Whale optimization with random contraction and Rosenbrock method for COVID-19 disease prediction

Abstract

Coronavirus Disease 2019 (COVID-19), instigated by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has hugely impacted global public health. To identify and intervene in critically ill patients early, this paper proposes an efficient, intelligent prediction model based on the machine learning approach, which combines the improved whale optimization algorithm (RRWOA) with the k-nearest neighbor (KNN) classifier. In order to improve the problem that WOA is prone to fall into local optimum, an improved version named RRWOA is proposed based on the random contraction strategy (RCS) and the Rosenbrock method. To verify the capability of the proposed algorithm, RRWOA is tested against nine classical metaheuristics, nine advanced metaheuristics, and seven well-known WOA variants based on 30 IEEE CEC2014 competition functions, respectively. The experimental results in mean, standard deviation, the Friedman test, and the Wilcoxon signed-rank test are considered, proving that RRWOA won first place on 18, 24, and 25 test functions, respectively. In addition, a binary version of the algorithm, called BRRWOA, is developed for feature selection problems. An efficient prediction model based on BRRWOA and KNN classifier is proposed and compared with seven existing binary metaheuristics based on 15 datasets of UCI repositories. The experimental results show that the proposed algorithm obtains the smallest fitness value in eleven datasets and can solve combinatorial optimization problems, indicating that it still performs well in discrete cases. More importantly, the model was compared with five other algorithms on the COVID-19 dataset. The experiment outcomes demonstrate that the model offers a scientific framework to support clinical diagnostic decision-making. Therefore, RRWOA is an effectively improved optimizer with efficient value.

1. Introduction

Coronavirus Disease 2019 (COVID-19) is a disease identified by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [1]. It manifested a rapid global spread, affecting many nations and posing a significant threat to public health. COVID-19 has a broad spectrum of clinical severity, spanning from asymptomatic or subclinical infections to life-threatening complications, classifying cases as mild (81 %), severe (14 %), or critical (5 %) [2], [3]. Due to the rapid progression of COVID-19, the demand for intensive care unit beds is increasing significantly, and the problem of inadequate medical resources is impacting the healthcare systems of various countries. As a result, early detection and management are more critical than rescue and therapy for critically sick patients in reducing COVID-19 development. A compelling novel prognostic model system would help monitor the status of patients with COVID-19 and potentially reduce mortality if developed. In recent years, machine learning-based artificial intelligence (AI) techniques have made their mark in the medical field. The rapid development of AI has led to the widespread use of machine learning techniques for diagnosing diseases, developing predictive models to aid clinical decision-making, and identifying critical factors associated with diseases [4], [5], [6]. On this basis, various models for COVID-19 are constantly being developed. It can be observed that metaheuristic algorithms (MAs) are active in this field [7], [8], [9].

The key point of using the model to predict COVID-19 is to effectively select the leading features that affect disease diagnosis. Popular feature selection techniques are mainly divided into three types [10]: filter-based [11], wrapper-based [12] and embedded-based [13]. The wrapper-based method considers the specific classifier and evaluates the quality of the feature subset according to the error on the classifier. A wrapper-based method is a common form of combining MAs with feature selection, and the results depend on both the algorithm and the classifier. The task of classification is to build a mapping from conditions to decision labels from a set of training samples. Among these, the nearest neighbor search is one of the most popular techniques [14]. The k-nearest neighbour (KNN) is often chosen due to its simplicity and accuracy [15]. Considering that no explicit knowledge of the underlying data is employed and no assumptions are made about it, it performs very well. Conveniently, KNN has only one parameter to tune. At the same time, its time complexity is lower than that of a support vector machine (SVM). Therefore, many applications of KNN combined with MAs in feature selection exist. In order to reduce the dimension of hyperspectral image data, Ghosh et al. [16] proposed a feature selection classification model combining self-adaptive differential evolution (SADE) and KNN for dimension reduction. Mohamed et al. [17] simulated the behavior of the predator, the parasite, and the host in nature to solve the problem of the dimensionality disaster of big data. The proposed feature selector worked with KNN to obtain a set of feature combinations.

Deterministic and evolutionary optimization approaches have been widely developed and applied for single-objective, multi-objective, and many-objective applications [18], [19]. MAs have been used to tackle a variety of optimization challenges [20], [21], [22]. MAs assume the optimization problem as a black box, so it does not rely on gradients like traditional methods [23]. Some MAs outperform gradient-based approaches in terms of efficiency [24], including Genetic Algorithm (GA) [25], Differential Evolution (DE) [26], Simulated Annealing (SA) [27] and Particle Swarm Optimization (PSO) [28], etc. Swarm intelligence algorithms (SIAs) stand out among MAs and have been essential in implementing many large-scale and real-world problems [24]. Because of their effectiveness, scholars are deeply involved in this field. In recent years, SIAs proposed include Slime Mould Algorithm (SMA) [29], Hunger Games Search (HGS) [30], Harris Hawks Optimization (HHO) [31] and Runge-Kutta Optimization Algorithm (RUN) [32]. These methods have yielded very good results in many areas, such as economic emission dispatch problem [33], image segmentation [34], [35], feature selection [36], [37], gate resource allocation [38], [39], bankruptcy prediction [40], [41], robust optimization [42], [43], medical diagnosis [44], [45], scheduling problems [46], [47], [48], and complex optimization problem [49].

Mirjalili adopted the Whale Optimizer (WOA) in 2016 as a relatively new swarm intelligence optimization approach [50], which simulates humpback whales in the prey search process with typical attack behaviors to solve the optimization problem. Compared with swarm intelligence optimization algorithms such as PSO, WOA can often obtain higher-quality solutions [51], [52]. The multiple updating solutions of WOA also help it create a better balance between exploration and exploitation [53]. Because of its powerful capabilities, WOA has been applied in various industries. Hussien et al. [54] used two transfer functions to map WOA to discrete space, and this binary version showed good performance in the traveling salesman problem. Aljarah et al. [55] applied WOA to the training process of the artificial neural network, and experiments proved that the optimizer could obtain better connection weights. Alameer et al. [56] used WOA to train a neural network to solve nonlinear regression problems.

Even though the preceding study on WOA has established its usefulness for parameter optimization, it must be acknowledged that the initial WOA's efficiency might still be enhanced [53]. According to the No-Free-Lunch theory (NFL) [57], WOA suffers from the problems common to MAs. It is crucial to fully utilize the exploration and exploitation capabilities and to get a balance between the two. In the case of WOA specifically, there are some problems with the exploration and exploitation process. Firstly, WOA has three search modes that can be employed to some extent for optimal solutions. However, as the iteration progresses, the individuals in the swarm continue to approach the food source, resulting in a gradual loss of population diversity, which is highly susceptible to local optima. Additionally, WOA's local search capability is poor since it depends on stochastic convergence factors to transition from exploration to exploitation. Next, WOA can converge quickly because of the small number of operators required and its simple structure. Nevertheless, WOA may prematurely converge to local optima when applied to challenging practical optimization tasks, wasting iterations. In response to the above problems, many scholars have improved WOA. These enhancements are essentially classified into three categories: integration with a variety of mechanisms, combining with effectual MAs, and algorithmic adjustment based on different dynamics [51]. In detail, Rana et al. [58] delved into the variants of WOA and their applications in various fields through 82 papers. When faced with a COVID-19 condition, early detection of the severity of the patient is crucial for the subsequent adoption of an appropriate treatment plan. The existing COVID-19 dataset is characterized by a large number of features and a small number of samples, and the original WOA tends to prematurely converge to local optimality in the face of high dimensional optimization situations. In order to make full use of the limited number of samples to filter out indicators with high classification accuracy, the original WOA is in urgent need of improvement.

As a result of those mentioned above, the questions to be answered in this research are as follows:

• •The exploration and exploitation ability of WOA needs to be enhanced.
• •A balance between the exploration and exploitation process needs to be maintained.
• •The predictive model for early identification and intervention in critically ill patients needs to be developed.

From this perspective, this paper introduces two strategies for the original WOA: Random Contraction Strategy (RCS) and the Rosenbrock Method (RM). The RCS strategy is modified from the contraction formula of SMA. Zheng et al. [59] have covered this aspect. The RCS strategy fully expands the position of the current solution and better explores a broader solution space, which can enhance the exploration ability of the algorithm. The RM strategy is a classic derivative-free local search technique. Its initial step size and termination conditions have been modified to make it better combined with the MAs, enhancing the exploitation ability. Li et al. [60] enhanced the performance of Harris Hawks Optimization (HHO) using an improved RM strategy. The balance of the two strategies is also crucial. Therefore, two control parameters are introduced to balance the two strategies. Due to its time consumption, the RM strategy is complementary to the original WOA.

To evaluate the performance of various aspects of RRWOA, some experiments are conducted based on 30 CEC2014 competition datasets [61] and compared with other classical and improved algorithms. At the same time, the improved method was also compared with commonly used WOA variants. In addition, a series of comprehensive comparisons are conducted through Wilcoxon signed-rank test [62] and the Friedman test [63], and the experimental results show that the optimizer is more effective than other models. Furthermore, to work on the COVID-19 dataset, a machine learning prediction model that combines the binary version of RRWOA (BRRWOA) with a KNN classifier is proposed. To verify the generality of the model, experiments are conducted on 15 UCI datasets with seven existing binary MAs and six binary WOA variants. It turns out that BRRWOA can balance the classification accuracy and the number of features to obtain optimal results. More importantly, the model was also applied to the COVID-19 dataset. The experimental results prove that the model can predict the critical features of this dataset. The main contributions of this paper are as follows:

• •The RCS and RM strategies are introduced into the original WOA, which is called RRWOA.
• •Comparisons with well-known optimizers and WOA variants serve to demonstrate the effectiveness of RRWOA.
• •Both the exploration and exploitation processes go smoothly with the developed RRWOA. The algorithm can assist the population in eliminating the local optimum while preserving its diversity during iteration. In order to achieve a higher degree of convergence accuracy, the proposed method can simultaneously tap potential regions as much as possible.
• •A machine learning prediction model is created using the binary form of RRWOA and the KNN classifier. The efficacy of the suggested approach is demonstrated by experiments using the COVID-19 dataset as well as the UCI datasets.

The rest of this paper is structured as follows: Section 2 introduces the WOA and the framework of RRWOA proposed in this paper. At the same time, Section 2 also proposes the binary version of RRWOA and the feature selection model. Section 3 will conduct comprehensive experiments with the proposed method and apply it to feature selection problems. Section 4 summarizes the work of this paper and points out future work directions.

2. Proposed whale optimization method (RRWOA)

Mirjalili [50] presented WOA as an effort to make a new swarm intelligence optimization in 2016. Because of its effectiveness and adaptability, WOA is being used in a wide range of fields.

Although WOA is a commonly utilized swarm intelligence optimization algorithm, there is still much opportunity for development. When faced with complex real-world problems, WOA still suffers from two common problems of MAs: slow convergence and low solution quality later in the iterative process. WOA converges slowly in the early stage of iteration but often loses diversity and falls into local optimum in the later stage of iteration. Furthermore, since WOA relies on stochastic convergence factors to transition from exploration to exploitation, its local search capability is also weak. Therefore, it is necessary to strengthen WOA.

Two strategies are introduced to WOA, namely the Random Contraction Strategy (RCS) and the modified Rosenbrock Method (RM). RCS can expand the exploration space and avoid the algorithm falling into local optimum, enhancing the exploration ability. The modified RM strategy can fully exploit the potential regions and enhance the exploitation capability of the algorithm.

2.1. An overview of WOA

WOA was inspired by the typical attack behavior of humpback whales during their prey hunt. Humpback whales are among the largest whales and have exciting social behaviors. They use a particular hunting method called the bubble-net feeding method [64]. The behavior of humpback whales encircling, locating, and feeding on prey was simulated to solve optimization problems. The detailed process of the WOA can refer to [50]. The specific formulas are shown in Table 1 .

Table 1.

Specific formulas in WOA.

Process	Formulas	No.
Encircling prey	$X(t+1)=X_{\mathit{best}}-A\times D$	(1)
	$D=|C\bullet X_{\mathit{best}}-X(t)|$	(2)
	$A=2\bullet a\bullet r-a$	(3)
	$C=2\bullet r$	(4)
Bubble-net attacking method	$X(t+1)=D^{\prime }\bullet e^{\mathit{fl}}\bullet cos(2\pi l)+X_{\mathit{best}}$	(5)
	$D^{\prime }=|X_{\mathit{best}}-X(t)|$	(6)
	$\begin{array}{c}X(t+1)=\left\{\begin{array}{c}{X}_{\mathit{best}}-A\times D,p<0.5\\ D^{\prime }\bullet e^{\mathit{fl}}\bullet cos(2\pi l)+X_{\mathit{best}},p\ge 0.5\end{array}\right)\end{array}$	(7)
Search for prey	$X(t+1)=X_{\mathit{rand}}-A\times D$	(8)
	$D=|C\bullet X_{\mathit{rand}}-X(t)|$	(9)

2.2. Random contraction strategy (RCS)

The contraction formula of SMA inspires the RCS strategy. Zheng et al. [59] were involved in the process of improving the Arithmetic Optimization Algorithm (AOA) [65]. SMA uses a contraction formula when approaching and wrapping food, as shown in Eq. (10).

$$X(t+1)=vc\bullet X(t)$$ (10)

where $\mathit{vc}$ is the contraction range control parameter that oscillates between [−1,1] and eventually approaches 0.

The shrinkage formula has shown good exploration performance when applied to SMA. $\mathit{vc}$ keeps approaching 0 as the iteration progresses. To fully expand the position of the current solution and better explore a wider solution space, the control parameter $\mathit{vc}$ has been modified as below:

$$X(t+1)=(2\bullet rand-1)\bullet X(t)$$ (11)

where $\mathit{rand}$ is a random number in [0,1], although the range is still [[−1,1], it will not gradually tend to 0 as the number of iterations increases. The control parameters are evenly distributed in [−1, 1], which ensures that each individual in the algorithm can evenly explore the solution space of [−$X(t)$, $X(t)$] in the later iteration.

This method enhances the exploration ability of the algorithm, helps the algorithm jump out of the local optimum, and obtains more diverse solutions. While ensuring the algorithm's flexibility, the greedy selection strategy is used to retain the one with the better fitness value among the candidate solution generated by RCS and the original solution.

2.3. The Rosenbrock method (RM)

RM is a traditional derivative-free local search approach with adaptive search direction and magnitude [66] that can quickly discover local optima on unimodal functions. RM has been used in ABC because of its effectiveness [67]. To fully exploit the potential of RM, Li et al. [60] proposed an improved RM method. The following will introduce the relevant content in turn.

2.3.1. The original RM

Classical RM has shown powerful capabilities in local search. The original RM mainly consists of two stages: search and update direction. The search phase mainly searches in the determined direction by adjusting the step size. The updating direction stage mainly updates the search direction through orthogonalization. The detailed process of the original RM can refer to [67]. The specific formulas are shown in Table 2 . The pseudocode of the original RM is shown in Algorithm 1.

Algorithm 1 Pseudo-code of RM —

Initialize the starting point $x_0$ and the step size ${\delta }_i$(i = 1, 2, …, n)

Initialize the initial orthonormal basis $d_i$(i = 1, 2, …, n), the step size adjustment factors $\alpha$ and $\beta$

Initialize the termination parameter $∊>0$ and $k=0$

Repeat

$x=x_k$

Repeat

Fori = 1:n

$y=x+d_i{\delta }_i$

Ify is better than x

$x=y$

${\delta }_i=\alpha {\delta }_i(\alpha >1)$

Else

${\delta }_i=\beta {\delta }_i(-1<\beta <0)$

End If

End For

Until in each direction, at least one successful and one failed motion, then set $k=k+1$, $x_k=x$, update the orthonormal basis $d_i$ and reset ${\delta }_i$

Until$\Vert x_{k+1}\Vert -\Vert x_k\Vert \le ∊$or${\delta }_{\mathit{min}}<∊$

Table 2.

Specific formulas in original RM.

Formulas	No.
$x_{k+1}-x_k={\sum }_{i=1}^n{\lambda }_i{d}_i$	(12)
$r_i=\left\{\begin{array}{c}{d}_i,{\lambda }_i=0\\ {\sum }_{i=1}^n{\lambda }_i{d}_i,{\lambda }_i\ne 0\end{array}\right)$	(13)
$q_i=\left\{\begin{array}{c}{r}_i,i=1\\ r_i-{\sum }_{j=1}^{i-1}\frac{q_j^T{r}_i}{q_j^T{q}_j}{q}_j,i\ge 2\end{array}\right)$	(14)
$d_i=\frac{q_i}{\Vert q_i\Vert }$	(15)

2.3.2. The modified RM

Although the classical RM has a very good effect on unimodal problems, it often falls into the local optimum when facing multimodal problems. This is against the direction of improvement of the optimizer, so it is inappropriate to incorporate RM directly into WOA. Inspired by the literature of Kawaguchi et al. [68] and Li et al. [60], the initial step size and termination conditions of the original RM are modified. The pseudocode of the modified RM is shown in Algorithm 2.

The initial step size of the original RM is a non-zero constant, which is not suitable for use in dynamic WOA. Because the solutions produced by each iteration of WOA are different and meaningful, it is inefficient and meaningless to use the same step size for these distinct solutions. To take full advantage of these solutions, it is necessary to change the setting of the initial step size. The core principle of the modification is to judge whether the area has further exploitation value according to the discrete degree of each dimension of the current position. The current position has a low degree of dispersion in a dimension, meaning that most individuals in that dimension are very concentrated. Therefore, the current area has the value of further exploitation, and smaller step size is more conducive to the current situation. Similarly, high dispersion of the current position in a dimension means that most individuals in that dimension are very dispersed. Therefore, larger step size is required to develop other regions. The relevant mathematical models are defined in Eq. (16) and Eq. (17).

$${\delta }_i=\sqrt{\frac{\sum _{k=1}^n{\left(X_{\mathit{ki}}-\overline{X_i}\right)}^2}{n}}+{∊}_1,i=1,2,...d$$ (16)

$$\overline{X_i}=\frac{{\sum }_{k=1}^n{X}_{\mathit{ki}}}{n}$$ (17)

where $n$ is the number of individuals and $d$ is the population dimension. $X_{\mathit{ki}}$ represents the value of the $k$-th individual in the $i$-th dimension and $\overline{X_i}$ is the average value of the $i$-th dimension, which is calculated according to Eq. (17). ${∊}_1=1.0e^{-150}$ is a tiny constant that guarantees that the initial step size is not zero.

Another modified aspect is the termination condition. The original RM only exits when there is at least one successful and one failed motion in each search direction or when a preset optimal value is satisfied. This condition is no problem when dealing with low-dimensional situations, but it is very harsh when dealing with high-dimensional situations. If the basic RM does not change, the search direction is not updated immediately throughout the iteration, and the correct direction information is not preserved. To circumvent these problems, it is also necessary to change the termination condition. In Algorithm 2, $z$ is a temporary variable and $k_1$ and $k_2$ are loop counters. The modified RM includes an inner loop and an outer loop. Two parameters control the inner loop ${∊}_1$ and ${∊}_2$. As previously stated, ${∊}_1$ can prevent division by zero mistakes and ensure that the initial step size is not 0. The absolute value of the relative change of the objective function needs to be smaller than the pre-set value ${∊}_2$. According to Kawaguchi's literature, ${∊}_2$ is set to $1.0e^{-4}$. The termination condition of the outer loop is that the number of loops exceeds $2N,$ or the minimum step size is small enough. It should be noted that, compared to the basic RM, the standard orthonormal basis $d_i$ does not reset the step size after updating. This avoids the extra time cost of resetting the step size frequently.

The modified RM can better fit into the dynamic WOA and exert its powerful performance.

Algorithm 2 Pseudo-code of modified RM —

Initialize the initial orthonormal basis $d_i$(i = 1, 2, …, n), the step size adjustment factors $\alpha$ and $\beta$

Initialize the termination parameter ${∊}_1$, ${∊}_2$, $N$, $k_1$ and $k_2$

Input: the population, population member $X_0$ (the starting point), the objective function $f$

Output: The best solution $X_{\mathit{best}}$ searched so far

Initialize the step size ${\delta }_i$(i = 1, 2, …, n) using Eq. (16), and let $X_k=X_0$

Repeat

$x=X_k$; $k_1=0$; $z=x$

Repeat

Fori = 1:n

$y=x+d_i{\delta }_i$

Ify is better than x

$x=y$

${\delta }_i=\alpha {\delta }_i(\alpha >1)$

Else

${\delta }_i=\beta {\delta }_i(-1<\beta <0)$

End If

End For

If$[abs(f(z)-f(x))/abs(f(x)+{∊}_1)]<{∊}_2$

$k_2=k_2+1$

Else

$k_2=0$

End If

$k_1=k_1+1$

Until$k_1\ge N$

If$f(x)<f(X_k)$

$X_k=x$

Update the orthonormal basis $d_i$

End If

Until$k_2\ge 2N$ or ${\delta }_{\mathit{min}}<{∊}_1$

2.4. Framework of RRWOA

The framework proposed in this study integrates the original WOA with RCS and RM strategies simultaneously. The proposed RRWOA can inherently achieve a better trade-off between exploration and exploitation. The RCS strategy can use the newly generated solution to explore the solution space further after the original exploitation, avoid falling into local optimum and increase the diversity of the population. The modified RM strategy can fully exploit the potential regions and improve the solution quality. Nevertheless, when to use both strategies is a question worth thinking about. It is not feasible to directly embed the two methods. RCS and RM have their characteristics and need to be embedded differently.

