares prediction (DNSGA-II-PLS) is proposed by this paper. First, based on the classical test examples, the feasibility of the dynamic benchmarking algorithm is measured. Then, the DNSGA-II-PLS algorithm is used to get the optimized solution of the dynamic multi-objective timing model and obtain the candidate timing scheme sets under different time windows, and the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) decision-making method is used to determine the final timing scheme. Then this paper tests the practicality of the DNSGA-II-PLS algorithm under the VISSIM and MATLAB integrated simulation platform.

Signal timing model of isolated intersection

Selection of performance indicators

Vehicle delay

The calculation of vehicle delays at intersections is currently widely using a mathematical analysis model in the Road Traffic Capacity Manual HCM2000.¹⁵ However, due to the special traffic characteristics of the mixed traffic flow of urban road signal intersections in China, there are bicycles and pedestrians in all directions, not just motor vehicles. According to statistics, motor vehicle traffic, non-motorized traffic, and pedestrian traffic account for one-third of urban traffic in China.¹⁶ Therefore, the mixed traffic flows interact with each other, resulting in more traffic conflict points.¹⁰ HCM2000 is based on a series of assumptions, and the mathematical model is used to calculate the delay, which is difficult to fully reflect the actual situation of China’s mixed traffic flow.

In the calculation method of intersection delay commonly used in China, it is generally divided into two steps. In the first step, the average signal control delay for each vehicle on each entrance lane is estimated separately. It is the weighted average of the vehicle delays at the intersection due to signal control in the entrance lane. In the second step, according to the average signal control delay of each vehicle on the entrance road, the average signal control delay per vehicle of intersection is calculated. It is the weighted average of signal control delays of the average per vehicle of each entrance.¹⁰ Every vehicle delay consists of the following three parts

d = d_{1} + d_{2} + d_{3}

(1)

where

d

is the average signal control delay per vehicle,

d_{1}

is the signal control delay assuming vehicles arrivals uniformly,

d_{2}

is the incremental delay to account for the effect of random arrivals and over-saturation queues, and

d_{3}

is the initial queue incremental delay, which accounts for the incremental delay caused by all accumulated vehicles in previous analysis period to initial queue at the start of analysis period. The calculation formula is as formulas (2) to (10)¹⁰

$d_{1}$ can be expressed by formula (2)

d_{1} = d_{s} \frac{t_{u}}{T} + f_{a} d_{u} \frac{T - t_{u}}{T}

(2)

where

d_{s}

is saturation delay, which can be expressed by formula (3)

d_{s} = 0.5 C (1 - λ)

(3)

where

d_{u}

is unsaturated delay, which can be expressed by formula (4)

d_{u} = 0.5 C \frac{{(1 - λ)}^{2}}{1 - \min [1, X] λ}

(4)

where

C

is the cycle length;

λ

is the green time ratio,

X

is the ratio of saturation for intersection;

T

is the duration of analysis period (h);

t_{u}

is the duration of the accumulated vehicle of

T

, and

t_{u}

can be expressed by formula (5)

t_{u} = \min [T, \frac{Q_{b}}{C_{p} (1 - \min [1, X])}]

(5)

where

Q_{b}

is the initial accumulated queue at the start of period

T

(pcu),

C_{p}

is the lane group capacity,

f_{a}

is the supplemental adjustment factor for traffic flow arriving during green light period, which can be expressed by formula (6)

f_{a} = \frac{1 - P}{1 - λ}

(6)

where

P

is the proportion of vehicles arriving on green light period to all period.

$d_{2}$ can be expressed by formula (7)

d_{2} = 900 [(X - 1) + \sqrt{{(X - 1)}^{2} + T \frac{8 e X}{C_{P}}}]

(7)

where

e

is a single intersection signal control type correction factor. Under normal circumstances, the timing signal

e

= 0.5 will be taken; sensing signal

e

changes with the saturation and green light time, and the proposed average

e

can be seen as Table 1.¹⁷

Table 1.

Recommended value for single-intersection signal control type correction factor.

Saturation $X$	$e$	Average of $e$
$\leq$ 0.5	0.04–0.23	0.13
0.6	0.13–0.28	0.20
0.7	0.22–0.34	0.38
0.8	0.32–0.39	0.35
0.9	0.41–0.45	0.43
>1.0	0.5	0.5

$d_{3}$ can be expressed by formula (8)

d_{3} = {\begin{array}{l} 3600 \frac{Q_{b}}{C_{p}} - 1800 T (1 - \min [1, X]), t_{u} = T \\ 1800 \frac{Q_{b} t_{u}}{T C_{p}} t_{u} < T \end{array}

(8)

On the basis of calculating the delay $d$ of every lane group in the imported lane, the average signal control delay $d_{A}$ can be approximately calculated by weighting $d$ , and $d_{A}$ can be expressed by formula (9)

d_{A} = \sum_{i} d_{i} q_{i} / \sum_{i} q_{i}

(9)

where

d_{i}

is the average signal control delay for every car on the ith lane of the imported road, and

q_{i}

is traffic flow ratio of ith lane.

On the basis of calculating every delay of import lane $d_{A}$ , the average signal control delay $\bar{d}$ can be obtained by weighting average $d_{A}$ , and $\bar{d}$ can be expressed by formula (10)

\bar{d} = \sum_{A} d_{A} q_{A} / \sum_{A} q_{A}

(10)

where

q_{A}

is the traffic flow rate of 15 min in the peak of the imported lane. If the traffic flow rate of the imported road is time-varying, the average signal control delay of the motor vehicle at time t is

\bar{d} (t) = \sum_{A} d_{A} q_{A} (t) / \sum_{A} q_{A} (t)

(11)

Non-motorized vehicle delay

At present, non-motorized vehicles in China are mainly bicycles (including electric bicycles), so this paper takes the average delay of the bicycles as the non-motor vehicle delay. Due to the interaction effect between the intersections, the bicycles arriving at each intersection have a cyclical fluctuation. The strength of this cyclical fluctuation is closely related to the distance between intersections. The greater the distance is, the smaller the intensity is. Besides, cyclical fluctuations can affect bicycle parking delays. The arrival of the downstream intersection can be predicted by the bicycle reaching and departure rule at the upstream intersection. Let the bicycle arrival flow at the downstream intersection be $q (t)$ and let the difference with green time at the upstream intersection be $Δ T$ . When the green light at the downstream intersection lights up, the vehicle will leave the intersection with the saturated flow rate $q_{s}^{b}$ . Due to the influence of the starting wave, the vehicle will continue to stop waiting at the tail of the queue for passage. After a lapse of m seconds, the starting wave will start to catch up with the parking wave. The vehicles arriving during the green light period pass through the intersection continuously. According to the principle that at intersection the number of vehicles entering and leaving is same, there exists¹⁰

