1 Introduction

Studying optimization issues is essential for solving many scientific and technical problems [1,2,3]. New techniques are being created and employed to address the current issues in science and engineering [4, 5]. For example, in NP-complete problems, when it is difficult to find the optimal solution to the problem, meta-heuristic is one of the fast feasible solutions [6, 7]. It is an effective way to provide an actionable answer in due course, but that doesn’t mean the method provided is the best. However, metaheuristic algorithms are not foolproof, and related studies have pointed out their shortcomings. According to the well-known “No-Free-Lunch Theorem” [8], it is difficult to find an algorithm that performs well in the face of all problems. Therefore, more effort is needed to develop algorithms that work well for most problems [9,10,11]. In recent decades, a rising number of academics have focused on how to balance exploitation and exploration to improve the performance of the meta-heuristic algorithm (MHA) [12]. Exploration refers to an algorithm’s ability to discover solutions, which are distributed in different regions of space [13], whereas exploitation emphasizes the idea of finding better solutions by speeding up the process of searching for the best solution in potential regions [14]. While exploitation strategies are known to increase the likelihood of getting stuck in a local optimum, they can also speed up convergence to a global optimum [15]. On the other hand, exploration and exploitation tend to increase the probability of finding a globally optimal region at the expense of the algorithm’s speed of convergence [16,17,18].

Although researchers continue to use new methods to balance exploitation and exploration [19,20,21,22,23,24,25,26,27,28,29,30,31], the question of how to more effectively balance exploitation and exploration remains a topical one [32]. Zheng proposed the water wave optimization (WWO) algorithm in 2015 [33]. WWO is an exploitation-biased algorithm, it can converge quickly in the early stages of an iteration but tends to enter a local optimum in the middle and late stages. Therefore, the population diversity will remain low from the beginning of the iteration on. Xue and Shen proposed the sparrow search algorithm (SSA) in 2020 [34]. SSA is an exploration biased algorithm, so the algorithm remains exploratory throughout the iterative process, but the probability of finding an optimal solution is greatly reduced due to the lack of exploitation capabilities. Because of this property, the algorithm is able to maintain a high level of population diversity over the course of the iterations [35, 36].

The optimization problem may be solved more precisely when exploitation and exploration are balanced, which is crucial for the improvement of MHA [37, 38]. The equilibrium between exploitation and exploration is generally done in the following ways. (1) Parameter adaptation [39, 40]: A state-of-the-art differential evolution algorithm (SCJADE) and (MASSA) are two of the representatives using a parameter adaptation strategy [41, 42]. SCJADE is based on a chaotic local search-based differential evolution algorithm (CJADE) and solves the problem of premature convergence of the original algorithm by improving the differential operation. MASSA is based on SSA and uses chaotic inverse learning techniques and adaptive strategies. (2) Hybrid computation [43, 44]: A genetic learning particle swarm optimization algorithm (GL-PSO) and (HIAO) are two of the representatives of hybrid computing [45, 46]. GL-PSO diversifies the search for particles through samples constructed by the genetic algorithm (GA), which helps particle swarm optimization (PSO) avoid premature convergence. In turn, pbest and gbest in PSO pass promising genetic information to the GA, effectively guiding it and thus achieving a balance between exploitation and exploration. HIAO is a hybrid algorithm based on (AIS) and (ACO), that combines a balance between development and exploration to solve real-world problems. (3) Population structure [47,48,49,50]: A distributed gravitational search algorithm with multi-layered information interaction (MGSA) is one of the representatives based on the population structure, MGSA is composed of four hierarchical population structures [51]. (4) Local search [52, 53]: A chaotic map incorporating grey wolf optimization algorithms (CGWO) is one of the representatives [54]. CGWO consists of the grey wolf optimizer (GWO) algorithm and the chaotic local search (CLS). Due to the integration of chaotic mapping, local search has a strong ability to escape from the local optimum, thus accelerating the convergence of the algorithm.

The effectiveness of the MGSA algorithm, which is based on a hierarchical population structure, has been established. Inspired by MGSA, a new swarm exploration mechanism-based distributed water wave optimization (DWSA) algorithm with distributed population structure is proposed, combining with the problem that using sparrow search hunting mechanisms to improve the water wave algorithm (WWOSSA) [55] is still insufficient in exploration ability and easy to fall into the local optimum. The following four points serve as a summary of DWSA’s four primary contributions: (1) Based on the distributed structured SSA algorithm and the hierarchical structured WWO algorithm, a multi-part distributed structured DWSA method is created. (2) A new type of population structure to balance exploitation and exploration. (3) A new elite perturbation mechanism is introduced in order to achieve a balance of exploitation and exploration around the best points. (4) To verify the effectiveness of DWSA, we used the 30, 50, and 100 dimension benchmark functions from IEEE CEC2017, real-world scenarios from IEEE CEC2011, a 2-D convergence trajectory figure, and population diversity data. The experimental findings show that, in comparison to the original method and other widely used algorithms, DWSA is more adaptable and competitive in solving various challenges.

