In this paper, a multiobjective root system growth algorithm-based
For multiobjective optimization problems (MOPs), many objectives should be optimized simultaneously under some constraints. Owing to the problems’ particularity, researchers prefer to find an approximation of the Pareto optimal set to support the effectiveness of their proposed approaches. At present, MOPs have made some achievements in academia and industry [
In recent years, a variety of MOEAs have been proposed. All of them can be divided into three classes: (1) non-dominant-sorting-based EAs [
In the existing literature [
The constructed and updated Pareto nondominant set can be approximately represented by the real optimal frontier. This method will obtain an acceptable result when the advantage relationship in the target space is obvious, while it will show poor performance when the advantage relationship is weak or when the multimodal and deceptive problems are optimal in isolation [
In view of this situation, Pareto dominance processing technology is divided into the following types: Randomized schemes, such as roulette selection [ Pareto-based relaxed domination methods, such as Reference-point-based dominance methods [ In the Optimality criterion in [ Average ranking (AR) [
In this paper,
With the test against several multiobjective benchmark functions, the results demonstrate that the proposed
In [
Yuan et al. [
Portfolio management refers to the investment manager’s diversification management of assets according to asset selection theory and portfolio theory to achieve the purpose of diversifying risks and improving efficiency. The application of EAs in solving MOPOP has attracted wide attention, yet their method is usually to convert MOPOP into single-objective POP by weighted sum. Therefore, OPF problem is to schedule the promised generator set to meet system load requirements with minimal costs under some constraints. OPF problem is a multiobjective nonlinear constrained optimization problem of competition. The experimental results show that
The remainder of this paper is organized as follows. Section
Therefore, the goal is to search for a vector
Consider the following functions:
Such criterion promises the most feasible solution in the same nondominant rank.
GM is defined as the sum of all the individual target values of the difference:
According to Pareto dominance, the smaller
The framework combines each goal with the information of all individuals to obtain ranking values. By comparing with the solution pairs, the sum of the good or bad parts of the solution pairs is calculated, as shown in Figure
A sketch map on global margin ranking.
This section describes the
The root system is defined as a group of tips, which are expressed as
In initialization,
In the initial stage, the nutritional value of each root tip is set as 0. During root growth, if its new position is better than the last one, the tip will receive nutrients from the environment. Inversely, the tips will lose nutrients and the nutrient content will be decreased by one.
Then, the auxin concentration of auxin
All root taps are classified according to the auxin concentration values. The stronger tip has a higher probability of becoming the main root. The size of main roots is limited as follows:
The threshold
The number of branching
The new branching tips foraging in the new regions will grow.
The fitness
Trajectory of the root is influenced by different tropisms. In the RSGA model, two typical orientations are realized, i.e., hydrotropism and gravitropism.
Half of the main roots will grow to the optimum position, and the water content in the roots will be the largest, which is given by the following formula:
Lower auxin concentration indicates that the root tip does not acquire as much nutrients as possible during its foraging and therefore is not active and continuous growing is not possible.
Iteration = 0; Initialize Evaluate the fitness of the group; Evaluate the global general (GG) of each solution by equations (
Compute the auxin concentration values of the root tip group by equation ( Sort the root tip group in order of descending accumulated auxin concentration; Divide the root tip group into two subgroups, i.e. the main root group and the lateral root group of equation (
Determine the branch number of each branching root tip by equation (
Sort the main root subgroup in order of increasing cumulative auxin concentration; The first half main roots are hydrotropic according to equation (
All the lateral root tips perform random walk operation according to equation (
Remove the dead root tips (auxin concentration 0) from the root tip group;
Gather the new generated solution as offspring group
Combine parent group and offspring group Non-dominated sort on Pareto dominance { Add nondominated front Add the solutions of last nondominated front
Iteration = iteration + 1;
The algorithm can be divided into six parts. The first part is to initialize the population and calculate the fitness and
In the two-objective test experiment, population size is set as 200, and the number of function evaluations (FEs) is set as 40000. In the three-objective test experiment, population size is set as 300, and the number of function evaluations (FEs) is set as 90000. The number of independent runs of the experiment is 10 times. For
Table
Test results of two objective functions.