\int_{Δ T - t_{R}^{b}}^{Δ T} q (t) d t + \int_{Δ T}^{Δ T + m} q (t) d t = \int_{Δ T - t_{R}^{b}}^{Δ T + m} q (t) d t = m \cdot q_{s}^{b}

(12)

where

t_{R}^{b}

is the signal red light time. Using the above formula can obtain continuous parking time m during the green light time.

when $t \in (Δ T - t_{R}^{b}, Δ T + m)$ , the delay of $q (t) d t$ is

(Δ T - t) + \frac{1}{q_{s}^{b}} \int_{Δ T - t_{R}^{b}}^{t} q (t) d t

(13)

When $t \in (Δ T + m, Δ T + t_{G}^{b})$ , the delay of $q (t) d t$ is 0, $t_{G}^{b}$ is the green light time.

Let $q (t) = q_{s}^{b}$ (constant). In Chinese mixed traffic conditions, when the vehicle arrived at an even time, the average delay of bike can be expressed by formula (14)

{\bar{d}}_{b} = \frac{q_{s}^{b} \cdot {(t_{R}^{b})}^{2}}{2 (q_{s}^{b} - q_{b}) (t_{G}^{b} + t_{R}^{b})}

(14)

Meanwhile, if bicycle flow arriving at the intersection is constantly changing, the average delay of bike at t can be expressed as

{\bar{d}}_{b} (t) = \frac{q_{s}^{b} \cdot {(t_{R}^{b})}^{2}}{2 (q_{s}^{b} - q_{b} (t)) (t_{G}^{b} + t_{R}^{b})}

(15)

Pedestrian delay

This article assumes that the average pedestrian arrival time at the intersection is uniform, and the average waiting time of pedestrians is only related to the red light time of signal, it can be expressed as¹⁰

{\bar{T}}_{w} = \frac{1}{2} \cdot \frac{{(t_{R})}^{2}}{C}

(16)

where

t_{R}

is the red light time for pedestrians crossing street.

Dynamic multi-objective signal optimization model for mixed traffic flow at intersections

Objective function. The above three evaluation indicators, which are the average delay of vehicles, the average delay of non-motor vehicles, and the pedestrian waiting time, reflect the service status of intersections from different angles. The three indicators represent the benefits of motor vehicles, motor vehicle benefits, and pedestrian benefits in mixed traffic flow environment, respectively. Optimal signal timing scheme should make the three indicators as optimal as possible at the same time, and this problem can be essentially attributed to the dynamic multi-objective optimization problem. Currently, most domestic solution methods are to solve the single-objective problem by weighting multiple targets, but it is easy to cause the partial optimality of some indicators, and the optimal signal timing scheme cannot be achieved for all the targets at the same time. In order to solve this problem, this paper first designs a dynamic multi-objective optimization model for intersection signal control. The control variables include the green signal-to-signal ratio and the period of the intersections. The objective function takes the average delay of the vehicles at the intersection, the traffic capacity of vehicles, and the average delay of non-motor vehicles (three indicators) into account. The objective function is described as following:

\min {\begin{matrix} f_{1} (t) = \bar{d} (t) \\ f_{2} (t) = {\bar{d}}_{b} (t) \\ f_{3} (t) = {\bar{T}}_{w} (t) \end{matrix}

(17)

Constraint condition

The actual signal cycle must be satisfied with formula (18)

C_{\min} < C < C_{\max}

(18)

where

C_{\min}

is the minimum long cycle of intersection and

C_{\max}

is the maximum long cycle of intersection. This paper takes 100 s and 200 s, respectively.

2. The signal cycle should be more than the sum of the green time of every phase

\sum_{i = 1}^{n} g_{i} < C, i = 1, 2, \dots, n

(19)

3. The maximum saturation of every motor vehicle flow should be less than 1

X_{i} = \frac{q_{i}}{q_{si} λ_{i}} < 1

(20)

where $q_{i}$ is the ith vehicle actual flow rate, $q_{si}$ is the ith saturation flow rate, and $λ_{i}$ is the i-phase green ratio.

Prediction model

When the algorithm detects the change of the environment, how to use the historical information of the population to predict the position of the optimal Pareto optimal solution in the new environment is the focus of the dynamic multi-objective evolutionary algorithm based on the prediction model. The working principle of partial least squares prediction model used in this paper is as follows:

When changes in the external environment is detected, the best population $Q_{t}, \dots Q_{1}$ recorded in historical time can provide the optimal POS information for the time of $(t + 1)$ , which means that there exists a functional relationship between the optimal $POS (Q_{t +1})$ of $(t + 1)$ and the best population $Q_{t}, \dots Q_{1}$ of the historical time

Q_{t + 1} = F (Q_{1}, \dots Q_{t}, t)

(21)

To predict a new solution $x_{t + 1}$ at time $(t + 1)$ , we need to use the optimal solution $(x_{t}, x_{t - 1}, \dots, x_{t - K + 1})$ at $K$ moments before time $(t + 1)$ . Figure 1 shows the working mechanism of the prediction model when $K =3$ .

Figure 1.

Forecast model.

For an individual $x_{t}$ in the population $Q_{t}$ stored at time $t$ , the parent individual $x_{t -1}$ in the last time $t -1$ can be defined as the most similar individuals in the population $Q_{t -1}$ .

x_{t - 1} = \arg \min_{y \in Q_{t - 1}} | | y - x_{t} | |_{2}

(22)

After constructing a time series for each individual in the population and for individuals most similar to historical moments, this paper uses partial least squares regression to predict the individuals at the next moment.