The rest of this essay is organized as follows: In Sect. 2, we describe the population structure of the different algorithms and the different characteristics corresponding to the different structures. In Sect. 3, we describe the population structure and characteristics of the WWO and SSA algorithms, respectively. In Sect. 4, after analyzing the advantages and disadvantages of the WWOSSA algorithm, we propose a new distributed structure for the DWSA algorithm, that can better balance the relationship between exploitation and exploration. In Sect. 5, we present the experimental results and compare the results obtained using DWSA with those obtained by the three original algorithms and other mainstream algorithms. Finally, in Sect. 6, we present a summary of the DWSA algorithm and future research directions.

2 Related Works

MHA is a search operation performed by iterating over multiple individuals in a population. The different population structures determine the different ways of exchanging information among individuals, which leads to differences in the exploitation and exploration abilities of algorithms. Researchers have conducted comprehensive studies on the population structure of MHA [56]. For example, the panmictic population structure, in which an individual has an equal chance of interacting with others who have equal opportunities. The cellular structures, where individuals in a population are only able to communicate with individuals in their intended neighborhood. The distributed population structure, commonly referred to as the island model in evolutionary computing, is the most popular parallel computing architecture. It uses several sub-populations for the entire individuals, and within each sub-population, usually in a pan-mathematical manner. In contrast, the hierarchical structure allows for more frequent information sharing between the layers, enhancing the overall algorithm’s search efficiency.

The differences between the four population structures can be summarized as follows: (a) A panmictic structure allows for random interactions between individuals, and therefore population diversity is difficult to maintain. (b) A distributed structure consists of two or more sub-populations. The effective exchange of information among sub-populations maintains the diversity of the population. (c) A hierarchical structure that places more emphasis on information exchange between levels, where individual interactions can be effectively improved according to the different levels. (d) A cellular structure can slow down the rate of information flow between individuals, thus improving the problem of premature convergence.

Exploitation suggests frequent information exchange among populations, whereas exploration suggests infrequent information exchange within populations. The ability of hierarchical population structures to encourage the sharing of information among populations in order to facilitate exploitation. The distributed population structure encourages the exchange of information between sub-populations, thus facilitating exploration. In an effort to balance exploitation and exploration, algorithm designers are increasingly considering the population structure of algorithms while designing new ones.

The SSA algorithm consists of a distributed population structure of discoverers and followers. The sine cosine algorithm (SCA) [57], based on a sine and cosine function mathematical model, is a distributed population structure with two components that search the solution space together, thus achieving a balance between exploitation and exploration. The dynamic neighborhood learning-based gravitational search algorithm (DNLGSA) is a distributed structure composed of the dynamic neighborhood learning (DNL) strategy and the original gravitational search algorithm (GSA). The original GSA algorithm is responsible for exploitation, while DNL enhances exploration, and strikes a balance between exploration and exploitation [58]. The WWO algorithm consists of three operations, which are used to update the water wave transmission, that is, the layered population structure.

Cuckoo search (CS) is an algorithm inspired by the cuckoo search process. It is a hierarchical population structure with three layers. The first layer is to dump cuckoo eggs in random nests [59]. The second layer is the continuation of high-quality eggs, which are in the best nest site, to the next generation; The third level is to track the number of available host nests. The WWOSSA algorithm is an algorithm composed of two population structures. It combines the WWO algorithm, which emphasizes exploitation, with the SSA algorithm, which emphasizes exploration. This combination improves the performance of the algorithm and balances the connection between exploitation and exploration.