The first is the RCS strategy. As discussed in previous sections, the original WOA used random values to control exploration and exploitation, which did not work very well. Because of the defects of the original WOA, it is not appropriate to use random values for the control parameter of RCS. This study discusses the usage of an iteration-related parameter to regulate RCS. The specific definition can be seen in Eq. (18).

$$b=1-\frac{\mathit{FEs}}{\mathit{MaxFEs}}$$ (18)

where $\mathit{FEs}$ is the current number of evaluations and $\mathit{MaxFEs}$ is the total number of evaluations. The number of evaluations instead of the number of iterations is employed here to ensure the algorithm's fairness [69] and avoid using time for space.

The RM strategy also needs careful consideration. As mentioned above, the modified RM strategy can significantly improve the exploitation ability of the algorithm. The original WOA is prone to settling for a local optimum in a promising location and failing to discover the ideal answer. The modified RM can help the original WOA jump out of the local optimum and improve the exploitation ability. However, although the modified RM has improved the basic RM to a certain extent, it will still bring a certain time consumption. Given this, the modified RM strategy is more suitable as a complementary algorithm to the original one. When WOA performs well, RM does not need to be called. When the exploitation capability of WOA is weak, RM will be called. Therefore, a control signal characterizing exploitation capacity is used to determine whether the RM strategy needs to be invoked, specifically presented in Eq. (19).

$$\mathit{pro}{b}_i=\frac{P_{\mathit{no}}}{n}\times rand(0,1)$$ (19)

where $n$ is the population size, $P_{\mathit{no}}$ is the number of individuals for which no better than optimal solution has been found so far and $\mathit{pro}{b}_i$ represents the probability that the $i$-th individual executes the RM strategy, which is located between (0,1). In a sense, this control signal is similar to the roulette wheel selection mechanism. The larger the $P_{\mathit{no}}$, the fewer times the optimal individual is updated, that is, the weaker the exploitation ability, and the greater the possibility that the RM strategy is invoked. According to [60], the threshold of $\mathit{prob}$ is set to 0.8. When $\mathit{prob}$ is greater than 0.8, the RM strategy is executed.

After executing the original WOA, the solution space is further explored using RCS according to the control parameter $b$. Then use the control signal $\mathit{prob}$ to execute the RM strategy to enhance the exploitation performance of the algorithm. It is worth noting that both WOA and the two strategies use a greedy selection strategy [70] to update the optimal position. The pseudocode of the specific algorithm is shown in Algorithm 3.

Algorithm 3 Pseudo-code of RRWOA —

Stage 1. Initialization

Generate the WOA population Xi (i = 1,2,….,n)

Calculate the objective function of each search agent

Determine the best search agent $X_{\mathit{best}}$

Stage 2. RRWOA operators

While not meet stop conditions

Fori = 1:n

Update a, A, C, l and p

Ifp < 0.5

If|A| < 1

Calculate $X_{\mathit{new}}$ using Eq. (1)

Else If$|A|\ge 1$

Select a random search agent $X_{\mathit{rand}}$

Calculate $X_{\mathit{new}}$ using Eq. (8)

End If

Else If$p\ge 0.5$

Calculate $X_{\mathit{new}}$ using Eq. (5)

End If

If$f(X_{\mathit{new}})<f(X_i)$

$X_i=X_{\mathit{new}}$

End If

Calculate $b$ and $\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}$ using Eq. (18) and Eq. (19), respectively

If$\mathit{rand}<b$

Calculate $X_{new1}$ using Eq. (11)

If$f(X_{new1})<f(X_i)$

$X_i=X_{new1}$

End If

End If

If$\mathit{prob}>0.8$

Calculate $X_{new2}$ using Algorithm 2

If$f(X_{new2})<f(X_i)$

$X_i=X_{new2}$

End If

End If

End For

Examine for and modify any search agents that extend beyond the solution space.

Calculate the objective function of each search agent

Update $X_{\mathit{best}}$ if a better solution exists

End While

Stage 3. Return${\mathit{X}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}$

2.5. Binary RRWOA

Feature selection is a typical application scenario of MAs. Various MAs have been widely used in feature selection. WOA and its variants have many applications in the field of feature selection. Samy et al. [71] applied binary WOA to five color image datasets. Zheng et al. [13] combined the WOA packaging method with the maximum Pearson maximum distance filtering method and applied it to the UCI datasets. Guha et al. [72] introduced chaos and fitness-dependent death mechanism into WOA. The suggested technique employed a wrapping procedure to achieve high accuracy in classification as well as a filter mechanism to further narrow the selected subset with minimal computing cost.

As a result, the suggested approach must be used in feature selection. Feature selection is commonly recognized as a binary optimization problem. The suggested RRWOA is appropriate for situations involving continuous optimization. As a result, an enhanced Binary RRWOA, denoted as BRRWOA, was created. In this approach, an individual's location is represented by a collection of decimals between [0,1]. On this basis, a single location defined by the specific transformation mechanism $X$ in the continuous motion space can be converted into a binary position $X^{\prime }=\{x_1^{\prime },x_2^{\prime },...,x_i^{\prime },...,x_n^{\prime }\}$ of length $n$, where $x_i^{\prime }\in \{0,1\}$, $n$ represents the feature number. Each bit corresponds to the i-th feature. “1″ indicates that the feature is selected, whereas “0″ indicates that it is not [36]. Each binary whale represents a subset of features. Discretize the original continuous individuals according to Eq. (20):

$$\begin{array}{c}{X}_d(t+1)=\left\{\begin{array}{c}1,sigmoid(X_d(t))\le rand\\ 0,others\end{array}\right)\end{array}$$ (20)

where $X_d(t+1)$ represents the binary position of the $d$-th dimension in the t + 1 iteration and $\mathit{rand}$ is a random number in [0,1]. The specific expression of the sigmoid is in Eq. (21).

$$\mathit{sigmoid}(x)=\frac{1}{1+e^{-2x}}$$ (21)

To demonstrate the advantage of BRRWOA in feature selection, a classifier must be utilized after feature selection. The k-nearest neighbor (KNN) [73] classification approach is widely used in feature selection. There is a clear relationship between the classification error margin of KNN and a precise measurement of distance in limited samples. KNN calculates distance using Euclidean, Manhattan, Mahalanobis, and other distance metrics. The Euclidean distance, as stated in Eq. (22) is used in this article.

$$D(x,y)=\sqrt{\sum _{j=1}^n{\left(x_j-y_j\right)}^2}$$ (22)

where $D(x,y)$ represents the distance between two samples, $x_j$ represents the $j$-th dimension of the sample in the training set, $y_j$ represents the $j$-th dimension of the sample in the test set and $n$ represents the total dimension of the current test sample.

The project has been relentless in its quest of machine learning forecast accuracy. The purpose of feature selection is to choose a feature subset from the original dataset with a minimal number of features and then utilize the features in the feature subset to execute further machine learning procedures. As a result, the algorithm must take into account two factors: the number of feature subsets and the accuracy rate [74]. The target algorithm value is larger when the number of chosen features is low and the model prediction accuracy is high. As a result, striking a balance between the two is critical. This is referred to as combinatorial optimization. To attain better outcomes concurrently, a fitness function defined by a linear combination of error frequency and the number of features is implemented. Because classification performance is much more significant, the former weight is set to 0.95 and the latter to 0.05 [75]. The specific mathematical expression can be seen in Eq. (23).

$$\mathit{fitness}=\theta \bullet E+(1-\theta )\bullet \frac{R}{D}$$ (23)

where $\theta$ is the classification accuracy weight mentioned above, $E$ represents the classification error rate of KNN, $R$ is the length of the selected feature subset, and $D$ is the total number of features in the dataset.

The flowchart of the machine learning model combined with BRRWOA is shown in Fig. 1 . First, the input dataset is processed and normalized to [−1,1]. Then, for the fairness of the experiment, the dataset is partitioned with the dataset using tenfold cross-validation. The number of characteristics in the database is adjusted to the dimension of the crowd in BRROWA prior randomized implementing the binary community members. As previously said, “1″ indicates that the feature is selected, whereas “0″ indicates that the feature is not chosen. The classification technique is used to assess the accuracy of specified characteristics. Following that, the fitness is computed using Eq (23). Again when the findings are obtained, the population is immediately updated utilizing the main component of BRRWOA. After the halting condition has been fulfilled, the best individual is found just after set number of repetitions. The classification technique is used to assess the accuracy of classification in this example. Finally, the best feature subset is returned.

Fig. 1.

The flowchart of BRRWOA based on KNN for feature selection.

2.6. Computational complexity

The computational complexity of RRWOA depends on the population size ($n$), dimension size ($d$), and the maximum number of iterations ($\mathit{MaxFEs}$), which are derived from WOA and two improved strategies. The computational complexity of the original WOA comes from initialization, calculation of fitness value, and population update. The computational complexity of initialization and calculation of fitness values are both $O(n\times d)$. The computational complexity of population update is $O(MaxFEs\times n\times d)$. Consequently, the computational complexity of the original WOA is $O((2+MaxFEs)\times n\times d)$.

The added RCS strategy and RM strategy both modify the $d$-dimensional data of $n$ individuals in the $\mathit{MaxFEs}$ evaluation, so the complexity is $O(MaxFEs\times n\times d)$. According to the above analysis, the complexity of RRWOA should be $O((2+3\times MaxFEs)\times n\times d)$.

3. Experimental design and analysis

We applied it to the competition functions and feature selection to verify the superiority of RRWOA in terms of basic performance and practical application. Firstly, the experiments based on 30 CEC2014 [61] competition functions mainly include four parts: the mechanism comparison experiment, the comparison experiment with the common MAs, the comparison experiment with the advanced MAs, and the comparison experiment with the famous variants of WOA. The mechanism comparison experiment is mainly to test whether the effect of adding two mechanisms is better than adding one alone. The remaining three comparative experiments demonstrate the superiority of the proposed method in this field. Furthermore, the feature selection experiments of RRWOA consist of experiments based on 15 UCI datasets and experiments based on the COVID-19 dataset. The former verifies the generality of the proposed algorithm, and the latter highlights its practical application ability.

Experiments must ensure fairness, and the hardware environment in which they run must be the same. So, all tests were performed on Windows Server 2012 R2. The system's processor is an Intel(R) Xeon(R) Silver 4110 CPU @ 2.10 GHz with 32 GB of memory. Experimental programming uses MATLAB R2014a.

The experimental results select average (Avg) and standard deviation (Std) for quantitative analysis to judge the superiority of the proposed algorithm. Avg is used to evaluate the algorithm's global search ability and the solution's quality, while Std is used to evaluate the algorithm's stability [76]. The best results for each function are marked in bold to make the experimental results stand out. Secondly, to facilitate the comparison between the algorithms, the Wilcoxon signed-rank test [62] is used to examine whether the performance between the algorithms is significant. The test's statistical significance threshold was specified at 0.05. Furthermore, the marks “+/=/−” show that RRWOA is superior to, equal to, or inferior to other techniques, accordingly. Finally, the Friedman test [63] is further used to demonstrate whether the proposed method is significantly different from other methods, as an extension of the Wilcoxon signed-rank test.

3.1. Benchmark functions comparison experiment

Experiments designed based on 30 IEEE CEC2014 [61] competition functions can preliminarily verify the superiority of the proposed RRWOA. The specific contents of the 30 competition functions are shown in Table A1 in Appendix A. These benchmark functions are divided into 4 categories, including three unimodal functions (F1–F3), thirteen multimodal functions (F4–F16), six hybrid functions (F17–F22), and eight composition functions (F23–F30). The unimodal functions have just one optimum solution, which may be used to test the algorithm's exploitation capabilities. Multimodal functions have numerous locally optimum solutions, which might show the algorithm's capacity to explore the solution space outside of the local ideal. The capacity of RRWOA to manage exploration and exploitation may be highlighted through hybrid and composition functions.

Experiments must guarantee fairness [77], so they are all carried out under the same framework. The population size is set to 30. The population dimension is 30 and the number of evaluations is 300,000. It is worth emphasizing that the judgment condition utilized here is not the number of iterations, but rather the ability to record the conduct of each application to the fitness function. At the same time, each experiment was replicated 30 times to limit the impact of the algorithm's inadvertent errors on the assessment findings.

3.1.1. Mechanism comparison experiment

In the previous section, two strategies were introduced into the original WOA: the RCS and the RM. A comparative experiment on the corresponding variants of RRWOA is required to prove that the two strategies' combined effect is optimal. The details of the variants are shown in Table 3 , where the first row of the table lists the strategy used, the first column describes the name of the variant, “Y” indicates that the WOA adopted the strategy, and “N” indicates that the strategy was not adopted.

Table 3.

Various WOA variants with two strategies.

	RCS	RM
WOA	N	N
RCSWOA	Y	N
RMWOA	N	Y
RRWOA	Y	Y

According to the above settings, the Avg and Std values of each competition function are shown in Table A2. Among them, the optimal result of each function is shown in bold. It can be seen that RRWOA performs better in most functions. RRWOA ranked first in 18 functions and ranked first overall. Combining the contents of Table A2 and Table A1, RRWOA converges to the global optimum on F7, F12–F14, and fully approximates the global optimum on F5, F6, F8, F15, F16, and F19. It can be seen that the two combined strategies can improve the performance of the algorithm and explore the location of the global optimal value. It can also be seen from Table A2 that RRWOA significantly outperforms RCSWOA on most functions. This shows that the RM strategy can exert its exploitation ability well and assist the method in shifting away from the local optimum. While RMWOA is better than RRWOA in F1, F8, F9, and other functions, it is weaker than RRWOA in F23–F30. F1–F3 are unimodal functions with only one global optimum, so the exploitation ability of the algorithm can be well-tested. F4–F16 are multimodal functions that can well detect the ability of the algorithm to jump out of the local optimum, that is, the ability to explore. F23–F30 are composition functions that detect the algorithm's ability to balance the exploration and exploitation steps. The performance of the RM strategy on unimodal and composition functions demonstrates that, despite its high exploitation capacity, it will imbalance the algorithm's stages. Therefore, RRWOA, which combines the RM strategy with the RCS strategy and utilizes the exploration ability of the RCS to balance the exploration and exploitation process, is the best optimizer. The RM strategy still performs well on multimodal functions, which originates from the invocation of control signals. The RM strategy will only be invoked when the algorithm has not updated the optimal value for a long time. Table A3 in appendix A shows the results of the Wilcoxon signed-rank test. To better visualize the results, a bar graph with p-values ranging from 0 to 0.1 was plotted in Fig. 3 against a threshold of 0.05. It can be observed that most of the values are less than 0.05, which shows that RRWOA has a significant performance improvement compared with other combinatorial mechanisms.

Fig. 3.

Bar graph of P-values for Wilcoxon test on 30 functions for comparison of mechanism experiments.

To more intuitively show the superiority of RRWOA, the convergence curves of RRWOA and its variants RCSWOA, RMWOA and WOA are given in Fig. 2. The unimodal function can detect the exploitation ability of the algorithm. In both F2 and F3, RRWOA can converge to a better position. This stems from the exploitation ability of the RM strategy, and the exploitation ability of RRWOA is stronger than that of RMWOA. F5, F6, F10, and F11 are multimodal functions in which RRWOA performs better. It shows that the RCS strategy can improve the exploration ability of the algorithm well, and the control signal also plays a corresponding role. F17, F18, and F21 are hybrid functions. It can be seen that RRWOA can better balance the exploration and exploitation process than adding a specific strategy alone. From this, it can be inferred that the combined strategy of RCS and RM is the most effective for improving the performance of WOA. RCS can explore as vast a region as possible, and RM can fully mine local optima. The two work together with half the effort. Therefore, RRWOA is the best variant in this experiment.

Fig. 2.

Convergence curves of 9 benchmark functions (First row: F2, F3, F5; second row: F6, F10, F11; third row: F17, F18, F21).

3.1.2. Comparison with basic methods

It is necessary to compare the newly proposed method with the base method. The following methods were compared with RRWOA under CEC2014: HGS [30], Harris Hawks Optimization (HHO) [31], RUN [32], DE [26], Salp Swarm Algorithm (SSA) [78], PSO [79], Grey Wolf Optimizer (GWO) [80], Sine Cosine Algorithm (SCA) [81] and Moth-flame Optimization Algorithm (MFO) [82]. Table 4 lists all parameter settings involved in the algorithm.

Table 4.

Parameters setting for common basic algorithms.

Method	Other parameters
RRWOA	$\mathit{prob}=0.8;k2=30;a1=[2,0];a2=[-2,-1];f=1$
HGS	$l=0.03;LH=100$
HHO	∼
RUN	$r=1\mathrm{or}-1\mathrm{or}0;a=20;b=12$
DE	$\mathit{Scaling}\mathit{Factor}=\left[0.2,0.8\right];CrossoverProbability=0.2$
SSA	$c1\in \left[0,1\right];c2\in \left[0,1\right]$
PSO	$c1=2;c2=2;vMax=6$
GWO	$a\in \left[2,0\right]$
SCA	$a=2$
MFO	$b=1;t=\left[-1,1\right];a=\left[-1,-2\right]$

Table A4 shows the mean and standard deviation of RRWOA and other well-known MAs. At the end of the table is the mean rank value given by the Friedman test. It can be found that RRWOA ranks 1st on 18 test functions and 2nd on 4 test functions. Combining the results of all test functions, RRWOA is also the best optimizer. Compared to other primitive MAs, RRWOA can find global optimum on F2, F7, and F12–F14. Even if the global optimal position is not reached on F4–F6, F8, F15, F16, and F19, the vicinity of the global optimal position is fully explored. This stems from the combined effect of the RCS strategy and RM strategy. RCS can fully expand the position of the original solution and increase the diversity of the solution. The RM strategy, as a supplementary strategy for the exploitation part, can give full play to the exploitation performance and find the optimal value of the potential area. The two complement each other and together improve the performance of WOA.

The convergence curve can better highlight the convergence performance of the algorithm. Fig. 4 presents some convergence images of RRWOA with conventional MAs. F2 is a representative of a unimodal function. It can be seen from the figure that RRWOA can converge to a position that other algorithms cannot reach, standing for very strong exploitation capabilities. The search and update direction of the RM strategy helps the algorithm to dig as much as possible in the potential area and enhance the local search ability. F5, F6, F11, F12, and F16 are representatives of multimodal functions. It can be seen from the figure that the curve of RRWOA converges quickly and well. This shows that the RCS strategy can speed up the convergence speed and assist the method in shifting away from the local optimum and approach the global optimum. The RCS strategy expands the search range as much as possible at the original solution's position and increases the population's diversity. F18 is the representative of the hybrid function, and F29 and F30 are the representatives of the composition function. Compared with other algorithms, RRWOA can better balance the exploration and exploitation process of the algorithm. Although F29 converges to the same position as HHO, the convergence speed of RRWOA is much faster, which still reflects the superiority of the algorithm.

Fig. 4.

Convergence curves of 9 benchmark functions (First row: F2, F5, F6; second row: F11, F12, F16; third row: F18, F29, F30).

To supplement Table A4, Table A5 in Appendix A presents the p-values for the results of the Wilcoxon signed-rank test. The p-value can judge whether RRWOA is significantly better than the contrasting algorithm. As can be seen from the end of Table A4, RRWOA has 16, 21, 19, 20, 22, 25, 22, 29, and 27 cases, respectively, which are better than HGS, HHO, RUN, DE, SSA, PSO, GWO, SCA, MFO. The p-value corresponding to the corresponding value is less than 0.05, and the optimization effect is significant. In addition, there are 14, 9, 8, 3, 5, 2, 8, 1, and 3 cases of RRWOA equal to HGS, HHO, RUN, DE, SSA, PSO, GWO, SCA, and MFO, respectively; RRWOA has 0, 0, 3, 7, 3, 3, 0, 0, and 0 cases lower than HGS, HHO, RUN, DE, SSA, PSO, GWO, SCA, and MFO, respectively. Combined with Table A4 and Fig. 4, even if there is no obvious optimization on some functions, the value of RRWOA is similar to the optimal value of the comparison algorithm. MAs have a certain degree of randomness. Based on all the results, the optimization effect of RRWOA is significant.

3.1.3. Comparison with advanced methods

In this part, to further demonstrate the powerful performance of RRWOA in global search, it is compared with other academically well-known advanced evolutionary algorithms, such as adaptive SCA integrated with PSO (ASCA-PSO) [83], enhanced SSA (ESSA) [84], chaos-induced and mutation-driven SSA (CMSSA) [85], modified SCA (MSCA) [86], improved GWO (IGWO) [87], oppositional based learning GWO (OBLGWO) [88], HHO with gaussian mutation (GCHHO) [89], HHO integrated with DE (HHODE) [90] and chaos-enhanced MFO (CMFO) [91]. Table 5 lists all parameter settings involved in the algorithm.

Table 5.

Parameters setting for well-known advanced algorithms.

Method	Other parameters
ASCA-PSO	$\mathit{Vmax}=4;wMax=0.9;wMin=0.2;c1=2;c2=2;a=2;$
ESSA	$c1\in \left[0,1\right];c2\in \left[0,1\right];$
CMSSA	$c1\in \left[0,1\right];c2\in \left[0,1\right];$
MSCA	$\mathit{wMax}=0.9;wMin=0.4;∊=30;\lambda =0.01;\beta =1.5;a=2;$
IGWO	$a\in \left[2,0\right];$
OBLGWO	$a\in \left[2,0\right];$
GCHHO	∼
HHODE	$\mathit{CR}=0.9;\beta =0.5;$
CMFO	$b=1;t=\left[-1,1\right];a\in \left[-1,-2\right];$

As in the experiments in Section 3.2, the mean and standard deviation of RRWOA and the advanced algorithms are shown in Table A6. At the end of Table A6 are the mean ranking values given by the Friedman test. It can be seen that the proposed RRWOA achieves a mean of 1.7333, far exceeding the second algorithm with a mean of 3. Specifically, RRWOA converges to the global optimum on F7, F12–F14 and fully approximates the global optimum on F5, F6, F8, F15, F16, and F19. From the data in bold, RRWOA ranks first on 24 functions, accounting for the vast majority. In the optimization process, RCS and RM strategies play an important role.

