Optimization is an eminent method in making opinions and in investigating physical systems. A variety of optimization problems arise in almost all fields of human acts. Mathematically, optimization is a practice of finding the best solution out of the set of feasible solutions. In the last few years, many nature inspired optimization algorithms have been evolved. These include Particle Swarm Optimization (PSO) Algorithm, Genetic Algorithm, Evolutionary Algorithm, Ant Colony Optimization Algorithm, Firefly Algorithm, etc. Attributes like simple application, computing efficiency, derivation free mechanism, flexibility, local optimal avoidance, and global search make nature inspired algorithms an admirable choice of researchers. Existing nature inspired optimization algorithms are proficient for several real life and test optimization problems. However, because of the no free lunch theorem [1], there exists no universal algorithm that can be considered most appropriate and finest of all for any of the optimization problem. Besides developing new nature inspired algorithms, another prevailing concept is hybridising distinct algorithms in order to merge their strengths, minimizing weaknesses, and thus, enhancing the search process.

The PSO is amongst the utmost favourable optimization algorithms often employed in hybrid procedures by the researchers considering simplicity, smaller count of parameters involved, convergence speed, and capability of searching global optima. On various problems of different domains, PSO works successfully and this makes us select the PSO as one of the algorithms for our hybrid to propose.

The PSO algorithm acquires memory, and the collaborative swarm interactions enhances the search procedure. The high exploitation ability of the PSO, which intends to locate the best solution within a limited region of the search domain, gives the PSO an edge over other optimization algorithms. Whereas, low exploration ability results in a lack of assurance of proper sampling of the search domain and thus enhances the chances of rejecting a domain containing high quality solutions. Perfect harmony between exploration and exploitation abilities in the course of selection of the best solution is needed. High exploitation capacity makes the PSO get trapped in local minima when its initial location is far off from the global minima [2-5]. The intent of this study is to reform this drawback of the PSO of getting trapped in local minima.

In this direction, we require an algorithm with a positive acceptable exploration capacity. We utilized the recently developed metaheuristic Grey Wolf Optimizer (GWO), emulating the seeking and hunting techniques of grey wolves for this purpose. In our way, the GWO has been utilized to assist the PSO in a manner to unite their strengths and lessen their weaknesses. The proposed hybrid has two driving parameters to adjust and assign the preference to PSO or GWO. This optimization algorithm is simple to use and quite efficient, with less randomness and varying individuals in assigning local and global search processes.In the literature, many existing hybrids utilize other metaheuristics with the PSO for this downside. Zhang et al. [6] proposed a hybrid variant by connecting the PSO algorithm with the Back Propagation (BP) algorithm. This variant makes use of dominant global search capacity and persuasive local search capacity of the PSO and the BP algorithms, respectively. The hybrid was proposed to train the weights of feed forward neural networks. In this proposed hybrid, a unique selection strategy of inertial weight was used in which at an initial stage, the inertial weight reduces rapidly to attain global minima and then the inertial weight was reduced smoothly around global optima in order to acquire higher accuracy. Moreover, a heuristic way was utilized to transit the search procedures amongst the involved methods. The authors demonstrated that their hybrid variant outperformed the BP and the adaptive PSO algorithm regarding the quality of solution, convergence speed, and stability.

Combining the strengths of Nelder - Mead Simplex Method variant and the PSO, another hybrid variant was put forward by Ouyang et al. [7] to solve the non-linear system of equations. Authors claimed that their proposed hybrid variant overcomes the strain of selection of initial guess of the Simplex Method and imprecision of the PSO because of easily getting captured into local minima.

Another hybrid was presented by Mirjalili and Hashim’s [8] by combining the PSO with the Gravitational Search Algorithm (GSA). The main intention was to assimilate the exploitation capacity of the PSO and the exploration capacity of the GSA. The execution of this combination was verified on some standard functions and compared to the PSO and GSA. The hybrid proved to possess a finer potential to get away from local optima along with rapid convergence in functional optimization.