3 Exploitation and Exploration in WWOSSA

The WWO algorithm with a hierarchical population structure is described below. Think of a water wave as a body with the following three characteristics: position X, wavelength \(\lambda\), and wave height H. Propagation, breaking, and refraction are the three operations available to WWO in the interim. While H diminishes as the water waves go along, the propagation operation controls how the water waves update. The breaking operation divides the water waves into a series of separate waves with the aim of finding more ideal solutions around the current best solution. The wave height falls by 1 with each propagation. The propagation of waves stops when the wave height is equal to zero. The algorithm refracts water waves to prevent stagnation, and the wave height is reformulated as \(H_\textrm{max}\). To prevent the search from becoming static, the water waves are updated by refraction operations, thus increasing the diversity of the population. The equation reads as follows. Where the ideal answer for the current situation is represented by \(x_\textrm{best}^{t}\).

$$\begin{aligned} x_{i}^{t+1}=N\left( \frac{x_\textrm{best}^{t}(d)+x_{i}^{t}(d)}{2}, \frac{x_\textrm{best}^{t}(d)+x_{i}^{t}(d)}{2}\right) . \end{aligned}$$
(1)

Remark 1

WWO is an algorithm with a hierarchical population structure consisting of three layers. Each of the three operations corresponds to a level of the hierarchy. To make full use of the entire search space, the hierarchy attempts to divide the entire population into three layers. The WWO algorithm prioritizes exploitation more than other algorithms because of its hierarchical population structure.

The SSA algorithm with a distributed population structure is discussed in this section. The algorithm mostly mimics the foraging habits of groups of sparrows. The foraging process of a sparrow flock can be seen as a model for information exchange between discoverers and followers with a warning mechanism superimposed. The sparrows who find better food are the discoverers, the others are the followers. A certain percentage of sparrows among the finders and followers are selected for scouting and warning work. Once danger is found, they will abandon their food, and safety comes first. The well placed sparrows in the population are chosen as the finder and the rest as more followers. The discoverer location formula has been updated as follows. \(x_{i,j}^{t}\) where i represents the i-th sparrow in the population, t is the iteration time, and j is the latitude value. \(R2\in [0, 1]\) is alarm value. \(ST\in [0.5, 1.0]\) is safety threshold.

$$\begin{aligned} x_{i,j}^{t+1} = {\left\{ \begin{array}{ll} x_{i,j}^{t}\cdot \textrm{exp}\left( \frac{-i}{\alpha \cdot iter_\textrm{max}} \right) , &{} R2<ST\\ x_{i,j}^{t}+P\cdot N, &{} R2 \ge ST.\\ \end{array}\right. } \end{aligned}$$
(2)

The formula for expressing the update of follower positions in a sparrow flock is as follows. Among the producers, the individual with the best position is denoted by \(x_{g}\). The current global worst location is denoted by \(x_\textrm{worst}\). The current global best location is denoted by \(x_\textrm{best}\).

$$\begin{aligned} x_{i,j}^{t+1} = {\left\{ \begin{array}{ll} Q\cdot \textrm{exp}\left( \frac{x_\textrm{worst}^{t}-x_{i,j}^{t}}{i^{2}} \right) , &{} i>\frac{n}{2} \\ x_{g}^{t+1}+\left\| x_{i,j}^{t}-x_{g}^{t+1} \right\| \cdot A^{+}\cdot N, &{} i\le \frac{n}{2}.\\ \end{array}\right. } \end{aligned}$$
(3)

A random selection of individuals in the sparrow flock are tasked with vigilance, warning when danger is detected, and abandoning their current position to move to a new one.

Remark 2

SSA is a distributed population structure.

SSA is a distributed population structure, which consists of two components: discoverer and follower. Information sharing between discoverers and followers can increase population variety and make exploration smoother. The SSA algorithm prioritizes exploration more than other algorithms because of its distributed population structure.

According to the descriptions of the WWO algorithm and the SSA algorithm in this section, it can be found that the WWO algorithm with a hierarchical population structure is a more exploitation-oriented algorithm, while the SSA algorithm with a distributed population structure is a more exploratory algorithm. In a previous study, we introduced the biased exploration SSA algorithm to enhance the exploration of the WWOSSA algorithm. A parameter O is also introduced to improve the under-exploration of the WWO algorithm by adjusting the value of the parameter O, so that the algorithm WWOSSA can better balance the relationship between exploitation and exploration.

$$\begin{aligned} {\left\{ \begin{array}{ll} E={E}',{E}'\le E\\ O= O+1, \textrm{otherwise}.\\ \end{array}\right. } \end{aligned}$$
(4)

\({E}'\) is WWO\(_{-}Optvalue\). and E is Optvalue.