Let the size of the population be $N$ and the dimension of the decision vector be $P$ . In order to improve the computational efficiency, the partial least squares prediction mechanism selects the $n (n \leq N)$ individual fitting prediction models according to non-inferior rankings every time. Taking $Q_{t}$ as the dependent variable and $Q_{t - 1}$ as the independent variable, the normalized array of observation variables of dependent variables and independent variables can be expressed as

Q_{t - 1} = [\begin{matrix} x_{t - 1}^{11} & \dots & x_{t - 1}^{1 p} \\ ⋮ & ⋮ \\ x_{t - 1}^{n 1} & \dots & x_{t - 1}^{np} \end{matrix}], Q_{t} = [\begin{matrix} x_{t}^{11} & \dots & x_{t}^{1 p} \\ ⋮ & ⋮ \\ x_{t}^{n 1} & \dots & x_{t}^{np} \end{matrix}]

(23)

Conducting partial least squares regression on $Q_{t - 1}$ and $Q_{t}$ yields the regression equation $PLS$ : $x_{t} = PLS (x_{t - 1})$ , where $x_{t} = (x_{t}^{1}, \dots, x_{t}^{p}), x_{t - 1} = (x_{t - 1}^{1}, \dots, x_{t - 1}^{p})$ , then the solution at time $t + 1$ can be predicted as $x_{t +1} = PLS (x_{t})$ based on the above PLS model.

Optimization algorithm

For dynamic evolutionary algorithms, the algorithm needs to know if there is any change in the external environment in time in order to implement corresponding response measures. The similarity test is the tool used to determine whether the environment changes. There are two kinds of dynamic environment detection methods: one is to select several individuals randomly from the parent population, and if any one of them has a change in the objective function or constraint, it is considered that the external environment has changed. However, this method is very sensitive to the noise in the environment when it evaluates the function, and can be easily misjudged. Therefore, this paper selects another similarity detection operator based on population statistics¹⁸

ε (t) = \frac{1}{n_{ε}} \sum_{j = 1}^{n_{ε}} ‖ \frac{f_{j} (X, t) - f_{j} (X, t - 1)}{R (t) - U (t)} ‖

(24)

where

m

is the number of objective functions,

R (t) = {(r_{1}, r_{2}, \dots, r_{m})}^{T}

U (t) = {(u_{1}, u_{2}, \dots, u_{m})}^{T}

are the maximum and minimum vectors, respectively, obtained from each objective function at time

t

r_{i}

is the maximum value obtained from the population by the ith objective function at time

t

, and

u_{i}

is the minimum value obtained from the population by the ith objective function at time

t

n_{g}

is the size of the target population to be detected and is adjusted according to the size of the population. When the calculated similarity value

n_{g}

is greater than the given threshold

δ

, that is,

ε (t) > δ

, the environment is considered to have changed significantly, then we run the partial least square prediction mechanism.

Dynamic multi-objective evolutionary algorithm based on partial least squares prediction model

This paper has improved the NSGA-II¹² algorithm, adding the above prediction model. The algorithm framework is as follows:

Step 1: Population size pop, evolutionary algebra gen = 1, environmental change parameters $n_{t}, τ_{t}$ , threshold $δ$ , initial time $t_{begin}$ , the maximum number of environmental changes $Tim e_{\max}$ ;

Step 2: Let time = 0, current time $t = t_{begin}$ . Initialize randomly to get the initial population $P_{gen}$ of size pop.

Step 3: The crossover and mutation operation of $P_{gen}$ was carried out to obtain the population $P_{cross, mutation}$ , and the non-inferior ranking of $P_{gen} \cup P_{offspring}$ was performed, and the sub-generation population $P_{offspring}$ with the size of pop was selected. Let $gen = gen + 1$ , $P_{gen} = P_{offspring}$ ;

Step 4: If $Time = Tim e_{\max}$ , output the optimal Pareto optimal solution $Q_{0}, Q_{1}, \dots, Q_{Tim e_{\max}}$ for each environment state, and end; otherwise, turn to Step 5.

Step 5: If gen is the integer multiple of $τ_{t}$ , we randomly select $n_{ε} =10$ individuals from the current $Q_{t}$ for the test population, to calculate the similarity detection operator $ε (t)$ value. If $ε (t) \geq δ$ , it is considered that the environment changed significantly. We record the optimal solution $Q_{t}$ in the current environment, and let $Time = Time + 1$ . If $t = t_{begin} + \frac{1}{n_{t}} ⌊ \frac{gen}{τ_{t}} ⌋$ , turn to Step 6; otherwise turn to Step 3.

Step 6: If $t < 2$ , then turn to Step 7; otherwise turn to Step 8;

Step 7: We mutate some individuals of $Q_{Time}$ , and make the modified $Q_{t}^{mutation}$ as the initial solution of the new moment, that is, to make $P_{gen} = Q_{Time}^{mutation}$ , then turn to Step 3.

Step 8: We use the partial least squares prediction mechanism to predict the best POS position $P^{Q_{Time + 1}} = PLS (Q_{Time})$ in the new time, which is used as the initial solution for the new moment solution, that is, $P_{gen} = P^{Q_{Time}}$ , then turn to Step 3.

Figure 2 shows the flow chart of dynamic multi-objective optimization algorithm (DNSGA-II-PLS) based on prediction mechanism proposed in this paper.

Figure 2.

Flowchart of DNSGA-II-PLS algorithm.

Experiments and results

Test function

In this section, two dynamic multi-objective optimization problems proposed by Farina et al.¹⁸, FDA3 and FDA1, are used to test the efficiency of the proposed algorithm. FDA1 belongs to the first-class question, and FDA3 belongs to the second-class question; and the DMOP2¹⁹ of dynamic multi-objective optimization problem (DMOP) series put forward by the Goh and Tan belongs to the third-class problem. In addition, to study the algorithms processing capacity of discontinuous problems, this paper makes the discontinuous frontier dynamic multi-objective optimization problem²⁰ put forward by Deb dynamic, in order to get discontinuous frontier dynamic multi-objective optimization problems (DCP).

The dynamic multi-objective optimization problem varies with time (environment), time parameter $t$ is defined as follows

t = \frac{1}{n_{t}} ⌊ \frac{τ}{τ_{t}} ⌋

(25)

n_{t}

and

τ_{t}

represent the intensity of environmental change and the frequency of environmental change, respectively. In this paper, to study the effects of various dynamic changes in uncertain environments, different combinations (

n_{t}

and

τ_{t}

) are given.