Problem |
|
NSGA-II | MOEA/D | MOPSO | rNSGA-II | RPDNSGA-II | ||
---|---|---|---|---|---|---|---|---|
ZDT1 | Converge | Avg |
|
9.8750 |
8.3829 |
1.0000 |
9.9750 |
9.9477 |
Metric | Std |
|
9.50 |
1.02 |
0.00 |
4.86 |
5.15 | |
Diversity | Avg |
|
2.3791 |
2.6362 |
2.0794 |
2.9883 |
4.9906 | |
Metric | Std |
|
6.99 |
8.43 |
8.36 |
6.08 |
5.35 | |
IGD | Avg |
|
2.3791 |
2.6362 |
2.0510 |
2.9883 |
4.9906 | |
Metric | Std |
|
6.99 |
8.43 |
8.48 |
6.08 |
5.35 | |
HV | Avg | 7.2140 |
7.2154 |
|
1.0000 |
4.1705 |
7.1954 | |
Metric | Std | 1.88 |
1.44e − 4 |
|
0.00e + 0 | 5.36e − 3 | 4.90e − 4 | |
|
||||||||
ZDT2 | Converge | Avg |
|
1.0000 |
4.5020 |
1.0000 |
9.9696 |
9.8232 |
Metric | Std |
|
0.00 + 0 | 2.69 |
0.00 |
4.85 |
3.46 | |
Diversity | Avg | 2.3924 |
2.9150 |
|
3.0864 |
3.3474 |
1.5993 | |
Metric | Std | 4.93 |
1.57 |
|
8.04 |
5.59 |
2.65 | |
IGD | Avg | 2.3924 |
2.9150 |
|
3.0864 |
3.3474 |
1.5993 | |
Metric | Std | 2.43 |
1.57 |
|
8.04 |
5.59 |
2.65 | |
HV | Avg |
|
4.4460 |
4.4628 |
1.0000 |
1.8738 |
4.2840 | |
Metric | Std |
|
4.58 |
1.66 |
0.00 |
3.13 |
3.34 | |
|
||||||||
ZDT3 | Converge | Avg |
|
8.7650 |
2.0135 |
1.0000 |
7.8232 |
8.1864 |
Metric | Std |
|
6.57 |
1.08 |
0.00 |
1.13 |
1.05 | |
Diversity | Avg | 1.2395 |
4.4731 |
|
1.9225 |
5.2856 |
4.2074 | |
Metric | Std | 3.43 |
8.16 |
|
6.79 |
7.40 |
4.90 | |
IGD | Avg | 1.2374 |
4.4731 |
|
1.8946 |
5.2856 |
4.2074 | |
Metric | Std | 1.02 |
8.16 |
|
6.63 |
7.40 |
4.90 | |
HV | Avg | 6.7935 |
6.3195 |
6.0250 |
1.0000 |
5.3392 |
6.5979 | |
Metric | Std | 1.33 |
6.82 |
4.07 |
0.00 |
1.21 |
8.87 | |
|
||||||||
ZDT4 | Converge | Avg |
|
9.9500 |
1.0000 |
1.0000 |
1.0000 |
9.6527 |
Metric | Std |
|
4.08 |
0.00 |
0.00 |
0.00 |
4.01 | |
Diversity | Avg |
|
3.0137 |
4.3406 |
1.4861 |
2.8996 |
1.1594 | |
Metric | Std |
|
5.73 |
1.50 |
5.97 |
3.95 |
3.16 | |
IGD | Avg |
|
3.0137 |
4.3406 |
1.4453 |
2.8996 |
1.1594 | |
Metric | Std |
|
5.73 |
1.50 |
5.84 |
3.95 |
3.16 | |
HV | Avg |
|
7.2019 |
7.1808 |
1.0000 |
4.2437 |
6.1428 | |
Metric | Std |
|
1.01 |
2.12 |
0.00 |
3.32 |
3.55 | |
|
||||||||
ZDT6 | Converge | Avg |
|
9.5450 |
9.1000 |
2.0450 |
9.7400 |
9.9829 |
Metric | Std |
|
9.55 |
1.34 |
3.96 |
8.22 |
3.86 | |
Diversity | Avg |
|
1.8732 |
1.6607 |