4 Proposed DWSA

4.1 Motivation

In the WWOSSA algorithm, a hybrid population structure consisting of two components is introduced. The hierarchical population structure of the WWO algorithm has strong exploitability, which enables the algorithm to converge quickly. However, due to insufficient exploration, the algorithm can easily fall into a local optimum as the iteration progresses. The SSA algorithm with a distributed population structure has a strong exploration capability, which can complement the under-explored feature of the WWO algorithm.

Despite the improved performance of WWOSSA, there are still some problems. For example, the WWOSSA algorithm is still designed with a priority on exploitation rather than balancing exploitation and exploration. Due to its limited capacity to explore, such an algorithm can fast converge at the start of an iteration, but also has a tendency to settle into a local optimum as the iteration goes on. To improve WWOSSA’s exploration capabilities, we attempted to redesign the algorithm. In this study, the algorithm is modified in two areas to further the algorithm’s exploration, striving to strike a balance between exploitation and exploration.

4.2 Elite Perturbation Mechanism

In the population of the WWOSSA algorithm, the individual with the best information is selected, and this individual is known as the elite individuals in the population. The elite individual can influence the population’s overall trend toward convergence, causing the algorithm to converge quickly, but it can also lead to the algorithm’s insufficient exploration. As inspired by the CJADE algorithm, we want to do perturbations near the optimal individuals to further enhance the exploration of the algorithm and increase the probability of the algorithm to have better individuals.

In the CJADE algorithm, the difference strategy is chosen to explore individuals near the elite. However, the difference strategy is the upper limit minus the lower limit, which leads to low efficiency because the range is too large and random.

In this study, the original difference strategy was modified, and two individuals \(x_{r1}\) and \(x_{r2}\) are randomly selected from the population for subtraction. This reduced the search range while ensuring randomness, thus improving the search efficiency. The new differentiation strategy enhances exploration near elite individuals, thus balancing the relationship between exploitation and exploration near elite individuals.

$$\begin{aligned} \begin{aligned} x_{k^\prime }=x_k+Q\cdot \left( x_{r2}-x_{r1}\right) .\\ \end{aligned} \end{aligned}$$
(5)
Fig. 1
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DWSA distributed population structure

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4.3 Description of DWSA

A new distributed population structure DWSA is based on the WWOSSA algorithm in order to better balance the algorithm’s exploitation and exploration. Based on the original hybrid population structure, two parts are redesigned, namely the optimal individual component and the information exchange component. They are described in the following way:

\(\bullet\) Optimal individual component The operation \(\psi\) represents the selection of the best individual from the WWO algorithm. The operation \(\varsigma\) represents the selection of the best individual from the SSA algorithm. In this component, the continuously optimized individual \(x_{k}\) is selected to update the optimal individual \(X_\textrm{best}\) in the WWOSSA population. This hastens the algorithm’s convergence more effectively. The best individual component is influenced by the information interchange component. When the information exchange component generates a temporary individual \(x_{k'}\) and outperforms \(x_k\), \(x_k\) will be replaced by \(x_{k'}\) individuals. The update is done as follows:

$$\begin{aligned}{} & {} \psi =N\left( \frac{x_\textrm{best}^{t}(d)+x_{i}^{t}(d)}{2}, \frac{x_\textrm{best}^{t}(d)+x_{i}^{t}(d)}{2}\right) \end{aligned}$$
(6)
$$\begin{aligned}{} & {} \varsigma = {\left\{ \begin{array}{ll} x_{i,j}^{t}\cdot exp\left( \frac{-i}{\alpha \cdot iter_\textrm{max}} \right) , &{} R2<ST\\ x_{i,j}^{t}+P\cdot N, &{} R2 \ge ST\\ \end{array}\right. } \end{aligned}$$
(7)
$$\begin{aligned}{} & {} x_{\textrm{best}',j}^{t+1} = {\left\{ \begin{array}{ll} \varsigma , &{} z=1000 \\ \psi , &{} z \ne 1000 \\ \end{array}\right. } \end{aligned}$$
(8)
$$\begin{aligned}{} & {} x_k(t)=\left\{ \begin{array}{rl} x_{k'}(t), &{} \text{ If } f(x_{k'}(t)) \le f(x_k(t))\\ x_k(t), &{} \text{ If } f(x_{k'}(t)) > f(x_k(t)),\\ \end{array} \right. \end{aligned}$$
(9)

where t is the number of iterations and f is the fitness function for the optimization problem. The information exchange component generates \(x_{k'}\).

