Brain storm optimization (BSO) algorithm is a simple and effective evolutionary algorithm. Some multiobjective brain storm optimization algorithms have low search efficiency. This paper combines the decomposition technology and multiobjective brain storm optimization algorithm (MBSO/D) to improve the search efficiency. Given weight vectors transform a multiobjective optimization problem into a series of subproblems. The decomposition technology determines the neighboring clusters of each cluster. Solutions of adjacent clusters generate new solutions to update population. An adaptive selection strategy is used to balance exploration and exploitation. Besides, MBSO/D compares with three efficient state-of-the-art algorithms, e.g., NSGAII and MOEA/D, on twenty-two test problems. The experimental results show that MBSO/D is more efficient than compared algorithms and can improve the search efficiency for most test problems.
Multiobjective optimization problems (MOPs) [
Multiobjective evolutionary algorithms (MOEAs) make use of the population evolution to achieve a set of nondominated solutions which are optimal solutions of the current population, which are effective methods of solving MOPs. In decades, many MOEAs [
In 2011, Shi proposes the brain storm optimization (BSO) [
In this paper, the decomposition technology is fused together with multiobjective brain storm optimization algorithm (MBSO/D) to solve MOPs. The main motivation of MBSO/D is improve the performance of multiobjective BSO algorithms (MOBSOs) by using the decomposition technology. The MBSO/D mainly includes three parts to solve effectively MOPs. First, the current solutions are automatically clustered as each subproblem is optimized and the size of each cluster is the same; second, the decomposition technology determines the neighboring clusters of each cluster, and parent solutions are selected from adjacent clusters to produce the new solutions, which can improve the search efficiency; third, an adaptive selection strategy is used to balance exploration and exploitation.
We have done some work [
The organization of the remaining paper is reproduced below. The related works about the MOP and BSO are briefly reviewed in Section
The related definitions of MOP, BSO, and aggregation function are stated in this section.
The description of a MOP by mathematic formulation is as follows [
Humans may work together to solve some problems which cannot be solved by one person. Shi proposes the brain storm optimization algorithm [
This paper combines the decomposition technology and multiobjective brain storm optimization algorithm (MBSO/D) to address MOPs. The major parts of this algorithm are that three strategies based on decomposition (cluster strategy, update strategy selection strategy) are developed. Three strategies will be pointed out in this section.
The main motivation of this paper is to design a MOEA to achieve a set of solutions which evenly distribute on the true PF. BSO algorithm uses the group information to address the problems, which can help MOEAs to improve their performance. MOEAs use the neighboring information to optimal population, which can improve the performance of MOEAs. In this paper, this optimization idea is utilized to enhance the performance of multiobjective BSO algorithm. First, new solutions update the population by using the updated strategy of MOEA/D [
The MOEA/D decomposes a MOP into a series of subproblems by weight vectors and aggregate functions. For each solution, it and some of its neighbor solutions are selected as parents to generate a new solution. Then, some of its neighbor solutions are updated by aggregate functions and the new solution. So, all subproblems are simultaneously optimized in a population evolution. Each aggregate function makes some solutions converge to the corresponding weight vector, which can improve the convergence of the algorithm. In addition, the diversity of solutions is maintained by the uniformly distributed weight vectors. The main advantages of MOEA/D are that the diversity of obtained solutions can be determined by the given weight vectors; neighbor’s information is used to generate new offspring, which can improve the search efficiency.
In this work, each given weight vector fixes a cluster, and the size of each cluster is made the same. The best solution of each cluster is determined by corresponding weight vector
An effective selection strategy can help crossover operators to perform search work more effectively. In this BSO algorithm, according to an adaptive selection probability, a new solution will be produced by one cluster or three clusters. The adaptive selection probability is calculated by the following formulation:
Based on the selection probability
In this subsection, the pseudocode for the multiobjective brain storm optimization algorithm based on decomposition (MBSO/D) is displayed as shown in Pseudocode
the initial population into best solution of each cluster; determine
In MBSO/D, firstly, solutions in initial population are randomly clustered into N clusters with size K and the best solution of each cluster is determine by (
In this MBSO/D, the MOPs are solved by updating the neighboring solutions of each solution. We use the aggregation function
This multiobjective BSO algorithm compares with other multiobjective BSO algorithms, e.g., MBSO-C [
In MBSO-C and MBSO-DE, after the solutions are cluttered by some clustering methods, some solutions of the population are replaced by the newly generated solutions. In these MBSOs, the population is firstly updated by the newly generated solutions; then, solutions are automatically clustered as each subproblem is optimized. In MBSO-C and MBSO-DE, the newly generated solutions compare with other solutions of the same cluster to update this cluster, which may reduce the pressure of convergence; in this algorithm, aggregate function is used to update the population, which may enhance pressure of convergence.
