4. Simulation results and analysis

In this section, three ZDT and two DTLZ benchmark functions are employed to test the proposed of AGMOPSO. This section compares the proposed AGMOPSO with four stateof-the-art MOPSO algorithms—adaptive gradient MOPSO (AMOPSO) [41], crowded distance MOPSO (cdMOPSO) [32], pdMOPSO [31] and NSGA-II [11].

#### 4.1. Performance metrics

To demonstrate the performance of the proposed AGMOPSO algorithm, two different quantitative performance metrics are employed in the experimental study.

1. Inverted generational distance (IGD):

$$IGD(\mathcal{F}^\*, \mathcal{F}) = \sum\_{\mathbf{x} \in \mathcal{F}^\*} \text{mindis}(\mathbf{x}, \mathcal{F}) / |\mathcal{F}^\*| \tag{18}$$

where mindis(x, F) is the minimum Euclidean distance between the solution x and the solutions in F. A smaller value of IGD(F\* , F) demonstrates a better convergence and diversity to the Pareto-optimal front.

2. Spacing (SP):

$$SP = \sqrt{\frac{1}{K - 1} \sum\_{i=1}^{K} \left(\overline{d} - d\_i\right)}\tag{19}$$

Figure 3. The Pareto front with non-dominated solutions obtained by the two multiobjective algorithms for ZDT4

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Figure 4. The Pareto front with non-dominated solutions obtained by the two multiobjective algorithms for ZDT6

Figure 5. The Pareto front with non-dominated solutions obtained by the two multiobjective algorithms for DTLZ2

function.

function.

function.

where di is the minimum Euclidean distance between ith solution and other solutions, K is the number of non-dominated solutions, d is the average distance of the all Euclidean distance di.

#### 4.2. Parameter settings

All the algorithms have three common parameters: the population size N, the maximum number of non-dominated solutions K and iterations T. Here, N = 100, K = 100 and T = 3000.

#### 4.3. Experimental results

The experimental performance comparisons of the cdMOPSO algorithm on ZDTs and DTLZs are shown in Figures 2–6. Seen from Figures 2–6, the non-dominated solutions obtained by the proposed AGMOPSO algorithm can approach to the Pareto Front appropriately and maintain

Figure 2. The Pareto front with non-dominated solutions obtained by the two multiobjective algorithms for ZDT3 function.

4.1. Performance metrics

84 Optimization Algorithms - Examples

1. Inverted generational distance (IGD):

in F. A smaller value of IGD(F\*

Pareto-optimal front.

4.2. Parameter settings

4.3. Experimental results

function.

2. Spacing (SP):

To demonstrate the performance of the proposed AGMOPSO algorithm, two different quanti-

mindis xð Þ ; <sup>F</sup> <sup>=</sup> <sup>F</sup><sup>∗</sup> j j, (18)

, F) demonstrates a better convergence and diversity to the

vuut , (19)

x∈F<sup>∗</sup>

1 K � 1

where mindis(x, F) is the minimum Euclidean distance between the solution x and the solutions

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

d � di � �

X K

i¼1

where di is the minimum Euclidean distance between ith solution and other solutions, K is the number of non-dominated solutions, d is the average distance of the all Euclidean distance di.

All the algorithms have three common parameters: the population size N, the maximum number of non-dominated solutions K and iterations T. Here, N = 100, K = 100 and T = 3000.

The experimental performance comparisons of the cdMOPSO algorithm on ZDTs and DTLZs are shown in Figures 2–6. Seen from Figures 2–6, the non-dominated solutions obtained by the proposed AGMOPSO algorithm can approach to the Pareto Front appropriately and maintain

Figure 2. The Pareto front with non-dominated solutions obtained by the two multiobjective algorithms for ZDT3

tative performance metrics are employed in the experimental study.

IGD <sup>F</sup><sup>∗</sup> ð Þ¼ ; <sup>F</sup> <sup>X</sup>

SP ¼

Figure 3. The Pareto front with non-dominated solutions obtained by the two multiobjective algorithms for ZDT4 function.