9.5418 |
2.2503 |
5.7035 | |
Metric | Std |
|
9.17 |
7.58 |
2.96 |
5.43 |
2.87 | |
IGD | Avg |
|
1.8732 |
1.6607 |
9.3956 |
2.2503 |
5.7035 | |
Metric | Std |
|
9.17 |
7.58 |
2.96 |
5.43 |
2.87 | |
HV | Avg | 2.9843 |
3.8983 |
|
3.4808 |
2.0532 |
3.8554 | |
Metric | Std | 4.32 |
2.98 |
|
1.22 |
3.40 |
3.95 |
Obtained results on ZDT1: (a) PF obtained by
Obtained results on ZDT2: (a) PF obtained by
Obtained results on ZDT3: (a) PF obtained by
Obtained results on ZDT4: (a) PF obtained by
Obtained results on ZDT6: (a) PF obtained by
In Table
Figures
Table
Test results on three-objective functions.
Problem |
|
NSGA-II | MOEA/D | MOPSO | rNSGA-II | RPDNSGA-II | ||
---|---|---|---|---|---|---|---|---|
DTLZ1 | Converge | Avg | 4.9323 |
5.2815 |
|
4.8607 |
6.1465 |
4.8924 |
Metric | Std | 4.12 |
4.43 |
|
1.14 |
3.25 |
4.06 | |
Diversity | Avg |
|
2.6981 |
1.3155 |
2.9024 |
2.0744 |
1.5770 | |
Metric | Std |
|
2.94 |
2.32 |
6.94 |
2.51 |
1.74 | |
IGD | Avg |
|
1.9798 |
1.1472 |
7.6192 |
1.7254 |
1.3049 | |
Metric | Std |
|
2.09 |
1.91 |
4.36 |
2.03 |
1.42 | |
HV | Avg |
|
4.6727 |
6.5240 |
1.0000 |
5.2542 |
5.6318 | |
Metric | Std |
|
3.92 |
3.04 |
0.00 |
3.70 |
3.48 | |
|
||||||||
DTLZ2 | Converge | Avg | 7.9873 |
7.1867 |
|
8.5567 |
3.1040 |
1.6016 |
Metric | Std | 6.32 |
8.71 |
|
5.94 |
2.43 |
3.91 | |
Diversity | Avg |
|
5.8855 |
5.9884 |
8.3690 |
5.0484 |
4.6103 | |
Metric | Std |
|
1.65 |
1.88 |
3.28 |
1.14 |
1.37 | |
IGD | Avg |
|
5.8855 |
5.9884 |
8.3690 |
5.0484 |
4.6103 | |
Metric | Std |
|
1.65 |
1.88 |
3.28 |
1.14 |
1.37 | |
HV | Avg |
|
5.4209 |
5.4279 |
4.8186 |
1.3043 |
5.6408 | |
Metric | Std |
|
1.78 |
1.74 |
5.18 |
7.24 |
1.19 | |
|
||||||||
DTLZ3 | Converge | Avg |
|
5.2444 |
4.8054 |
2.7995 |
6.3365 |
5.7198 |
Metric | Std |
|
2.51 |
1.62 |
1.08 |
2.61 |
1.34 | |
Diversity | Avg |
|
5.1430 |
6.8767 |
2.4610 |
5.3446 |
8.3516 | |
Metric | Std |
|
5.96 |
7.45 |
5.51 |
5.62 |
8.37 | |
IGD | Avg |
|
4.4432 |
6.1109 |
1.4399 |
4.7071 |
4.4227 | |
Metric | Std |
|
5.08 |
6.91 |
6.48 |
4.91 |
4.28 | |
HV | Avg |
|
2.2438 |
2.2523 |
1.0000 |
2.2284 |
2.2596 | |
Metric | Std |
|
2.90 |
2.91 |
0.00 |
2.88 |
2.92 |
True PF on DTLZ functions. (a) True PF on DTLZ1. (b) True PF on DTLZ2. (c) True PF on DTLZ3.