The convergence curve in Fig. 5 vividly demonstrates the significant superiority of RRWOA compared to other algorithms. F1 and F3 are unimodal functions, proving the RM strategy's superior performance in the algorithm. Compared with other algorithms, the convergence accuracy of RRWOA is significantly improved, and more potential regions can be reached. F10, F11, and F16 are multimodal functions in which the superior performance of RRWOA is more prominent. The proposed algorithm can reach much better positions with a powerful ability to escape from local optima. F18–F20 are hybrid functions, and F29 is a composition function. The performance of RRWOA is still superior on these two types of functions, proving that the algorithm can achieve a better balance during exploration and exploitation. In addition to the role of the two strategies, the selected control parameters also play a corresponding role. The RCS parameters decrease with increasing iterations and eventually tend to zero. It shows that RCS is more likely to be executed in the early stage of algorithm iteration, that is, the exploration stage, which increases the exploration ability of the algorithm. The RM as a supplementary strategy is controlled by the control signal and will only be executed when the algorithm has not updated the global optimal for a long time. Together, these two parameters control the exploration and exploitation process, enhancing the performance of WOA. However, it can also be seen from Fig. 5 that the improvement of the convergence accuracy is accompanied by a slowdown of the convergence speed, which is due to the time consumption caused by the RM strategy. In addition, since the RM strategy appears as a supplementary strategy to the original algorithm, it will only be executed when the algorithm has not updated the global optimum for a long time.

Fig. 5.

Convergence curves of 9 benchmark functions (First row: F1, F3, F10; second row: F11, F16, F18; third row: F19, F20, F29).

Likewise, the p-values for the Wilcoxon signed-rank test results are given in Table A7 in Appendix A. 26, 29, 22, 28, 25, 23, 18, 21, and 28 cases of RRWOA are superior to ASCA-PSO, ESSA, CMSSA, MSCA, IGWO, OBLGWO, GCHHO, HHODE, and CMFO, respectively. The p-value corresponding to the value is less than 0.05, and the overall performance is excellent. In addition, RRWOA has 4, 1, 8, 2, 3, 7, 12, 9, and 2 cases equal to ASCA-PSO, ESSA, CMSSA, MSCA, IGWO, OBLGWO, GCHHO, HHODE, and CMFO, respectively; RRWOA has 0, 0, 0, 0, 2, 0, 0, 0 and 0 cases lower than ASCA-PSO, ESSA, CMSSA, MSCA, IGWO, OBLGWO, GCHHO, HHODE, and CMFO, respectively. The experimental results are sufficient to demonstrate that the performance of RRWOA has significant advantages.

3.1.4. Comparison with improved WOA variants

Experiments are carried out in this section to support the superior performance of RRWOA. RRWOA is compared not only with other MAs but also with many variants of WOA. These algorithms are derived from better-performing, reproducible variants developed in recent years. This section uses A-C parametric WOA (ACWOA) [92], chaotic WOA (CWOAI) [93], improved WOA (IWOA) [94], levy flight trajectory-based WOA (LWOA) [95], modified WOA (MWOA) [96], opposition-based WOA (OBWOA) [97] and double adaptive random spare reinforced WOA (RDWOA) [24]. Table 6 lists all parameter settings involved in the algorithm.

Table 6.

Parameters setting for famous WOA variants.

Method	Other parameters
ACWOA	$a1=[2,0];a2=[-2,-1];b=1;$
CWOAI	$\mathit{cindex}=5;a1=[2,0];a2=[-2,-1];b=1;$
IWOA	$\mathit{CR}=0.1;a1=[2,0];a2=[-2,-1];b=1;$
LWOA	$a1=[2,0];a2=[-2,-1];b=1;$
MWOA	$a1=[2,0];a2=[-2,-1];b=1;$
OBWOA	$a1=[2,0];a2=[-2,-1];b=1;$
RDWOA	$a1=[2,0];a2=[-2,-1];b=1;$

A p-value less than 0.05 in the Wilcoxon signed-rank test indicates that the method's performance varies substantially from other techniques. The better the procedure, the lower the ranking value of the Freidman test findings. Table A8 demonstrates that RRWOA has the lowest mean, showing that the approach suggested in this research is statistically better than all of the evaluated algorithms. Looking closely at Table A8, the proposed RRWOA achieves first place on 25 test functions, achieving outstanding results. The std values on F23–F25 and F27–F30 are all 0, which proves the algorithm's stability. Specifically, RRWOA converges to the global optimum on F2, F7, F12–F14 and fully approximates the global optimum on F3–F6, F8, F15, F16, and F19. This proves that RRWOA has excellent global search ability. This stems from the fact that the RCS strategy can fully expand the position of the current solution and assist the method in shifting away from the local optimum. The RM strategy, on the other hand, makes full use of potential areas through search update direction, improving the quality of solutions.

The related convergence curves in Fig. 6 show that RRWOA performs very well, whether in unimodal, multimodal, hybrid, or composition functions. The proposed algorithm can quickly reach the potential area and obtain the optimal fitness value. RRWOA substantially improves the convergence speed even though the results on F28 are the same as RDWOA. It can be seen that the two strategies added can not only enhance the algorithm's performance and the population's diversity but also achieve a balance between exploration and exploitation. This stems from two strategies and two control parameters. The control parameter of the RCS decreases with iteration, while the control signal of the RM allows the RM strategy to come into play when it is most needed.

Fig. 6.

Convergence curves of 9 benchmark functions (First row: F1, F3, F6; second row: F11, F16, F18; third row: F28, F29, F30).

Similarly, the p-values for the Wilcoxon signed-rank test results are given in Table A9 in Appendix A. There were 25, 28, 24, 26, 29, 23, and 21 cases of RRWOA, respectively, all of which were superior to ACWOA, CWOAI, IWOA, LWOA, MWOA, OBWOA, and RDWOA. The p-value corresponding to the value is less than 0.05, and the overall performance is excellent. In addition, RRWOA has 4, 2, 4, 2, 1, 7, and 8 cases equal to ACWOA, CWOAI, IWOA, LWOA, MWOA, OBWOA, and RDWOA, respectively; there are 1, 0, 2, 2, 0, 0 and 1 cases of RRWOA lower than ACWOA, CWOAI, IWOA, LWOA, MWOA, OBWOA, and RDWOA, respectively. The experimental results are sufficient to demonstrate that the performance of RRWOA has significant advantages so that it can be considered for many applications in the future such as recommender system [98], [99], image denoising [100], human activity recognition [101], drug discovery [102], [103], location-based services [104], [105], road network planning [106], disease identification and diagnosis [107], [108], and colorectal polyp region extraction [109].

3.2. The experiments for feature selection

In addition to the basic experiments on the competition functions, the algorithm also has a very good application in the feature selection of machine learning. Feature selection is a common case for finding feasible solutions to problems. Applying the proposed algorithm to feature selection can effectively demonstrate the practicability of our algorithm. The binary form of RRWOA and the corresponding fitness function has been introduced in Section 2.

To verify the generality of the proposed algorithm, we compared BRRWOA with the binary versions of 7 algorithms on 15 datasets extracted from the UCI data repository [110]. Next, to highlight the practical application capability of RRWOA, BRRWOA was applied to the real medical dataset of COVID-19 and compared with the binary versions of five algorithms.

To ensure the unbiasedness of the experimental results, the 10-fold cross-validation technique was used to evaluate the test results [111], the number of iterations was set to 50, and the population scale was set to 20. The experimental operating environment is MATLAB R2014a. During the testing process, the test data set is divided into 10 equal parts, one of which is randomly selected as the test set, and the remaining 9 parts are used as the training set.

3.2.1. Standard of evaluation of experiments

To show the optimization impact of the techniques in the feature selection procedure, the following requirements are utilized to assess each optimization method [87].

Fitness average: The average fitness of each algorithm on the selected data set. The calculation is as below:

$$\begin{array}{c}Mean\_fit=\frac{1}{K}{\sum }_{i=1}^{K}Fitness(i)\end{array}$$ (24)

where $\mathit{Fitness}(i)$ is the fitness value obtained from each run of the algorithm. $K$ is the number of times the algorithm is executed.

Average classification error rate: The average of each algorithm's classification error margins on the specified dataset. The computation procedure is as follows:

$$\begin{array}{c}Mean\_err=\frac{1}{K}{\sum }_{i=1}^K(1-Accuracy(i))\end{array}$$ (25)

where $\mathit{Accuracy}(i)$ is the classification accuracy obtained for each run of the algorithm, and $K$ is the number of times that the algorithm is executed.

The average number of characteristics chosen: The number of chosen characteristics averaged over K times is determined as follows:

$$\begin{array}{c}Mean\_num=\frac{1}{K}{\sum }_{i=1}^{K}Size(i)\end{array}$$ (26)

where $\mathit{Size}(i)$ indicates the number of features chosen in the optimization algorithm's i-th run.

Average duration: Average time in seconds for each program run.

$$\begin{array}{c}Mean\_time=\frac{1}{K}{\sum }_{i=1}^{K}Time(i)\end{array}$$ (27)

where $\mathit{Time}(i)$ is the time for each run of the algorithm.

3.2.2. Experimental results and analysis on UCI datasets compared with basic methods

To verify the generality of RRWOA, we extracted 15 datasets from the UCI data repository [110]. The specific information on the datasets comes from http://archive.ics.uci.edu/ml/index.php. In its widest sense, benchmarking is the comparing of one or more items to a competent model throughout a wide range of performance indicators [112], [113], [114].

Table 7 shows the information on the datasets in detail, including the number of test samples, the number of test features, and sample categories.

Table 7.

Information on the datasets (text taken from website for formal descriptions).

Dataset	Samples	Features	Classes	Abstract descriptions
Breastcancer	699	10	2	Original Wisconsin Breast Cancer Database
Exactly	1000	13	2	Please refer to https://archive.ics.uci.edu/
HeartEW	270	13	2	Please refer to https://archive.ics.uci.edu/
M-of-n	100	13	2	Artificial problem representing M-of-N rules
Congress	435	16	2	1984 United Stated Congressional Voting Records; Classify as Republican or Democrat
Hepatitis	155	19	2	From G. Gong: CMU; Mostly Boolean or numeric-valued attribute types; Includes cost data (donated by Peter Turney)
USA	2336	10	2	The samples are obtained from Compustat North America, WHARTON RESEARCH DATA SERVICE.
Wine	178	13	3	Using chemical analysis determine the origin of wines.
cmc	1473	9	3	Dataset is a subset of the 1987 National Indonesia Contraceptive Prevalence Survey.
Lymphography	148	18	4	This lymphography domain was obtained from the University Medical Centre, Institute of Oncology, Ljubljana, Yugoslavia. (Restricted access)
vehicle	946	18	4	3D objects within a 2D image by application of an ensemble of shape feature extractors to the 2D silhouettes of the objects.
heart	270	13	5	This dataset is a heart disease database similar to a database already present in the repository (Heart Disease databases) but in a slightly different form.
glass	214	10	6	From USA Forensic Science Service; 6 types of glass; defined in terms of their oxide content (i.e. Na, Fe, K, etc).
Vote	435	16	2	1984 United Stated Congressional Voting Records; Classify as Republican or Democrat.
Zoo	101	17	7	Artificial, 7 classes of animals.

As can be seen from Table 7, these datasets have 100–2336 instances, 9-19 features, and 2–7 classes. These datasets represent different types of data and are extensive. It is enough to verify the generality of RRWOA.

To evaluate the effectiveness of BRRWOA, we selected several high-performance optimization algorithms for testing on the UCI datasets, including BWOA [115], BMFO [116], BGSA [117], BPSO [118], BBA [119], BSSA [78], and BRUN. The corresponding parameter settings are listed in Table 8 . These optimization algorithms are all suitable for feature selection and are therefore compared with the proposed algorithm.

Table 8.

Parameter settings for the 7 contrast classifiers (Number of attributes in the specific dataset = Dimension, population size = 20).

Algorithm	Values
BRRWOA	$\mathit{prob}=0.8;k2=30;b=1;a1=[2,0];a2=[-2,-1];$
BWOA	a = [0,2]
BMFO	a = 2; b = 1
BGSA	$\mathit{wMax}=20;wMin=1e-10;$
BPSO	$\mathit{Max}=0.9;Min=0.4;$
BBA	a = 0.5; r = 0.5
BSSA	a = 2
BRUN	r = 1 or −1 or 0; a = 20; b = 12

These algorithms are tested on 15 UCI datasets with 50 iterations. To ensure fairness, 10-fold cross-validation is performed. The classification results are evaluated according to the average fitness value, average error rate, average number of features, and average running time. The specific results are shown in Table A10, Table A11, Table A12, Table A13.

The number of selected features and the final classification accuracy are important parameters for feature selection. Balancing these two parameters is critical. It can be seen from Eq. (23) that the fitness value function balances these two parameters. Therefore, this is an important indicator to measure the result of feature selection. As can be seen from Table A10, the proposed BRRWOA obtained the smallest fitness value in 11 datasets and ranked second in three and third in one of the remaining four datasets. Combining all the results, BRRWOA wins the average ranking of 1.3333, far exceeding the second overall ranking of 2.7333, and wins the first. This shows that BRRWOA has the best-combined effect on the number of features and classification accuracy obtained in the experiment. The RCS strategy and the RM strategy can achieve a good balance between exploration and exploitation and obtain the optimal fitness value.

As seen from Table A11, except for the Breastcancer, glass, and Vote datasets, BRRWOA obtained the lowest classification error rate in the rest of the datasets. According to Eq. (25), the classification error rate is derived from the classification accuracy. The lower the classification error rate, the higher the classification accuracy. Although it is not ranked first in those three datasets, it is also ranked second and third, ranking first overall. Furthermore, in the six datasets of Exactly, M-of-n, hepatitis, Wine, Lymphography, and Zoo, BRRWOA has an error rate of 0 and a standard deviation of 0, proving that it has good robustness while ensuring high accuracy. In datasets that do not reach the minimum error rate, BRRWOA also reduces the value of this parameter as much as possible. It can be seen that the combination of the RCS strategy and RM strategy is very effective.

Table A12 can be analyzed in conjunction with Table A11. As seen from Table A12, although BRRWOA only ranks first in 5 datasets, it ranks either second or third in the remaining datasets, and the overall ranking is still first. Combining with Table A11, although the binary algorithm participating in the comparison can obtain the least number of features, it will bring a high classification error rate at the same time. For example, although BBA achieves the least number of features on Breastcancer, Exactly, cmc, and heart, the error rate also increases in the corresponding shares. The classification accuracy of the algorithm is more important than the number of features, so in terms of the overall effect, BRRWOA performs better. Although BRRWOA does not obtain the smallest features on some datasets, it is sufficiently approximated, proving its good feature selection ability. In Exactly with 1000 samples and M-of-n with 100 samples, BRRWOA obtained an Std of 0, proving its stability in multiple sample data. Although the std of USA, Wine, and glass are not 0, they are all less than 1, which shows the stability of BRRWOA.

Table A13 shows the specific results of the algorithm's average time. It can be observed that BRRWOA is time-consuming compared to other methods. In addition to the RCS strategy, the most time is spent on the RM strategy. Even if the initial step size and termination condition of the original RM is modified, each iteration increases the time consumption of the algorithm. However, considering other indicators, BRRWOA is still better than other algorithms in feature selection.

To more intuitively and vividly show the superiority of BRRWOA in feature selection, the convergence curve of the best fitness value calculated by the KNN classifier is drawn. Fig. 7 is precisely the set of fitness curves for all datasets. The abscissa represents the number of iterations of the algorithm, and the ordinate corresponds to the average fitness value after the algorithm is independently executed 10 times. The figure shows that the fitness values of BRRWOA on Breastcancer, HeartEW, Congress, hepatitis, USA, cmc, vehicle, and heart are much smaller than other algorithms, and an excellent convergence effect is obtained. The performance on Exactly, M-of-n, and Wine is similar to some algorithms, but they also find the optimal relative value and improve the convergence speed. The performance of Lymphography, glass, Vote, and Zoo is not the best, but they all achieve good results in classification error rate or the number of features and are generally helpful. These results are attributed to the RCS strategy and the RM strategy. The RCS strategy can better expand the solution space, and the RM strategy exploits potential regions as much as possible through search and update direction as a supplementary strategy for exploitation performance. The control parameters that balance these two strategies also play a very large role.

Fig. 7.

Convergence curves of BRRWOA and other algorithms for 15 data sets.

3.2.3. Experimental results and analysis of UCI datasets compared with WOA variants

To further verify the performance of the proposed method, this section embeds the WOA variants of Section 3.1.4 into the model through the same transfer function for experiments. BRRWOA is compared to BACWOA, BCWOAI, BIWOA, BLWOA, BMWOA, BRDWOA, excluding OBWOA, which does not apply to the model. Similarly, the parameter settings of the experiments are shown in Table 6, and the details of the UCI datasets are displayed in Table 7. Meanwhile, the number of iterations is 50, and 10-fold cross-validation is performed. The classification results are evaluated according to the average fitness value, average error rate, average number of selected features, and average running time. The specific results are shown in Table A14, Table A15, Table A16, Table A17.

As can be seen from Table A14, BRRWOA can achieve first place in all 15 datasets compared with the binary versions of the six WOA variants. Combining all the results, BRRWOA wins an average ranking of 1, far exceeding the second overall ranking of 2.8667, and wins the first. The average ranking first means that the proposed algorithm performs well in terms of classification accuracy and the number of selected features and can achieve a good balance among them. Therefore, the results also confirm that the RCS strategy and the RM strategy help the algorithm optimization, allowing the algorithm to obtain the optimal fitness value.

As seen from Table A15, except for the cmc dataset, the classification error rate of BRRWOA can reach the lowest level. Looking closely, the proposed method has both avg and std of 0 on the six datasets, namely Exactly, M-of-n, hepatitis, Wine, Lymphography, and Zoo. Combined with Table A11 in the previous section, the performance of BRRWOA is both excellent and stable. At the same time, the results of other algorithms on datasets such as Breastcancer, glass, and Vote are ten times worse than BRRWOA. The table results highlight the algorithm's superiority in situations where classification accuracy is far more critical than the number of selected features. All in all, adding the RCS strategy and RM strategy helps the algorithm optimize.

In the case of ensuring the model's classification accuracy, the final selected feature subset cannot be ignored, shown in Table A16. BRRWOA can get first place in all 12 datasets and second place in the remaining three. The proposed method ranks first overall, far exceeding the second place, with an average ranking of 2.8. Combined with Table A15, although BMWOA can obtain fewer selected features on Breastcancer and USA, the accuracy is far less than BRRWOA; although BACWOA can obtain fewer selected features on cmc, the error rate is higher than the proposed method. Therefore, considering all the results, the proposed method is still quite competitive. Similarly, in Exactly with 1000 samples and M-of-n with 100 samples, BRRWOA obtains a standard deviation of zero. Therefore, the robustness of the proposed method is good.

As can be seen in Table A17, BIWOA undoubtedly wins first place in terms of time, and the proposed algorithm ranks last. However, in the results of Table A14, Table A15, Table A16, BIWOA ranks last. Therefore, although the proposed method is time-consuming, it outperforms other algorithms in optimization ability. Overall, BRRWOA is still competitive. As mentioned above, the time consumption comes from the loop iteration of the RM strategy. In the future, parallel or multi-group mechanisms can be considered to alleviate this consumption.

Likewise, Fig. 8 plots the associated fitness convergence curves during training stage. The abscissa represents the number of iterations of the algorithm, and the ordinate corresponds to the average fitness value after the algorithm is independently executed 10 times. It can be observed that BRRWOA converges fast and well on all 15 datasets. Under the limited number of iterations, it not only speeds up the convergence speed but also obtains a better fitness value; that is, a good balance between the classification accuracy and the number of selected features is obtained. Returning to the algorithm itself, optimization performance improvement comes from the RCS and the RM strategies. RCS can jump out of local optimum and enhance exploration ability; RM strategy can fully study potential areas and enhance exploitation ability. In a word, the balance between the two helps the algorithm's performance improves.

Fig. 8.

Convergence curves of BRRWOA and other WOA variants for 15 data sets.

3.2.4. Experiments on the COVID-19 dataset

There are significant efforts to develop AI-assisted medical diagnosis systems for the early detection of diseases [70], [120], [121]. In this study, clinical characteristics, blood routines, inflammatory markers, coagulation function indicators, blood biochemical and arterial blood gas analysis indicators were collected retrospectively from non-severe COVID-19 patients, and severe COVID-19 patients through Wenzhou Medical University affiliated Yueqing Hospital electronic medical record system from January 21 to March 20, 2020.

In our study, we included 51 participants: 30 patients with non-severe COVID-19 and 21 patients with severe COVID-19 [122]. The ethics committee of Yueqing Hospital of Wenzhou Medical University approved the research (No. 202000002 Ethics), and all participants in COVID-19 signed an informed consent form. COVID-19 patients' clinical characteristics, blood routines, inflammatory markers, coagulation function indicators, blood biochemical, and arterial blood gas analysis indicators are described in Table 9 .

Table 9.

List of the features used in this study and their definitions number [122].

