A newly hybrid nature inspired algorithm called HPSOGWO is presented with the combination of Particle Swarm Optimization (PSO) and Grey Wolf Optimizer (GWO). The main idea is to improve the ability of exploitation in Particle Swarm Optimization with the ability of exploration in Grey Wolf Optimizer to produce both variants’ strength. Some unimodal, multimodal, and fixed-dimension multimodal test functions are used to check the solution quality and performance of HPSOGWO variant. The numerical and statistical solutions show that the hybrid variant outperforms significantly the PSO and GWO variants in terms of solution quality, solution stability, convergence speed, and ability to find the global optimum.
In recent years, several numbers of nature inspired optimization techniques have been developed. These include Particle Swarm Optimization (PSO), Gravitational Search algorithm (GSA), Genetic Algorithm (GA), Evolutionary Algorithm (EA), Deferential Evolution (DE), Ant Colony Optimization (ACO), Biogeographically Based Optimization (BBO), Firefly algorithm (FA), and Bat algorithm (BA). The common goal of these algorithms is to find the best quality of solutions and better convergence performance. In order to do this, a nature inspired variant should be equipped with exploration and exploitation to ensure finding global optimum.
Exploitation is the convergence capability to the most excellent result of the function near a good result and exploration is the capability of a variant to find whole parts of function area. Finally the goal of all nature inspired variants is to balance the capability of exploration and exploitation capably in order to search best global optimal solution in the search space. As per Eiben and Schippers [
As per the above, the existing nature inspired variants are capable of solving several numbers of test and real life problems. It has been proved that there is no population-based variant, which can perform generally enough to find the solution of all types of optimization problems [
The Particle Swarm Optimization is one of the most usually used evolutionary variants in hybrid techniques due to its capability of searching global optimum, convergence speed, and simplicity.
There are several studies in the text which have been prepared to combine Particle Swarm Optimization variant with other variants of metaheuristics such as hybrid Particle Swarm Optimization with Genetic Algorithm (PSOGA) [
Ahmed et al. [
Mirjalili and Hashim’s [
Zhang et al. [
Ouyang et al. [
Experimental results show that the hybrid variant has precision, high convergence rate, and great robustness and it can give suitable results of nonlinear equations.
Yu et al. [
Yu et al. [
Abd-Elazim and Ali [
Grey Wolf Optimizer is recently developed metaheuristics inspired from the hunting mechanism and leadership hierarchy of grey wolves in nature and has been successfully applied for solving optimizing key values in the cryptography algorithms [
Mittal et al. [
S. Singh and S. B. Singh [
N. Singh and S. B. Singh [
In this study, we present a newly hybrid variant combining PSO and GWO variants named HPSOGWO. We use twenty-three unimodal, multimodal, and fixed-dimension multimodal functions to compare the performance of hybrid variant with both standard PSO and standard GWO.
The rest of the paper is structured as follows. The Particle Swarm Optimization (PSO) and Grey Wolf Optimizer (GWO) algorithm are discussed in Sections
Initialization Initialize // Evaluate the fitness of agents by using ( while ( for each search agent Update the velocity and position by using ( end for Update Evaluate the fitness of all search agents Update positon first three agents
end while return // first best search agent position
The PSO algorithm was firstly introduced by Kennedy and Eberhart in [
This approach is learned from animal’s behavior to calculate global optimization functions/problems and every partner of the swarm/crowd is called a particle. In PSO technique, the position of each partner of the crowd in the global search space is updated by two mathematical equations. These mathematical equations are
The above literature shows that there are many swarm intelligence approaches originated so far, many of them inspired by search behaviors and hunting. But there is no swarm intelligence approach in the literature mimicking the leadership hierarchy of grey wolves, well known for their pack hunting. Motivated by various algorithms, Mirjalili et al. [
Grey wolf belongs to Canidae family. Grey wolves are measured as apex predators, meaning that they are at the top of the food chain. Grey wolves mostly prefer to live in a pack. The leaders are a female and a male known as alphas. The alpha (
The second top level in the hierarchy of grey wolves is beta (
The third level ranking grey wolf is omega (
If a wolf is not an alpha (
In addition, three main steps of hunting, searching for prey, encircling prey, and attacking prey, are implemented to perform optimization.