Obtained results on DTLZ1: (a) PF obtained by
Obtained results on DTLZ2: (a) PF obtained by
Obtained results on DTLZ3: (a) PF obtained by
It can be seen from Figure
From Figure
In order to demonstrate the difference in the time complexity of these algorithms, Figures
CPU time of MOEAs on two-objective benchmark functions.
CPU time of MOEAs on three-objective benchmark functions.
Figure
Compared to Figure
Markowitz [
The assumptions of several portfolio models introduced are too harsh to meet the actual needs of the securities market, which leads to the deviation between the results of the model operation and the actual situation. On the basis of revising the hypothesis of the classical portfolio model, semivariance is introduced to replace variance, which makes the model more reasonable, more in line with the investment and financing environment of China’s financial market, and provides more effective auxiliary tools for investors.
Different from the 2-objective model, the 3-objective portfolio model includes other objective: minimizing expected cost.
The MV model has the function as follows:
The weight of each asset portfolio should be the sum of 1.
In [
The indirect method is utilized to represent transaction cost:
Therefore, the return-risk-cost portfolio model is constructed as follows:
Budget constraints are given as follows:
It should be noted that when appropriate indicators are available, the number of model objectives can also increase.
The experiment uses daily historical data of 12 kinds of assets from Shanghai Stock Exchange, which are collected at the monthly rate of each stock from January 2010 to December 2016.
The frontier of
PF obtained by
PF obtained by
Comparison of performance on MV portfolio model.
MV portfolio model |
|
NSGA-II | MOEA/D | |
---|---|---|---|---|
Hypervolume indicator | Max |
|
3.95 |
3.44 |
Min |
|
2.04 |
9.93 | |
Avg |
|
2.45 |
1.57 | |
Std |
|
4.77 |
7.82 |
Comparison of performance on return-risk-cost portfolio model.
Return-risk-cost portfolio model |
|
NSGA-II | MOEA/D | |
---|---|---|---|---|
Hypervolume indicator | Max |
|
5.41 |
6.37 |
Min |
|
9.86 |
7.02 | |
Avg |
|
1.07 |
2.83 | |
Std |
|
2.59 |
3.91 |
Figure
In Figure
From Figure
As is visible from Figure
OPF’s main objective is to optimize restriction variables’ setting while fulfilling several inequality constraints and equality. Generally speaking, OPF issues can be expressed mathematically as follows:
In this paper, there are three competing objective functions regarding the OPF problem, namely, total fuel cost, total power loss, and total emission cost, while satisfying several inequality constraints and equality.
The problem is usually expressed as follows.
From (
The inequality constraints are the power system handling.
To verify the numerical correctness and the effectiveness of
The standard IEEE 30 bus system has been applied as a test system [
Characteristics of the generation units.