ID	Features	Abbreviation
F1	Gender	Gender
F2	Age	Age
F3	White blood cell	WBC
F4	Neutrophil percentage	NEU%
F5	Lymphocyte percentage	LY%
F6	Monocyte percentage	MONO%
F7	Eosinophils percentage	EOS%
F8	Basophils percentage	BA%
F9	Neutrophil count	NEU
F10	Lymphocyte count	LY
F11	Monocyte count	MONO
F12	Eosinophils count	EOS
F13	Basophils count	BA
F14	Red blood cell	RBC
F15	Haemoglobin	HB
F16	Hematocrit	HCT
F17	mean corpuscular hemoglobin	MCH
F18	Mean corpuscular volume	MCV
F19	Mean corpuscular hemoglobin concentration	MCHC
F20	Red blood cell distribution width	RDW
F21	Red blood cell distribution width-SD	RDW-SD
F22	Mean platelet volume	MPV
F23	Blood platelet	PLT
F24	Plateletcrit	PCT
F25	Platelet distribution width	PDW
F26	Platelet-large cell ratio	P-LCR%,
F27	Procalcitonin	PCT
F28	C-reactive protein	CRP
F29	Erythrocyte sedimentation rate	ESR
F30	Prothrombin time	PT
F31	International normalized ratio	INR
F32	Prothrombin time activity	PTA
F33	Fibrinogen	FIB
F34	Activated partial thromboplastin time	APTT
F35	Thrombin time	TT
F36	d-dimer	D-D
F37	Total bilirubin	TBIL
F38	Direct bilirubin	DBIL
F39	Alanine aminotransferase	ALT
F40	Total protein	TP
F41	Albumin	ALB
F42	Globulin	GLB
F43	Albumin/Globulin ratio	A/G
F44	Alkaline phosphatase	ALP
F45	Gamma-Glutamyltransferase	GGT
F46	Aspartate aminotransferase	AST
F47	Creatine kinase	CK
F48	Lactate dehydrogenase	LDH
F49	Creatine kinase isoenzymes	CK-MB
F50	Potassium ion	K+
F51	Sodium ion	Na+
F52	Chloride ion	Cl−
F53	Blood urea nitrogen	BUN
F54	Creatinine	Cr
F55	Uric acid	UA
F56	Inorganic phosphorus	P+
F57	Blood magnesium	Mg2+
F58	Calcium ion	Ca2+
F59	Troponin I	TnI
F60	Hydrogen ion concentration	PH
F61	Partial Pressure of Carbon Dioxide	PaCO2
F62	Partial pressure of oxygen	PaO2
F63	Oxygen saturation	SaO2%
F64	Hemoglobin percentage	Hb
F65	Oxyhemoglobin percentage	HbO2%
F66	Carboxyhaemoglobin percentage	COHb%
F67	Deoxyhemoglobin percentage	deoxyHb %
F68	Methaemoglobin percentage	MetHb %
F69	Potassium ion concentration	K+
F70	Sodium ion concentration	Na+
F71	Chloride ion concentration	Cl-
F72	Calcium ion concentration	Ca2+
F73	Glucose concentration	GLU
F74	Lactic Acid	LAC
F75	Anion gap	AG
F76	Buffer bases	BB
F77	Bases excess	BE
F78	Standard bicarbonate	SB
F79	Actual bicarbonate	AB
F80	Smoking history	SH
F81	Drinking history	DH
F82	Hypertension	HP
F83	Diabetes mellitus	DM
F84	Chronic kidney disease	CKD
F85	Chronic liver disease	CLD
F86	Chronic heart disease	CHD
F87	Chronic lung disease	CLD
F88	Malignant tumor	MT
F89	Fever	Fever
F90	Cough	Cough
F91	Sputum	Sputum
F92	Fatigue	Fatigue
F93	Myalgia	Myalgia
F94	Headache	Headache
F95	Nausea/Vomiting	Nausea/Vomiting
F96	Diarrhea	Diarrhea
F97	Dyspnea	Dyspnea
F98	Respiratory failure	Respiratory failure
F99	Acute kidney injury	AKI
F100	Acute myocardial injury	AMI
F101	Acute liver injury	ALI
F102	Secondary infection	SI

Throughout this study, SPSS version 24.0 was used to analyze the data. Continuous variables with a normally distributed distribution are represented by means ($\overline{\mathrm{\chi }}$) and standard deviations (SD). The non-severe COVID-19 was compared to severe COVID-19 using an independent samples t-test. In cases where the distribution of the continuous variables did not fit a normal distribution, the Mann-Whitney U test was used for statistical analysis, and the continuous variables were presented as medians and interquartile ranges. An analysis of categorical variables was conducted using the Chi-square test. The statistical results of blood routine, biochemical parameters, coagulation function indexes, arterial blood gas analysis, and clinical parameters in Non-severe COVID-19 patients and severe COVID-19 patients are shown in Table 10 . An analysis of the statistics of past medical history, symptoms, and comorbidities in non-severe COVID-19 patients and severe COVID-19 patients is shown in Table 11 . P < 0.05 was considered statistically significant.

Table 10.

Blood routine, biochemical parameters, coagulation function indexes, arterial blood gas analysis, and clinical parameters in Non-severe COVID-19 patients and severe COVID-19 patients.

Index	Non-severe (n = 30)	Severe (n = 21)	p-value
Age	42.30 ± 11.529	61.43 ± 17.642	0.000
WBC	4.54 (3.593, 5.188)	4.06 (3.600, 6.430)	0.818
NEU%	55.33 ± 9.980	69.50 ± 14.097	0.000
LY%	33.25 ± 7.901	21.20 ± 10.542	0.000
MONO%	10.35 ± 3.188	8.71 ± 4.042	0.111
EOS%	0.60 (0.175, 1.200)	0.100 (0.000, 0.300)	0.006
BA%	0.20 (0.075, 0.300)	0.10 (0.000, 0.300)	0.188
NEU	2.26 (1.805, 3.040)	3.22 (2.275, 4.475)	0.043
LY	1.50 ± 0.449	0.95 ± 0.464	0.000
MONO	0.48 ± 0.178	0.40 ± 0.191	0.151
EOS	0.03 (0.010, 0.053)	0.01 (0.000, 0.010)	0.006
BA	0.01 (0.008, 0.010)	0.01 (0.000, 0.010)	0.174
RBC	4.58 ± 0.471	4.44 ± 0.657	0.357
HB	138.53 ± 14.706	132.67 ± 18.682	0.216
HCT	41.83 ± 3.867	40.25 ± 4.915	0.205
MCH	30.28 ± 1.469	29.65 ± 1.215	0.112
MCV	91.543 ± 4.081	90.348 ± 3.971	0.303
MCHC	330.87 ± 10.766	323.76 ± 21.225	0.122
RDW	12.08 ± 0.584	12.48 ± 0.553	0.019
RDW-SD	40.19 ± 2.466	41.27 ± 1.741	0.091
MPV	10.43 ± 0.946	10.42 ± 0.922	0.972
PLT	205.50(149.000, 241.250)	165.00(130.500, 197.500)	0.087
PCT	0.2200 (0.150, 0.243)	0.17 (0.140, 0.255)	0.250
PDW	12.00 (11.750, 14.250)	12.00 (11.000, 13.500)	0.763
P-LCR%,	28.10 ± 7.087	28.14 ± 7.350	0.983
PCT	0.20 (0.200, 0.229)	0.20 (0.200, 0.200)	0.036
CRP	4.00 (3.000, 16.525)	55.30 (22.550, 120.000)	0.000
ESR	21.15 (9.000, 21.362)	29.82(27.000,29.818)	0.000
PT	13.00 (12.650, 13.082)	13.80(13.300, 5.171)	0.000
INR	1.01 (1.000, 1.032)	1.09 (1.045, 1.155)	0.001
PTA	99.00 (95.660, 100.000)	88.28 (82.500, 93.000)	0.001
FIB	4.17 ± 0.869	4.966 ± 1.647	0.053
APTT	38.48 ± 4.309	43.65 ± 8.779	0.019
TT	16.20 (15.700, 16.525)	16.52 (15.950, 17.150)	0.116
D-D	0.40 (0.283, 0.669	0.87 (0.465, 1.250)	0.001
TBIL	9.40 (6.700, 11.625)	10.000 (6.500, 15.150)	0.503
DBIL	4.76 (3.700, 4.761)	7.68 (3.850, 9.1000)	0.054
ALT	21.00 (13.750, 25.500)	27.00 (18.500, 79.000)	0.018
TP	66.65 ± 4.330	67.86 ± 7.685	0.520
ALB	41.53 ± 2.574	35.90 ± 4.649	0.000
GLB	25.13 ± 3.278	31.97 ± 7.271	0.000
A/G	1.68 ± 0.241	1.187 ± 0.358	0.000
ALP	61.17 ± 15.572	71.24 ± 27.633	0.142
GGT	25.50 (16.500, 42.000)	47.00(22.500, 130.000)	0.034
AST	21.50 (16.750, 28.000)	40.00 (28.000, 105.000)	0.000
CK	60.50 (48.000, 93.750)	155.00 (83.500, 309.500)	0.000
LDH	244.37 ± 61.066	382.95 ± 152.780	0.001
CK-MB	18.00 (14.750, 26.000)	18.00 (13.000, 27.500)	0.723
K+	4.23 ± 0.475	4.07 ± 0.618	0.291
Na+	139.26 ± 1.959	133.84 ± 4.024	0.000
Cl−	100.18 ± 2.918	95.55 ± 3.399	0.000
BUN	3.71 ± 0.999	4.64 ± 1.976	0.057
Cr	60.57 ± 12.291	70.30 ± 15.878	0.017
UA	264.83 ± 69.146	210.33 ± 87.732	0.017
P+	1.02 ± 0.167	0.95 ± 0.251	0.303
Mg2+	0.91 ± 0.072	0.93 ± 0.087	0.472
Ca2+	2.19 ± 0.071	2.09 ± 0.097	0.000
TnI	0.01 (0.010, 0.016)	0.01 (0.010, 0.039)	0.173
PH	7.43 (7.410, 7.450)	7.46 (7.435, 7.480)	0.003
PaCO2	37.55 ± 4.507	32.09 ± 4.204	0.000
PaO2	95.150 (88.725, 112.000)	63.90 (58.350, 71.050)	0.000
SaO2%	98.03 ± 1.002	92.73 ± 4.204	0.000
Hb	14.05 (12.800, 16.275)	12.60 (11.950, 15.700)	0.080
HbO2%	96.55 (95.975, 97.100)	93.50 (89.450, 94.600)	0.000
COHb%	1.00 (0.975, 1.200)	1.00 (0.900,1.200)	0.632
DeoxyHb %	1.95 ± 0.979	6.84 ± 4.286	0.000
MetHb %	0.50 (0.400, 0.600)	0.40 (0.350,0.600)	0.409
K+	3.32 ± 0.2999	3.36 ± 0.4236	0.699
Na+	137.00 (135.000, 138.000)	131.00(129.000, 134.000)	0.000
Cl-	107.37 ± 3.873	103.33 ± 3.367	0.000
Ca2+	1.12 ± 0.030	1.08 ± 0.059	0.015
GLU	7.10 (6.100, 10.200)	8.70 (7.350, 14.600)	0.010
LAC	1.65 (1.300, 2.025)	2.20(1.700, 2.750)	0.007
AG	4.27 ± 1.407	4.857 ± 2.199	0.253
BB	1.00 (0.000, 2.000)	0.00 (-2.000, 1.000)	0.079
BE	1.00 (0.000, 2.000)	−1.00 (-3.000, 1.000)	0.050
SB	25.450 (24.475, 26.025)	24.100 (23.000, 25.500)	0.063
AB	24.58 ± 2.721	22.56 ± 3.214	0.019

Table 11.

Past medical history, symptoms and comorbidities in Non-severe COVID-19 patients and severe COVID-19 patients.

Index	Non-severe (n = 30)	Severe (n = 21)	χ2 Value	p-value
Gender	16/14	14/7	0.907	0.341
SH (Yes/No)	29/1	14/7	6.291	0.012
DH (Yes/No)	0/30	4/17	3.845	0.050
HP (Yes/No)	0/30	4/17	14.878	0.000
DM (Yes/No)	5/25	6/15	0.451	0.502
CKD (Yes/No)	0/30	0/21	0.000	1
CLD (Yes/No)	3/27	0/21	0.791	0.374
CHD (Yes/No)	0/30	2/19	/	0.165
CLD (Yes/No)	0/30	0/21	0.000	1
MT (Yes/No)	0/30	0/21	0.000	1
Fever (Yes/No)	21/9	20/1	3.519	0.061
Cough (Yes/No)	21/5	21/0	2.723	0.099
Sputum (Yes/No)	20/10	21/0	6.721	0.010
Fatigue (Yes/No)	5/25	20/1	30.516	0.000
Myalgia (Yes/No)	6/24	14/7	11.286	0.001
Headache (Yes/No)	4/26	5/16	0.351	0.553
Nausea/Vomiting (Yes/No)	6/24	12/9	7.462	0.006
Diarrhea (Yes/No)	8/22	15/6	9.996	0.002
Dyspnea (Yes/No)	2/28	21/0	43.461	0.000
Respiratory failure (Yes/No)	0/30	21/0	51.000	0.000
AKI (Yes/No)	0/30	0/21	0.000	1
AMI (Yes/No)	0/30	4/17	3.845	0.050
ALI (Yes/No)	2/28	6/15	2.978	0.084
SI (Yes/No)	0/30	9/12	12.803	0.000

To evaluate the effectiveness of BRRWOA, we again selected several high-performance optimization algorithms, including BWOA, BMFO, BPSO, BBA, and BSMA [123]. The corresponding parameter settings are listed in Table 12 .

Table 12.

Parameter settings for the five contrast classifiers.

Algorithm	Common parameters	Values
BRRWOA	Dimension = Number of attributes in the specific dataset; population size = 20	$\mathit{prob}=0.8;k2=30;b=1;a1=[2,0];a2=[-2,-1];$
BWOA	Dimension = Number of attributes in the specific dataset; population size = 20	a = [0,2]
BMFO	Dimension = Number of attributes in the specific dataset; population size = 20	a = 2; b = 1
BPSO	Dimension = Number of attributes in the specific dataset; population size = 20	$\mathit{Max}=0.9;Min=0.4;$
BBA	Dimension = Number of attributes in the specific dataset; population size = 20	a = 0.5; r = 0.5
BSMA	Dimension = Number of attributes in the specific dataset; population size = 20	a = [0,2]

Similar to the experiments on UCI datasets, the classification results are evaluated according to the average fitness, average error rate, and the average number of selected features. The specific results are shown in Table 13 .

Table 13.

Comparison of classification results between the proposed BRRWOA and five binary MAs on the COVID-19 dataset.

Method	Fitness		Error		Features
	Avg	Std	Avg	Std	Avg	Std
BRRWOA	1.0588E−02	1.1600E−03	0.0000E+00	0.0000E+00	21.6	2.3664
BWOA	1.3333E−02	2.7224E−03	0.0000E+00	0.0000E+00	27.2	5.5538
BMFO	2.1078E−02	1.3671E−03	0.0000E+00	0.0000E+00	43	2.7889
BPSO	1.1716E−02	1.6575E−03	0.0000E+00	0.0000E+00	23.9	3.3813
BBA	1.5539E−02	3.1940E−03	7.3333E−02	1.5540E−01	39.6	5.8348
BSMA	1.7059E−02	1.3632E−03	0.0000E+00	0.0000E+00	34.8	2.7809

From Table 13, it can be concluded that the proposed BRRWOA performs the best on all three evaluation metrics. The average fitness is an indicator that comprehensively measures the classification accuracy and the number of selected features. The proposed algorithm obtains the smallest average fitness, which proves that it can reduce the number of features while ensuring classification accuracy. Meanwhile, BRRWOA has an average error rate of 0 % or a 100 % accuracy rate in predicting the COVID-19 dataset. However, the table shows that such results can also be obtained for BWOA, BMFO, BPSO, and BSMA. This may be because COVID-19 is a small sample dataset. Nevertheless, considering the number of selected features, the proposed algorithm can reduce the number of features as much as possible while maintaining a 100 % accuracy rate. This strongly highlights the performance of the proposed BRRWOA.

To visualize the stability of the proposed BRRWOA on the COVID-19 dataset, Fig. 9 shows the boxplots of BRRWOA and the other five algorithms on three evaluation metrics. As mentioned above, the test data will be divided into ten equal parts: nine are used for training, and one is used for testing. On this basis, Fig. 9 is drawn from the results of 10-fold cross-validation in 51 pieces of data from 51 participants. It can be seen from the figure that the proposed method not only has an excellent performance in the evaluation indicators but also has less fluctuation and more stable performance. Even though the proposed algorithm has a similar performance in terms of accuracy rate, it still performs stably in terms of the number of features, proving its superior performance. On closer inspection, BBA is higher than the proposed method in classification error rate. This is because its results in 10-fold cross-validation results are not stable enough. The average classification error rate of BRRWOA can reach 0, while BBA cannot.

Fig. 9.

Box line plots of the classification performance of the 6 methods in terms of fitness, error, and number of features selected.

To more intuitively and vividly show the superiority of BRRWOA in feature selection, Fig. 10 shows the convergence curve of the best fitness value calculated by the KNN classifier during training stage. It can be seen from the figure that the proposed algorithm is significantly better than the comparison algorithms in terms of convergence speed and convergence accuracy. BWOA, BMFO, and BBA all get stuck in local optima. Although BPSO and BSMA can jump out of the local optimum, the convergence accuracy is far less than BRRWOA.

Fig. 10.

Convergence curve of BRRWOA and other algorithms for the COVID-19 dataset.

For a real medical dataset, in addition to some evaluation metrics, its effect in practical applications is more worth looking forward to. To investigate whether the subset of features selected by BRRWOA is of substantial help in medical diagnosis, ten times ten cross-validation was performed on the results of the features selected for classification. The number of clinical indicators selected in all experiments was counted, and the results are shown in Fig. 11 . In the figure, 46, 60, 61, 67, and 74 were selected the most times, representing aspartate aminotransferase (AST), hydrogen ion concentration (PH), partial pressure of carbon dioxide (PaCO₂), deoxyhemoglobin percentage (deoxyHb) and lactic acid (LAC) respectively. AST, PH, PaCO₂, deoxyHb, and LAC were selected 39, 43, 37, 41, and 44 times, respectively, proving that these indicators are important factors in distinguishing mild and severe.

Fig. 11.

Selected features of the BRRWOA method during 10 times 10-fold cross-validation.

According to the current clinical data, patients with COVID-19 usually experience abnormal liver function tests, including the most commonly utilized AST and alanine aminotransferase (AST) liver functions. The prevalence of AST elevations ranged between 4 % and 53 % in the Chinese cohorts and 58 % in the US cohort [124]. In addition, many studies confirmed that in extreme COVID-19 patients, AST was substantially higher than that in non-severe patients and can therefore be used as a predictor of the severity of the disease [125], [126], [127]. The underlying pathogenesis of COVID-19-related liver functional abnormality is incompletely understood; SARS-CoV-2 direct invasion in hepatocytes, microthrombotic endothelialitis, immunological dysregulation, drug-induced liver damage, and hepatic ischemia due to hypoxia and MOF might all have an impact [128].

Blood pH measures the blood's acid-base status and maintains in the typical range of 7.35 to 7.45 to ensure the normal physiological activity of the body. However, if the acid-base regulatory capacity is overloaded or the regulatory mechanism is impaired, acid-base disorders will clinically manifest as acidosis or alkalosis. Acid-base abnormalities are prevalent in critically sick individuals, and they indicate the intensity of the core pathogenic process, while severe alterations of acid-base balance may lead to severe multiorgan consequences [129]. In theory, SARS-CoV-2 invasion of the lungs and kidneys might cause acid-base imbalances leading to pneumonia and renal damage [130], [131]. Alfano et al. confirmed the presence of acid-base disorders in 79.7 % of COVID-19 patients, with metabolic alkalosis (33.6 %) being the main alteration, which was followed by respiratory alkalosis (30.3 %) [132]. It is also worth mentioning that a retrospective-prospective multicentric study suggested that blood pH showed an independent association with clinical outcomes in critically ill patients with COVID-19 [133].

PaCO₂, with a normal value of 33–46 mmHg, refers to the tension by the physical dissolution of CO₂ in plasma. It reflects the ventilation of the lungs, which increases with hyperventilation and decreases with hyperventilation. Several pulmonary diseases such as pneumonia, pulmonary embolism, asthma, or interstitial lung disease may produce the reduction of PaCO₂; the leading physiologic cause is related to hyperventilation [134]. Based on the above, Klann et al. proposed an AI prediction model for COVID-19, using PaCO₂ as one of the predictors of patient prognosis [135]. Likewise, Pulgar-Sánchez et al. hinted that PaCO₂ had a greater predictive relevance for severe COVID-19 pneumonia [136].

DeoxyHb is the unbound form of hemoglobin (Hb) with oxygen, and the normal concentration of is 2 g/dl. Elevated deoxyHb is closely associated with hypotonic hypoxia, characterized by a decrease in arterial partial pressure of oxygen (PaO₂) and oxygen saturation (SaO₂). COVID-19 patients exhibited major oxygen metabolism problems, as evidenced by considerable declines in levels in SaO₂, PaO₂, and PaO₂/FiO₂. When comparing died patients to survivors, SaO₂ is 1.12 times lower (4.88 % lower), while PaO₂ is 1.08 times lower (4.88 % lower) (2.5 mm lower) [137]. Many studies indicated that hypoxia in COVID-19 is predictive of mortality, and low SaO₂ and PaO₂/FiO₂ ratio are independent factors associated with mortality [138], [139]. In this investigation, we discovered that deoxyHb levels in severe COVID-19 patients were considerably greater than in non-severe instances, indicating that deoxyHb might be used to assess COVID-19 severity.

