Manta ray foraging and Gaussian mutation-based elephant herding optimization for global optimization

Optimization has long been a fundamental tool for applications in mathematics, medicine, engineering, the military, and many other fields. In general, optimization is an approach used to determine the best performance of a system that can be achieved based on optimized parameters to make the final solution as perfect and efficient as possible [1]. In optimization problems, the final goal is usually considered the maximization or minimization of some objective function (the function used to evaluate the nature or quality of the resulting solution). In this case, several steps are involved, such as defining a mathematical model for the system, identifying variable, defining the type of system needed, specifying constraints, and then finding the optimization response [2].

Traditional optimization techniques include the dynamic programming approach (DPA), stochastic search, the steepest descent method, and Newton’s method. Most of these methods are nonpolynomial and can find optimal solutions smoothly. Unfortunately, they are usually limited by the specific problem and the size of the given dataset. Furthermore, they are often highly complex and specialized in nature. In addition, the execution times and running costs are very high [3]. Therefore, such approaches are only suitable for solving small-scale problems. Currently, most real-world optimization problems are complex, highly nonlinear, and large-scale spatial, and the information metrics used are essentially multifaceted. On this basis, traditional methods are not suitable for such situations. The popular trend is to combine the multifaceted nature of the subject system with various proposed heuristics to solve difficult problems. Heuristics methods include nearest neighbor algorithms, evolutionary algorithms, insertion algorithms, memetic algorithms, dynamic relaxation, etc. By introducing greedy mechanisms, the number of search steps is fixed, and the number of iterations is reduced. With these advantages, heuristic methods are able to find near-optimal solutions with less time costs than those of other methods [4]. However, most of them are designed for specific problems. Due to the introduction of greedy mechanisms, these methods often fall into local optima when solving complex problems. Soon after the development of heuristic methods, metaheuristics, which are problem-independent adaptive strategies, emerged to provide a set of guidelines or strategies for developing heuristics. In this term, “meta” means using a higher level, usually one level higher than that of the given heuristic, to solve a variety of intractable and complex hard problems [5]. Specifically, metaheuristics perform the search process by using multiple agents, which are essentially systems that find the best solution based on constraints and rules over multiple iterations. The iterative process is stopped when a predefined criterion is satisfied, at which point the solution found is determined to be the best solution and the system reaches a fused state. Metaheuristics are not dependent on gradient information, are easy to execute and are highly flexible because they treat the optimization process as a black box [6]. As research has progressed, metaheuristics have been developed as powerful tools and are now widely used in various fields.

In general, metaheuristic algorithms are classified into four categories, namely, evolutionary-based algorithms, physics-based algorithms, population-based algorithms, and human-based algorithms. Inspired by Darwinian evolutionary theory, evolutionary-based algorithms were introduced in the early 1970s and usually draw from natural rules to find optimal solutions. By setting one or more operators (crossover, variation, or selection operators), such algorithms evaluate search agents during optimization [7]. Evolutionary-based algorithms include genetic algorithms (GA) [8], biogeography-based optimization algorithms (BBO) [9], differential evolutionary algorithms (DE) [10], etc. Physics-based algorithms are modelled according to physical rules. For example, the gravitational search algorithm (GSA) [11] and the equilibrium optimizer (EO) [12] are all physics-based approaches. Population-based algorithms simulate the group behaviour of animals, performing hunting based on cooperation. In population-based algorithms, multiple agents perform the search process together and share useful information with each other, thus improving operational efficiency. Representative algorithms include the artificial bee colony (ABC) algorithm [13], the sailfish optimizer (SFO) [14], whale optimization algorithm (WOA) [15], Harris hawks algorithm (HHO) [16], etc. Human-based algorithms generally simulate the special behaviour of humans. Such algorithms include the harmony search algorithm (HS) [17], teaching-learning-based optimization (TLBO) [18].

Elephant herding optimization (EHO) is a population-based algorithm proposed by Wang et al. [19] . Inspired by the nomadic herding behaviour of elephant populations, EHO is designed to solve global optimization problems. In EHO, an entire population is divided into different clans. Each clan is led by a female matriarch, while the male elephants live independently when they reach adulthood. EHO sets up two operators, a clan updating operator and a separation operator, to update the positions of the agents. A clan updating operator is implemented for each individual, and the next position of each individual is updated according to the current position in relation to the position of the clan leader. Subsequently, the agent with the worst position is replaced by the separation operator to speed up convergence and enhance the diversity of the population. In terms of advantages, EHO has fewer controlled parameters, is easier to implement, and has better performance than other optimization approaches, so it attracts much attention from mathematicians and engineers [20].