G1 | G2 | G3 | G4 | G5 | G6 | |
---|---|---|---|---|---|---|
|
||||||
|
150 | 150 | 150 | 150 | 150 | 150 |
|
5 | 5 | 5 | 5 | 5 | 5 |
|
||||||
|
||||||
A | 10 | 10 | 20 | 10 | 20 | 10 |
B | 200 | 150 | 180 | 100 | 180 | 150 |
C | 100 | 120 | 40 | 60 | 40 | 100 |
|
||||||
|
||||||
Α | 4.091 | 2.543 | 4.258 | 5.326 | 4.258 | 6.131 |
Β | −5.554 | −6.407 | −5.094 | −3.550 | −5.094 | −5.555 |
|
6.490 | 5.638 | 4.586 | 3.380 | 4.586 | 5.151 |
Δ | 2.0 |
5.0 |
1.0 |
2.0 |
1.0 |
1.0 |
|
2.857 | 3.333 | 8.000 | 2.000 | 8.000 | 6.667 |
In this section, we consider the following two test system scenarios: Two objectives: the emission cost, the emission loss, and the loss cost are considered separately. Three objectives: emissions, costs, and losses.
The new algorithm is compared with three MOEAs in three aspects: minimum cost, minimum emission, and minimum loss. Figures
First, two competing objectives need to be considered: fuel costs and emissions. From Figure
The objective function of cost loss is shown in Figure
From Table
For the emission loss objective function, we can come to know that the algorithm achieves performance ranking similar to that of the fuel cost emission objective function.
Pareto fronts obtained by
Pareto fronts obtained by
Pareto fronts obtained by
The best compromise solutions for cost, emission, and loss.
|
MOABC | NSGA-II | MOPSO | |
---|---|---|---|---|
PG1 | 18.9832 | 17.8933 | 35.6514 | 21.9132 |
PG2 | 33.085 | 27.0342 | 53.9965 | 15.2151 |
PG3 | 67.0249 | 70.8835 | 47.6396 | 90.2566 |
PG4 | 82.1065 | 85.2331 | 45.7462 | 84.3363 |
PG5 | 29.0455 | 27.0337 | 55.0045 | 7.0304 |
PG6 | 53.2144 | 53.3102 | 46.0104 | 65.4609. |
f1 fuel cost | 610.0513 | 614.0154 | 622.449 | 631.3809 |
f2 (emission) | 0.2219 | 0.2376 | 0.2323 | 0.2448 |
f3 (loss) | 2.1488 | 2.1790 | 3.9912 | 2.8491 |
The best solutions for cost and emission.
|
MOABC | NSGA-II | MOPSO | |||||
---|---|---|---|---|---|---|---|---|
Best f1 | Best f2 | Best f1 | Best f2 | Best f1 | Best f2 | Best f1 | Best f2 | |
PG1 | 11.66 | 40.03 | 10.72 | 7.37 | 34.82 | 41.67 | 25.88 | 55.67 |
PG2 | 30.23 | 47.52 | 28.51 | 41.98 | 54.88 | 47.64 | 31.88 | 45.82 |
PG3 | 54.33 | 55.82 | 55.75 | 60.73 | 50.03 | 49.79 | 61.84 | 55.22 |
PG4 | 103.63 | 42.04 | 104.83 | 100.88 | 45.02 | 42.99 | 100.83 | 39.98 |
PG5 | 42.62 | 51.28 | 43.99 | 41.51 | 54.72 | 51.80 | 44.88 | 54.30 |
PG6 | 37.94 | 53.62 | 36.50 | 34.99 | 43.88 | 51.60 | 37.98 | 47.99 |
f1 (cost) | 605.62 | 640.02 | 609.23 | 640.93 | 609.60 | 641.64 | 606.98 | 639.36 |
f2 (emission) | 0.2526 | 0.2031 | 0.2537 | 0.1998 | 0.2287 | 0.1898 | 0.2482 | 0.1937 |
The best solutions for cost and loss.