LAC is an intermediate product in the metabolism of glucose. LAC accumulates in the blood when there is a deficit in oxygen supply or blood perfusion in the tissues. Therefore, changes in blood LAC can reflect tissue hypoxia and the severity of the disease [140]. McElvaney et al. found that severe patients with COVID-19 had elevated levels of LAC compared to stable patients [141]. Another study by Izcovich et al. noted that high LAC with the definition of more than 1.5–2.2 mmol/L provides valuable prognostic information on severe COVID-19 disease outcomes [142]. In accordance with these findings, our study found that severe COVID-19 patients had significantly higher LAC levels than non-severe patients, and we, therefore, assume that LAC concentration can function as an effectual biomarker for severe COVID-19.

Although the proposed model can improve the diagnostic performance of COVID-19, it still has some limitations. First of all, there are not enough samples in the dataset. Despite the fact that the number of severe and non-severe patients in the samples is relatively balanced, the number of 51 samples is still insufficient. After that, we will collect as much detailed data as possible for further analysis. Then, considering the time consumption, the feature selection experiment sets a general 50 iterations to obtain the experimental results, and the results of fewer or more iterations can be considered in the future. At the same time, it can be observed that the addition of RM strategy is the reason for the significant increase of time consumption. Therefore, parallel structures such as CUDA can be utilized to speed up the prediction process.

4. Conclusions and future directions

WOA is a commonly used meta-heuristic algorithm proposed recently, with a wide range of applications in all walks of life. This paper proposes a new variant of WOA, which is called RRWOA. This variation enhances the performance of the original WOA by incorporating the RCS strategy and the RM strategy. The RCS strategy can fully expand the current solution's position and improve the algorithm's exploration ability. As a supplement to the exploitation capability, the RM strategy can approach the optimal value as much as possible. At the same time, the settings of the two control parameters enable RRWOA to obtain a better balance between exploration and exploitation. To verify the primary performance of RRWOA, four experiments are conducted separately. Firstly, the mechanism comparison experiment verifies that the proposed algorithm is the best. Next, RRWOA is compared with nine classical MAs, including HGS, HHO, RUN, DE, SSA, PSO, GWO, SCA, and MFO. The proposed method ranks first on 18 tested functions. Subsequently, RRWOA is compared with nine advanced MAs, including ASCA-PSO, ESSA, CMSSA, MSCA, IGWO, OBLGWO, GCHHO, HHODE, and CMFO. The proposed method achieves a mean of 1.7333, far exceeding the second place with a mean of 3. Finally, to verify the superiority of RRWOA in the field of WOA, it competes with ACWOA, CWOAI, IWOA, LWOA, MWOA, OBWOA, and RDWOA. The proposed method won first place on 25 test functions, converged to the global optimum on five functions, and fully approximated the global optimum on eight functions. The results show that RRWOA has a strong global optimization ability. Furthermore, to verify the performance of RRWOA in practical applications, a binary version of the algorithm called BRRWOA is developed. This binary version experiments with seven existing binary MAs on 15 UCI datasets, and the results show that BRRWOA can balance the classification accuracy and the number of features to obtain the most relevant results. Going a step further, BRRWOA is compared with five binary MAs on the real medical dataset of COVID-19. The results show that BRRWOA still performs stably in actual medical scenarios, and mining AST, PH, PaCO₂, deoxyHb, and LAC are essential features of this dataset.

Although RRWOA has improved the problem of WOA to a certain extent, there are still deficiencies. Most prominently, the RM strategy brings performance improvements, the time consumption it brings is also obvious. In order to solve this problem, multiple groups or parallel strategies can be considered in the future. In addition, this paper adopts a model that combines the method with KNN in feature selection. Later, it can be considered to build a diagnostic system for COVID-19, in which multiple classifiers can be integrated for prediction. Likewise, in terms of practical applications, more data samples can be collected to build more efficient and reliable frameworks. Finally, besides COVID-19, the model can be extended to other fields such as CT image segmentation.

CRediT authorship contribution statement

Meilin Zhang: Software, Visualization, Investigation, Formal analysis, Methodology, Validation, Writing - original draft, Writing - review & editing. Qianxi Wu: Software, Visualization, Investigation, Writing - original draft, Writing - review & editing. Huiling Chen: Conceptualization, Methodology, Formal analysis, Investigation, Funding acquisition, Supervision, Project administration, Resources, Software, Visualization, Writing - original draft, Writing - review & editing. Ali Asghar Heidari: Software, Visualization, Investigation, Formal analysis, Project administration, Writing - original draft, Writing - review & editing. Zhennao Cai: Software, Visualization, Investigation, Writing - original draft, Writing - review & editing. Jiaren Li: Conceptualization, Methodology, Formal analysis, Investigation, Funding acquisition, Supervision, Visualization, Writing - original draft, Writing - review & editing. Elsaid Md. Abdelrahim: Software, Visualization, Investigation, Project administration, Resources, Validation, Writing - review & editing. Romany F. Mansour: Software, Visualization, Investigation, Resources, Validation, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A.

See Table A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9, Table A10 .

Table A1.

Summary of the CEC’14 test functions (Search Range: [−100, 100]^D).

Class	No.	Functions	$F_i^{\ast }=F_i(x^{\ast })$
Unimodal Functions	1	Rotated High Conditioned Elliptic Function	100
	2	Rotated Bent Cigar Function	200
	3	Rotated Discus Function	300
Simple Multimodal Functions	4	Shifted and Rotated Rosenbrock’s Function	400
	5	Shifted and Rotated Ackley’s Function	500
	6	Shifted and Rotated Weierstrass Function	600
	7	Shifted and Rotated Griewank’s Function	700
	8	Shifted Rastrigin’s Function	800
	9	Shifted and Rotated Rastrigin’s Function	900
	10	Shifted Schwefel’s Function	1000
	11	Shifted and Rotated Schwefel’s Function	1100
	12	Shifted and Rotated Katsuura Function	1200
	13	Shifted and Rotated HappyCat Function	1300
	14	Shifted and Rotated HGBat Function	1400
	15	Shifted and Rotated Expanded Griewank’s plus Rosenbrock’s Function	1500
	16	Shifted and Rotated Expanded Scaffer’s F6 Function	1600
Hybrid Functions	17	Hybrid Function 1 (N = 3)	1700
	18	Hybrid Function 2 (N = 3)	1800
	19	Hybrid Function 3 (N = 4)	1900
	20	Hybrid Function 4 (N = 4)	2000
	21	Hybrid Function 5 (N = 5)	2100
	22	Hybrid Function 6 (N = 5)	2200
Composition Functions	23	Composition Function 1 (N = 5)	2300
	24	Composition Function 2 (N = 3)	2400
	25	Composition Function 3 (N = 3)	2500
	26	Composition Function 4 (N = 5)	2600
	27	Composition Function 5 (N = 5)	2700
	28	Composition Function 6 (N = 5)	2800
	29	Composition Function 7 (N = 3)	2900
	30	Composition Function 8 (N = 3)	3000

Table A2.

Results of mechanism comparison in CEC2014.

	F1		F2			F3
	Avg	Std	Avg	Std		Avg	Std
RRWOA	1.0331E+07	1.0607E+07	9.2467E+03	1.3617E+04		2.8472E+03	2.2693E+03
RCSWOA	4.3673E+07	2.8189E+07	4.3824E+09	4.1397E+09		4.4913E+04	1.1879E+04
RMWOA	6.9532E+06	2.6939E+06	2.0754E+04	1.2454E+04		4.2253E+03	8.0993E+03
WOA	1.1804E+08	5.0026E+07	1.6332E+09	5.0147E+08		8.0944E+04	3.4542E+04
	F4		F5			F6
Avg	Std	Avg	Std		Avg	Std
RRWOA	5.7479E+02	2.4147E+02	5.2000E+02	5.6475E−03		6.1379E+02	3.3066E+00
RCSWOA	8.1896E+02	9.5046E+01	5.2031E+02	1.3557E−01		6.4059E+02	2.7987E+00
RMWOA	4.7536E+02	1.1825E+01	5.2002E+02	1.8184E−02		6.1490E+02	2.3314E+00
WOA	9.0628E+02	1.5561E+02	5.2078E+02	7.7831E−02		6.3864E+02	2.8543E+00
	F7		F8			F9
Avg	Std	Avg	Std		Avg	Std
RRWOA	7.0004E+02	4.5456E−02	8.2309E+02	2.2797E+01		1.0438E+03	1.8014E+01
RCSWOA	7.1445E+02	1.4641E+01	9.6609E+02	1.6674E+01		1.1094E+03	1.9045E+01
RMWOA	7.0004E+02	2.6846E−02	8.0946E+02	3.7346E+00		9.7922E+02	1.3867E+01
WOA	7.1186E+02	5.2870E+00	1.0230E+03	3.4551E+01		1.1466E+03	3.8748E+01
	F10		F11			F12
Avg	Std	Avg	Std		Avg	Std
RRWOA	1.4416E+03	1.9856E+02	3.5618E+03	4.5995E+02		1.2002E+03	3.0005E−02
RCSWOA	5.8040E+03	7.5413E+02	6.8993E+03	1.1424E+03		1.2027E+03	4.9067E−01
RMWOA	1.6072E+03	2.1914E+02	3.6847E+03	3.2527E+02		1.2002E+03	4.8774E−02
WOA	5.7689E+03	6.6697E+02	6.8905E+03	8.5194E+02		1.2023E+03	7.3408E−01
	F13		F14			F15
Avg	Std	Avg	Std		Avg	Std
RRWOA	1.3007E+03	9.7364E−01	1.4003E+03	1.8166E−01		1.5082E+03	2.1222E+00
RCSWOA	1.3010E+03	7.6608E−01	1.4100E+03	9.5923E+00		4.2961E+03	1.2820E+03
RMWOA	1.3004E+03	5.8148E−02	1.4002E+03	2.8771E−02		1.5077E+03	1.6521E+00
WOA	1.3005E+03	1.3995E−01	1.4048E+03	5.5956E+00		1.7511E+03	1.9368E+02
	F16		F17			F18
Avg	Std	Avg	Std		Avg	Std
RRWOA	1.6112E+03	6.0801E−01	3.8833E+06	2.1724E+06		3.3037E+03	2.0991E+03
RCSWOA	1.6124E+03	4.7342E−01	5.3243E+06	4.5984E+06		4.8128E+03	2.7577E+03
RMWOA	1.6111E+03	2.5503E−01	4.6432E+06	3.0095E+06		5.9066E+03	3.6006E+03
WOA	1.6125E+03	7.0233E−01	1.7548E+07	1.0820E+07		5.1031E+05	4.8082E+05
	F19		F20			F21
Avg	Std	Avg	Std		Avg	Std
RRWOA	1.9082E+03	1.6937E+00	2.6048E+04	2.2138E+04		6.1396E+05	5.6668E+05
RCSWOA	2.0916E+03	6.5443E+01	2.7322E+04	1.4748E+04		1.2183E+06	1.1743E+06
RMWOA	1.9091E+03	1.4382E+00	1.0706E+04	8.1530E+03		1.3322E+06	1.0583E+06
WOA	1.9946E+03	4.2891E+01	1.0724E+05	5.9849E+04		9.5788E+06	5.2314E+06
	F22		F23			F24
Avg	Std	Avg	Std		Avg	Std
RRWOA	2.9039E+03	2.3305E+02	2.5000E+03	0.0000E+00		2.6000E+03	2.4735E−04
RCSWOA	3.3695E+03	3.7626E+02	2.5000E+03	0.0000E+00		2.6000E+03	0.0000E+00
RMWOA	2.5986E+03	1.7728E+02	2.6179E+03	2.7663E+00		2.6258E+03	7.4055E+00
WOA	3.0651E+03	2.6455E+02	2.7046E+03	2.2944E+01		2.6086E+03	3.6204E+00
	F25		F26			F27
Avg	Std	Avg	Std		Avg	Std
RRWOA	2.7000E+03	0.0000E+00	2.7701E+03	4.8077E+01		2.9000E+03	1.5158E−13
RCSWOA	2.7000E+03	0.0000E+00	2.7901E+03	3.1446E+01		2.9000E+03	0.0000E+00
RMWOA	2.7076E+03	2.9566E+00	2.7004E+03	8.7240E−02		3.4247E+03	1.7484E+02
WOA	2.7194E+03	1.8269E+01	2.7204E+03	4.1974E+01		3.7200E+03	3.8685E+02
	F28		F29			F30
Avg	Std	Avg	Std		Avg	Std
RRWOA	3.0000E+03	1.5158E−13	3.1000E+03	1.5158E−13		3.2000E+03	1.5158E−13
RCSWOA	3.0000E+03	0.0000E+00	3.1000E+03	0.0000E+00		3.2000E+03	0.0000E+00
RMWOA	3.8924E+03	1.6758E+02	3.4461E+06	4.4389E+06		1.1076E+04	1.8176E+03
WOA	5.3674E+03	5.7760E+02	8.9138E+06	7.8084E+06		2.4671E+05	1.6315E+05

	Overall Rank
Rank	+/=/−	ARV
RRWOA	1	∼	1.4667
RCSWOA	3	16/14/0	2.8667
RMWOA	2	11/13/6	1.9667
WOA	4	26/4/0	3.5

Table A3.

P-value of Wilcoxon test obtained from comparison with RRWOA’s variants on 30 functions.

Function	RCSWOA	RMWOA	WOA
F1	5.8594E−03	6.9531E−01	1.9531E−03
F2	1.9531E−03	3.7109E−02	1.9531E−03
F3	1.9531E−03	7.6953E−01	1.9531E−03
F4	8.3984E−02	2.7344E−02	4.8828E−02
F5	1.9531E−03	1.9531E−02	1.9531E−03
F6	1.9531E−03	4.3164E−01	1.9531E−03
F7	1.9531E−03	6.2500E−01	1.9531E−03
F8	1.9531E−03	9.7656E−03	1.9531E−03
F9	1.9531E−03	1.9531E−03	1.9531E−03
F10	1.9531E−03	1.3086E−01	1.9531E−03
F11	1.9531E−03	4.9219E−01	1.9531E−03
F12	1.9531E−03	1.9531E−02	1.9531E−03
F13	2.7539E−01	2.7539E−01	5.5664E−01
F14	9.7656E−03	3.7109E−02	1.6016E−01
F15	1.9531E−03	8.4570E−01	1.9531E−03
F16	1.9531E−03	6.9531E−01	1.9531E−03
F17	5.5664E−01	6.9531E−01	9.7656E−03
F18	2.7539E−01	4.8828E−02	1.9531E−03
F19	1.9531E−03	5.5664E−01	1.9531E−03
F20	7.6953E−01	8.3984E−02	3.9063E−03
F21	3.2227E−01	1.3086E−01	1.9531E−03
F22	1.9531E−03	3.7109E−02	2.7539E−01
F23	1.0000E+00	1.9531E−03	1.9531E−03
F24	5.0000E−01	1.9531E−03	1.9531E−03
F25	1.0000E+00	1.9531E−03	3.1250E−02
F26	3.7500E−01	9.7656E−03	1.0547E−01
F27	1.0000E+00	1.9531E−03	1.9531E−03
F28	1.0000E+00	1.9531E−03	1.9531E−03
F29	1.0000E+00	1.9531E−03	1.9531E−03
F30	1.0000E+00	1.9531E−03	1.9531E−03

Table A4.

Results of basic comparison in CEC2014.

	F1		F2		F3
	Avg	Std	Avg	Std	Avg	Std
RRWOA	3.3494E+05	7.6810E+04	2.0000E+02	8.5265E−14	3.4154E+02	7.8831E+01
HGS	7.9016E+06	4.3333E+06	4.5926E+08	7.3064E+08	7.4279E+03	7.3472E+03
HHO	1.1385E+07	3.7842E+06	1.1230E+07	2.0742E+06	6.7521E+03	2.7843E+03
RUN	2.7280E+05	1.2242E+05	1.5327E+04	8.2844E+03	3.0479E+02	4.2032E+00
DE	1.8145E+07	3.0660E+06	1.1910E+03	3.0517E+03	4.2189E+02	1.2308E+02
SSA	1.3912E+06	7.6127E+05	1.3230E+04	1.1750E+04	1.1900E+03	5.3436E+02
PSO	6.4451E+06	2.9379E+06	7.1456E+07	4.7864E+06	6.6526E+02	7.0966E+01
GWO	5.4572E+07	2.6589E+07	2.1611E+09	1.9876E+09	3.6106E+04	8.0374E+03
SCA	2.2748E+08	4.5180E+07	1.6548E+10	3.4308E+09	3.8503E+04	5.6706E+03
MFO	9.0863E+07	1.0457E+08	1.4265E+10	8.1889E+09	8.9055E+04	3.2190E+04
	F4		F5		F6
Avg	Std	Avg	Std	Avg	Std
RRWOA	4.1357E+02	2.7844E+01	5.2000E+02	5.8943E−04	6.1065E+02	1.8129E+00
HGS	5.3410E+02	4.7803E+01	5.2013E+02	5.0502E−02	6.1843E+02	3.2171E+00
HHO	5.5224E+02	3.8001E+01	5.2024E+02	1.8654E−01	6.3139E+02	3.9128E+00
RUN	4.8119E+02	2.1683E+01	5.2052E+02	3.8764E−01	6.2566E+02	4.1629E+00
DE	5.1593E+02	2.4166E+01	5.2060E+02	4.4788E−02	6.1825E+02	1.7716E+00
SSA	4.9134E+02	3.4293E+01	5.2008E+02	7.1047E−02	6.2128E+02	3.8911E+00
PSO	4.5661E+02	2.8228E+01	5.2097E+02	3.4936E−02	6.2572E+02	5.0708E+00
GWO	6.5468E+02	9.8734E+01	5.2096E+02	3.3002E−02	6.1144E+02	3.2850E+00
SCA	1.4280E+03	2.4602E+02	5.2095E+02	4.9704E−02	6.3390E+02	2.9856E+00
MFO	1.0411E+03	2.7648E+02	5.2034E+02	1.8439E−01	6.2314E+02	2.2583E+00
	F7		F8		F9
Avg	Std	Avg	Std	Avg	Std
RRWOA	7.0000E+02	2.3388E−03	8.1055E+02	1.3620E+01	1.0307E+03	2.1576E+01
HGS	7.0095E+02	3.0580E−01	8.0424E+02	1.6379E+00	1.0261E+03	2.7304E+01
HHO	7.0110E+02	1.4655E−02	9.0522E+02	1.3761E+01	1.0849E+03	2.0812E+01
RUN	7.0002E+02	1.6852E−02	9.0845E+02	8.2846E+00	1.0690E+03	1.3646E+01
DE	7.0000E+02	7.5791E−14	8.0094E+02	7.4426E−01	1.0120E+03	4.4296E+00
SSA	7.0001E+02	5.9161E−03	9.1301E+02	2.8730E+01	1.0337E+03	2.3550E+01
PSO	7.0156E+02	6.6550E−02	9.5213E+02	2.4446E+01	1.0745E+03	2.5918E+01
GWO	7.1597E+02	1.2591E+01	8.7515E+02	1.5321E+01	1.0096E+03	3.2327E+01
SCA	8.3716E+02	1.7776E+01	1.0327E+03	1.5358E+01	1.1730E+03	2.1958E+01
MFO	8.2214E+02	7.4190E+01	9.4541E+02	3.6780E+01	1.1319E+03	5.4349E+01
	F10		F11		F12
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.2630E+03	2.6005E+02	3.2205E+03	3.4646E+02	1.2001E+03	4.4095E−02
HGS	1.2420E+03	1.5476E+02	3.6848E+03	4.7343E+02	1.2002E+03	4.0960E−02
HHO	2.6950E+03	6.2432E+02	5.5516E+03	6.4219E+02	1.2017E+03	4.9643E−01
RUN	2.5926E+03	3.8030E+02	3.9782E+03	6.8618E+02	1.2015E+03	1.0231E+00
DE	1.0285E+03	3.6736E+01	5.8694E+03	2.3124E+02	1.2009E+03	1.3366E−01
SSA	4.3529E+03	1.1704E+03	4.5352E+03	4.8207E+02	1.2004E+03	2.8747E−01
PSO	5.1377E+03	5.6029E+02	5.5415E+03	6.5316E+02	1.2022E+03	2.9878E−01
GWO	2.9719E+03	3.5228E+02	3.7337E+03	5.3285E+02	1.2015E+03	1.1990E+00
SCA	7.1201E+03	2.5416E+02	7.9234E+03	4.5680E+02	1.2025E+03	1.0591E−01
MFO	4.7509E+03	6.5027E+02	5.3255E+03	5.8670E+02	1.2004E+03	2.2116E−01
	F13		F14		F15
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.3004E+03	5.5758E−02	1.4002E+03	2.9386E−02	1.5062E+03	6.0377E−01
HGS	1.3008E+03	1.1902E−01	1.4010E+03	3.2891E−01	1.5137E+03	4.9217E+00
HHO	1.3005E+03	1.4623E−01	1.4003E+03	5.4172E−02	1.5384E+03	8.3716E+00
RUN	1.3004E+03	7.4455E−02	1.4003E+03	2.8816E−02	1.5419E+03	1.2613E+01
DE	1.3003E+03	4.1535E−02	1.4003E+03	6.2732E−02	1.5120E+03	8.6811E−01
SSA	1.3005E+03	9.5117E−02	1.4003E+03	2.4299E−01	1.5081E+03	3.5149E+00
PSO	1.3004E+03	6.9521E−02	1.4003E+03	1.5025E−01	1.5157E+03	9.2476E−01
GWO	1.3005E+03	4.7809E−01	1.4028E+03	5.0064E+00	1.5508E+03	7.4435E+01
SCA	1.3029E+03	2.8900E−01	1.4419E+03	4.0924E+00	6.3576E+03	6.9826E+03
MFO	1.3019E+03	1.0336E+00	1.4464E+03	3.7522E+01	4.6940E+04	8.5131E+04
	F16		F17		F18
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.6101E+03	5.5790E−01	6.1452E+05	3.8868E+05	3.6253E+03	2.1144E+03
HGS	1.6118E+03	3.0937E−01	7.6094E+05	3.8014E+05	1.8073E+04	1.0421E+04
HHO	1.6122E+03	3.9037E−01	2.3399E+06	1.9550E+06	8.7407E+04	3.4861E+04
RUN	1.6111E+03	3.4371E−01	6.0821E+04	1.7462E+04	4.2939E+03	3.2365E+03
DE	1.6115E+03	2.2412E−01	1.4750E+06	7.2145E+05	7.0184E+03	3.7069E+03
SSA	1.6116E+03	3.9706E−01	8.2987E+04	8.8030E+04	4.6966E+03	2.8065E+03
PSO	1.6124E+03	4.0270E−01	2.2658E+05	1.3176E+05	8.5809E+05	2.8074E+05
GWO	1.6112E+03	5.6662E−01	1.0091E+06	1.6483E+06	1.1179E+07	2.3695E+07
SCA	1.6128E+03	1.5899E−01	6.2836E+06	3.8086E+06	1.5025E+08	7.1308E+07
MFO	1.6125E+03	6.5847E−01	9.6488E+06	1.9368E+07	1.6618E+08	5.2549E+08
	F19		F20		F21
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.9068E+03	1.6787E+00	7.6751E+03	4.3068E+03	2.5820E+05	2.8069E+05
HGS	1.9411E+03	3.9199E+01	5.4238E+03	2.7468E+03	2.4199E+05	1.5853E+05
HHO	1.9289E+03	3.0444E+01	1.5622E+04	7.7983E+03	6.3308E+05	2.3684E+05
RUN	1.9227E+03	1.7590E+01	2.2007E+03	4.3281E+01	1.5008E+04	7.2232E+03
DE	1.9081E+03	4.7573E−01	4.1076E+03	1.4741E+03	2.0993E+05	9.5417E+04
SSA	1.9139E+03	2.0055E+00	2.3720E+03	8.2652E+01	4.9292E+04	4.0390E+04
PSO	1.9168E+03	3.1960E+00	2.3290E+03	8.2965E+01	7.4292E+04	4.8111E+04
GWO	1.9382E+03	2.5032E+01	1.5248E+04	6.4880E+03	4.3309E+05	3.0838E+05
SCA	1.9980E+03	1.8267E+01	1.5806E+04	4.3693E+03	1.2100E+06	6.5513E+05
MFO	1.9778E+03	6.0250E+01	1.1583E+05	7.8074E+04	2.1695E+06	2.9244E+06
	F22		F23		F24
Avg	Std	Avg	Std	Avg	Std
RRWOA	2.6024E+03	1.5446E+02	2.5000E+03	0.0000E+00	2.6000E+03	0.0000E+00
HGS	2.8054E+03	1.7987E+02	2.5000E+03	0.0000E+00	2.6000E+03	9.0500E−04
HHO	3.0187E+03	2.7631E+02	2.5000E+03	0.0000E+00	2.6000E+03	3.0357E−05
RUN	2.8810E+03	1.9489E+02	2.5000E+03	0.0000E+00	2.6000E+03	3.3333E−07
DE	2.3541E+03	5.2756E+01	2.6152E+03	4.7935E−13	2.6258E+03	1.9206E+00
SSA	2.5736E+03	1.1774E+02	2.6153E+03	5.4000E−02	2.6416E+03	8.5896E+00
PSO	3.0707E+03	1.9305E+02	2.6149E+03	1.9160E−01	2.6301E+03	9.2419E+00
GWO	2.5797E+03	2.2135E+02	2.6351E+03	9.4414E+00	2.6000E+03	8.8685E−04
SCA	3.0240E+03	1.1431E+02	2.6665E+03	6.7288E+00	2.6001E+03	5.3511E−02
MFO	2.9943E+03	1.9110E+02	2.6596E+03	3.6974E+01	2.6859E+03	4.8187E+01
	F25		F26		F27
Avg	Std	Avg	Std	Avg	Std
RRWOA	2.7000E+03	0.0000E+00	2.7601E+03	5.1475E+01	2.9000E+03	0.0000E+00
HGS	2.7000E+03	0.0000E+00	2.7306E+03	4.7909E+01	2.9000E+03	0.0000E+00
HHO	2.7000E+03	0.0000E+00	2.8000E+03	0.0000E+00	2.9000E+03	0.0000E+00
RUN	2.7000E+03	0.0000E+00	2.7006E+03	9.3366E−02	2.9000E+03	0.0000E+00
DE	2.7076E+03	1.1937E+00	2.7003E+03	4.6343E−02	3.2139E+03	6.1013E+01
SSA	2.7123E+03	2.9322E+00	2.7006E+03	1.7643E−01	3.3829E+03	1.5980E+02
PSO	2.7159E+03	7.3675E+00	2.7903E+03	3.1592E+01	3.4400E+03	3.0643E+02
GWO	2.7077E+03	4.4574E+00	2.7602E+03	5.1340E+01	3.3004E+03	1.2669E+02
SCA	2.7255E+03	3.4734E+00	2.7024E+03	7.9902E−01	3.6243E+03	3.4858E+02
MFO	2.7138E+03	8.3993E+00	2.7027E+03	1.4284E+00	3.6168E+03	2.0143E+02
	F28		F29		F30
Avg	Std	Avg	Std	Avg	Std
RRWOA	3.0000E+03	0.0000E+00	3.1000E+03	0.0000E+00	3.4837E+03	8.9698E+02
HGS	3.0000E+03	0.0000E+00	6.1119E+03	3.8908E+03	4.3350E+03	1.8695E+03
HHO	3.0000E+03	0.0000E+00	3.1000E+03	0.0000E+00	4.0758E+03	2.7694E+03
RUN	3.0000E+03	0.0000E+00	8.4530E+05	2.6563E+06	1.3922E+04	1.2678E+04
DE	3.6332E+03	2.3690E+01	7.5000E+05	2.3554E+06	5.8420E+03	1.1055E+03
SSA	3.8714E+03	2.3211E+02	3.8183E+06	6.2700E+06	1.2708E+04	4.0431E+03
PSO	7.4821E+03	7.6806E+02	2.6221E+04	2.7235E+04	1.4156E+04	4.3539E+03
GWO	4.0502E+03	4.2058E+02	9.3204E+04	1.8121E+05	4.5721E+04	4.2007E+04
SCA	4.9431E+03	4.9693E+02	1.0104E+07	4.9430E+06	2.9067E+05	9.4611E+04
MFO	3.8962E+03	1.3034E+02	4.0073E+06	4.2301E+06	4.4477E+04	3.0659E+04

