In the last two decades, swarm intelligence optimization algorithms have been widely studied and applied to multiobjective optimization problems. In multiobjective optimization, reproduction operations and the balance of convergence and diversity are two crucial issues. Imperialist competitive algorithm (ICA) and sine cosine algorithm (SCA) are two potential algorithms for handling single-objective optimization problems, but the research of them in multiobjective optimization is scarce. In this paper, a fusion multiobjective empire split algorithm (FMOESA) is proposed. First, an initialization operation based on opposition-based learning strategy is hired to generate a good initial population. A new reproduction of offspring is introduced, which combines ICA and SCA. Besides, a novel power evaluation mechanism is proposed to identify individual performance, which takes into account both convergence and diversity of population. Experimental studies on several benchmark problems show that FMOESA is competitive compared with the state-of-the-art algorithms. Given both good performance and nice properties, the proposed algorithm could be an alternative tool when dealing with multiobjective optimization problems.
With the continuous innovation of technology and the rapid development of industrial production, the multiobjective optimization problem (MOP) has gradually become a research focus in the current scientific and engineering fields [
The subobjectives in a multiobjective optimization problem are usually mutually constrained. There is no absolute optimal solution, and only the dominance of the solutions can be adopted to evaluate the pros and cons of the solutions. Therefore, intelligent algorithms are usually used to approximate to the Pareto front, and a set of Pareto optimal solution sets of algorithm is received. The swarm intelligence evolution algorithm is optimized in the form of swarms. In the past few decades, swarm intelligence evolution algorithm has been continuously developed and extended, such as genetic algorithm (GA) [
The first type is domination-based algorithms. This type of algorithm is represented by methods such as NSGAII [
The second type is the MOEA algorithm based on decomposition. MOEA based on decomposition (MOEA/D) [
The third type of algorithm is the MOEA algorithm based on evaluation indicators. This type of algorithm is represented by algorithms such as IBEA [
In the past two decades, a variety of intelligent optimization algorithms have been introduced to deal with multiobjective optimization problems and have achieved the expected results, such as particle swarm optimization, grey wolf optimization [
In order to prevent the population from falling into the local optimal solution, mutation operation is a commonly used strategy in multiobjective evolutionary algorithms. However, the use of mutation operations may change the original trajectory of some excellent solutions. Therefore, in order to improve the global search ability and improve the diversity of the proposed algorithm, a hybrid generation method is introduced. The sine cosine algorithm (SCA) [
In addition, archive strategy is also adopted by many algorithms [
In this paper, a fusion multiobjective empire split algorithm (FMOESA) is proposed. Inspired by the imperialist competitive algorithm, when dealing with multiobjective optimization, the excellent solutions are saved through the behavior of empire splitting. The approach of colonial individuals to empire individuals was also exploited when producing offspring individuals. In addition, in order to balance convergence and diversity, a new individual power assessment mechanism is also proposed. Finally, in order to verify the effectiveness of the proposed algorithm, FMOESA is compared with four state-of-the-art MOEAs on various well-known benchmark MOPs. The experimental results show that FMOESA is capable of obtaining high-quality solutions and the power assessment mechanism works well on maintaining good balance of population diversity and convergence.
The main contributions of this paper are highlighted as follows. A new reproduction of offspring is introduced, which combines three operators: ICA, SCA, and GA. For empire individuals, the offspring are generated by performing sine and cosine operators with other empire individuals. For colonial individuals, the offspring are generated through crossover and mutation operations with their empire. In other words, the idea of colonial individuals approaching empire individuals through crossover and mutation operations is similar to that in ICA. A novel power evaluation mechanism is proposed to identify individual performance. When evaluating the performance of an individual, convergence and diversity are considered simultaneously. This evaluation mechanism is not only used to distinguish between imperial and colonial individuals but also to eliminate redundant individuals in the population. This strategy can balance the convergence and diversity of the entire population. Archive strategy is not adopted in this paper. The outstanding individuals selected after each iteration constitute a candidate solution set, and, at the same time, they are used as the parent population of the next iteration to directly guide the process of the next iteration. This operation saves computing resources and reduces unnecessary computational complexity.