\(\bullet\) Information exchange component Although \(x_{k}\) can speed up the algorithm’s convergence in the optimal individual portion, it can also cause the algorithm to reach a local optimum too soon. We configure an information exchange component and a best individual component. Information is exchanged between these two components so that the best individual part establishes a balance between exploitation and exploration, thus avoiding local optimization. In the information exchange component, we generate \(x_{k'}\) based on \(x_{k}\) to balance \(x_{k}\). Its formula is shown as:

$$\begin{aligned} U_\textrm{max}\left( t \right) = \left\{ \begin{array}{rl} x_{k'}(t), &{} \text{ If } f(x_{k'}(t)) \le f(U_\textrm{max}\left( t \right) )\\ U_\textrm{max}\left( t \right) , &{} \text{ If } f(x_{k'}(t)) > f(U_\textrm{max}\left( t \right) ),\\ \end{array} \right. \end{aligned}$$
(10)

where \(x_{k}\) is the best individual in population. \(x_{k^\prime }\) is a temporary individual. If the fitness value of \(x_{k^\prime }\) is higher than \(x_{k}\), \(x_{k^\prime }\) replaces \(x_{k}\), else \(x_{k}\) is retained and will continue to exist in the following iteration. In W, we assume the individual with the greatest fitness as \(W_\textrm{max}\). If the fitness of \(x_{k^\prime }\) is better than that of \(W_\textrm{max}\), replace \(W_\textrm{max}\) with \(x_{k^\prime }\). Thus, the historical information is recorded.

As seen in Fig. 1, the DWSA is a distributed structure made up of four sections. DWSA consists of WWO with a hierarchical structure and SSA with a distributed structure. WWO and SSA are located in the first and second components of the distributed structure, and the optimal individual and information exchange components are located in the third and fourth components, respectively. Figure 1a shows that the optimal individual is generated by the WWO, passed to the optimal individual component, and then the optimal individual component interacts with the information exchange component. Figure 1b shows that the optimal individual is generated by the SSA, passed to the optimal individual component, and then the optimal individual component interacts with the information exchange component.

It is evident that the information exchange component and the optimal individual component have two-way information flow. It should be emphasized that between the information exchange component and the optimal individual component, the red arrows indicate the influence of the optimal individual component on the information exchange component, and if the fitness of \(x_{k^\prime }\) is better than \(W_\textrm{max}\), \(x_{k^\prime }\) is used instead of \(W_\textrm{max}\). The black arrow indicates that the optimal individual component is updated by the information exchange component. Among them, the information exchange component emphasizes the ability to explore, while the optimal individual component emphasizes the ability to exploit. The algorithm strikes a balance between exploitation and exploration by interacting with the information between the two components.

The main process of the proposed DWSA algorithm is shown in Algorithm 1. From the pseudo-code, we can find that the DWSA has only two improvements over the WWOSSA: doing perturbations near elite individuals, and information exchange between individuals in the population. Small improvements bring big improvements, which are based on the characteristics of the population structure.

5 Analysis and Results from Experiments

The performance of the DWSA algorithm is examined in this work through comparison tests on two benchmark functions using IEEE CEC2017 and IEEE CEC2011. The IEEE CEC2017 benchmark set consists of 29 problems of varying difficulty, with (F1 and F2) being two unimodal functions, (F3–F9) being seven simple multimodal functions, (F10–F19) being ten hybrid functions, and (F20–F29) being ten combinatorial functions. The 22 real-world issues in the IEEE CEC2011 benchmark set include engineering and static economic scheduling issues. The DWSA is contrasted with a number of different algorithms, which are classified into two kinds, in this article. The first type of algorithms includes WWOSSA, WWO, and SSA since the suggested DWSA is improved based on the WWOSSA. The improvement can be demonstrated to be successful if DWSA is superior to WWOSSA, WWO, and SSA. The second category of algorithms includes DNLGSA, CGWO, SCA, and CS. The comparison with the second type of algorithm also allows for an evaluation of the effectiveness of the DWSA algorithm. In addition, the DWSA algorithm contains a parameter Q that can affect the performance of the algorithm. We therefore added a few experiments in order to identify a better value for Q before the other experiments.

5.1 Criteria for Performance Evaluation

Following is a description of the algorithm performance evaluation tools [60]:

(1) W/T/L: The percentage of functions for which DWSA performs noticeably better than other algorithms is expressed as W, the percentage of functions for which DWSA performs similarly to other algorithms is expressed as T, and the percentage of functions for which DWSA performs noticeably worse is expressed as L.

(2) Convergence curve: Where the y-axis shows the average error value, the x-axis shows the number of iterations.