|
MOABC | NSGA-II | MOPSO | |||||
---|---|---|---|---|---|---|---|---|
Best f1 | Best f3 | Best f1 | Best f3 | Best f1 | Best f3 | Best f1 | Best f3 | |
PG1 | 14.99 | 3.99 | 9.26 | 4.02 | 18.98 | 3.58 | 67.00 | 9.45 |
PG2 | 28.99 | 26.22 | 32.61 | 11.29 | 19.98 | 22.88 | 37.54 | 9.68 |
PG3 | 54.79 | 106.00 | 58.03 | 108.99 | 39.22 | 71.94 | 67.26 | 108.78 |
PG4 | 110.46 | 56.99 | 96.21 | 67.76 | 119.69 | 100.02 | 100.03 | 70.63 |
PG5 | 30.21 | 5.01 | 45.83 | 11.87 | 41.12 | 5.20 | 36.54 | 72.01 |
PG6 | 39.03 | 86.98 | 40.03 | 79.88 | 45.51 | 81.81 | 28.25 | 34.23 |
f1 (cost) | 606.16 | 624.88 | 610.12 | 627.34 | 609.24 | 634.26 | 606.81 | 672.88 |
f3 (loss) | 2.4554 | 1.5202 | 2.5021 | 1.5421 | 3.4197 | 1.6740 | 2.5414 | 1.5632 |
The best solutions for emission and loss.
|
MOABC | NSGA-II | MOPSO | |||||
---|---|---|---|---|---|---|---|---|
Best f2 | Best f3 | Best f2 | Best f3 | Best f2 | Best f3 | Best f2 | Best f3 | |
PG1 | 44.92 | 9.00 | 33.60 | 11.98 | 43.11 | 62.48 | 32.04 | 11.01 |
PG2 | 45.22 | 4.98 | 58.98 | 6.23 | 40.42 | 142.98 | 45.34 | 76.78 |
PG3 | 55.66 | 98.45 | 50.53 | 69.54 | 56.33 | 106.22 | 60.99 | 22.87 |
PG4 | 40.92 | 69.65 | 33.52 | 94.12 | 39.99 | 61.55 | 42.99 | 62.80 |
PG5 | 47.02 | 4.47 | 61.33 | 6.28 | 40.49 | 4.99 | 48.78 | 81.78 |
PG6 | 53.25 | 102.62 | 47.00 | 94.99 | 35.00 | 35.80 | 51.45 | 59.99 |
f2 (emission) | 0.2003 | 0.2587 | 0.2004 | 0.3100 | 0.2398 | 0.2134 | 0.2061 | 0.2603 |
f3 (loss) | 4.1198 | 1.5935 | 4.7623 | 1.6345 | 4.3061 | 2.2314 | 4.1351 | 1.6501 |
The best compromise solutions for emission and cost.
|
MOABC | NSGA-II | MOPSO | |
---|---|---|---|---|
PG1 | 25.0102 | 24.8816 | 35.1988 | 24.3028 |
PG2 | 39.0058 | 38.4973 | 54.4014 | 36.7983 |
PG3 | 56.4611 | 38.1053 | 49.0943 | 54.3984 |
PG4 | 72.0211 | 74.8915 | 46.0004 | 71.9823 |
PG5 | 51.3025 | 46.1210 | 56.1235 | 51.8923 |
PG6 | 45.5280 | 41.8988 | 43.1914 | 44.04728 |
f1 (cost) | 614.8302 | 615.5069 | 625.9913 | 615.2733 |
f2 (emission) | 0.1998 | 0.2000 | 0.1988 | 0.2014 |
The best compromise solutions for loss and cost.
|
MOABC | NSGA-II | MOPSO | |
---|---|---|---|---|
PG1 | 4.3822 | 8.5941 | 3.5519 | 10.5911 |
PG2 | 24.9932 | 28.1041 | 12.8737 | 19.6955 |
PG3 | 78.9744 | 73.6835 | 87.8831 | 81.9942 |
PG4 | 104.1988 | 104.6922 | 92.0908 | 95.3140 |
PG5 | 8.9811 | 16.5047 | 28.0134 | 11.9914 |
PG6 | 61.9852 | 49.8818 | 62.7941 | 71.4352 |
f1 (cost) | 620.5899 | 622.2713 | 622.2812 | 628.2924 |
f3 (loss) | 1.6791 | 1.6854 | 1.7991 | 1.5963 |
The best compromise solutions for loss and emission.