In this section, the performance of MBSO/D will be verified by comparing it with existing multiobjective optimization algorithms, e.g., NSGAII [
Three performance metrics are adopted to quantify the performances of algorithms: generational distance (GD) [
All algorithms are implemented by using the MATLAB language and run independently for thirty times with the maximal number of function evaluations 100 000 on all test problems. For fair comparisons, the population size and the maximal number of function evaluations of the compared algorithms are the same as this work, and other parameters of NSGAII and MOEA/D are the same as the original literature. In MBSO/D,
The statistical results of the GD, IGD, and HV metrics obtained by each MOEA are posted in Tables
IGD, GD, and HV obtained by IMOEA/DA, MOEA/D, and NSGAII.
Problems | IGD | GD | HV | ||||
---|---|---|---|---|---|---|---|
mean | std | mean | std | mean | std | ||
F1 | MBSO/D | | 0.0068 | | 0.0020 | | 0.0073 |
MOEA/D | 0.0750(+) | 0.0513 | 0.0150(+) | 0.0082 | 0.7622(+) | 0.0515 | |
NSGAII | 0.0951(+) | 0.0355 | 0.0135(+) | 0.0028 | 0.7298(+) | 0.0517 | |
| |||||||
F2 | MBSO/D | | 0.0001 | | 0.0003 | | 0.0004 |
MOEA/D | 0.0137(+) | 0.0032 | 0.0050(+) | 0.0006 | 0.8538(+) | 0.0036 | |
NSGAII | 0.0095(+) | 0.0005 | 0.0101(+) | 0.0006 | 0.8614(+) | 0.0007 | |
| |||||||
F3 | MBSO/D | | 0.0101 | 0.0168 | 0.0043 | 0.8231 | 0.0123 |
MOEA/D | 0.1194(+) | 0.0939 | 0.0145(-) | 0.0067 | 0.7538(+) | 0.0604 | |
NSGAII | 0.0351(+) | 0.0255 | | 0.0015 | | 0.0195 | |
| |||||||
F4 | MBSO/D | | 0.0001 | | 0.0001 | | 0.0001 |
MOEA/D | 0.0084(+) | 0.0009 | 0.0040(+) | 0.0002 | 0.5300(+) | 0.0023 | |
NSGAII | 0.0056(+) | 0.0003 | 0.0050(+) | 0.0003 | 0.5348(+) | 0.0004 | |
| |||||||
F5 | MBSO/D | | 0.0718 | 0.4799 | 0.0896 | 0.0767 | 0.0599 |
MOEA/D | 0.4342(+) | 0.1547 | 0.3598(-) | 0.1601 | | 0.0926 | |
NSGAII | 0.4643(+) | 0.1087 | | 0.1325 | 0.0782(=) | 0.0817 | |
| |||||||
F6 | MBSO/D | | 0.0677 | 0.3879 | 0.9174 | | 0.0811 |
MOEA/D | 0.2116(+) | 0.1383 | 0.1467(-) | 0.0554 | 0.3911(+) | 0.0946 | |
NSGAII | 0.1935(+) | 0.0897 | | 0.0100 | 0.3900(+) | 0.0889 | |
| |||||||
F7 | MBSO/D | | 0.0024 | 0.0222 | 0.0097 | | 0.0048 |
MOEA/D | 0.0794(+) | 0.1676 | 0.0166(-) | 0.0127 | 0.6165(+) | 0.1386 | |
NSGAII | 0.0782(+) | 0.1203 | | 0.0016 | 0.6066(+) | 0.1002 | |
| |||||||
F8 | MBSO/D | | 0.0080 | 0.0443 | 0.0093 | 0.6444 | 0.0198 |
MOEA/D | 0.0939(+) | 0.0114 | | 0.0025 | | 0.0158 | |
NSGAII | 0.1482(+) | 0.0242 | 0.7099(+) | 0.6017 | 0.5616(+) | 0.0427 | |
| |||||||
F9 | MBSO/D | | 0.0135 | 0.1484 | 0.1113 | | 0.0276 |
MOEA/D | 0.1039(+) | 0.0448 | | 0.0375 | 0.9067(+) | 0.0638 | |
NSGAII | 0.1666(+) | 0.0709 | 0.9917(+) | 0.8967 | 0.7647(+) | 0.1496 | |
| |||||||
F10 | MBSO/D | | 0.0772 | 7.3988 | 3.3987 | | 0.1025 |
MOEA/D | 0.3597(+) | 0.2113 | | 0.1425 | 0.3100(+) | 0.1442 | |
NSGAII | 0.3505(+) | 0.0682 | 3.4759(-) | 3.5608 | 0.1811(+) | 0.0624 | |
| |||||||
DTLZ1 | MBSO/D | | 0.0001 | | 0.0001 | | 0.0001 |