	Overall Rank
Rank	+/=/−	ARV
RRWOA	1	∼	2.1
HGS	4	16/14/0	4.2
HHO	6	21/9/0	5.3667
RUN	2	19/8/3	3.5667
DE	3	20/3/7	3.9333
SSA	5	22/5/3	4.5667
PSO	8	25/2/3	6.4
GWO	7	22/8/0	6.3667
SCA	10	29/1/0	9.3
MFO	9	27/3/0	8.3667

Table A5.

P-value of Wilcoxon test obtained from comparison with basic methods on 30 functions.

Function	F1	F2	F3	F4	F5	F6
HGS	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
HHO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
RUN	1.6016E−01	1.9531E−03	8.3984E−02	1.9531E−03	1.9531E−03	1.9531E−03
DE	1.9531E−03	1.9531E−03	4.8828E−02	1.9531E−03	1.9531E−03	1.9531E−03
SSA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
PSO	1.9531E−03	1.9531E−03	1.9531E−03	6.4453E−02	1.9531E−03	1.9531E−03
GWO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	4.3164E−01
SCA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
MFO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
Function	F7	F8	F9	F10	F11	F12
HGS	1.9531E−03	8.3984E−02	8.4570E−01	7.6953E−01	4.8828E−02	1.9531E−03
HHO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
RUN	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	9.7656E−03	1.9531E−03
DE	6.2500E−02	3.9063E−03	3.7109E−02	5.8594E−03	1.9531E−03	1.9531E−03
SSA	1.9531E−03	1.9531E−03	1.0000E+00	1.9531E−03	1.9531E−03	1.9531E−03
PSO	1.9531E−03	1.9531E−03	3.9063E−03	1.9531E−03	1.9531E−03	1.9531E−03
GWO	1.9531E−03	1.9531E−03	1.0547E−01	1.9531E−03	3.7109E−02	3.7109E−02
SCA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
MFO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	5.8594E−03
Function	F13	F14	F15	F16	F17	F18
HGS	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	2.7539E−01	1.9531E−03
HHO	6.4453E−02	3.9063E−03	1.9531E−03	1.9531E−03	2.7344E−02	1.9531E−03
RUN	2.7344E−02	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	7.6953E−01
DE	4.8828E−02	1.9531E−03	1.9531E−03	1.9531E−03	9.7656E−03	2.7344E−02
SSA	9.7656E−03	2.7344E−02	4.3164E−01	1.9531E−03	5.8594E−03	8.3984E−02
PSO	7.6953E−01	5.8594E−03	1.9531E−03	1.9531E−03	2.7344E−02	1.9531E−03
GWO	6.9531E−01	1.9531E−03	1.9531E−03	3.9063E−03	1.0000E+00	6.4453E−02
SCA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
MFO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	3.2227E−01	2.7344E−02
Function	F19	F20	F21	F22	F23	F24
HGS	1.9531E−03	4.9219E−01	6.9531E−01	2.7344E−02	1.0000E+00	6.2500E−02
HHO	1.9531E−03	9.7656E−03	1.9531E−02	9.7656E−03	1.0000E+00	1.2500E−01
RUN	1.9531E−03	1.9531E−03	1.9531E−03	5.8594E−03	1.0000E+00	1.5625E−02
DE	8.3984E−02	1.9531E−02	9.2188E−01	1.9531E−03	1.9531E−03	1.9531E−03
SSA	1.9531E−03	1.9531E−03	5.8594E−03	9.2188E−01	1.9531E−03	1.9531E−03
PSO	1.9531E−03	1.9531E−03	3.7109E−02	1.9531E−03	1.9531E−03	1.9531E−03
GWO	1.9531E−03	1.9531E−03	3.2227E−01	6.9531E−01	1.9531E−03	1.9531E−03
SCA	1.9531E−03	1.9531E−03	3.9063E−03	3.9063E−03	1.9531E−03	1.9531E−03
MFO	1.9531E−03	1.9531E−03	8.3984E−02	3.9063E−03	1.9531E−03	1.9531E−03
Function	F25	F26	F27	F28	F29	F30
HGS	1.0000E+00	8.2031E−01	1.0000E+00	1.0000E+00	1.2500E−01	3.7500E−01
HHO	1.0000E+00	1.2500E−01	1.0000E+00	1.0000E+00	1.0000E+00	1.0000E+00
RUN	1.0000E+00	8.3984E−02	1.0000E+00	1.0000E+00	3.1250E−02	1.9531E−03
DE	1.9531E−03	3.7109E−02	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
SSA	1.9531E−03	8.3984E−02	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
PSO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
GWO	7.8125E−03	3.6133E−01	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
SCA	1.9531E−03	8.3984E−02	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
MFO	1.9531E−03	8.3984E−02	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03

Table A6.

Results of advanced comparison in CEC2014.

	F1		F2		F3
	Avg	Std	Avg	Std	Avg	Std
RRWOA	1.1629E+07	8.5711E+06	3.0328E+04	1.0652E+04	3.1345E+03	2.0746E+03
ASCA-PSO	4.9513E+07	1.8020E+07	2.4377E+08	2.0207E+08	6.2976E+04	1.4619E+04
ESSA	8.3173E+08	1.7572E+08	5.1813E+10	1.0559E+10	7.5737E+04	6.9314E+03
CMSSA	8.6665E+08	2.7638E+08	5.1948E+10	7.5363E+09	7.7218E+04	8.4127E+03
MSCA	1.6003E+08	1.9482E+08	1.3554E+10	5.6052E+09	3.8885E+04	1.7174E+04
IGWO	5.3668E+07	1.4907E+07	1.9377E+08	7.0510E+07	3.0114E+04	6.2675E+03
OBLGWO	8.1675E+07	3.8506E+07	2.6743E+08	1.0028E+08	4.9841E+04	1.5635E+04
GCHHO	1.9589E+07	9.9173E+06	7.4711E+06	9.3215E+06	9.8399E+03	3.1098E+03
HHODE	1.0376E+08	3.5918E+07	3.3271E+09	2.2742E+09	3.6203E+04	8.9037E+03
CMFO	1.6509E+08	8.7818E+07	9.3500E+09	2.9696E+09	8.7953E+04	2.4117E+04
	F4		F5		F6
Avg	Std	Avg	Std	Avg	Std
RRWOA	4.9621E+02	2.8600E+01	5.2000E+02	2.1924E−03	6.1384E+02	3.5487E+00
ASCA-PSO	5.7741E+02	5.5349E+01	5.2101E+02	4.3605E−02	6.2928E+02	3.3697E+00
ESSA	9.1397E+03	1.6979E+03	5.2105E+02	5.3942E−02	6.4159E+02	1.5682E+00
CMSSA	9.7175E+03	2.3523E+03	5.2097E+02	8.6016E−02	6.3815E+02	1.7268E+00
MSCA	1.8276E+03	1.6732E+03	5.2001E+02	1.8810E−02	6.2396E+02	3.6928E+00
IGWO	6.3066E+02	7.4658E+01	5.2101E+02	6.7892E−02	6.2487E+02	3.6073E+00
OBLGWO	6.1625E+02	3.9314E+01	5.2108E+02	4.1044E−02	6.2232E+02	2.2002E+00
GCHHO	5.7568E+02	3.3044E+01	5.2009E+02	8.4513E−02	6.3050E+02	4.2082E+00
HHODE	9.5520E+02	1.7511E+02	5.2090E+02	1.0923E−01	6.2565E+02	2.4950E+00
CMFO	2.1553E+03	9.4636E+02	5.2055E+02	2.3961E−01	6.3623E+02	4.0054E+00
	F7		F8		F9
Avg	Std	Avg	Std	Avg	Std
RRWOA	7.0011E+02	2.9336E−01	8.1834E+02	1.1357E+01	1.0425E+03	2.5151E+01
ASCA-PSO	7.0510E+02	3.7680E+00	9.9041E+02	2.7191E+01	1.1176E+03	1.4923E+01
ESSA	1.1781E+03	6.9908E+01	1.1155E+03	1.3393E+01	1.2509E+03	1.4623E+01
CMSSA	1.1151E+03	7.1067E+01	1.0940E+03	2.5597E+01	1.2352E+03	3.0306E+01
MSCA	8.4060E+02	7.6832E+01	8.8881E+02	2.1171E+01	1.1299E+03	4.0197E+01
IGWO	7.0305E+02	8.1180E−01	9.1821E+02	1.0644E+01	1.0560E+03	3.1882E+01
OBLGWO	7.0364E+02	1.7790E+00	9.6507E+02	4.4069E+01	1.1215E+03	4.5140E+01
GCHHO	7.0102E+02	7.9067E−02	9.3993E+02	2.1622E+01	1.0768E+03	1.4234E+01
HHODE	7.1893E+02	9.9552E+00	9.6751E+02	2.0378E+01	1.1241E+03	2.5716E+01
CMFO	7.8419E+02	5.1075E+01	9.8026E+02	4.8652E+01	1.1365E+03	5.9099E+01
	F10		F11		F12
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.3555E+03	2.0927E+02	3.3827E+03	4.7715E+02	1.2002E+03	5.9058E−02
ASCA-PSO	5.9558E+03	5.8530E+02	7.2853E+03	1.2681E+03	1.2031E+03	2.9591E−01
ESSA	8.2318E+03	2.9642E+02	8.8939E+03	2.9075E+02	1.2033E+03	2.9463E−01
CMSSA	7.5974E+03	7.7587E+02	8.4853E+03	7.3448E+02	1.2032E+03	6.0743E−01
MSCA	2.2363E+03	3.7423E+02	4.8172E+03	6.0605E+02	1.2002E+03	8.0225E−02
IGWO	4.3379E+03	6.4157E+02	5.5717E+03	5.6902E+02	1.2016E+03	4.7864E−01
OBLGWO	6.2225E+03	1.5093E+03	7.4798E+03	1.5856E+03	1.2031E+03	4.4621E−01
GCHHO	3.4651E+03	7.9432E+02	4.9992E+03	6.3671E+02	1.2009E+03	3.8914E−01
HHODE	5.2981E+03	8.4591E+02	6.4196E+03	4.6648E+02	1.2024E+03	5.1547E−01
CMFO	6.0695E+03	7.9580E+02	6.8585E+03	8.6760E+02	1.2017E+03	8.2151E−01
	F13		F14		F15
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.3005E+03	2.2342E−01	1.4002E+03	4.3570E−02	1.5094E+03	2.0254E+00
ASCA-PSO	1.3007E+03	1.2259E−01	1.4019E+03	3.1719E+00	1.5659E+03	1.3141E+02
ESSA	1.3058E+03	4.8930E−01	1.5573E+03	3.8881E+01	1.2863E+05	3.4816E+04
CMSSA	1.3063E+03	7.6256E−01	1.6022E+03	3.0698E+01	9.3930E+04	6.3537E+04
MSCA	1.3028E+03	1.1016E+00	1.4411E+03	1.6336E+01	2.6968E+04	4.4110E+04
IGWO	1.3008E+03	1.2373E−01	1.4004E+03	2.2812E−01	1.5315E+03	8.3898E+00
OBLGWO	1.3007E+03	1.4346E−01	1.4006E+03	3.5910E−01	1.5290E+03	6.5733E+00
GCHHO	1.3006E+03	1.3216E−01	1.4003E+03	1.9229E−01	1.5476E+03	1.6574E+01
HHODE	1.3007E+03	4.6462E−01	1.4095E+03	8.5225E+00	1.6440E+03	6.2862E+01
CMFO	1.3028E+03	6.4730E−01	1.4359E+03	1.3421E+01	1.0747E+04	4.5945E+03
	F16		F17		F18
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.6112E+03	4.0415E−01	3.3253E+06	1.8177E+06	3.5626E+03	3.4536E+03
ASCA-PSO	1.6130E+03	2.5750E−01	2.6446E+06	1.9129E+06	5.4199E+06	1.6252E+06
ESSA	1.6132E+03	2.3480E−01	3.1199E+07	1.6948E+07	1.4583E+09	4.0834E+08
CMSSA	1.6128E+03	2.4171E−01	6.2036E+07	3.7606E+07	1.0226E+09	1.0791E+09
MSCA	1.6128E+03	5.5739E−01	1.4304E+07	1.0753E+07	4.8973E+08	1.0297E+09
IGWO	1.6126E+03	7.8830E−01	4.1538E+06	2.6615E+06	3.8464E+05	3.2919E+05
OBLGWO	1.6128E+03	6.5831E−01	4.6120E+06	1.9190E+06	7.3479E+05	6.5104E+05
GCHHO	1.6122E+03	5.9773E−01	2.1875E+06	1.6343E+06	4.3123E+03	3.4900E+03
HHODE	1.6126E+03	6.2809E−01	4.1612E+06	2.5160E+06	1.2188E+06	1.6487E+06
CMFO	1.6129E+03	6.0662E−01	1.8270E+07	1.6646E+07	1.0827E+07	1.6442E+07
	F19		F20		F21
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.9074E+03	1.4580E+00	1.2673E+04	7.2221E+03	1.3293E+06	1.0927E+06
ASCA-PSO	1.9225E+03	1.8013E+01	4.1788E+04	1.6301E+04	8.3767E+05	8.0817E+05
ESSA	2.2391E+03	7.7656E+01	9.3255E+04	3.3670E+04	9.7601E+06	4.8057E+06
CMSSA	2.2491E+03	9.1370E+01	1.3885E+05	8.1149E+04	1.7982E+07	1.4506E+07
MSCA	1.9978E+03	7.4813E+01	6.0430E+04	3.4068E+04	5.9029E+06	4.2313E+06
IGWO	1.9261E+03	2.3980E+01	2.6608E+04	9.8757E+03	5.8821E+05	5.7621E+05
OBLGWO	1.9274E+03	2.1854E+01	3.5830E+04	1.1102E+04	8.0158E+05	6.5942E+05
GCHHO	1.9260E+03	2.2010E+01	1.7898E+04	9.9006E+03	7.1019E+05	6.8803E+05
HHODE	1.9957E+03	3.3923E+01	3.0941E+04	1.3964E+04	6.2223E+05	7.8718E+05
CMFO	2.0276E+03	7.9067E+01	8.3510E+04	5.3239E+04	5.9501E+06	4.5621E+06
	F22		F23		F24
Avg	Std	Avg	Std	Avg	Std
RRWOA	2.8460E+03	1.8811E+02	2.5000E+03	0.0000E+00	2.6000E+03	0.0000E+00
ASCA-PSO	2.9172E+03	1.8405E+02	2.6404E+03	8.7849E+00	2.6503E+03	8.9629E+00
ESSA	3.8106E+03	1.1615E+02	2.5000E+03	6.8565E−03	2.6000E+03	2.0577E−02
CMSSA	3.2411E+03	2.5970E+02	2.5000E+03	0.0000E+00	2.6000E+03	0.0000E+00
MSCA	3.1245E+03	3.0913E+02	2.6612E+03	5.1259E+01	2.6465E+03	1.1629E+01
IGWO	2.7028E+03	1.5640E+02	2.6352E+03	6.7955E+00	2.6001E+03	5.7800E−02
OBLGWO	2.7317E+03	2.2492E+02	2.6195E+03	4.2756E+01	2.6029E+03	9.0742E+00
GCHHO	3.0230E+03	1.4942E+02	2.5000E+03	0.0000E+00	2.6000E+03	1.2859E−03
HHODE	2.8106E+03	1.7812E+02	2.5000E+03	0.0000E+00	2.6000E+03	1.5659E−03
CMFO	3.1194E+03	3.6485E+02	2.7111E+03	4.0577E+01	2.6815E+03	1.3089E+01
	F25		F26		F27
Avg	Std	Avg	Std	Avg	Std
RRWOA	2.7000E+03	0.0000E+00	2.7302E+03	4.8146E+01	2.9000E+03	4.3432E−09
ASCA-PSO	2.7241E+03	7.9292E+00	2.7007E+03	2.2607E−01	3.6071E+03	2.6004E+02
ESSA	2.7000E+03	1.5535E−04	2.7048E+03	4.8433E−01	2.9000E+03	1.9369E−03
CMSSA	2.7000E+03	0.0000E+00	2.7052E+03	6.1025E−01	2.9000E+03	0.0000E+00
MSCA	2.7128E+03	7.8071E+00	2.7011E+03	9.1494E−01	3.6873E+03	1.4195E+02
IGWO	2.7159E+03	8.3939E+00	2.7009E+03	1.2382E−01	3.1864E+03	1.5192E+02
OBLGWO	2.7000E+03	0.0000E+00	2.7007E+03	1.5424E−01	3.3340E+03	3.6251E+02
GCHHO	2.7000E+03	0.0000E+00	2.7105E+03	3.1443E+01	2.9000E+03	0.0000E+00
HHODE	2.7000E+03	0.0000E+00	2.7010E+03	8.7879E−01	2.9844E+03	2.6674E+02
CMFO	2.7411E+03	1.2464E+01	2.7049E+03	2.7676E+00	3.7532E+03	3.6925E+02
	F28		F29		F30
Avg	Std	Avg	Std	Avg	Std
RRWOA	3.0000E+03	0.0000E+00	3.1000E+03	0.0000E+00	3.2000E+03	1.0055E−12
ASCA-PSO	4.6921E+03	6.1804E+02	3.8296E+06	5.9854E+06	7.9155E+04	4.0749E+04
ESSA	3.0000E+03	2.4143E−03	1.6346E+04	1.0606E+04	3.9785E+03	5.6719E+02
CMSSA	3.0000E+03	0.0000E+00	3.1000E+03	8.5748E−13	3.2000E+03	2.1437E−13
MSCA	3.3745E+03	8.4321E+01	4.4536E+03	4.1630E+03	4.4115E+03	7.0335E+02
IGWO	4.2393E+03	3.8500E+02	1.3080E+06	4.0093E+06	7.4821E+04	2.8130E+04
OBLGWO	3.5372E+03	5.9011E+02	8.2663E+06	2.9282E+06	4.9307E+04	4.1239E+04
GCHHO	3.0000E+03	0.0000E+00	1.8398E+06	3.8730E+06	1.9014E+04	9.4855E+03
HHODE	3.0000E+03	1.5158E−13	3.0244E+06	6.3834E+06	1.1682E+05	8.1005E+04
CMFO	5.3449E+03	4.7737E+02	5.0421E+07	4.6800E+07	1.6670E+05	1.3271E+05