A newly hybrid PSO variant was developed by Yu et al. [9] by merging the modified velocity model with Space Transformation Search (STS). The proposed model involves monitoring fitness values of pbest and gbest. In the case of pbest, gbest shows no improvements, then being considered as getting trapped in local optima, they are given some disturbances to break away the local optima. The variant was verified on eight classical problems and compared with the standard PSO and STS-PSO, and results revealed that the suggested hybrid variant performed well in determining solutions of both unimodal and multi-model problems.

Esmin et al. [10] suggested another hybrid variant of the PSO with the Genetic Algorithm (GA). Both algorithms were merged with the GA mutation technique. The suggested hybrid acquires an automatic balance maintaining potential between global and local search. The hybrid was validated on a variety of benchmark functions compared to the standard PSO. It performed significantly well with regard to the quality of solution, firmness of solution, convergence speed, and attainment of global optima.

A multi-objective model to minimize water shortages and maximize hydropower generation of the Tao River in China, utilizing an adjustable PSO-GA hybrid algorithm, was conducted by Chang et al. [11]. This hybrid merges the potencies of both PSO and GA to balance the process of sharing good knowledge and natural selection, enabling robustness and effective search procedure. The hybrid was validated by comparing it with the involved techniques.The results showed this hybrid as a promising algorithm with rapid convergence speed. The authors also mentioned that this method has the potential for significant application in large water resource systems.

In another study, Yu et al. [12] recommended a new hybrid variant of the PSO by combining it with the Differential Evolution (DE). In this variant, the PSO and the DE were mixed with the help of a symmetric parameter. In this hybrid, when the population gets clustered around a local optima, the current population conducts adaptive mutation. This hybrid, along with maintaining diversity, relishes the advantages of both of the involved algorithms. The conduct of this hybrid variant was then verified on a number of benchmark test problems and compared with the PSO, the DE, and their variants. The authors have proved that this hybrid variant outperformed all of them regarding the quality of solution, frequency of quality solutions, and works effectively.

Abd-Elazim and Ali [13] developed a novel hybrid variant for tuning static var compensator for multi-machine power system by merging the PSO with the Bacterial Foraging Optimization Algorithm (BFOA) named Bacterial Swarm Optimizer (BSO). In this proposed variant, the personal and global best locations of the PSO algorithm orient the search directions of each tumble behavioural bacterium. The hybrid BSO outperformed both the standard PSO and BFOA. Simulation results validate the proposed tuning approach for static var compensator in comparison to tuning with standard PSO and BFOA. Furthermore, results demonstrate the effectiveness of the suggested technique in improving the stability of power systems varying over a varied range of loading conditions.

With the aim of discovering the optimum solution, Kumar and Vidhyarthi [14] developed another hybrid of the PSO, blending it with the GA. They started the search procedure with the PSO algorithm in order to reduce the search domain and then progressed with the GA method. In the suggested hybrid, the PSO provides diversification and the GA enhances intensification. The proposed PSO-GA blend was validated for task scheduling of basic linear algebra problems, namely LU decomposition and Gauss-Jordan elimination, and also on other heuristics with known solutions. The results revealed that the proposed hybrid performed quite effectively for scheduling problems.

Further, another penalty guided PSO-GA hybrid was proposed by Garg [15] by fusing the strengths of both for constrained optimization problems. In the PSO, the whole population continued to take part in each iteration, whereas the GA intends to transfer only satisfactory individuals of the population to the succeeding generation. In this hybrid, two algorithms were combined in such a way that the GA technique formulated the next generation to be used by the PSO at each iteration by executing crossover, mutation, and selection.To validate the proposed technique, an engineering design problem was investigated. The proposed algorithm was compared with other evolutionary algorithms and the experimental results revealed superiority of the proposed technique and proved the approach more potent in engineering problems.

Ghasemi et al. [16] proposed a novel approach for finding peak particle velocity in open pit mines due to bench blasting. In this approach, the Artificial Neuro Fuzzy Inference System (ANFIS) is merged in a way that they, with the help of the PSO algorithm, optimize the ANFIS arrangement. To validate the effectiveness of the proposed approach, a model based on the Support Vector Regression was developed. The obtained results revealed that the proposed ANFIS-PSO algorithm provides better results in this way; they reduced the gap between the results secured with the ANFIS training and the real results.