Figure 4. The Pareto front with non-dominated solutions obtained by the two multiobjective algorithms for ZDT6 function.

Figure 5. The Pareto front with non-dominated solutions obtained by the two multiobjective algorithms for DTLZ2 function.

Figure 6. The Pareto front with non-dominated solutions obtained by the two multiobjective algorithms for DTLZ7 function.


a greater diversity than other compared algorithms. Experimental results in Figures 2–4 show that the proposed AGMOPSO algorithm is superior to the cdMOPSO algorithm in diversity performance and can approach the Pareto Front. In addition, the results in Figures 5 and 6 show that the proposed AGMOPSO algorithm can obtain a better performance on the three-

Function Index AGMOPSO AMOPSO pdMOPSO cdMOPSO NSGA-II ZDT3 Best 0.023475 0.097811 0.099654 0.10356 0.081569

ZDT4 Best 0.030914 0.039825 0.069564 0.139577 0.031393

ZDT6 Best 0.010981 0.008739 0.009935 0.012396 0.006851

DTLZ2 Best 0.1438 0.0943 0.0569 0.0932 0.021456

DTLZ7 Best 0.1958 0.1047 0.0932 0.1347 0.0632

Worst 0.087874 0.416626 0.487126 0.87449 0.106568 Mean 0.067451 0.245931 0.198551 0.59684 0.092216 std 0.012873 0.050937 0.079442 0.22468 0.008415

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Worst 0.078011 0.193765 0.233794 0.300951 0.044254 Mean 0.049923 0.078821 0.186698 0.204573 0.038378 std 0.001092 0.004517 0.063757 0.095562 0.003837

Worst 0.100551 0.088535 0.023766 0.040205 0.010127 Mean 0.034127 0.040251 0.010683 0.034569 0.008266 std 0.009756 0.007341 0.003021 0.003884 0.000918

Worst 0.6893 0.8947 0.6991 0.5897 0.7314 Mean 0.0398 0.4631 0.4721 0.3562 0.4162 std 0.00764 0.03401 0.02964 0.01772 0.03655

Worst 0.9032 0.9355 0.8361 0.9307 0.7466 Mean 0.0502 0.0493 0.4459 0.5972 0.4191 std 0.01097 0.03201 0.00896 0.2133 0.00796

In order to show the experimental performance in details, the experimental results, which contain the best, worst, mean and standard deviations of IGD and SP based on the twoobjective of ZDTs and the three-objective of DTLZs are listed in Tables 2 and 3, respectively. Moreover, the experimental results in Tables 2 and 3 include the details of the four evolutionary algorithms. To illustrate the significance of the findings, the comparing results for the

1. Comparison of IGD index: From Table 2, the proposed AGMOPSO algorithm is superior to other MOPSO algorithms in terms of the results of IGD. Firstly, in the two-objective of ZDTs instances, the AGMOPSO can have better mean deviations of IGD than other four evolutionary algorithms on ZDT3 and ZDT4. It is indicated that the MOG method has

objective benchmark problems with accurate convergence and the preferable diversity.

performance index is analyzed as follows:

Table 3. Comparisons of different algorithms for SP.

Table 2. Comparisons of different algorithms for IGD.


std 0.012873 0.050937 0.079442 0.22468 0.008415

ZDT4 Best 0.030914 0.039825 0.069564 0.139577 0.031393

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Table 3. Comparisons of different algorithms for SP.

Figure 6. The Pareto front with non-dominated solutions obtained by the two multiobjective algorithms for DTLZ7

Function Index AGMOPSO AMOPSO pdMOPSO cdMOPSO NSGA-II ZDT3 Best 0.00149 0.00425 0.2019 0.003109 0.005447

ZDT4 Best 3.0194 2.7133 3.3980 4.9760 0.00462

ZDT6 Best 0.2046 0.0936 2.2310 0.000897 0.01119

DTLZ2 Best 0.0477 0.0519 0.1330 0.0322 0.07830

DTLZ7 Best 0.05766 0.02044 0.00796 0.00701 0.00614

Table 2. Comparisons of different algorithms for IGD.