(3) Box-and-whisker diagrams: The maximum value is represented by the line above the blue box, while the minimum value is represented by the line below the blue box. The first and third quartiles are represented by the upper and lower edges of the box, respectively. The median is represented by the red line, and extreme values are represented by the red “\(+\)” symbol. In addition, the larger the gap between the maximum and minimum, the more unstable the performance of the algorithm.

5.2 Experiment Setup

The number of evaluations with the same termination condition is set at \(10^{4} * D\), where D is the dimension of the benchmark function. We also conducted tests at IEEE CEC2017 on 30, 50, and 100 different dimensions. In IEEE CEC2011, because the dimensions are different for different questions, we use the pre-defined dimensions for each question. In the following tables,  the best values among all compared algorithms for each tested problem are highlighted in bold. Finally, MATLAB is used to conduct all tests on a computer with an Intel(R) Core(TM) i5-7400 processor running at 3.00 GHz and 12 GB of RAM. To ensure the accuracy of the experimental results, the experiment is carried out 51 times independently.

Table 1 Experimental results of IEEE CEC2107 on 30 dimensions
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Table 2 Experimental results of IEEE CEC2017 on 50 dimensions
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Table 3 Experimental results of IEEE CEC2017 on 100 dimensions
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Table 4 Experimental results of IEEE CEC2011
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Table 5 Experimental results of IEEE CEC2017 on 30 dimensions
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Table 6 Experimental results of IEEE CEC2017 on 50 dimensions
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Table 7 Experimental results of IEEE CEC2017 on 100 dimensions
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Table 8 Experimental results of IEEE CEC2011
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Fig. 2
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Convergence graphs comparison on IEEE CEC2017

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Fig. 3
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Box-and-whisker diagrams comparison on IEEE CEC2017

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Convergence graphs comparison on IEEE CEC2011

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Fig. 5
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Box-and-whisker diagrams comparison on IEEE CEC2011

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Bar graph of CPU running time consumed by all tested algorithms on IEEE CEC2017 functions with 30, 50, and 100 dimensions on 1 running time

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Fig. 7
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Convergence graphs comparison on IEEE CEC2017

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Fig. 8
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Box-and-whisker diagrams comparison on IEEE CEC2017

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Convergence graphs comparison on IEEE CEC2011

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Box-and-whisker diagrams comparison on IEEE CEC2011

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5.3 Discussion of DWSA with the Original Algorithm for Statistical Testing

To verify the performance of DWSA, we test it on the IEEE CEC2017 and IEEE CEC2011 test sets. In this section, we compared the WWOSSA, WWO, and SSA algorithms to DWSA. We analyzed and compared a number of algorithms at IEEE CEC2017 with dimensions of 30, 50, and 100, respectively, to study the performance change of DWSA from medium dimension to high dimension to enormous dimension. According to Table 1, it is known that the number of DWSA wins compared to WWOSSA is 14, compared to WWO is 16, and compared to SSA is 22. According to Table 2, it is known that the number of DWSA wins compared to WWOSSA is 15, compared to WWO is 14, and compared to SSA is 23. According to Table 3, it is known that the number of DWSA wins compared to WWOSSA is 17, compared to WWO is 16, and compared to SSA is 16. It is evident that DWSA outperforms the other three algorithms on the 30, 50, and 100 dimensions of IEEE CEC2017, demonstrating DWSA’s appropriateness for functions with medium, high, and large dimensions. According to Table 4, it is known that the number of DWSA wins compared to WWOSSA is 10, compared to WWO is 10, and compared to SSA is 15. This demonstrates that DWSA is very competitive in IEEE CEC2017 and IEEE CEC2011.

The convergence graphs of the average optimal solutions produced by four algorithms on the problems F9, F13, and F17 with 30 dimensions, F13, F17, and F28 with 50 dimensions, and F13, F17, and F25 with 100 dimensions are shown in Figs. 2 and 4 shows the convergence plots of the average optimal solutions obtained for the four algorithms in F2, F4, and F10 on IEEE CEC2011.

Figure 3 shows the box-and-whisker plots of four algorithms, DWSA, WWOSSA, WWO, and SSA, for obtaining optimal solutions in different dimensions and for different functional problems (see Fig. 4). Figure 5 shows box-and-whisker plots of the optimal solutions obtained for DWSA, WWOSSA, WWO, SSA, and the four algorithms in F2, F7, and F22. In contrast to other algorithms with various dimensions, DWSA has practically the lowest distribution and the least error of optimal solutions, suggesting its higher performance and stability, which is illustrated in Figs. 3 and 5.