Performance metrics

In this paper, we present two performance metrics, which are Inverted Generation Distance ( $IGD$ ) and Spacing ( $Sp$ ).²¹ The definitions of these two performance metrics are as follows:

Coverage metric

PF represents the true Pareto front of uniform distribution. $A$ is the approximate Pareto front edge obtained by the algorithm, and the average distance of PF to $A$ is defined as follows

IGD (A, PF) = \frac{\sum_{v \in PF} d (v, A)}{| A |}

(26)

d (v, A)

means that the distance between the non-dominated individual of

A

and the individual in the nearest

POF

IG D_{i}

means the measure of time

t

, and the average

IGD

of the algorithm is defined as follows

\bar{IGD} = \frac{\sum_{i = 1}^{T_{\max}} IG D_{i}}{T_{\max}}

(27)

where

T_{\max}

is the maximum number of time step. The convergence metric measures the distance between the approximate Pareto front edge and the real Pareto front edge obtained by the algorithm. Therefore, the lower the

IGD

metric is, the better the convergence of the algorithm is.

2. Distribution metric

Diversity metrics $A$ is the approximate Pareto front edge obtained by the algorithm, and the $Sp$ definition is as follows

Sp = \sqrt{\frac{1}{| A | - 1} \sum_{i = 1}^{| A |} {(\bar{d} - d_{i})}^{2}}

(28)

where

d_{i} = \min {\sum_{m = 1}^{k} | f_{m} (a_{i}) - f_{m} (a_{j}) |}, a_{i}, a_{j} \in A

i, j = 1, 2, \dots, | A |

${\bar{d}}_{i}$ is the average of $d_{i}$ , and $K$ is the number of objective functions. $S p_{i}$ represents the measurement metric value of time $t$ , and the average $Sp$ definition in the whole operation time of the algorithm is as follows

\bar{Sp} = \frac{\sum_{i = 1}^{T_{\max}} S p_{i}}{T_{\max}}

(29)

$Sp$ is used to measure the diversity of the approximate Pareto front edge obtained by the algorithm. If the value of $Sp$ is 0, the Pareto front edge is evenly distributed in the target space.

Experimental comparison with other dynamic multi-objective algorithms

In order to verify the superiority of the proposed algorithm in this paper, three representative dynamic multi-objective optimization algorithms are selected for comparison. They are dynamic multi-objective evolutionary algorithm based on vector auto-regression prediction (DNSGA-II-AR),²¹ presented by Aimin Zhou et al., dynamic multi-objective evolutionary algorithm DNSGA-II-PREM²² based on simple prediction, and mutation-based DNSGA-II-MUT²³ of Deb et al. The contrast algorithm and the algorithm proposed in this paper are all based on the dynamic improvement of NSGA-II, but the difference is in the dynamic processing mechanism after the environment changes. Based on this, the comparison is more conducive to the analysis of the performance of the proposed partial least squares prediction mechanism.

Comparison of the distribution of Pareto solutions obtained by the algorithm

For these four algorithms, the parameters are set as follows: the population size pop is 100, the crossover probability $P_{c}$ = 0.9, and the mutation probability $P_{m}$ = 0.1. To study the effects of various dynamic changes in an uncertain environment, two different $(n_{t}, τ_{t})$ are given. We conduct 20 independent experiments on each combination of problems, tracking 50 environmental changes uniformly. Tables 3 and 4 give the statistical results of the convergence metric $IGD$ and the distribution metric $Sp$ , respectively.

As can be seen from Table 2, DNSGA-II-PLS obtains a good convergence compared with other algorithms. Table 3 also shows that DNSGA-II-PLS has good tracking performance without loss of distribution. At the same time, DNSGA-II-PLS also has some advantages in terms of stability. It also has superior performance for different testing problems even when optimizing discrete and dynamic multi-objective problems. In Table 2 and Table 3, the grey shade values are the better results obtained from different methods. It can be seen from Table 2 and Table 3.

Table 2.

Metric of $IGD$ .

			PLS	AR	PREM	MUT
FDA1	$(n_{t} = 5, τ_{t} = 15)$	Mean	0.036255	0.107558	0.698511	0.947356
		std	0.00584	0.116839	0.03345	0.125755
	$(n_{t} = 10, τ_{t} = 30)$	Mean	0.006114	0.019189	0.163323	0.175461
		std	0.000629	0.052026	0.00284	0.030412
DMOP2	$(n_{t} = 5, τ_{t} = 15)$	Mean	0.883632	0.976647	1.649786	1.381601
		std	0.032472	0.166097	0.035137	0.08407
	$(n_{t} = 10, τ_{t} = 30)$	Mean	0.025719	0.03026	0.211727	0.241482
		std	0.00166	0.066554	0.00247	0.032263
FDA3	$(n_{t} = 5, τ_{t} = 15)$	Mean	0.074431	0.081976	0.1078	0.118029
		std	0.008225	0.053054	0.010357	0.01139
	$(n_{t} = 10, τ_{t} = 30)$	Mean	0.068364	0.06597	0.13512	0.187501
		std	0.013813	0.029324	0.004038	0.144542
DCP	$(n_{t} = 5, τ_{t} = 15)$	Mean	0.253436	0.309539	0.476451	0.765179
		std	0.014597	0.026515	0.015771	0.026189
	$(n_{t} = 10, τ_{t} = 30)$	Mean	0.222113	0.241727	0.313682	0.519001
		std	0.019453	0.066554	0.009623	0.00641

Table 3.

Metric of $Sp$ .