Worst 0.00697 0.00832 0.4265 0.028986 0.006105 Mean 0.00433 0.00632 0.3052 0.003063 0.005834 std 0.00297 0.00527 0.1003 0.007131 0.000202

Worst 5.1522 5.0543 4.9760 6.3610 0.11166 Mean 3.7933 3.8943 4.0330 5.9120 0.016547 std 1.5133 2.7401 1.6510 4.5180 0.031741

Worst 0.7834 0.9154 2.8790 0.003627 0.01498 Mean 0.4878 0.5433 2.4690 0.002988 0.01286 std 0.0242 0.0236 0.8169 0.0001543 0.001004

Worst 0.3913 0.3425 0.3690 0.2067 0.2740 Mean 0.1058 0.1878 0.2070 0.1015 0.1059 std 0.0060 0.0132 0.0413 0.0134 0.008383

Worst 0.32803 0.10295 0.07678 0.05439 0.03208 Mean 0.01985 0.04573 0.04831 0.02856 0.01799 std 0.00139 0.00312 0.00289 0.00165 0.00129

function.

86 Optimization Algorithms - Examples

a greater diversity than other compared algorithms. Experimental results in Figures 2–4 show that the proposed AGMOPSO algorithm is superior to the cdMOPSO algorithm in diversity performance and can approach the Pareto Front. In addition, the results in Figures 5 and 6 show that the proposed AGMOPSO algorithm can obtain a better performance on the threeobjective benchmark problems with accurate convergence and the preferable diversity.

In order to show the experimental performance in details, the experimental results, which contain the best, worst, mean and standard deviations of IGD and SP based on the twoobjective of ZDTs and the three-objective of DTLZs are listed in Tables 2 and 3, respectively.

Moreover, the experimental results in Tables 2 and 3 include the details of the four evolutionary algorithms. To illustrate the significance of the findings, the comparing results for the performance index is analyzed as follows:

1. Comparison of IGD index: From Table 2, the proposed AGMOPSO algorithm is superior to other MOPSO algorithms in terms of the results of IGD. Firstly, in the two-objective of ZDTs instances, the AGMOPSO can have better mean deviations of IGD than other four evolutionary algorithms on ZDT3 and ZDT4. It is indicated that the MOG method has played a vital role on the algorithm. Meanwhile, compared with NSGA-II [11], the proposed AGMOPSO has better IGD index performance of accuracy and stability for the twoobjective of ZDTs (except ZDT4). Second, in the three-objective of DTLZs instances, the AGMOPSO is superior to other four algorithms in terms of the mean deviations value of IGD. According to the comparisons between the AGMOPSO and other four evolutionary algorithms, it is demonstrated that the proposed AGMOPSO is the closest to the true front and nearly enclose the entire front, which means the proposed AGMOPSO algorithm achieves the best convergence and divergence.

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2. Comparison of SP index: The comparison of SP among the proposed AGMOPSO algorithm and other compared algorithms was shown in Table 3. Firstly, in the two-objective of ZDTs instances, the AGMOPSO can have better mean deviations and best deviations of SP than other four evolutionary algorithms ZDT3 and ZDT4. Meanwhile, compared with NSGA-II [11], the proposed AGMOPSO has better SP index performance of diversity for the two-objective of ZDTs (except ZDT6). From the results in Table 3, the comparison of the SP between the proposed AGMOPSO algorithm illustrate that the MOG method can have better effect on the diversity performance than other existing methods. Secondly, in the three-objective of DTLZs instances, the proposed AGMOPSO algorithm has the best SP performance on the DTLZ2 and DTLZ7 than the other four compared algorithms. In addition, to verify the effect of the MOG method, the proposed AGMOPSO can obtain a set of non-dominated solutions with greater diversity and convergence than NSGA-II on instances (except ZDT4 and ZDT6). Therefore, the proposed AGMOPSO algorithm can obtain more accurate solutions with better diversity on the most ZDTs and DTLZs.