The remainder of this paper is organized as follows. Section
In general, multiobjective optimization problems (MOP) can be defined as
A solution
A set
A set that contains all the Pareto optimal objective vectors is called the Pareto front, denoted by
The traditional imperialist competitive algorithm (ICA) is mainly used to deal with single-objective optimization problems. It is a sociopolitical evolutionary algorithm proposed by Atashpaz-Gargari and Lucas [ Step 1: form an empire. The imperialist competitive algorithm first generates an initial country by a random method, and each country represents a solution to the problem. These countries are divided into two categories: colonial countries and colonies according to the size of the power (the quality of the solution). The former several countries became colonial countries. Then the remaining countries were assigned as colonies to the colonial countries in order according to the power of the colonial countries. The more powerful the colonial countries, the more colonies were assigned. The colonial countries and the colonies they belong to are collectively called empires. Step 2: assimilation and revolution. After the formation of the empire, the economic, cultural, and language attributes of the colonies will inevitably tend to belong to the colonial countries. This process is called assimilation. The goal of assimilation is to improve the quality of the solution of all countries, which can increase the influence of colonial countries on colonies. In order to be consistent with history, the process of colonial tending to empire always has a certain deviation. In extreme cases, there may even be a reverse deviation, that is, the colonial revolution. If the colony is undergoing assimilation and revolution, in the process, the power surpassed the colonial country; then the colony will replace the colonial country and establish a new empire. Step 3: colonial competition. There is competition between empires. The decline of the power between empires will make the colonies of weak empires deprived of powerful empires until the weak empires disappear. At the same time, new empires will continue to appear in the competition. After generations of assimilation, revolution, and competition, ideally only one empire will remain, and all countries are members of that empire. Through such a series of evolutionary operations, the algorithm finally finds the global optimal solution.
The standard sine cosine algorithm (SCA) was proposed by Australian scholar Mirjalili in 2016 [
In (
As shown in (
The theoretical advantages of SCA in solving the global optimal solution of an optimization problem with sufficient candidate solution set size and number of iterations are as follows: SCA is the same as other swarm intelligence optimization algorithms. Compared with the algorithm of a single candidate solution, the candidate solution set of a certain size achieves a stronger search ability and the ability to escape the local optimal trap, and SCA has sufficient random search ability. The optimal candidate solution is always retained during the iteration process and used as a basis for updating the candidate solution set, and there is a tendency to move to the optimal solution space during the search process. The randomness of its global search and local development has stronger adaptability and stability than searching and developing in two stages.
In this section, the details of the proposed fusion multiobjective empire split algorithm (FMOESA) are documented. Particularly, the framework of the proposed algorithm is presented first in Section
The first part of the algorithm is the initialization phase (lines 2–8 in Algorithm
Initialize Execute the opposition-based learning operation Evaluate the fitness value of each individual Evaluate the power of each individual Select empires and then assign colonies to them Evaluate the fitness value of each individualV Evaluate the power of each individual in Redistributing the empire and their colonies according to the power Sort the colonies in reverse order Implementing empire reduction strategy until the number of empires is
The second part of the algorithm is the iterative update phase (lines 10–26 in Algorithm
Most of the traditional algorithms adopt the initialization method to generate the initial population. Due to the lack of prior knowledge, the probability of searching the population to a better area is greatly reduced. Opposition-based learning (OBL) [
In this paper, a reverse population is obtained based on OBL after randomly generating an initial population. In the original population and the reverse population, the optimal
In the algorithm, population
Randomly choose an to Produce offspring
Through the fusion reproduction strategy, sufficient information was exchanged not only between empires but also between colonies and empires. The former focuses on diversity, while the latter focuses on convergence.
The empire power evaluation mechanism includes two different evaluation methods. The first case is the assessment of the empire’s power during the initialization phase. The second case is the assessment of the empire’s power during the update iteration.
In the initialization phase, convergence is generally poor, so the assessment of individual power is mainly based on convergence. For the
In the second stage, not only does the entire population need to improve convergence, but also diversity is important. Therefore, when evaluating individual power, it is necessary to consider both convergence and diversity. During the update iteration phase, for the
It is worth noting that before calculating individual power, the convergence index (
In the population, each individual has his own role: empire or colony. In the FMOESA, individuals are assigned to corresponding roles according to their nondominated level and power. First, all nondominated solutions in the population are selected as empires. Then, the remaining individuals are sorted in descending order according to the value of power. Finally, allocate colonial individuals among the remaining populations. The specific method is as follows: we pick the individual with the largest power at a time and then assign it to the nearest empire, until (
It should be noted that the number of colonies contained in each empire is not fixed. That is, it is possible that some empires do not have colonial individuals, and some empires contain more than one colony individual.
In the later stages of the algorithm, the number of empires may exceed
Calculate the empire’s power according to equation ( Remove
In order to demonstrate the effectiveness of the proposed framework, we compare its results with respect to those obtained by MOEA/D [
Parameter settings of all compared algorithms.
Algorithm | Parameter settings |
---|---|
MOEA/D | |
dMOPSO | |
NSGAII | |
MOEA/D-STM | |
MOEA/D-ACD |
We adopted eight test problems whose Pareto fronts have different characteristics. In these functions, convexity, concavity, disconnections, and multifrontality are included. The 2-objective test suite of Zitzler-Deb Thiele (ZDT) [
The general parameter settings are as follows: ZDT:
In our experimental study, the widely used metrics inverted generational distance (IGD) [
The IGD metric is defined as follows. If
If
The hypervolume (HV) metric is defined as the volume of the hypercube enclosed in the objective space by the reference point and every vector in the Pareto approximation set
The HV metric is applied to access both convergence and maximum spread of the solutions for the Pareto approximation set obtained with any MOESs. In addition, lager values of this measure value that the solutions are closer to the true PF and that the solutions cover a wider extension of it. In this paper, the HV metric is calculated with respect to a given reference point
Table
IGD results on the eight problems.