			PLS	AR	PREM	MUT
FDA1	$(n_{t} = 5, τ_{t} = 15)$	Mean	0.020696	0.020331	0.035014	0.021922
		std	0.005615	0.00364	0.007977	0.004656
	$(n_{t} = 10, τ_{t} = 30)$	Mean	0.011611	0.024192	0.00348	0.029072
		std	0.005873	0.012656	7.92E-05	0.006964
DMOP2	$(n_{t} = 5, τ_{t} = 15)$	Mean	0.015321	0.019554	0.015929	0.016347
		std	0.00205	0.00323	0.002572	0.001647
	$(n_{t} = 10, τ_{t} = 30)$	Mean	0.027509	0.033599	0.034964	0.017998
		std	0.007151	0.005889	0.007686	0.003619
FDA3	$(n_{t} = 5, τ_{t} = 15)$	Mean	0.041759	0.060536	0.055094	0.043995
		std	0.007659	0.011613	0.00585	0.002099
	$(n_{t} = 10, τ_{t} = 30)$	Mean	0.050877	0.068659	0.061063	0.051523
		std	0.006634	0.012873	0.007082	0.009545
DCP	$(n_{t} = 5, τ_{t} = 15)$	Mean	0.009787	0.010493	0.010726	0.008539
		std	0.000947	0.003217	0.000947	0.000659
	$(n_{t} = 10, τ_{t} = 30)$	Mean	0.004257	0.005183	0.00845	0.006112
		std	0.000481	0.000878	0.000422	0.000209

FDA1: The optimal frontier $POF$ for FDA1 does not change over time (environment), $POF$ is $1 - \sqrt{f_{1}}$ for any environment, but $POS$ varies with the environment. DNSGA-II-MUT, DNSGA-II-PREM, and DNSGA-II-AR are not good at the best solution set’s tracking performance at $(n_{t}, τ_{t})$ of 5 and 15, but DNSGA-II-PLS can track the optimal solution set well. It’s $IGD$ is decreased by 95% relative to DNSGA-II-AR, decreased by 70% compared with DNSGA-II-PREM, and decreased by 96% compared with DNSGA-II-PREM, indicating that DNSGA-II-PLS can quickly respond to environmental changes and quickly converge to a small neighborhood of the optimal solution. It can be seen from the $Sp$ metric table that the distribution performance of the four algorithms is not much different, indicating that the proposed prediction mechanism does not reduce the distribution performance of the algorithm.

DMOP2: $POF$ , which is the optimal frontier of DMOP2, is $1 - f_{1}^{H (t)}$ , but the $POS$ is the same in any environment. It can be seen from Table $IGD$ that compared with the three other algorithms, the $IGD$ metric of DNSGA-II-PLS are reduced by 46%, 11%, and 74%, respectively, especially when the environment setting $(n_{t}, τ_{t})$ is (5,10). Figure 3 shows the comparison between $POF$ and the real $POF$ obtained from each algorithm when $(n_{t}, τ_{t})$ is (5,100). In order to visualize, this paper chooses the 50 solutions with the least crowding degree from the Pareto optimal solutions, and draws a graph. From the figure, it can be clearly seen that compared with other algorithms, the tracking performance of DNSGA-II-PLS is better, whose true $POF$ of the dissociation problem is more similar, and the solution distribution is relatively more uniform. The effect of DNSGA-II-AR is slightly inferior to that of DNSGA-II-PLS. DNSGA-II-MUT has the worst performance and cannot track the movement of $POF$ at most times.

FDA3: FDA3’s $POS$ changes based on the sinusoidal function, so the POS changes with the environment, $POF$ also changes with the environment, its value is $(1 + G (t)) \times (1 - \sqrt{f_{1} / (1 + G (t)})$ . From the $IGD$ Table, the effects of DNSGA-II-PLS and DNSGA-II-AR have no significant difference, but the result is significantly better than DNSGA-II-PREM and DNSGA-II-MUT, as shown in Figure 4.

Figure 3.

Four algorithms of Pareto solution set for DMOP2.

Figure 4.

Four algorithms of Pareto solution set for FDA3.

DCP: The DCP’s $POS$ varies with the environment. The $POF$ also varies with the environment, and it is discontinuous. And the $POF$ is located on $1 - f_{1}^{a} - f_{1} \cdot \sin (2 q π f_{1} t)$ . This test function allows you to test the algorithm’s ability to track discontinuous $POFs$ . Table $IGD$ shows DNSGA-II-PLS has better tracking capabilities and the $IGD$ value is less than the other three algorithms. Figure 5 shows the $POFs$ converged from the four algorithms when t = 0.4, 0.6, 0.8, and 1, respectively. In the figure, the curve segment is the true $POF$ of the problem and the dot is the $POF$ obtained from the algorithm’s evolution. Obviously, the DNSGA-PLS’s convergence effect is better.

Figure 5.

Four algorithms of Pareto solution set for DCP.

Based on the above analysis of the non-dominated results obtained from the four problems, we can clearly see that the quality of DNSGA-II-PLS solution is the best compared with the other three algorithms, and the uniformity is in no way inferior. Moreover, DNSGA-II-PLS converges rapidly in response to environmental changes under the conditions of more frequent and severe environmental changes.

Comparing the performance box diagram of the algorithm

Figures 6 and 7 are the $\bar{IGD}$ , $\bar{Sp}$ box figures’ comparing performance from 20 times operations of four algorithms that run independently, in terms of $\bar{IGD}$ metric, DNSGA-II-PLS’s performance on four test functions is all very good, which has significant performance advantages on the convergence accuracy, but DNSGA-II-AR is second only to DNSGA-II-PLS considering performance, DNSGA-II-MUT is the worst, and $\bar{IGD}$ is significantly higher than the other three algorithms. In terms of the $\bar{Sp}$ metric, the four algorithms are close to each other, and no algorithm is significantly better than other algorithms, indicating that the four solutions are evenly distributed.

Figure 6.

Average $IGD$ box diagram.

Figure 7.

Average $Sp$ box diagram.

Analysis of algorithm complexity

The complexity of the algorithm is an important criterion for measuring the practicality of the algorithm. The parameters that determine the complexity in DNSGA-II-PLS are: The target number of problems $M$ , the population size $pop$ , and the number of time steps $T_{\max}$ . According to Deb’s thesis analysis, as for a population of size $n$ , when the number of targets is $m$ , the non-dominated ordering’s computational complexity is $O (m \cdot n^{2})$ , the computational complexity of NSGA2 congestion distance’s estimation assignment is $O (m \cdot n \cdot \log n)$ , the computational complexity of congestion ordering is $O (n \cdot \log (n))$ , the complexity of crossover operator is $O (n)$ , and the computational complexity of mutation operator is $O (n)$ . Moreover, the computational complexity of partial least squares prediction mechanism is $O (m \cdot n^{2})$ , then the total computational complexity of the algorithm is

\begin{matrix} T_{\max} \times G_{\max} \times (O (m \cdot n^{2}) + O (m \cdot n \cdot \log n) \\ + O (n \cdot \log (n)) + O (n) + O (n) + O (m \cdot n^{2})) \end{matrix}

(30)

G_{\max}

is the maximum algebra of the algorithm. According to the calculation rule of the operator

O

, the above equation is simplified to

O (m \cdot n^{2})

, which shows that the computational complexity of the algorithm mainly concentrates on the non-dominated ranking and prediction mechanism.