MOEA/D | 0.0314(+) | 0.0016 | 0.0075(+) | 0.0002 | 0.1295(+) | 0.0011 | |
NSGAII | 0.0356(+) | 0.0500 | 0.0183(+) | 0.0611 | 0.1335(+) | 0.0204 | |
| |||||||
DTLZ2 | MBSO/D | | 0.0037 | | 0.0009 | | 0.0027 |
MOEA/D | 0.0813(+) | 0.0053 | 0.0209(+) | 0.0010 | 0.6673(+) | 0.0107 | |
NSGAII | 0.0692(+) | 0.0021 | 0.0234(+) | 0.0013 | 0.7011(+) | 0.0057 | |
| |||||||
DTLZ3 | MBSO/D | | 0.0515 | | 0.0017 | | 0.0577 |
MOEA/D | 0.0807(+) | 0.0048 | 0.0204(+) | 0.0009 | 0.6709(+) | 0.0115 | |
NSGAII | 0.0692(+) | 0.0024 | 0.0231(+) | 0.0146 | 0.7113(+) | 0.0062 | |
| |||||||
DTLZ4 | MBSO/D | | 0.0027 | | 0.0008 | | 0.0018 |
MOEA/D | 0.0822(+) | 0.0053 | 0.0202(+) | 0.0010 | 0.6788(+) | 0.0155 | |
NSGAII | 0.1299(+) | 0.1628 | 0.0218(+) | 0.0037 | 0.6748(+) | 0.0853 | |
| |||||||
DTLZ5 | MBSO/D | 0.0186 | 0.0015 | 0.0077 | 0.0047 | 0.4281 | 0.0012 |
MOEA/D | 0.0121(-) | 0.0030 | | 0.0001 | 0.4174(+) | 0.0083 | |
NSGAII | | 0.0003 | 0.0011(-) | 0.0002 | | 0.0003 | |
| |||||||
DTLZ6 | MBSO/D | 0.0207 | 0.0003 | 0.0035 | 0.0040 | | 0.0002 |
MOEA/D | | 0.0038 | | 0.0001 | 0.4190(+) | 0.0092 | |
NSGAII | 0.0555(+) | 0.0260 | 0.0668(+) | 0.0249 | 0.3745(+) | 0.0291 | |
| |||||||
DTLZ7 | MBSO/D | | 0.0005 | | 0.0005 | | 0.0013 |
MOEA/D | 0.1558(+) | 0.0239 | 0.0079(+) | 0.0010 | 0.9319(+) | 0.0031 | |
NSGAII | 0.1124(+) | 0.0935 | 0.0157(+) | 0.0132 | 1.0256(+) | 0.0024 |
“+” means that MBSO/D outperforms its competitor algorithm, “-” means that MBSO/D is worse than its competitor algorithm, and “=” means that the competitor algorithm has the same performance as MBSO/D.
GD and
Problems | GD | | |||
---|---|---|---|---|---|
best | mean | best | mean | ||
ZDT1 | MBSO/D | 0.0009 | 0.0009 | 0.0563 | 0.0569 |
MBSO-DE | | 0.0011 | | 0.1257 | |
MBSO-C | 0.0695 | 0.0912 | 0.5105 | 0.5529 | |
| |||||
ZDT2 | MBSO/D | | 0.0008 | | 0.0274 |
MBSO-DE | | 0.0008 | 0.0997 | 0.1253 | |
MBSO-C | 0.0725 | 0.0905 | 0.4898 | 0.5588 | |
| |||||
ZDT3 | MBSO/D | | 0.0012 | | 0.4187 |
MBSO-DE | 0.0011 | 0.0012 | 0.4126 | 0.4188 | |
MBSO-C | 0.0443 | 0.0589 | 0.5708 | 0.6364 | |
| |||||
ZDT4 | MBSO/D | | 0.0023 | | 0.1460 |
MBSO-DE | 2.9322 | 13.8379 | 1.2958 | 1.3968 | |
MBSO-C | 6.4966 | 15.2905 | 0.8362 | 0.9699 | |
| |||||
ZDT6 | MBSO/D | | 0.0009 | | 0.0082 |
MBSO-DE | 0.0037 | 0.0040 | 0.5322 | 0.5346 | |
MBSO-C | 0.0580 | 0.0813 | 0.6969 | 0.7425 |
This subsection presents the comparison results on IGD, GD, and HV in seventeen problems. Table
Moreover, MBSO/D outperforms NSGAII and MOEA/D in solving DTLZ1 and DTLZ3; this emphasizes that the selection strategy and crossover operators have the advantage in solving multiple local fronts problems. NSGAII is better at solving DTLZ5. The reason for the higher mean IGD value for MBSO/D is because the update strategy is not suitable for MOPs with degenerated PF. According to the median values of IGD metric, Figure
According to the median values of IGD metric, nondominated solutions obtained by MBSO/D on seven DTLZ problems and ten F1-F10 problems.