Function | FMOESA | MOEA/D | dMOPSO | NSGAII | STM | ACD | |
---|---|---|---|---|---|---|---|
ZDT1 | Mean | 4.664 | 7.806 | 4.922 | 1.907 | 6.429 | |
Std | 5.834 | 8.352 | 2.585 | 3.583 | 5.734 | ||
3.013 | 4.363 | 2.147 | 2.827 | 6.083 | |||
ZDT2 | Mean | 2.984 | 6.926 | 4.354 | 1.847 | 1.548 | |
Std | 2.475 | 1.038 | 1.352 | 3.452 | 3.154 | ||
1.193 | 8.068 | 5.093 | 1.748 | 9.273 | |||
ZDT3 | Mean | 1.312 | 1.094 | 7.223 | 2.245 | 1.072 | |
Std | 5.587 | 7.298 | 7.734 | 5.827 | 6.863 | ||
9.834 | 4.002 | 1.105 | 9.641 | 8.267 | |||
ZDT6 | Mean | 2.315 | 1.928 | 2.606 | 1.509 | 2.101 | |
Std | 1.894 | 6.547 | 2.521 | 1.696 | 4.028 | ||
1.945 | 1.356 | 9.137 | 3.905 | 7.302 | |||
DTLZ1 | Mean | 4.119 | 3.201 | 1.920 | 8.202 | 2.868 | |
Std | 1.439 | 1.810 | 6.101 | 2.037 | 5.827 | ||
2.938 | 8.076 | 4.466 | 3.082 | 6.286 | |||
DTLZ2 | Mean | 4.579 | 4.751 | 5.424 | 7.442 | 4.661 | |
Std | 1.326 | 1.498 | 2.337 | 3.001 | 2.319 | ||
2.270 | 5.386 | 7.156 | 1.090 | 3.917 | |||
DTLZ4 | Mean | 1.486 | 1.517 | 5.373 | 1.582 | 5.704 | |
Std | 1.435 | 1.984 | 7.267 | 3.916 | 2.397 | ||
5.681 | 3.569 | 8.324 | 1.729 | 9.143 | |||
DTLZ7 | Mean | 1.926 | 1.256 | 6.809 | 5.321 | 1.275 | |
Std | 2.686 | 5.016 | 6.278 | 5.011 | 6.764 | ||
3.672 | 2.579 | 6.788 | 5.156 | 3.582 | |||
+/=/− | 5/0/3 | 5/1/2 | 6/0/2 | 6/2/0 | 5/0/3 |
As shown, FMOESA is the most effective algorithm in terms of the number of the best results it obtains. MOEA/D performs very competitively to FMOESA.
For the two-objective test function, FMOESA obtains the best performance on ZDT2 and ZDT3. The performance of dMOPSO is poor compared to that of its competitors, and it performs best only on ZDT1 and ZDT6. Our proposed FMOESA obtains significantly better performance than NSGAII and MOEA/D-STM. Except for ZDT6, the performance of MOEA/D and MOEA/D-ACD is worse than FMOESA.
For the three-objective test function, FMOESA and MOEA/D have comparable performance. FMOESA shows clear improvements over the other four compared algorithms on DTLZ2 and DTLZ7. For MOEA/D, it obtains the best performance on DTLZ1 and DTLZ4. FMOESA performs slightly worse than NSGAII on DTLZ1 and DTLZ4. Except for the similar performance on DTLZ4 and DTLZ7, FMOESA performs better than MOEA/D-STM on other functions. MOEA/D-ACD and MOEA/D have similar performance on most of the test functions.