The encircling behavior of each agent of the crowd is calculated by the following mathematical equations:
In order to mathematically simulate the hunting behavior, we suppose that the alpha (
When
Many researchers have presented several hybridization variants for heuristic variants. According to Talbi [
In this text, we hybridize Particle Swarm Optimization with Grey Wolf Optimizer algorithm using low-level coevolutionary mixed hybrid. The hybrid is low level because we merge the functionalities of both variants. It is coevolutionary because we do not use both variants one after the other. In other ways, they run in parallel. It is mixed because there are two distinct variants that are involved in generating final solutions of the problems. On the basis of this modification, we improve the ability of exploitation in Particle Swarm Optimization with the ability of exploration in Grey Wolf Optimizer to produce both variants’ strength.
In HPSOGWO, first three agents’ position is updated in the search space by the proposed mathematical equations (
In this section, twenty-three benchmark problems are used to test the ability of HPSOGWO. These problems can be divided into three different groups: unimodal, multimodal, and fixed-dimension multimodal functions. The exact details of these test problems are shown in Tables
Unimodal benchmark functions.
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Multimodal benchmark functions.
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Fixed-dimension multimodal benchmark functions.
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The PSO, GWO, and HPSOGWO pseudocodes are coded in MATLAB R2013a and implemented on Intel HD Graphics, 15.6′′ 3 GB Memory, i5 Processor 430 M, 16.9 HD LCD, Pentium-Intel Core™, and 320 GB HDD. Number of search agents is 30, maximum number of iterations is 500,
In this paper, our objective is to present the best suitable optimal solution as compared to other metaheuristics. The best optimal solutions and best statistical values achieved by HPSOGWO variant for unimodal functions are shown in Tables
PSO, GWO, and HPSOGWO numerical results of unimodal benchmark functions.
Problem number | PSO | GWO | HPSOGWO | |||
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PSO, GWO, and HPSOGWO statistical results of unimodal benchmark functions.
Problem number | PSO | GWO | HPSOGWO | |||
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Firstly, we tested the ability of HPSOGWO, PSO, and GWO variant that were run 30 times on each unimodal function. The HPSOGWO, GWO, and PSO algorithms have to be run at least more than ten times to search for the best numerical or statistical solutions. It is again a general method that an algorithm is run on a test problem many times and the best optimal solutions, mean and standard deviation of the superior obtained results in the last generation, are evaluated as metrics of performance. The performance of proposed hybrid variant is compared to PSO and GWO variant in terms of best optimal and statistical results. Similarly the convergence performances of HPSOGWO, PSO, and GWO variant have been compared on the basis of graph; see Figures
Convergence curve of PSO, GWO, and HPSOGWO variants on unimodal functions.
Further, we noted that the unimodal problems are suitable for standard exploitation. Therefore, these results prove the superior performance of HPSOGWO in terms of exploiting the optimum.
Secondly, the performance of the proposed hybrid variant has been tested on six multimodal benchmark functions. In contrast to the multimodal problems, unimodal benchmark problems have many local optima with the number rising exponentially with dimension. This makes them appropriate for benchmarking the exploration capability of an approach. The numerical and statistical results obtained from HPSOGWO, PSO, and GWO algorithms are shown in Tables
PSO, GWO, and HPSOGWO numerical results of multimodal benchmark functions.
Problem number | PSO | GWO | HPSOGWO | |||
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PSO, GWO, and HPSOGWO statistical results of multimodal benchmark functions.
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Experimental results show that the proposed variant finds a superior quality of solution without trapping in local maximum and to attain faster convergence performance; see Figures
Convergence curve of PSO, GWO, and HPSOGWO variants on multimodal functions.
Thirdly, the suitable solutions of fixed-dimension multimodal benchmark functions are illustrated in Tables
PSO, GWO, and HPSOGWO numerical results of fixed-dimension multimodal benchmark functions.
Problem number | PSO | GWO | HPSOGWO | |||
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Min | Max | Min | Max | Min | Max | |
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PSO, GWO, and HPSOGWO statistical results of fixed-dimension multimodal benchmark functions.
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Convergence curve of PSO, GWO, and HPSOGWO variants on fixed-dimension multimodal functions.
Finally, the accuracy of the newly hybrid approach has been verified using starting and ending time of the CPU (TIC and TOC), CPU time, and clock. These results are provided in Tables
Time-consuming results of unimodal benchmark functions.