The comparison above demonstrates that DWSA performs far better than the other three original algorithms. It further demonstrates the improvement of algorithmic performance and the importance of balance between exploitation and exploration.

Fig. 6 shows the time required to run each algorithm on IEEE CEC2017 for 1 time on 29 problems for the comparison. Although DWSA takes 13\(\%\) more time compared to WWOSSA, the performance is improved by 52\(\%\), thus demonstrating that the benefits of the algorithm’s improved performance significantly outweigh the increased time cost.

5.4 Discussion of DWSA with Other Mainstream Algorithm for Statistical Testing

To verify the performance of DWSA, we test it on the IEEE CEC2017 and IEEE CEC2011 test sets. In this section, we compare the DNLGSA, CGWO, SCA, and CS algorithms. We analyzed and compared a number of algorithms at IEEE CEC2017 with dimensions of 30, 50, and 100, respectively, to study the performance change of DWSA from medium dimension to high dimension to enormous dimension. According to Table 5, it is known that the number of DWSA wins compared to DNLGSA is 28, compared to CGWO is 15, compared to SCA is 29, and compared to CS is 20. According to Table 6, it is known that the number of DWSA wins compared to DNLGSA is 28, compared to CGWO is 17, compared to SCA is 29, and compared to CS is 21. According to Table 7, it is known that the number of DWSA wins compared to DNLGSA is 23, compared to CGWO is 19, compared to SCA is 27, and compared to CS is 18. It is evident that DWSA outperforms the other four algorithms on the 30, 50, and 100 dimensions of IEEE CEC2017, demonstrating DWSA’s appropriateness for functions with medium, high, and large dimensions. According to Table 8, it is known that the number of DWSA wins compared to DNLGSA is 17, compared to CGWO is 10, compared to SCA is 20, and compared to CS is 12. This demonstrates that DWSA is very competitive in IEEE CEC2017 and IEEE CEC2011.

The convergence graphs of the average optimal solutions produced by five algorithms on the problems F1, F17, and F29 with 30 dimensions, F1, F14, and F26 with 50 dimensions, and F1, F14, and F17 with 100 dimensions are shown in Fig. 7. Figure 9 shows the convergence plots of the average optimal solutions obtained for the five algorithms in F11, F18, and F20 on IEEE CEC2011.

Figure 8 shows the box-and-whisker plots of five algorithms, DWSA, DNLGSA, CGWO, SCA, and CS, for obtaining optimal solutions in different dimensions and for different functional problems. Figure 10 shows box-and-whisker plots of the optimal solutions obtained for DWSA, DNLGSA, CGWO, SCA, and CS the five algorithms in F11, F18, and F20 (see Fig. 9). In contrast to other algorithms with various dimensions, DWSA has practically the lowest distribution and the least error of optimal solutions, suggesting its higher performance and stability Figs. 8, and  10.

The comparison above demonstrates that DWSA performs far better than the other four mainstream algorithms. It further demonstrates the improvement of algorithmic performance and the importance of exploiting and exploring balance.

Table 9 Discussion of the parameter Q
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5.5 Discussion of the Parameter Q

The parameter Q represents the step size, which has an impact on the algorithm by adjusting the value of the step size. When the value of Q is too large, a larger surrounding will be selected for exploration. When the value of Q is too small, a smaller surrounding is selected for exploration. Finding better values is hindered by both too high and too low Q values. In the comparison experiments, we set the values of Q to 0.5, 0.7, 1.0, 1.4, 1.8, 1.9, 2.0, and 4.0. The value of the appropriate distance is found experimentally.

We use the IEEE CEC2017 test set on 30 dimensions in the comparative experiment and average the final findings of 51 tests. The Wilcofferson test is used to validate the outcomes of eight sets of trials. Wilcofferson will compare each issue in the test set separately (there are 29 in total) and determine the value of W/T/L; the greater the value of W and the lower the value of L, the better the performance may be proved. As can be seen from Table 9, when Q = 1.8, the ranking value is the smallest. As a result, the value of Q is adjusted to 1.8 in following studies.