	Overall Rank
Rank	+/=/−	ARV
RRWOA	1	∼	1.7333
ASCA-PSO	6	26/4/0	5.9667
ESSA	10	29/1/0	8.2
CMSSA	8	22/8/0	7.2667
MSCA	6	28/2/0	5.9667
IGWO	3	25/3/2	4.2667
OBLGWO	5	23/7/0	5.1333
GCHHO	2	18/12/0	3
HHODE	4	21/9/0	4.7667
CMFO	9	28/2/0	7.9

Table A7.

P-value of Wilcoxon test obtained from comparison with advanced methods on 30 functions.

Function	F1	F2	F3	F4	F5	F6
ASCA-PSO	3.9063E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
ESSA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
CMSSA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
MSCA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	4.8828E−02	3.9063E−03
IGWO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
OBLGWO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
GCHHO	1.3086E−01	1.9531E−03	1.9531E−03	3.9063E−03	1.9531E−03	1.9531E−03
HHODE	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
CMFO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
Function	F7	F8	F9	F10	F11	F12
ASCA-PSO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
ESSA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
CMSSA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
MSCA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	3.9063E−03	7.6953E−01
IGWO	1.9531E−03	1.9531E−03	4.3164E−01	1.9531E−03	1.9531E−03	1.9531E−03
OBLGWO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
GCHHO	3.9063E−03	1.9531E−03	1.9531E−02	1.9531E−03	1.9531E−03	1.9531E−03
HHODE	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
CMFO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
Function	F13	F14	F15	F16	F17	F18
ASCA-PSO	3.7109E−02	3.9063E−03	1.9531E−03	1.9531E−03	6.9531E−01	1.9531E−03
ESSA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
CMSSA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
MSCA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	3.9063E−03	1.9531E−03
IGWO	3.7109E−02	1.9531E−03	1.9531E−03	3.9063E−03	3.7500E−01	1.9531E−03
OBLGWO	6.4453E−02	1.9531E−03	1.9531E−03	1.9531E−03	4.8828E−02	1.9531E−03
GCHHO	1.6016E−01	3.7109E−02	1.9531E−03	2.7344E−02	2.7539E−01	2.3242E−01
HHODE	2.7344E−02	5.8594E−03	1.9531E−03	5.8594E−03	3.7500E−01	1.9531E−03
CMFO	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−02	1.9531E−03
Function	F19	F20	F21	F22	F23	F24
ASCA-PSO	1.9531E−03	1.9531E−03	3.7500E−01	4.9219E−01	1.9531E−03	1.9531E−03
ESSA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
CMSSA	1.9531E−03	1.9531E−03	3.9063E−03	1.9531E−03	1.0000E+00	1.0000E+00
MSCA	1.9531E−03	1.9531E−03	9.7656E−03	4.8828E−02	1.9531E−03	1.9531E−03
IGWO	1.9531E−03	5.8594E−03	4.8828E−02	4.8828E−02	1.9531E−03	1.9531E−03
OBLGWO	1.9531E−03	3.9063E−03	3.7500E−01	2.3242E−01	3.9063E−03	1.0000E+00
GCHHO	1.9531E−03	1.3086E−01	2.7539E−01	8.3984E−02	1.0000E+00	1.5625E−02
HHODE	1.9531E−03	9.7656E−03	6.4453E−02	9.2188E−01	1.0000E+00	3.9063E−03
CMFO	1.9531E−03	1.9531E−03	1.9531E−02	6.4453E−02	1.9531E−03	1.9531E−03
Function	F25	F26	F27	F28	F29	F30
ASCA-PSO	1.9531E−03	1.0000E+00	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
ESSA	1.9531E−03	1.0000E+00	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
CMSSA	1.0000E+00	1.0000E+00	5.0000E−01	1.0000E+00	5.0000E−01	1.0000E+00
MSCA	1.9531E−03	1.0000E+00	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
IGWO	1.9531E−03	1.0000E+00	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
OBLGWO	1.0000E+00	1.0000E+00	1.5625E−02	6.2500E−02	1.9531E−03	1.9531E−03
GCHHO	1.0000E+00	7.6953E−01	5.0000E−01	1.0000E+00	1.9531E−03	3.9063E−03
HHODE	1.0000E+00	1.0000E+00	3.7500E−01	1.0000E+00	1.2500E−01	3.9063E−03
CMFO	1.9531E−03	1.0000E+00	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03

Table A8.

Comparison of RRWOA and improved WOA variants in CEC2014.

	F1		F2		F3
	Avg	Std	Avg	Std	Avg	Std
RRWOA	3.1598E+05	9.5132E+04	2.0000E+02	5.0131E−14	3.4843E+02	4.5477E+01
ACWOA	1.2136E+08	5.4960E+07	6.7549E+09	3.4062E+09	5.3476E+04	4.2582E+03
CWOAI	9.0401E+07	8.7162E+07	2.3647E+09	3.3020E+09	7.0327E+04	3.0245E+04
IWOA	4.1633E+06	1.7029E+06	4.1290E+06	1.2956E+07	5.5111E+03	3.3605E+03
LWOA	4.1880E+06	1.6233E+06	4.8042E+05	1.5638E+05	9.5895E+02	2.1795E+02
MWOA	1.3946E+09	4.5834E+08	5.7238E+10	3.7353E+10	1.7535E+05	1.3499E+05
OBWOA	8.9936E+07	6.3671E+07	1.3272E+10	6.5760E+09	3.8410E+04	1.1393E+04
RDWOA	6.6814E+06	2.3816E+06	2.7528E+07	4.0261E+07	4.9733E+03	3.5162E+03
	F4		F5		F6
Avg	Std	Avg	Std	Avg	Std
RRWOA	4.2112E+02	3.2402E+01	5.2000E+02	1.0651E−04	6.1197E+02	2.8124E+00
ACWOA	1.1803E+03	2.5667E+02	5.2078E+02	2.1089E−01	6.3408E+02	2.9253E+00
CWOAI	9.2619E+02	3.8721E+02	5.2025E+02	1.0199E−01	6.3786E+02	2.6645E+00
IWOA	5.0486E+02	2.8918E+01	5.2015E+02	1.4396E−01	6.2029E+02	2.6247E+00
LWOA	5.0108E+02	2.9764E+01	5.2047E+02	1.3363E−01	6.3050E+02	4.3427E+00
MWOA	1.2463E+04	7.7800E+03	5.2126E+02	7.9083E−02	6.4363E+02	3.6925E+00
OBWOA	1.0761E+03	5.5794E+02	5.2077E+02	1.8650E−01	6.3420E+02	2.7214E+00
RDWOA	5.3473E+02	4.5294E+01	5.2014E+02	1.2327E−01	6.2172E+02	4.1474E+00
	F7		F8		F9
Avg	Std	Avg	Std	Avg	Std
RRWOA	7.0000E+02	3.8962E−03	8.1433E+02	1.3579E+01	1.0391E+03	2.7168E+01
ACWOA	7.3534E+02	2.1323E+01	9.8999E+02	2.2359E+01	1.1524E+03	2.1624E+01
CWOAI	7.3890E+02	4.9928E+01	9.8249E+02	3.1949E+01	1.1358E+03	3.6405E+01
IWOA	7.0011E+02	6.1796E−02	8.3612E+02	9.6122E+00	1.0425E+03	2.9703E+01
LWOA	7.0063E+02	1.0085E−01	8.6851E+02	1.3853E+01	1.1023E+03	4.2416E+01
MWOA	1.2020E+03	2.7503E+02	1.1710E+03	6.0615E+01	1.2605E+03	6.4009E+01
OBWOA	7.6206E+02	5.1334E+01	9.7520E+02	1.7188E+01	1.1264E+03	1.9778E+01
RDWOA	7.0096E+02	1.7062E−01	8.4472E+02	8.8103E+00	1.0910E+03	3.0656E+01
	F10		F11		F12
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.2252E+03	1.4347E+02	3.2817E+03	2.8350E+02	1.2001E+03	4.1977E−02
ACWOA	4.7710E+03	5.0179E+02	5.9668E+03	8.4404E+02	1.2016E+03	5.6250E−01
CWOAI	5.2973E+03	8.2400E+02	6.3267E+03	7.8829E+02	1.2016E+03	3.9342E−01
IWOA	1.4476E+03	2.5049E+02	4.5473E+03	8.1232E+02	1.2004E+03	1.5755E−01
LWOA	2.3018E+03	3.2265E+02	4.9508E+03	6.4899E+02	1.2007E+03	1.2256E−01
MWOA	9.1458E+03	6.5610E+02	9.1517E+03	1.0315E+03	1.2049E+03	8.2456E−01
OBWOA	4.8767E+03	7.4535E+02	6.2135E+03	1.1447E+03	1.2018E+03	6.1857E−01
RDWOA	1.6645E+03	1.2931E+02	4.8568E+03	6.8094E+02	1.2006E+03	2.7635E−01
	F13		F14		F15
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.3003E+03	6.3561E−02	1.4002E+03	4.1495E−02	1.5054E+03	1.6832E+00
ACWOA	1.3015E+03	1.0101E+00	1.4172E+03	9.6624E+00	1.9154E+03	5.2444E+02
CWOAI	1.3008E+03	5.7675E−01	1.4045E+03	6.3478E+00	2.4681E+03	1.5729E+03
IWOA	1.3005E+03	1.6464E−01	1.4003E+03	4.3123E−02	1.5231E+03	8.8488E+00
LWOA	1.3005E+03	1.3508E−01	1.4003E+03	6.3254E−02	1.5216E+03	6.1786E+00
MWOA	1.3056E+03	2.0484E+00	1.5841E+03	9.0629E+01	2.4378E+05	4.3908E+05
OBWOA	1.3015E+03	1.2501E+00	1.4313E+03	2.0584E+01	2.2579E+03	1.0480E+03
RDWOA	1.3005E+03	1.2465E−01	1.4003E+03	5.4421E−02	1.5211E+03	9.0932E+00
	F16		F17		F18
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.6104E+03	5.6175E−01	6.1073E+05	3.4684E+05	3.3157E+03	2.5571E+03
ACWOA	1.6122E+03	4.0609E−01	1.9137E+07	1.1506E+07	3.6298E+07	3.2595E+07
CWOAI	1.6131E+03	3.3108E−01	5.4593E+06	3.2673E+06	1.4912E+07	2.9322E+07
IWOA	1.6118E+03	3.3325E−01	7.7861E+05	4.5842E+05	9.5329E+03	9.9328E+03
LWOA	1.6124E+03	4.2084E−01	4.9136E+05	3.3369E+05	1.1774E+04	7.9587E+03
MWOA	1.6140E+03	2.6138E−01	7.9420E+07	1.0574E+08	2.6444E+09	2.4039E+09
OBWOA	1.6123E+03	4.0044E−01	9.5930E+06	6.1009E+06	2.7184E+08	8.4771E+08
RDWOA	1.6116E+03	4.5327E−01	6.3647E+05	4.9274E+05	8.1878E+03	5.5948E+03
	F19		F20		F21
Avg	Std	Avg	Std	Avg	Std
RRWOA	1.9068E+03	1.4834E+00	1.9614E+04	1.5677E+04	2.2711E+05	1.3149E+05
ACWOA	2.0088E+03	3.5075E+01	3.2958E+04	8.1508E+03	5.5718E+06	5.3984E+06
CWOAI	1.9863E+03	2.7293E+01	7.9388E+04	4.3941E+04	2.4413E+06	2.2756E+06
IWOA	1.9258E+03	2.9215E+01	6.4453E+03	4.5360E+03	5.7221E+05	5.1700E+05
LWOA	1.9137E+03	1.3854E+00	2.9568E+03	5.8562E+02	1.8285E+05	1.2373E+05
MWOA	2.4219E+03	1.7259E+02	6.6466E+05	1.0815E+06	3.5758E+07	2.6839E+07
OBWOA	2.0373E+03	5.2208E+01	6.1297E+04	5.8641E+04	2.0570E+06	2.7052E+06
RDWOA	1.9109E+03	1.5053E+00	7.4066E+03	3.7978E+03	5.3846E+05	3.4064E+05
	F22		F23		F24
Avg	Std	Avg	Std	Avg	Std
RRWOA	2.7351E+03	1.2444E+02	2.5000E+03	0.0000E+00	2.6000E+03	0.0000E+00
ACWOA	3.1477E+03	1.7208E+02	2.5397E+03	8.3793E+01	2.6000E+03	1.5546E−06
CWOAI	2.9747E+03	2.9175E+02	2.6400E+03	5.5875E+01	2.6038E+03	3.7125E+00
IWOA	2.6645E+03	1.3544E+02	2.6156E+03	5.3386E−01	2.6095E+03	2.9474E+01
LWOA	2.9527E+03	2.7784E+02	2.6154E+03	1.2384E−01	2.6057E+03	8.6223E+00
MWOA	5.9279E+03	5.9324E+03	2.9015E+03	2.0361E+02	2.7211E+03	7.2922E+01
OBWOA	3.2034E+03	3.0373E+02	2.5000E+03	2.1437E−13	2.6025E+03	2.9940E+00
RDWOA	2.7915E+03	2.0385E+02	2.5000E+03	0.0000E+00	2.6000E+03	1.0057E−04
	F25		F26		F27
Avg	Std	Avg	Std	Avg	Std
RRWOA	2.7000E+03	0.0000E+00	2.7900E+03	3.1541E+01	2.9000E+03	0.0000E+00
ACWOA	2.7000E+03	0.0000E+00	2.7106E+03	3.1415E+01	3.6513E+03	4.2923E+02
CWOAI	2.7052E+03	1.0906E+01	2.7404E+03	5.1337E+01	3.7415E+03	4.0050E+02
IWOA	2.7141E+03	6.3027E+00	2.7006E+03	1.5137E−01	3.5651E+03	2.4631E+02
LWOA	2.7168E+03	4.4440E+00	2.7005E+03	1.0034E−01	3.5959E+03	3.4939E+02
MWOA	2.7359E+03	3.5559E+01	2.7678E+03	7.5794E+01	4.3354E+03	2.6909E+02
OBWOA	2.7000E+03	0.0000E+00	2.7801E+03	4.1939E+01	2.9000E+03	1.5158E−13
RDWOA	2.7000E+03	0.0000E+00	2.7005E+03	1.1885E−01	2.9000E+03	0.0000E+00
	F28		F29		F30
Avg	Std	Avg	Std	Avg	Std
RRWOA	3.0000E+03	0.0000E+00	3.1000E+03	0.0000E+00	3.2000E+03	0.0000E+00
ACWOA	3.3708E+03	8.2217E+02	2.5893E+07	2.6969E+07	2.7504E+05	2.6312E+05
CWOAI	5.2192E+03	6.8819E+02	1.0093E+07	3.4070E+06	1.2878E+05	7.2209E+04
IWOA	4.3544E+03	3.3906E+02	5.9901E+06	4.1316E+06	9.3459E+03	3.1965E+03
LWOA	4.8431E+03	3.2884E+02	6.0598E+06	4.1833E+06	1.3931E+04	1.1345E+04
MWOA	7.3184E+03	1.1332E+03	2.7246E+08	1.5496E+08	3.5470E+06	2.1373E+06
OBWOA	3.6676E+03	1.0794E+03	4.7392E+06	8.6640E+06	1.2469E+04	2.9311E+04
RDWOA	3.0000E+03	0.0000E+00	2.5774E+06	4.1429E+06	8.0420E+03	1.3892E+03

	Overall Rank
Rank	+/=/−	ARV
RRWOA	1	∼	1.4333
ACWOA	6	25/4/1	5.5
CWOAI	7	28/2/0	5.8667
IWOA	3	24/4/2	3.2667
LWOA	4	26/2/2	3.7667
MWOA	8	29/1/0	7.9333
OBWOA	5	23/7/0	5.3
RDWOA	2	21/8/1	2.5

Table A9.

P-value of Wilcoxon test between RRWOA and improved WOA variants on 30 functions.

Function	F1	F2	F3	F4	F5	F6
ACWOA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
CWOAI	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
IWOA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
LWOA	1.9531E−03	1.9531E−03	1.9531E−03	3.9063E−03	1.9531E−03	1.9531E−03
MWOA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
OBWOA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
RDWOA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
Function	F7	F8	F9	F10	F11	F12
ACWOA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
CWOAI	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
IWOA	1.9531E−03	9.7656E−03	1.0000E+00	3.7109E−02	5.8594E−03	1.9531E−03
LWOA	1.9531E−03	1.9531E−03	5.8594E−03	1.9531E−03	1.9531E−03	1.9531E−03
MWOA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
OBWOA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
RDWOA	1.9531E−03	3.9063E−03	5.8594E−03	1.9531E−03	1.9531E−03	1.9531E−03
Function	F13	F14	F15	F16	F17	F18
ACWOA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
CWOAI	1.9531E−03	5.8594E−03	1.9531E−03	1.9531E−03	1.9531E−03	5.8594E−03
IWOA	4.8828E−02	5.8594E−03	1.9531E−03	1.9531E−03	5.5664E−01	1.3086E−01
LWOA	3.9063E−03	5.8594E−03	1.9531E−03	1.9531E−03	3.2227E−01	1.9531E−02
MWOA	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03	3.9063E−03	1.9531E−03
OBWOA	5.8594E−03	3.9063E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
RDWOA	3.9063E−03	6.4453E−02	1.9531E−03	1.9531E−03	9.2188E−01	1.3672E−02
Function	F19	F20	F21	F22	F23	F24
ACWOA	1.9531E−03	8.3984E−02	1.9531E−03	1.9531E−03	5.0000E−01	3.9063E−03
CWOAI	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−02	3.9063E−03	1.9531E−03
IWOA	1.9531E−03	3.7109E−02	2.7344E−02	3.7500E−01	1.9531E−03	1.9531E−03
LWOA	1.9531E−03	1.9531E−03	6.2500E−01	4.8828E−02	1.9531E−03	1.9531E−03
MWOA	1.9531E−03	3.9063E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
OBWOA	1.9531E−03	3.7109E−02	9.7656E−03	1.9531E−03	1.0000E+00	1.9531E−03
RDWOA	1.9531E−03	8.3984E−02	9.7656E−03	6.2500E−01	1.0000E+00	1.9531E−03
Function	F25	F26	F27	F28	F29	F30
ACWOA	1.0000E+00	7.8125E−03	1.9531E−03	5.0000E−01	3.9063E−03	7.8125E−03
CWOAI	5.0000E−01	6.2500E−02	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
IWOA	1.9531E−03	3.9063E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
LWOA	1.9531E−03	3.9063E−03	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
MWOA	1.9531E−03	2.7539E−01	1.9531E−03	1.9531E−03	1.9531E−03	1.9531E−03
OBWOA	1.0000E+00	1.0000E+00	1.0000E+00	2.5000E−01	2.5000E−01	1.2500E−01
RDWOA	1.0000E+00	3.9063E−03	1.0000E+00	1.0000E+00	7.8125E−03	1.9531E−03

Table A10.