The Figure 8 shows the time needed for four different algorithms running 500 generations on the same computer device. It can be seen that DNSGA-II-PLS (223.9 s) and DNSGA-II-AR (225.3 s) algorithms run faster than the other two algorithms, which is related to the complexity of the algorithm’s prediction mechanism. The fitting of the partial least squares regression model and the time series autoregressive model is a rather time-consuming process. However, when we contrast the $IGD$ tables, under the circumstance of DNSGA-II-PLS increasing by 2.7% versus DNSGA-II-PREM (218.2 s), the average $IGD$ metric decreased by 50%, and under the circumstance of DNSGA-II-MUT (214.1 s) increasing by 4.6%, the $IGD$ metric decreased by 75% on an average, but the time consumption and $IGD$ metric of DNSGA-II-PLS are smaller than that of DNSGA-II-AR.

Figure 8.

Average running time of four algorithms.

In a word, compared with the other three algorithms in terms of tracking performance, the DNSGA-PLS algorithm proposed in this paper performs better on the four types of testing problems described above. Meanwhile, it improves the convergence precision while has no loss of solution distribution. Although the amount of time spent increased slightly, the convergence accuracy improved greatly.

Case study and simulation

Basic information of single intersection

In order to test the intersection control method’s, which is based on partial least squares prediction of dynamic multi-objective evolutionary algorithm, signal control effect under the condition of mixed traffic flow, using the actual case of intersection in literature,¹⁰ that is, taking Nanjing Taiping North Road–Zhujiang Road’s junction’s intersection points’ intersection as the research object, the basic situation of the intersection is shown in Figure 9. The north–south intersection is Taiping North Road, and the east–west direction is Zhujiang Road. There are two entrances and two straight-ahead lanes in all directions, with one motor vehicle left-turn lane and right-turn lane, respectively, and one non-motorized vehicle inlet and outlet, respectively.

Figure 9.

Basic situation map of Taiping North Road–Zhujiang Road intersection in Nanjing.

The intersection is using five-phase signal timing control, the cycle duration is 136 s, and the yellow light phase of each phase is 3 s. The signal phase diagram is shown in Figure 10, regardless of the impact of the import gradient.

Figure 10.

The signal phase diagram.

According to the data in the literature, the traffic flow at the intersections (pcu/h) and the pcu data at the peak hours (17: 00–18: 00) are shown in Table 4, and Table 5 shows green time ratio of the traffic flow at the intersections.

Table 4.

Traffic flow of intersection at peak time.

	East		South		West		North
	Flow	Surplus vehicle	Flow	Surplus vehicle	Flow	Surplus vehicle	Flow	Surplus vehicle
vehicle
straight	717	2	881	2	817	2	856	2
Turn left	157	1	179	1	153	1	172	1
Turn right	592	1	218	1	204	1	252	1
Non-motor vehicle
straight	931		1045		848		1242
Turn left	464		515		397		505
Turn right	130		172		102		121
pedestrian	312		529		269		721

Table 5.

Green time ratio of the traffic flow at the intersections.

	East imports	South imports	West imports	North imports
Vehicle
Straight	$λ_{1}$	$λ_{3} + λ_{4}$	$λ_{1}$	$λ_{3} + λ_{4}$
Turn left	$λ_{2}$	$λ_{5}$	$λ_{2}$	$λ_{5}$
Turn right	$λ_{2} + λ_{3} + λ_{4} + λ_{5}$	$λ_{2} + λ_{4}$	$λ_{2} + λ_{4}$	$λ_{2} + λ_{4}$
Non-motor vehicle
Straight	$λ_{1}$	$λ_{3} + λ_{4}$	$λ_{1}$	$λ_{3} + λ_{4}$
Turn left	$λ_{2}$	$(λ_{5} \cdot C - 6) / C$	$λ_{2}$	$(λ_{5} \cdot C - 6) / C$
Turn right	1	1	1	1
Pedestrian	$λ_{1}$	$λ_{3} + λ_{4}$	$λ_{1}$	$λ_{3} + λ_{4}$

In order to realize the dynamic demand of road traffic, this paper uses the piecewise function to dynamically change the flow data at the intersection. The straight-ahead motor vehicle in the east is taken as an example to show the dynamic process. With half hour (0.5 h) as an optimization window, the traffic flow in each optimized window is p times of the peak traffic. If there are five optimization windows in this paper, this paper set the value of p as (0.85, 0.7, 1.15, 1.0.55), then the flow changes of straight-through vehicles in the east entrance are shown in Figure 11.

Figure 11.

Dynamic changes of traffic flow.

Analysis of algorithm operation results

Based on the selected performance indicators described in section “Selection of performance indicators” and the parameter settings which used to establish a three-objective optimization model for a single-intersection considering hybrid traffic flow environment described in section “Basic information of single intersection”, the dynamic multi-objective evolutionary algorithm based on partial least squares prediction model proposed in this paper are used to solve the multi-objective optimization problem, and the TOPSIS decision method are used to determine the optimal signal timing scheme. By optimizing the green signal-to-noise ratio and period of the five phases at the intersections, the average delay of the intersections, the average delay of non-motor vehicles, and the waiting time of the pedestrians can be optimized simultaneously. The population size of DNSGA-II-PLS is 100, and the evolutionary algebra for each optimization window is 100. The other parameters used in the proposed algorithm has the same value as the other three algorithms described in section “Experiments and results”.

As can be seen from Table 6, time window 3 shows that the delay of motor vehicles and non-motor vehicles in the optimized control scheme are significantly higher than those of other time windows. This is clearly illustrated in Figure 11, whose Pareto front is located at the upper right of the other time windows square. This is due to the time window p = 1.15, when the intersection is in a state of oversaturation, and at this time the geometry of the intersection cannot meet the traffic demand normally, resulting in prolonged delays of motor vehicles and non-motor vehicles.

Table 6.

Control results of intersection signal optimization.