Another algorithm was proposed by Javidrad and Nazari [17] by uniting the PSO with the Simulated Annealing (SA). They adapted a turn – based transformation approach to merge them. The process of optimization initiates with the PSO process and transformed to the SA process at the time when the PSO was not adept at getting a superior solution than its previous ones. In a similar way, the SA algorithm continued the optimization process and the turn transfer to the PSO when the SA could not find a better fitted solution. This transformation between the two algorithms continued till a stopping condition was achieved or a desired count of iterations were performed. The proposed PSO-SA algorithm showed better performance in terms of solutions to the problems considered compared to other evolutionary techniques. Moreover, the authors exhibit their algorithm as more reliable and effective for various optimization problems.

In another study, a three phase combination of the PSO and the GA methods was developed by Ali and Tawhid [18]. In the first phase, a stable fitness value was calculated by the PSO algorithm. The second phase involved the division of the population into sub-population using the GA crossover operation and thus increasing the diversity. Moreover, the GA’s mutation operation hampered trapping to local minima. The optimization process was shifted back to the PSO algorithm again in the final stage, and the PSO was executed till the stopping condition was met or the maximum count of iterations was attained. The algorithm was compared with the standard PSO for solving complex global optimization problems and with nine benchmark functions for validating the expertise in determining molecule’s potential energy function. The experimental results revealed that the suggested methodology is promising, efficient, and rapid in getting global minimum for molecular energy function.

Hasanipanah et al. [19] combined the Support Vector Regression with the PSO to anticipate air-over pressure generated by mine-blasting. The PSO method was used to compute hyper-parameters of each of the kernels and the Support Vector Regression procedure was tested with three distinct kernels. To check the accuracy of the suggested approach, multi-linear regression was utilized. The outcomes of the study revealed the reliability of the suggested algorithm in training the Support Vector Regression model.

To enhance the network lifetime of wireless sensor networks, a hybrid combining the PSO algorithm with the Ant Colony Algorithm (ACA) was developed by Kaur and Mahajan [20]. In this hybrid, the ACA was used first to evolve the population and then optimization was progressed with the PSO algorithm. At first, the clusters were constructed depending upon remaining energy, and then, the hybrid dependent clustering was carried out. The results of the study demonstrated enhanced network lifetime in comparison to the other techniques.

Furthermore, analogous to our proposed hybrid, other hybrid variants blend strengths of the PSO algorithm with the GWO algorithm in different ways. Chopra et al. [21] proposed a hybrid in which the PSO and the GWO algorithms were employed in succession. The population attained by one algorithm was utilized by the other in its upcoming iteration. The motive of the hybrid is to utilize the strengths of both methods. Kamboj [22] also presented a similar hybrid in which both the PSO and the GWO run in succession. Unlike Chopra’s hybrid in which the whole of the population was handed over from the first algorithm to the second at each iteration, in this, the whole population was renewed by one algorithm with the best ones of the other algorithm secured in the previous iteration. Another hybrid was proposed by Singh and Singh [23] in which instead of running the algorithms in succession, both algorithms work in parallel using the mixture of algorithm commanding equations. Şenel et al. [24] conducted another study in which a PSO-GWO hybrid variant was developed with the optimization control process under the PSO algorithm. In this approach, few individuals of the PSO algorithm were selected and replaced with the best individuals of the GWO algorithm. The GWO algorithm runs for a limited population and for a limited number of iterations in order to create partially better individuals. These partially improved individuals possess the potential of avoiding getting trapped in local minima. With intact PSO algorithm stability, this hybrid takes the support of the GWO for exploration. Teng et al. [25] proposed another hybrid variant combining the GWO with the PSO. To initiate the individual’s location, Tent Chaotic sequence was utilized to increase the diversification of the wolf pack, and nonlinearity of the control parameter was utilized for enhancing local and global exploration along with improving the convergence speed of the algorithm. The notion of the PSO utilizes the pre-eminent position of the individual and prevents the wolf pack from falling into local optima.