As shown in Table
Function | FMOESA | MOEA/D | dMOPSO | NSGAII | STM | ACD | |
---|---|---|---|---|---|---|---|
ZDT1 | Mean | 9.153 | 9.127 | 9.158 | 9.154 | 9.136 | |
Std | 1.332 | 1.163 | 1.302 | 1.429 | 1.227 | ||
1.112 | 2.627 | 2.347 | 1.352 | 1.945 | |||
ZDT2 | Mean | 8.204 | 8.226 | 8.328 | 8.317 | 8.247 | |
Std | 5.862 | 2.643 | 1.207 | 1.745 | 2.729 | ||
4.927 | 3.669 | 3.031 | 1.340 | 4.262 | |||
ZDT3 | Mean | 9.458 | 9.426 | 9.495 | 9.491 | 9.477 | |
Std | 7.704 | 1.752 | 1.372 | 1.168 | 1.387 | ||
3.684 | 3.061 | 7.797 | 6.908 | 7.265 | |||
ZDT6 | Mean | 5.643 | 7.362 | 9.001 | 8.692 | 7.996 | |
Std | 4.163 | 9.525 | 3.214 | 8.137 | 5.208 | ||
5.979 | 1.581 | 6.842 | 1.237 | 4.253 | |||
DTLZ1 | Mean | 9.798 | 9.756 | 9.728 | 0.00 | 9.767 | |
Std | 7.796 | 5.778 | 7.884 | 0.00 | 3.662 | ||
8.596 | 3.032 | 4.968 | 1.261 | 2.311 | |||
DTLZ2 | Mean | 9.255 | 9.285 | 9.258 | 8.795 | 9.247 | |
Std | 1.135 | 1.074 | 9.258 | 6.853 | 6.286 | ||
1.456 | 2.670 | 5.148 | 3.062 | 7.214 | |||
DTLZ4 | Mean | 9.273 | 9.260 | 9.265 | 8.807 | 9.268 | |
Std | 8.437 | 1.056 | 9.468 | 9.221 | 3.109 | ||
1.831 | 2.449 | 8.516 | 3.029 | 6.287 | |||
DTLZ7 | Mean | 4.757 | 4.744 | 4.790 | 4.821 | 4.779 | |
Std | 7.964 | 3.742 | 1.980 | 2.376 | 2.817 | ||
7.142 | 3.645 | 4.928 | 3.082 | 2.011 | |||
+/=/− | 5/2/1 | 6/0/2 | 5/2/1 | 6/1/1 | 7/0/1 |
To further compare the difference between FMOESA and the other compared algorithms, the Pareto fronts of some adopted test problems have been shown in Figures
The approximate front of FMOESA, MOEA/D, dMOPSO, NSGAII, MOEA/D-STM, and MOEA/D-ACD on ZDT2. (a) FMOESA, (b) MOEA/D, (c) dMOPSO, (d) NSGAII, (e) MOEA/D-STM, and (f) MOEA/D-ACD.
The approximate front of FMOESA, MOEA/D, dMOPSO, NSGAII, MOEA/D-STM, and MOEA/D-ACD on ZDT3. (a) FMOESA, (b) MOEA/D, (c) dMOPSO, (d) NSGAII, (e) MOEA/D-STM, and (f) MOEA/D-ACD.
The approximate front of FMOESA, MOEA/D, dMOPSO, NSGAII, MOEA/D-STM, and MOEA/D-ACD on DTLZ1. (a) FMOESA, (b) MOEA/D, (c) dMOPSO, (d) NSGAII, (e) MOEA/D-STM, and (f) MOEA/D-ACD.
The approximate front of FMOESA, MOEA/D, dMOPSO, NSGAII, MOEA/D-STM, and MOEA/D-ACD on DTLZ7. (a) FMOESA, (b) MOEA/D, (c) dMOPSO, (d) NSGAII, (e) MOEA/D-STM, and (f) MOEA/D-ACD.
As seen from the performance diagram of ZDT3, DTLZ1, and DTLZ7, our algorithm can well converge to its true PF, especially in DTLZ1, dMOPSO, and MOEA/D-STM may be trapped in local optimization and cannot well converge to the true PF. Besides, it can be observed that FMOESA produces better distribution than other algorithms, especially in ZDT2, ZDT3, and DTLZ7.
In this paper, a fusion multiobjective empire split algorithm has been proposed. In the proposed algorithm, the opposition-based learning is adapted in the initialization phase. In this way, the diversity of the initial population is enhanced, and the ability of the algorithm to approach the global optimal solution is enhanced. Then, a fusion reproduction strategy is utilized to produce high-quality offspring population. Inspired by the ICA, individuals with better convergence and diversity are identified as empire individuals in the selection mechanism. This process marked the demise of the old empires and the birth of new empires. Finally, a novel type of power evaluation mechanism was proposed to select candidate solutions with superior performance. For quantifying the performance of the proposed algorithm, a series of well-designed experiments are performed against peer competitors, which include four state-of-the-art MOEAs competitors and four multiobjective evolutionary algorithm competitors, upon a couple of widely used benchmark test suites with 2-, and 3-objective. It is demonstrated by the experimental results that FMOESA is quite competitive on the majority of all the test instances. FMOESA especially shows significant improvement over other peer algorithms in terms of the distribution of the obtained solutions.
In future research, we will investigate into more effective mechanisms for the selection process. Also, we will put efforts into the MOPs with irregularly shaped Pareto fronts.
The data used to support the findings of this study are included within the article. The test data are also available from the corresponding author upon request via email.
The author declares no conflicts of interest with respect to the research, authorship, and/or publication of this article.