Time window	$λ_{1}$	$λ_{2}$	$λ_{3}$	$λ_{4}$	$λ_{5}$	$C$	$\bar{d}$	${\bar{d}}_{b}$	${\bar{T}}_{w}$
1	0.305	0.181	0.112	0.213	0.187	108.210	35.500	26.588	24.632
2	0.291	0.152	0.101	0.283	0.161	140.505	38.740	30.800	28.754
3	0.323	0.152	0.125	0.238	0.162	130.954	153.711	44.758	43.015
4	0.314	0.163	0.102	0.235	0.184	126.885	54.211	31.318	30.265
5	0.255	0.157	0.103	0.347	0.138	152.846	37.261	31.377	29.435

Analysis of the relationship between optimization objectives

Figures 12 to 14, respectively, show comparisons between the optimization targets at the different time windows. We can see that the optimal allocation of time-set evaluation metric changes with the time window changes, suggesting that the proposed strategy of dynamically optimizing the timing is feasible. That is to say that the method can provide different timing schemes for different time windows to meet the control requirements of different traffic flows, thereby improving the service level of the intersection. Figure 12 shows that vehicle delays and pedestrian crossing times conflict with each other, with vehicle delays reduce increasing the waiting time for pedestrian crossings. Similarly, Figure 13 shows that reducing vehicle delays increases non-motor vehicle delays. Figure 14 shows that delay of non-motor vehicles and the waiting time for pedestrian crossing are a pair of goals of the same increase and decrease with no conflict between the two targets.

Figure 12.

Pareto frontier of motor vehicle delays and pedestrian crossing waiting time.

Figure 13.

Pareto frontier of motor vehicle delays and non-motor vehicle delays.

Figure 14.

Pareto frontier of non-motor vehicle delays and pedestrian crossing waiting time.

Decisions and contrast of timing scheme. It can be seen from the above analysis that the solution obtained by the multi-objective optimization algorithm is a set of non-inferior solutions. Therefore, when performing signal timing, the selection decision needs to be made from a number of feasible solutions. This paper uses interval TOPSIS decision-making method²⁴ for decision-making. The TOPSIS method is a common method for multi-objective decision-making. It is also called the ideal point method, which was first proposed by C. L. Hwang and K. Yoon in 1981, that the TOPSIS method’s satisfactory solution is closest to the ideal solution and far from the negative ideal solution. The comprehensive evaluation formula is as follows

C C_{i} = D_{i}^{-} / (D_{i}^{+} + D_{i}^{-})

(31)

where

D_{i}^{+} = \sqrt{\sum_{j = 1}^{n} {(Z_{ij} - Z_{j}^{+})}^{2}}, D_{i}^{-} = \sqrt{\sum_{j = 1}^{n} {(Z_{ij} - Z_{j}^{-})}^{2}}

The above are the candidate decision scheme $Z_{ij}$ ’s Euclidean distance of positive solution $Z_{j}^{+}$ and negative solution $Z_{j}^{-}$ , respectively.

Table 6 shows the results, which are made from the use of proposed algorithm, to solve the three-objective signal optimization control model of single-intersection in single-intersection under mixed traffic environment and then through TOPSIS decision-making, includes each parameter setting scheme ( $λ_{1}, λ_{2}, λ_{3}, λ_{4}, λ_{5}$ is green time ratio of 1-phase, 2-phase, 3-phase, 4-phase, and 5-phase, respectively. C is the cycle length) and corresponding control target value of each solution ( $\bar{d}$ : average vehicle delay; ${\bar{d}}_{b}$ : average non-motorized vehicle delay; and ${\bar{T}}_{w}$ : pedestrian delay).

In order to measure DNSGA-II-PLS and other existing intelligent dynamic optimization algorithms’ signal optimal control performance on the same intersection, we use the DNSGA-II-PLS to solve the signal optimization control model, and through TOPSIS decision to get the average of 10 independent operation results. Then we contrast the classical algorithm used in the section “Experiments and results”, and set the parameters as follows: Each algorithm population size is 100, and other parameters are the same as theirs in section “Experiments and results”. The experimental results are shown in Table 7.

Table 7.

Control results comparison crossing signal optimization.

	Comparison algorithm	$\bar{d}$	${\bar{d}}_{b}$	${\bar{T}}_{w}$
Time Window 1	DNSGA-II-PLS	35.5	26.588	25.084
	DNSGA-II-AR	45.783	36.384	34.978
	DNSGA-II-PREM	48.317	35.745	34.332
	DNSGA-II-MUT	50.963	35.139	33.720
Time Window 2	DNSGA-II-PLS	38.74	30.8	29.338
	DNSGA-II-AR	42.359	31.04	29.580
	DNSGA-II-PREM	44.02	31.296	29.839
	DNSGA-II-MUT	51.997	31.548	30.093
Time Window 3	DNSGA-II-PLS	153.711	44.758	43.435
	DNSGA-II-AR	166.024	43.72	42.387
	DNSGA-II-PREM	164.619	44.269	42.941
	DNSGA-II-MUT	166.672	43.641	42.307
Time Window 4	DNSGA-II-PLS	54.211	31.318	29.861
	DNSGA-II-AR	67.021	24.455	22.929
	DNSGA-II-PREM	63.353	24.629	23.105
	DNSGA-II-MUT	82.411	24.078	22.549
Time Window 5	DNSGA-II-PLS	37.261	31.377	29.921
	DNSGA-II-AR	37.787	30.746	29.283
	DNSGA-II-PREM	37.664	30.991	29.531
	DNSGA-II-MUT	38.046	30.702	29.239

DNSGA-II-PLS: dynamic multi-objective optimization algorithm; DNSGA-II-AR: Autonomous regression dynamic multi-objective evolutionary algorithm; DNSGA-II-PREM: dynamic multi-objective evolutionary algorithm; DNSGA-II-MUT: mutation-based dynamic multi-objective evolutionary algorithm.

It can be seen from the calculation results in Table 7 that the DNSGA-II-PLS algorithm has better performance metrics of $\bar{d}$ , ${\bar{d}}_{b}$ , and ${\bar{T}}_{w}$ traffic performance than those of other optimization algorithms. This indicates that compared with DNSGA-II-AR, DNSGA-II-PREM, and DNSGA-II-MUT, three kinds of dynamic algorithms, DNSGA-II-PLS proposed in this paper can to some extent improve the service level of urban road under the dynamic traffic demand, reduce the average delay of vehicles and the average delay of non-motor vehicles, improving the signal control efficiency in the intersection effectively. However, the above only validates the feasibility of the algorithm theoretically, and the simulation method is used to test the adaptability of the algorithm under actual road conditions below.