In a similar direction, we proposed a hybrid variant of the PSO and the GWO. The main features of our study are: (1) In our hybrid variant, two driving parameters are employed to adjust the control of the optimization process between the PSO and the GWO. (2) The aim of this approach is to merge the strengths of the PSO and the GWO in a way that improves local search ability and accelerates algorithm optimization. (3) Non-linear control parameter strategy is adopted to have coordination between the exploration and exploitation ability.

The remaining paper is organized as follows:

Section 2 illustrates the engaged optimization algorithms, depicts the proposed hybrid variant and illustrates the benchmark functions used to assess our approach. Section 3 represents the experimental outcomes, including discussion. Section 4 concludes the paper and outlines the subsequent scope.

The PSO, GWO, and SGWO pseudo codes were coded in visual basic. In all the functions, parameters of PSO and GWO were chosen as [23, 24]: w = 0.7, c₁ = c₂ = 2, population size = 30, number of iterations = 30.

Experiments were redone 25 times for each of the methods and for each of the benchmark functions. The methods were compared with regard to their best, worst, and average values as well as their mean absolute deviation (MAD) and standard deviation (Std. Dev.). The numerical and statistical results are illustrated in Table 2. The graphs of the values obtained in 25 runs for PSO, GWO, and SGWO for each of the test functions are shown in Fig. (2).

The obtained results revealed that in terms of average values, Std. Dev. and MAD, our hybrid SGWO outperformed both PSO and GWO notably, with only F₂ and F₃ supplying better results in terms of the average for the GWO, in terms of Std. Dev. and MAD, only F₂ supplied better results for the GWO. Also with regard to best value, the GWO gives better results than our hybrid for functions F₂, F₃, F_4, and F₅ only. However, our method performed superior to PSO in all respects. So, we can conclude that SGWO worked fairly well in comparison to both PSO and GWO. This certainly marks the success of our hybridization technique. The results indicate that our hybrid is directed towards improving the downside of the PSO of getting trapped in local minima by merging it with the GWO in this adjustable manner.

Table 1. Benchmark functions.

Function Name	Function Expression	Type	Range	fmin
Sphere function		Unimodal	[-100, 100]	0
Schwefel 2.22 function		Unimodal	[-10, 10]	0
Schwefel 1.2 function		Unimodal	[-100, 100]	0
Schwefel 2.21 function		Unimodal	[-100, 100]	0
Step 2 function		Unimodal	[-100, 100]	0
Quartic Noise		Unimodal	[-1.28, 1.28]	0
Schwefel 2.6 function		Unimodal	[-100, 100]	0
Matyas function		Unimodal	[-10, 10]	0
Booth function		Unimodal	[-10, 10]	0
Himmelblare function		Multimodal	[-5, 5]	0
Camel function Three hump		Multimodal	[-5, 5]	0

Fig. (2). (F1-F11) Comparison graphs of values obtained by benchmark functions.

In this paper, an adjustable hybrid of the PSO and the GWO has been proposed. Our strategy aims to prevent the PSO from getting trapped in local minima and merging the strengths of both. With this intention, two driving parameters are employed to adjust the control of the optimization process between the PSO and the GWO. We have analysed our algorithm on eleven benchmark functions and compared it with the PSO and the GWO. The numerical and statistical results of the experiments reveal that the proposed hybrid is better in comparison to both PSO and GWO in the searchability, quality of solution, stability of solution, and ability to seek global optima. The proposed hybrid can be used for solving various real life optimization problems. Though the SGWO algorithm refines result quality, the computational complexity also gets elevated. Thus, lowering the computational complexity would be another issue of future work. Moreover, we will apply the proposed hybrid for tuning the parameters of the fuzzy logic controller to carry out water quality estimation and prediction.

Not applicable.

The data supporting the findings of this study is available in this article.

None.