Fig. 11
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Search history of individuals of DWSA in 2 dimensions in IEEE CEC2017

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Fig. 12
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Diversity of DWSA, WWOSSA, WWO, and SSA in 30 dimensions in IEEE CEC2017

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5.6 Discussion of Search Trajectory

To test whether the DWSA may depart from the local optimum, we conduct one more experiment on it. Individual trajectories on the function graph as the number of iterations grows, using IEEE CEC2017 as the test function set, demonstrate DWSA’s ability to break out of the local optimum. In Fig. 11, the contour lines are fitness value contour lines, and the color of the line from blue to red indicates the fitness value from low to high. In F3 and F6, we recorded images for 2, 100, and 200 iterations, respectively. In F3 and F6, it is found that the populations maintain the trend of narrowing the search range and finally converge to the minimum range, which can prove that DWSA has a strong exploitation potential. In F4, we record images at 2, 100, and 200 iterations, respectively. It can be seen that the population can converge swiftly from the 2td iteration to the 200th iteration, but after 200 iterations, the population can still keep a certain search range, proving that the algorithm can still maintain a certain exploration ability in the late iteration. The convergence trajectories of DWSA on different problems show that the algorithm can demonstrate independent switching between exploitation and exploration for various function problems and various solution spaces, indicating that it achieves a balance between the two.

5.7 Population Diversity Discussion

One of the most important indicators of algorithmic exploitation and exploration is population diversity. In general, the algorithm tends to exploit more when the diversity value is lower, conversely, the algorithm prefers to explore more when the arithmetic diversity value is higher. The following equation can be used to determine population diversity:

$$p = \frac{1}{N}\sqrt {\sum\nolimits_{{i = 1}}^{N} {\left( {\left\| {x_{i} - x_{{{\text{mean}}}} } \right\|} \right)^{2} } } ,$$
(11)

where p is the population diversity, N is population size, \(X_{i}\) is the ith individual, and \(X_{mean}\) is the average of the population. Figure 12 depicts the diversity of DWSA, WWOSSA, WWO, and SSA in F7, F10, and F29 in IEEE CEC2017 with 30 dimensions. The four algorithms mentioned above are all mainstream algorithms with a wide variety of applications. From the figure, it can be seen that SSA diversity has been high, proving that the SSA algorithm is an algorithm that is biased toward exploring functions. WWO diversity decreases rapidly from the beginning of the iteration and remains a low value, proving that the WWO algorithm is an algorithm biased toward exploitation. As seen in F7 and F10, the diversity of DWSA is consistently higher than that of WWOSSA and WWO. However, the consistently high diversity of SSA proves that its exploitation capabilities are not very strong and demonstrates that DWSA is easier to balance between exploitation and exploration compared to other three algorithms. In F29, it can be seen that DWSA diversity rapidly decreases at the start, then sharply increases in the middle of the process before abruptly decreasing with iteration diversity, demonstrating that DWSA emphasizes exploration ability in the middle of the iteration.

6 Conclusion and Future Works

In this study, we propose an algorithm swarm exploration mechanism-based distributed water wave optimization DWSA. Based on the original WWOSSA algorithm, two parts are introduced, which together form the distributed population structure. This is to the key of the algorithm improvement. The creative addition of a part for information interchange and one for individual optimization. WWO is an algorithm that favors exploitation, while SSA is an algorithm that favors exploration. SSA and WWO are responsible for exploring and exploiting functions respectively, and SSA is called when WWO falls into the local optimum for 1000 times. The whole DWSA algorithm is a four-component distributed structure, where WWO and SSA occupy the first and second components of the distributed structure, the optimal individual component of the third component is used to record the optimal individuals, and the fourth component is perturbed around the elite individuals, thus improving the balance between exploitation and exploration.

We test the effectiveness of DWSA by comparing it to the original method and other well-known heuristics. Before all experiments, we discuss the value of the step size Q. The Wilcoxon test findings show that the best outcomes in this experiment occur when the value of Q is equal to 1.8. DWSA has a faster rate of global convergence and a greater ability for problem-solving in the real world, according to the experimental results of WWOSSA, WWO, and SSA. Moreover, the search trajectory graph of DWSA shows that the algorithm has the potential to jump out of the local optimum while maintaining a strong convergence ability, demonstrating the efficacy of our modified scheme. The results of comparing DWSA to other algorithms demonstrate DWSA’s competitiveness.

In summary, DWSA is an algorithm with strong performance improvement, which increases our interest in population-based evolutionary algorithms. For upcoming work, some significant research is listed below—(1) Further enhance the performance of the DWSA algorithm. (2) The scheme of population structure applied to more MHAs. (3) The population structure scheme requires theoretical study or demonstration. (4) The performance of some real-world applications, such as training neural networks, wave energy problem optimization, automotive internet challenges, and brain structure design, will be investigated.