Simulation of results

First, we set up an isolated intersection traffic network in the mixed traffic environment under the VISSIM platform. As shown in Figure 15, we complete the traffic input of each inlet road. The dynamic traffic flow given in Figure 11 are input into the VISSIM environment.

Figure 15.

Simulation diagram of isolated intersection road network in VISSIM.

In order to express the simulation results clearly, we set the simulation duration to 2.5 h. When the traffic flow change value at the whole intersection is over 100 pcu/h, the traffic status is considered to be changed. At this time, the dynamic multi-objective optimization algorithm is used to optimize real-time optimal Time allocation program. In order to test the effectiveness of the proposed timing optimization based on dynamic multi-objective decision-making, we first compare the effectiveness of different algorithms. The parameters of each algorithm are set as follows: the population size is 100, the optimal algebra for each time window is 100, and other parameters are the same. Table 8 shows the comparison of the control effects of different algorithms. The evaluation metrics used are average vehicle delay, average motor vehicle stop delay, average non-motor vehicle delay, and average non-motor vehicle stop times of the entire simulation period. It can be seen from the table that the DNSGA-II-PLS method proposed in this paper is superior to other algorithms in all evaluation dimensions although the control effect between different algorithms do not differ much.In table 8, the grey shad values are the better results obtained from different methods. It can be seen from table 8.

Table 8.

Comparsion of the optimizaiton results for different algorithms.

	Vehicle Delay	Vehicle stop times	Non-motor vehicle Delay	Non-motor stop times	Pedestrian waiting time
DNSGA-II-PLS	29.52	0.85	9.82	0.47	11.35
DNSGA-II-AR	31.64	0.94	9.55	0.49	14.28
DNSGA-II-PREM	32.11	0.86	9.64	0.48	16.80
DNSGA-II-MUT	34.61	0.95	13.30	0.68	19.06
Dynamic control method	29.52	0.85	9.82	0.47	11.35
TRRL timing control	35.45	1.04	19.67	0.64	20.37

DNSGA-II-PLS: dynamic multi-objective optimization algorithm; DNSGA-II-AR: autonomous regression dynamic multi-objective evolutionary algorithm; DNSGA-II-PREM: dynamic multi-objective evolutionary algorithm; DNSGA-II-MUT: mutation-based dynamic multi-objective evolutionary algorithm; TRRL: Transport and Road Research Laboratory.

In addition, Table 8 shows the comparison between the dynamic control method and the timing control method based on the DNSGA-II-PLS algorithm. The control method of the timing signal is the Transport and Road Research Laboratory (TRRL) method.¹⁸ The TRRL method is used to set the intersections according to the initial state of the intersections Control parameters, and the value does not change with the traffic flow changes.

Figure 16 compares the average delay time and number of stops for motorized and non-motorized vehicles at two different timing methods. In Figure 16(a), the abscissa indicates a different timing method, the vertical axis represents the average vehicle delay time, and d units is seconds (s). Obviously under real-time control, vehicle delay time reduced greatly, reducing by 17% relative to the timing control delay. The reduction in non-motor vehicle delays by 50%, significantly reducing the time it took for vehicles to cross the intersection. In Figure 16(b), the abscissa indicates different timing methods and the ordinate indicates the average number of stops. We can see that the dynamic timing method proposed in this paper relative to the TRRL timing control method also significantly reduces the number of stops. Motor vehicle stops reduced by 20% and non-motorized parking reduced by 30%. In general, an isolated intersection dynamic timing method presented in this paper improves the service level of the intersection effectively.

Figure 16.

Comparison of the average delay time and number of stops for motorized and non-motorized vehicles. (a) Delay comparison chart and (b) Parking comparison chart.

Conclusion

Aiming at the traffic jam problem caused by mixed traffic flow at single intersection, this paper establishes a dynamic multi-objective optimization model based on the delay of vehicles, the delay of non-motor vehicles, and the service level of urban roads, and proposes the dynamic multi-objective genetic algorithm DNSGA-II-PLS, which is based on partial least squares prediction model to solve the objective function. In this paper, through the comparison with three improved multi-objective evolutionary algorithms (dynamic multi-objective evolutionary algorithm, DNSGA-II-PREM, based on simple prediction; dynamic multi-objective evolutionary algorithm, DNSGA-II-AR, based on vector auto-regression prediction; and evolutionary algorithm, DNSGA-II-MUT), the DNSGA-II-PLS presented in this paper can show better convergence and distribution with less computation time. According to the final simulation, the proposed DNSGA-II-PLS algorithm can effectively improve the service level of urban roads at the intersection of dynamic traffic demand and reduce the delay of motor vehicles and non-motor vehicles. However, the test problems used in this paper are simpler, and the dynamic multi-objective in reality is far more complicated than this. Whether DNSGA-II-PLS can handle the real problems in other fields is still outstanding, which is the content of the next step. In addition, the prediction model adopted by DNSGA-II-PLS is more complicated, resulting in an increase of the overall time-consuming algorithm. How to simplify the partial least squares prediction mechanism and reduce the difficulty of model fitting under the condition of ensuring the prediction accuracy need to be further studied.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors would like to thank the Natural Science Foundation of China (61104166).

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Dynamic multi-objective optimization for mixed traffic flow based on partial least squares prediction model

Abstract

Keywords

Introduction

Signal timing model of isolated intersection

Selection of performance indicators

Vehicle delay

Non-motorized vehicle delay

Pedestrian delay

Dynamic multi-objective signal optimization model for mixed traffic flow at intersections

Constraint condition

Prediction model

Optimization algorithm

Dynamic multi-objective evolutionary algorithm based on partial least squares prediction model

Experiments and results

Test function

Performance metrics

Experimental comparison with other dynamic multi-objective algorithms

Comparison of the distribution of Pareto solutions obtained by the algorithm

Comparing the performance box diagram of the algorithm

Analysis of algorithm complexity

Case study and simulation

Basic information of single intersection

Analysis of algorithm operation results

Analysis of the relationship between optimization objectives

Simulation of results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References