In response to the shortcomings of EHO, researchers have proposed many improvement ideas and optimization methods. For instance, Muthusamy et al. [21] introduced the sine cosine algorithm (SCA) and opposition-based learning (OBL) to update the positions of agents in the original EHO algorithm. Through this operation, the convergence speed and the searchability of the algorithm were enhanced. Li and Wang [20] proposed a new improved EHO approach based on dynamic topology and BBO methods. The operators of this EHO variant were updated, and a new separation operator was set, thus ensuring that the evolutionary process of the examined population moved in a better direction. Ismaeel et al. [22] proposed three versions of an unbiased algorithm, EEHO15, EEHO20, and EEHO25, to overcome the defect of unreasonable convergence in the original EHO algorithm. Three similar variants were proposed by Elhosseini et al. [23], namely cultural-based, alpha-tuning, and biased initialization EHO, with the aim of strengthening the global optimization capability of EHO. Balamurugan et al. [24] proposed an opposition-based EHO (OEHO) algorithm. In their work, opposition-based learning was adopted to enhance the performance of EHO. Xu et al. [25] proposed a novel algorithm, LFEHO, which combined Levy flight with EHO to overcome the problems of the original EHO algorithm falling into local optima and exhibiting poor convergence performance. Improved EHO algorithms such as IMEHO were presented by Li et al. [26] The robustness and diversity of a given population are optimized by using a new global learning-based evolution strategy to update the velocities and positions of agents. Reference [27] extended and improved the original EHO algorithm by designing six strategies to update the agents. The information obtained from the previous iterations of individuals was adopted for weighing operations. Therefore, the obtained results were better than those output by the original EHO algorithm. In addition, engineers extended the application of EHO to solve practical engineering problems in real production cycles.

In addition, MRFO is a novel optimization algorithm proposed by Zhao et al. [28]. It is a promising and powerful optimizer inspired by the predatory behaviour of manta rays. MRFO has three foraging operators, namely, chain foraging, cyclone foraging, and tumble foraging operators. Recently, researchers have introduced MRFO to other fields. For example, Ekinci et al. [29] combined the opposition-based (OBL) with MRFO and hybrid simulated annealing algorithm to provides MRFO a better exploration capability. Shaheen et al. [30] optimized the MRFO algorithm by using adaptive penalty parameters and introduced it to handle a thermal scheduling problem. In addition, MRFO has been extended to many fields, such as feature selection (FS) [31], fuel power generation [32], system identification [33], photovoltaic system operation [34], and other areas.

According to the no-free lunch (NFL) theorem [35], any algorithm that is applicable for solving a given problem cannot provide the best solution for all problems. In other words, there is a need to improve the metaheuristic algorithms that exist in reality. Maintaining a balance between exploration and exploitation is the common and fundamental feature of any optimization algorithm and the most challenging problem in metaheuristics. Among them, exploration is the process of exploring optimal solutions over a wide range of unknown regions, while the exploitation of explored regions allows an algorithm to effectively find high-quality solutions. Therefore, it is necessary to fine-tune exploration and exploitation, as they are the two main factors used to obtain satisfactory results. In addition, it has been shown that although some algorithms are very effective in solving certain types of problems or large-scale problems, more significant optimization results can be achieved by combining the advantages of different algorithms in a hybrid manner [36]. By combining them, the convergence performance of the resulting hybrid algorithm is improved and the balance between exploration and exploitation is better maintained, thus allowing the algorithm to avoid falling into local optima. The above facts have motivated researchers to continuously develop hybrid metaheuristic algorithms with good exploration and exploitation capabilities and have made our research highly relevant.

Like most metaheuristic algorithms, EHO is prone to falling into local optima when dealing with complex multipeaked problems, which stems from the stochasticity of such problems and the imbalance between exploration and exploitation. Therefore, these shortcomings become major issues that affect the performance of EHO. At the same time, EHO also has drawbacks such as a poor exploitation capability and a low convergence rate. To summarize, there are three defects in the EHO algorithm that need to be solved. The rest of the paper is organized as follows: Sect. 2 reviews the original EHO and MRFO algorithms. Section 3 provides a detail of the proposed MGEHO method. Experiments and the analysis of the results are given in Sect. 4. Tests on practical engineering problems are conducted in Sect. 5. Section 6 summarizes the conclusions of this paper and plans for future work.

In EHO, the position update strategies of the clan leader and other members are different. Specifically, the position of the patriarch is determined by the action factor \(\beta\) and the central positions of all agents in the same clan during each iteration. When \(\beta\) is small, the patriarch moves near the origin. When \(\beta\) is large, the patriarch suddenly moves to the center of the clan. This kind of strategy lacks guidance towards the global optimal solution, so the algorithm may fall into a local optimum, which affects its convergence efficiency and reduces its exploration ability. Second, the search process of EHO is fixed and lacks an effective way to control the balance between exploration and exploitation, and this largely affects the performance of the algorithm. Third, the worst individuals in the clan are randomly replaced by the separation operator, which makes it difficult to ensure the improvement of population diversity.

Therefore, MGEHO is proposed for the purpose of improving the above shortcomings of EHO. In this section, three aspects of the newly developed strategy are explained in detail.

In this section, experimental results are presented. To verify the effectiveness and applicability of MGEHO, experiments are conducted on 33 classical benchmark functions, 32 CEC2014 and CEC2017 benchkmark functions. To ensure the fairness of the experiments, each algorithm is run 30 times independently on the benchmark functions to maintain stability. In addition, the population size N for all algorithms is set to 30, and the maximum number of iterations \({t_{\max }}\) is set to 500. Statistical tests are conducted to examine the differences between the different algorithms. The environmental conditions are an Intel (R) Core (TM) i5-9300H CPU @ 2.40 GHz, 16 GB of RAM, and the Windows 10 operating system, and the simulation experiments are set up on the MATLAB R2019b platform.