|
MOABC | NSGA-II | MOPSO | |
---|---|---|---|---|
PG1 | 20.0945 | 67.7844 | 29.7010 | 22.6015 |
PG2 | 29.9966 | 22.9971 | 32.2034 | 25.2043 |
PG3 | 80.3288 | 29.7523 | 76.7044 | 102.9912 |
PG4 | 65.1966 | 66.0088 | 66.0093 | 63.9801 |
PG5 | 17.0043 | 76.4533 | 34.1155 | 6.5088 |
PG6 | 71.2054 | 20.9199 | 48.0114 | 86.0299 |
f2 (emission) | 1.8956 | 2.0087 | 2.6788 | 1.9204 |
f3 (loss) | 0.1953 | 0.2078 | 0.2166 | 0.2525 |
In this instance, each algorithm needs to optimize three objectives. Figure
Judging from the data in Table
Referring to the data in Table
Pareto fronts obtained by CMOABC, MOPSO, MOABC, and NSGA-II for fuel cost, emission and loss: (a)
The best solutions for cost, emission, and loss.
|
MOABC | MOPSO | NSGA-II | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Best f1 | Best f2 | Best f3 | Best f1 | Best f2 | Best f3 | Best f1 | Best f2 | Best f3 | Best f1 | Best f2 | Best f3 | |
PG1 | 20.20 | 43.99 | 5.88 | 34.88 | 3.18 | 17.95 | 21.92 | 37.02 | 8.99 | 24.99 | 36.89 | 17.95 |
PG2 | 20.56 | 45.77 | 14.16 | 50.55 | 15.47 | 29.89 | 24.75 | 46.88 | 24.62 | 27.41 | 48.11 | 26.43 |
PG3 | 69.52 | 60.02 | 104.25 | 61.37 | 80.41 | 60.66 | 64.99 | 62.55 | 81.38 | 56.88 | 49.54 | 93.88 |
PG4 | 100.76 | 39.10 | 69.00 | 40.55 | 68.99 | 100.09 | 100.61 | 46.77 | 62.39 | 47.29 | 38.88 | 65.2 |
PG5 | 45.99 | 53.89 | 4.98 | 43.99 | 9.06 | 37.77 | 39.29 | 50.12 | 66.45 | 44.55 | 50.99 | 8.07 |
PG6 | 36.55 | 47.13 | 86.44 | 35.76 | 77.02 | 40.99 | 33.66 | 52.34 | 57.77 | 41.88 | 49.21 | 67.99 |
f1 (cost) | 610.99 | 650.99 | 637.44 | 639.99 | 625.78 | 612.98 | 615.99 | 640.10 | 661.27 | 615.23 | 639.12 | 643.02 |
f2 (emission) | 0.2503 | 0.2065 | 0.2499 | 0.2154 | 0.2507 | 0.2610 | 0.3799 | 0.2388 | 0.4046 | 0.2453 | 0.1989 | 0.2599 |
f3 (loss) | 2.0012 | 4.1707 | 1.5944 | 4.0062 | 1.6931 | 2.0002 | 2.1074 | 2.4576 | 1.5010 | 2.4002 | 2.3422 | 2.1055 |
In this paper, a multiobjective optimizer based on
The MATLAB data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work was supported by the National Key Research and Development Program of China under Grant No. 2017YFB1103603 and 2017YFB1103003, National Natural Science Foundation of China under Grant No. 61602343, 51607122, 61772365, 41772123, 61802280, 61806143, 51575158, and 61502318, Tianjin Province Science and Technology Projects under Grant No. 17JCYBJC15100 and 17JCQNJC04500, and Basic Scientific Research Business Funded Projects of Tianjin (2017KJ093 and 2017KJ094).