Problem | PSO | GWO | HPSOGWO | ||||||
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1.02847 | 0.0176203 | 1.052 | 1.01117 | 0.0526014 | 1.021 | 1.00342 | 0.0105 | 1.002 |
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1.0319 | 0.011 | 1.011 | 1.01083 | 0.0257021 | 1.041 | 1.00562 | 0.011 | 1.011 |
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1.00905 | 0.051 | 1.018 | 1.01281 | 0.002 | 1.025 | 1.0031 | 0.0211 | 1.017 |
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1.00417 | 0.00031 | 1.018 | 1.01785 | 0.00011 | 1.036 | 0.789512 | 0.00015 | 1.014 |
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1.0151 | 0.0523051 | 1.019 | 1.01501 | 0.0213012 | 1.025 | 1.00190 | 0.00010 | 1.012 |
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1.00567 | 0.0276408 | 1.115 | 1.01479 | 0.0136071 | 1.024 | 1.00471 | 0.0132091 | 1.010 |
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1.018 | 0.0765021 | 1.016 | 1.00507 | 0.0367013 | 1.037 | 1.00140 | 0.0116701 | 1.014 |
Time-consuming results of multimodal benchmark functions.
Problem | PSO | GWO | HPSOGWO | ||||||
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TIC and TOC | CPU time | Clock | TIC and TOC | CPU time | Clock | TIC and TOC | CPU time | Clock | |
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1.01232 | 0.0232001 | 1.017 | 1.00181 | 0.0260007 | 1.004 | 1.00211 | 0.0108021 | 1.000 |
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1.01371 | 0.200301 | 1.019 | 1.00617 | 0.0417009 | 1.010 | 1.00781 | 0.00321 | 1.008 |
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1.01709 | 0.0717003 | 1.011 | 1.01117 | 0.0837056 | 1.08 | 1.00125 | 0.0215302 | 1.009 |
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1.01731 | 0.0011 | 1.012 | 1.0160 | 0.0011 | 1.009 | 1.00157 | 0.006 | 1.002 |
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1.02019 | 0.0211005 | 1.014 | 1.00397 | 0.0417042 | 1.010 | 1.00719 | 0.0272012 | 1.007 |
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1.01214 | 0.431507 | 1.019 | 1.0157 | 0.0266007 | 1.007 | 1.01237 | 0.0117009 | 1.001 |
Time-consuming results of fixed-dimension multimodal benchmark functions.
Problem | PSO | GWO | HPSOGWO | ||||||
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TIC and TOC | CPU time | Clock | TIC and TOC | CPU time | Clock | TIC and TOC | CPU time | Clock | |
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1.01375 | 0.0317 | 1.087 | 1.00787 | 0.0415123 | 1.012 | 1.00423 | 0.0003 | 1.005 |
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1.02471 | 0.8783879 | 1.069 | 1.00581 | 0.0529014 | 1.017 | 1.01247 | 0.0126391 | 1.009 |
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1.01691 | 0.00097 | 1.015 | 0.897854 | 0.0372012 | 1.008 | 1.01219 | 0.00161 | 1.006 |
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1.01719 | 0.8788782 | 1.098 | 1.02139 | 0.0156071 | 1.019 | 1.00194 | 0.0136321 | 1.014 |
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1.03241 | 0.03715 | 1.045 | 1.00596 | 0.0166701 | 1.024 | 1.00279 | 0.00107 | 1.011 |
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1.00607 | 0.5198069 | 1.096 | 1.01685 | 0.0196041 | 1.036 | 1.0203 | 0.0017981 | 1.024 |
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1.01307 | 0.59741 | 1.045 | 1.01652 | 0.0413012 | 1.006 | 1.00734 | 0.00231 | 1.001 |
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1.00009 | 0.00077 | 1.011 | 1.01727 | 0.0146781 | 1.001 | 1.02751 | 0.00001 | 1.000 |
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0.906710 | 0.61981 | 0.856 | 1.00582 | 0.0712109 | 1.013 | 1.0071 | 0.02041 | 1.008 |
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1.03272 | 0.0335917 | 1.066 | 1.00522 | 0.0192509 | 1.010 | 1.01017 | 0.0087552 | 1.002 |
To sum up, all simulation results assert that the HPSOGWO algorithm is very helpful in improving the efficiency of the PSO and GWO in terms of result quality as well as computational efforts.
In this article, a newly hybrid variant is proposed utilizing strengths of GWO and PSO. The main idea behind developing is to improve the ability of exploitation in Particle Swarm Optimization with the ability of exploration in Grey Wolf Optimizer to produce both variants’ strength. Twenty-three classical problems are used to test the quality of the hybrid variant compared to GWO and PSO. Experimental solutions proved that hybrid variant is more reliable in giving superior quality of solutions with reasonable computational iteration as compared to PSO and GWO.
The authors declare no conflicts of interest.