The main goal of MOEAs is to obtain a set of solutions with good diversity and convergence. To test the algorithms' ability to accomplish this goal, Figure
Evolution of the mean of IGD metric values for F1, F2, DTLZ1, and DTLZ3.
In this subsection, we compare MBSO/D with two MBSOs (MBSO-C [
The mean and the best values of GD and
According to the median values of GD metric, nondominated solutions obtained by MBSO/D on five ZDT problems.
In this subsection, MOHS/D is used to compare with MBSO/D [
Table
IGD obtained by MBSO/D and MOHS/D on five ZDT problems and seven DTLZ problems.
Problems | IGD | |||
---|---|---|---|---|
Best | Median | Worst | ||
ZDT1 | MBSO/D | 1.62e-3 | | 2.04e-3 |
MOHS/D | 1.37e-3 | 1.86e-3 | 2.18e-3 | |
| ||||
ZDT2 | MBSO/D | 8.42e-4 | | 2.45e-3 |
MOHS/D | 2.26e-3 | 2.26e-3 | 3.01e-3 | |
| ||||
ZDT3 | MBSO/D | 6.14e-4 | | 1.26e-3 |
MOHS/D | 9.15e-4 | 1.19e-3 | 1.76e-3 | |
| ||||
ZDT4 | MBSO/D | 1.86e-4 | | 5.21e-3 |
MOHS/D | 1.76e-4 | 1.64e-4 | 4.23e-3 | |
| ||||
ZDT6 | MBSO/D | 1.62e-4 | | 3.41e-4 |
MOHS/D | 1.59e-4 | 1.91e-4 | 2.42e-4 | |
| ||||
DTLZ1 | MBSO/D | 1.77e-2 | 1.81e-2 | 1.92e-2 |
MOHS/D | 4.69e-03 | | 4.16e-2 | |
| ||||
DTLZ2 | MBSO/D | 5.12e-2 | 5.22e-2 | 5.44e-2 |
MOHS/D | 3.23e-3 | | 4.71e-3 | |
| ||||
DTLZ3 | MBSO/D | 6.13e-2 | | 8.26e-2 |
MOHS/D | 8.62e-2 | 1.33e-1 | 2.00e-1 | |
| ||||
DTLZ4 | MBSO/D | 6.23e-3 | 5.30e-2 | 5.74e-2 |
MOHS/D | 8.43e-3 | | 1.06e-2 | |
| ||||
DTLZ5 | MBSO/D | 5.49e-3 | 1.86e-2 | 2.61e-2 |
MOHS/D | 1.34e-3 | | 1.45e-3 | |
| ||||
DTLZ6 | MBSO/D | 1.57e-2 | | 2.51e-2 |
MOHS/D | 1.21e-2 | 2.21e-2 | 3.35e-2 | |
| ||||
DTLZ7 | MBSO/D | 6.25e-2 | 7.85e-2 | 9.21e-2 |
MOHS/D | 2.89e-2 | | 3.07e-2 |
In this paper, we proposed a multiobjective brain storm optimization algorithm, called MBSO/D, based on the idea of decomposition. In this approach, the update strategy of MOEA/D [
The 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 National Natural Science Foundation of China (nos. 61502290, 61806120, 61672334, 61673251, and 61401263), the Fundamental Research Funds for the Central Universities (GK201901010), China Postdoctoral Science Foundation (no. 2015M582606), and Natural Science Basic Research Plan in Shaanxi Province of China (no. 2016JQ6045).