Comparison between the proposed BRRWOA classification method and the other seven classification methods on average fitness values.

	Breastcancer		Exactly			HeartEW
	Avg	Std	Avg	Std		Avg	Std
BRRWOA	2.6457E−02	8.9228E−03	2.3077E−02	0.0000E+00		7.8604E−02	3.8730E−02
BWOA	2.8719E−02	9.3590E−03	2.3846E−02	1.6217E−03		9.6467E−02	3.5579E−02
BMFO	3.0675E−02	1.3037E−02	2.6923E−02	7.3142E−18		1.0350E−01	4.4140E−02
BGSA	3.2500E−02	1.2093E−02	2.3077E−02	0.0000E+00		8.4815E−02	3.2802E−02
BPSO	3.0611E−02	1.3865E−02	2.3462E−02	1.2163E−03		9.7735E−02	4.7125E−02
BBA	4.4035E−02	2.0491E−02	2.5650E−01	8.5813E−02		1.8487E−01	4.8676E−02
BSSA	3.3345E−02	1.5174E−02	2.7873E−02	4.8702E−03		8.5527E−02	3.0266E−02
BRUN	3.0980E−02	1.0248E−02	2.4615E−02	1.9861E−03		9.6524E−02	2.8814E−02
	M-of-n		Congress			Hepatitis
Avg	Std	Avg	Std		Avg	Std
BRRWOA	2.3077E−02	0.0000E+00	1.8693E−02	1.1403E−02		1.3684E−02	4.0768E−03
BWOA	2.5000E−02	2.0271E−03	2.6549E−02	1.6323E−02		2.6070E−02	2.3144E−02
BMFO	2.6923E−02	1.8131E−03	3.1003E−02	1.2887E−02		4.2156E−02	4.0118E−02
BGSA	2.3077E−02	0.0000E+00	2.1886E−02	1.3720E−02		1.8569E−02	1.8642E−02
BPSO	2.3462E−02	1.2163E−03	1.9261E−02	1.9395E−02		1.7105E−02	4.5156E−03
BBA	1.5642E−01	6.1602E−02	5.6481E−02	2.9416E−02		8.6989E−02	3.2904E−02
BSSA	2.8642E−02	6.0110E−03	2.6265E−02	1.6876E−02		3.2797E−02	2.8874E−02
BRUN	2.4231E−02	1.8579E−03	2.4340E−02	1.0119E−02		2.2780E−02	2.0661E−02
	USA		Wine			cmc
Avg	Std	Avg	Std		Avg	Std
BRRWOA	2.4368E−01	2.1511E−02	1.0769E−02	3.5344E−03		4.6703E−01	2.8764E−02
BWOA	2.4858E−01	1.7376E−02	1.4231E−02	3.1664E−03		4.7761E−01	2.4756E−02
BMFO	2.5075E−01	2.2432E−02	1.9615E−02	2.1833E−03		4.7204E−01	3.4368E−02
BGSA	2.4636E−01	1.9981E−02	1.2308E−02	3.5344E−03		4.8180E−01	1.9625E−02
BPSO	2.4612E−01	1.4433E−02	1.0769E−02	3.0339E−03		4.7093E−01	2.2218E−02
BBA	2.8412E−01	1.3758E−02	2.0769E−02	3.7157E−03		5.1860E−01	2.0783E−02
BSSA	2.4816E−01	1.0986E−02	1.2692E−02	4.4596E−03		4.7151E−01	3.2012E−02
BRUN	2.4794E−01	1.4517E−02	1.3846E−02	1.9861E−03		4.7148E−01	2.9732E−02
	Lymphography		Vehicle			Heart
Avg	Std	Avg	Std		Avg	Std
BRRWOA	1.5278E−02	3.7611E−03	1.9389E−01	2.1232E−02		6.7279E−02	4.2923E−02
BWOA	4.6486E−02	3.4768E−02	2.1073E−01	3.0425E−02		8.5912E−02	4.0583E−02
BMFO	4.4715E−02	3.0646E−02	2.2335E−01	1.4008E−02		8.9046E−02	2.7627E−02
BGSA	1.4167E−02	4.6204E−03	1.9824E−01	2.1990E−02		7.1952E−02	2.8398E−02
BPSO	2.7826E−02	2.3799E−02	2.0138E−01	3.1149E−02		7.7835E−02	3.7897E−02
BBA	1.0356E−01	4.5151E−02	2.6978E−01	1.9970E−02		1.5255E−01	5.7595E−02
BSSA	3.6452E−02	2.7957E−02	2.2394E−01	3.0106E−02		1.0081E−01	4.5048E−02
BRUN	3.0604E−02	2.7624E−02	2.0336E−01	2.9743E−02		7.2279E−02	3.2046E−02
	Glass		Vote			Zoo
Avg	Std	Avg	Std		Avg	Std
BRRWOA	1.2503E−01	4.6748E−02	2.0750E−02	1.6407E−02		1.0625E−02	2.6352E−03
BWOA	1.4206E−01	5.3097E−02	2.8198E−02	1.8551E−02		1.3750E−02	4.4683E−03
BMFO	1.2130E−01	4.7254E−02	3.7192E−02	2.3038E−02		1.6250E−02	2.4650E−03
BGSA	1.3085E−01	5.0577E−02	1.2440E−02	1.1513E−02		9.6875E−03	2.7362E−03
BPSO	1.3132E−01	5.1865E−02	2.6669E−02	2.7961E−02		1.0937E−02	2.2097E−03
BBA	1.7312E−01	6.5831E−02	4.4674E−02	2.1518E−02		1.7188E−02	4.9411E−03
BSSA	1.3068E−01	4.8147E−02	2.9650E−02	2.6319E−02		1.3438E−02	2.5727E−03
BRUN	1.2150E−01	6.1299E−02	2.7108E−02	2.2336E−02		1.3125E−02	2.8717E−03

	Overall Rank
Rank	+/=/−	ARV
BRRWOA	1	∼	1.3333
BWOA	5	15/0/0	5.2667
BMFO	7	14/0/1	5.9333
BGSA	2	10/2/3	2.7333
BPSO	3	14/1/0	3.0667
BBA	8	15/0/0	8
BSSA	6	15/0/0	5.4667
BRUN	4	14/0/1	4

Table A11.

Comparison between the developed BRRWOA and the other seven heuristics on the average KNN error values.

	Breastcancer		Exactly		HeartEW
	Avg	Std	Avg	Std	Avg	Std
BRRWOA	8.5513E−03	7.3600E−03	0.0000E+00	0.0000E+00	5.9259E−02	4.3474E−02
BWOA	8.5927E−03	7.3963E−03	0.0000E+00	0.0000E+00	7.0370E−02	4.0759E−02
BMFO	7.1429E−03	1.2141E−02	0.0000E+00	0.0000E+00	7.7778E−02	4.7655E−02
BGSA	1.4327E−02	1.1721E−02	0.0000E+00	0.0000E+00	6.2963E−02	3.5136E−02
BPSO	1.0000E−02	1.5134E−02	0.0000E+00	0.0000E+00	7.7778E−02	5.0753E−02
BBA	9.3867E−02	1.1597E−01	3.4996E−01	4.4803E−02	2.5926E−01	9.5629E−02
BSSA	1.2878E−02	1.7105E−02	1.0000E−03	3.1623E−03	5.9259E−02	3.1232E−02
BRUN	8.6335E−03	1.0094E−02	0.0000E+00	0.0000E+00	7.4074E−02	3.0241E−02
	M-of-n		Congress		Hepatitis
Avg	Std	Avg	Std	Avg	Std
BRRWOA	0.0000E+00	0.0000E+00	4.5455E−03	9.5827E−03	0.0000E+00	0.0000E+00
BWOA	0.0000E+00	0.0000E+00	9.1966E−03	1.1874E−02	6.6667E−03	2.1082E−02
BMFO	0.0000E+00	0.0000E+00	6.9767E−03	1.1234E−02	1.9167E−02	4.2682E−02
BGSA	0.0000E+00	0.0000E+00	1.1525E−02	1.6201E−02	6.2500E−03	1.9764E−02
BPSO	0.0000E+00	0.0000E+00	9.0909E−03	1.5891E−02	0.0000E+00	0.0000E+00
BBA	1.8087E−01	8.2438E−02	1.0365E−01	7.1402E−02	1.7191E−01	9.7873E−02
BSSA	1.0000E−03	3.1623E−03	6.9239E−03	1.5636E−02	1.2917E−02	2.7248E−02
BRUN	0.0000E+00	0.0000E+00	6.8710E−03	1.1064E−02	6.2500E−03	1.9764E−02
	USA		Wine		cmc
Avg	Std	Avg	Std	Avg	Std
BRRWOA	2.3756E−01	2.4212E−02	0.0000E+00	0.0000E+00	4.6295E−01	3.0814E−02
BWOA	2.4272E−01	1.9342E−02	0.0000E+00	0.0000E+00	4.7175E−01	2.3368E−02
BMFO	2.4184E−01	2.3261E−02	0.0000E+00	0.0000E+00	4.6764E−01	3.4955E−02
BGSA	2.3933E−01	2.3105E−02	0.0000E+00	0.0000E+00	4.7792E−01	2.1263E−02
BPSO	2.3802E−01	1.5339E−02	0.0000E+00	0.0000E+00	4.6706E−01	2.4693E−02
BBA	3.1721E−01	5.5278E−02	1.0621E−01	6.1240E−02	5.5470E−01	5.2156E−02
BSSA	2.4017E−01	1.3057E−02	0.0000E+00	0.0000E+00	4.6709E−01	3.6397E−02
BRUN	2.3889E−01	1.5660E−02	0.0000E+00	0.0000E+00	4.6706E−01	3.2139E−02
	Lymphography		Vehicle		Heart
Avg	Std	Avg	Std	Avg	Std
BRRWOA	0.0000E+00	0.0000E+00	1.7836E−01	2.1669E−02	4.8148E−02	4.6358E−02
BWOA	2.5833E−02	3.3380E−02	1.9259E−01	2.9633E−02	5.9259E−02	4.6849E−02
BMFO	1.9583E−02	3.1553E−02	2.0207E−01	1.4074E−02	6.2963E−02	3.0492E−02
BGSA	0.0000E+00	0.0000E+00	1.8324E−01	2.3324E−02	5.1852E−02	3.1232E−02
BPSO	1.2917E−02	2.7248E−02	1.8800E−01	3.4856E−02	5.9259E−02	3.9814E−02
BBA	2.3140E−01	1.3349E−01	2.9660E−01	4.3165E−02	2.5556E−01	6.8627E−02
BSSA	1.3810E−02	2.9135E−02	2.0561E−01	3.2259E−02	7.7778E−02	4.7655E−02
BRUN	1.2917E−02	2.7248E−02	1.8570E−01	3.5084E−02	4.8148E−02	3.5136E−02
	Glass		Vote		Zoo
Avg	Std	Avg	Std	Avg	Std
BRRWOA	1.0764E−01	5.0450E−02	1.0000E−02	1.6102E−02	0.0000E+00	0.0000E+00
BWOA	1.2147E−01	5.5096E−02	1.3563E−02	1.7514E−02	0.0000E+00	0.0000E+00
BMFO	9.7857E−02	5.0045E−02	1.6452E−02	2.3068E−02	0.0000E+00	0.0000E+00
BGSA	1.1201E−01	5.3956E−02	3.2258E−03	1.0201E−02	0.0000E+00	0.0000E+00
BPSO	1.1250E−01	5.6855E−02	1.6559E−02	2.8260E−02	0.0000E+00	0.0000E+00
BBA	3.4535E−01	1.3995E−01	1.6405E−01	1.2349E−01	5.1313E−02	7.5511E−02
BSSA	1.1300E−01	5.1219E−02	1.3118E−02	2.2764E−02	0.0000E+00	0.0000E+00
BRUN	1.0391E−01	7.0733E−02	9.7849E−03	2.1899E−02	0.0000E+00	0.0000E+00

	Overall Rank
Rank	+/=/−	ARV
BRRWOA	1	∼	1.3333
BWOA	6	12/3/0	4.2
BMFO	5	10/3/2	3.8667
BGSA	3	10/4/1	3.0667
BPSO	4	11/4/0	3.2
BBA	8	15/0/0	8
BSSA	7	13/2/0	4.6667
BRUN	2	9/4/2	2.2667

Table A12.

Comparison between the developed BRRWOA and the other seven metaheuristics on the average value of the selected features.

	Breastcancer		Exactly		HeartEW
	Avg	Std	Avg	Std	Avg	Std
BRRWOA	3.3	0.94868	6	0	5.8	1.3984
BWOA	3.7	1.1595	6.2	0.42164	7.7	1.767
BMFO	4.3	1.4181	7	0	7.7	0.82327
BGSA	3.4	0.84327	6	0	6.5	1.5811
BPSO	3.8	0.63246	6.1	0.31623	6.2	1.2293
BBA	3.2	1.5492	5.8	1.3984	5.9	1.2867
BSSA	3.8	1.3166	7	0.66667	7.6	1.2649
BRUN	4.1	0.8756	6.4	0.5164	6.8	2.0976
	M-of-n		Congress		Hepatitis
Avg	Std	Avg	Std	Avg	Std
BRRWOA	6	0	4.6	1.8974	5.2	1.5492
BWOA	6.5	0.52705	5.7	2.1628	7.5	2.4608
BMFO	7	0.4714	7.8	2.0976	9.1	1.3703
BGSA	6	0	3.5	2.2236	4.8	1.8135
BPSO	6.1	0.31623	3.4	1.9551	6.5	1.7159
BBA	6.7	2.2632	5.1	2.2828	8	1.8856
BSSA	7.2	0.91894	6.3	2.2136	7.8	2.4855
BRUN	6.3	0.48305	5.7	1.567	6.4	1.6465
	USA		Wine		cmc
Avg	Std	Avg	Std	Avg	Std
BRRWOA	3.6	0.84327	2.8	0.91894	4.9	0.56765
BWOA	3.6	0.84327	3.7	0.82327	5.3	1.1595
BMFO	4.2	1.0328	5.1	0.56765	5	0.94281
BGSA	3.8	0.78881	3.2	0.91894	5	0.8165
BPSO	4	0.94281	2.8	0.78881	4.9	1.3703
BBA	3.9	1.1005	5.5	1.354	3.5	1.7159
BSSA	4	0.94281	3.3	1.1595	5	0.94281
BRUN	4.2	0.63246	3.6	0.5164	5	0.94281
	Lymphography		Vehicle		Heart
Avg	Std	Avg	Std	Avg	Std
BRRWOA	5.5	1.354	8.8	1.1353	5.6	1.075
BWOA	7.9	2.9231	10	2.1082	7.7	1.4944
BMFO	9.4	1.4298	11.3	1.4181	7.6	1.1738
BGSA	5.1	1.6633	8.7	2.7508	5.9	1.5239
BPSO	5.6	1.5055	8.2	2.8983	5.6	1.2649
BBA	6.8	1.7512	8.8	2.201	4.9	1.5239
BSSA	8.4	2.6331	10.3	1.8288	7	0.8165
BRUN	6.6	1.8379	9.7	1.8886	6.9	1.912
	Glass		Vote		Zoo
Avg	Std	Avg	Std	Avg	Std
BRRWOA	4.1	0.73786	3.6	1.7764	3.4	0.84327
BWOA	4.8	1.0328	4.9	1.9692	4.4	1.4298
BMFO	5.1	1.4491	6.9	1.4491	5.2	0.78881
BGSA	4.4	0.84327	3	1.3333	3.1	0.8756
BPSO	4.4	1.5055	3.5	2.273	3.5	0.70711
BBA	4.2	1.1353	6.1	1.8529	6.6	2.1705
BSSA	4.2	1.0328	5.5	2.3214	4.3	0.82327
BRUN	4.1	1.1972	5.7	1.6364	4.2	0.91894

	Overall Rank
Rank	+/=/−	ARV
BRRWOA	1	∼	1.8667
BWOA	6	14/1/0	5.5333
BMFO	8	15/0/0	7.2667
BGSA	2	8/1/6	2.4667
BPSO	3	11/1/3	2.9333
BBA	4	11/0/4	4.0667
BSSA	7	15/0/0	5.6667
BRUN	5	14/1/0	4.7333

Table A13.

Comparison between the developed BRRWOA and the other seven metaheuristics on average running time.

	Breastcancer		Exactly		HeartEW
	Avg	Std	Avg	Std	Avg	Std
BRRWOA	4.8288E+02	3.4246E+01	7.8822E+02	6.9285E+01	4.8110E+02	5.0225E+01
BWOA	4.4163E+00	7.2395E−02	5.7078E+00	8.0595E−02	3.5502E+00	9.7552E−02
BMFO	4.4684E+00	6.4434E−02	5.7042E+00	1.2365E−01	3.5864E+00	6.7677E−02
BGSA	4.2825E+00	1.0829E−01	5.4809E+00	1.3541E−01	3.6261E+00	7.5790E−02
BPSO	4.1845E+00	1.1827E−01	5.4355E+00	8.8046E−02	3.4783E+00	7.8602E−02
BBA	4.2999E+00	6.9620E−02	5.6576E+00	1.0451E−01	3.6704E+00	7.8690E−02
BSSA	4.4980E+00	6.0707E−02	5.6909E+00	1.0295E−01	3.5478E+00	1.0112E−01
BRUN	7.7342E+00	1.2670E−01	9.7796E+00	2.2226E−01	6.2649E+00	1.5043E−01
	M-of-n		Congress		Hepatitis
Avg	Std	Avg	Std	Avg	Std
BRRWOA	1.1007E+03	9.1259E+01	8.2443E+02	8.4310E+01	7.9500E+02	1.0296E+02
BWOA	7.8536E+00	4.7918E−02	5.1741E+00	6.6221E−02	4.6589E+00	8.7735E−02
BMFO	7.8986E+00	7.8660E−02	5.0875E+00	5.5201E−02	4.7140E+00	4.9113E−02
BGSA	7.6942E+00	6.5451E−02	5.2564E+00	5.1817E−02	4.9596E+00	5.0390E−02
BPSO	7.6050E+00	9.2382E−02	5.1379E+00	9.1856E−02	4.7794E+00	3.4006E−02
BBA	7.8485E+00	7.3415E−02	5.4118E+00	8.1106E−02	5.2570E+00	6.1249E−02
BSSA	7.8888E+00	7.6709E−02	5.2872E+00	1.0298E−01	5.0874E+00	7.1330E−02
BRUN	1.3572E+01	1.8872E−01	9.0089E+00	1.7926E−01	8.2244E+00	1.3622E−01
	USA		Wine		cmc
Avg	Std	Avg	Std	Avg	Std
BRRWOA	1.5446E+03	1.7394E+02	4.6975E+02	4.4347E+01	3.5217E+02	4.8671E+01
BWOA	1.1913E+01	4.3582E−01	3.3713E+00	7.4340E−02	3.2700E+00	3.0743E−01
BMFO	1.6829E+01	1.7231E+00	3.3036E+00	7.4004E−02	4.9767E+00	1.4788E−01
BGSA	1.4433E+01	1.8582E+00	3.3236E+00	8.5184E−02	3.1125E+00	3.8437E−01
BPSO	1.4515E+01	1.6283E+00	3.2616E+00	1.0127E−01	3.0154E+00	2.5567E−01
BBA	1.3122E+01	1.2399E+00	3.3927E+00	8.2471E−02	3.1942E+00	2.4796E−01
BSSA	1.2714E+01	8.1199E−01	3.4083E+00	1.0760E−01	3.4287E+00	3.4324E−01
BRUN	1.9878E+01	7.4155E−01	5.8057E+00	9.9487E−02	5.5483E+00	5.2286E−01
	Lymphography		Vehicle		Heart
Avg	Std	Avg	Std	Avg	Std
BRRWOA	5.0775E+02	5.7672E+01	1.1254E+03	1.8560E+02	4.5566E+02	2.0712E+01
BWOA	3.2480E+00	1.1851E−01	1.2136E+01	2.3665E+00	3.5791E+00	4.3738E−02
BMFO	3.1498E+00	7.6017E−02	1.0975E+01	1.8227E+00	3.5607E+00	1.1353E−01
BGSA	3.3425E+00	9.0822E−02	9.5969E+00	2.1691E+00	3.5291E+00	7.0872E−02
BPSO	3.2413E+00	5.6365E−02	1.0015E+01	1.4998E+00	3.4345E+00	7.1530E−02
BBA	3.3801E+00	7.5562E−02	1.1844E+01	2.3910E+00	3.5986E+00	7.4721E−02
BSSA	3.2567E+00	1.1769E−01	1.0592E+01	2.8853E+00	3.4808E+00	6.3590E−02
BRUN	5.6945E+00	7.9012E−02	1.7535E+01	3.6745E+00	6.2025E+00	1.5762E−01
	Glass		Vote		Zoo
Avg	Std	Avg	Std	Avg	Std
BRRWOA	5.5436E+02	4.7003E+01	5.9493E+02	4.8700E+01	6.0001E+02	3.3711E+01
BWOA	4.6550E+00	7.0043E−02	3.5829E+00	6.5023E−02	3.3851E+00	1.1226E−01
BMFO	5.0870E+00	6.6798E−02	3.4803E+00	9.9217E−02	3.3336E+00	4.9957E−02
BGSA	5.0492E+00	4.8855E−02	3.6748E+00	1.1307E−01	3.3265E+00	7.6105E−02
BPSO	4.7891E+00	6.1560E−02	3.5296E+00	8.7849E−02	3.2492E+00	1.0189E−01
BBA	4.7285E+00	7.4549E−02	3.9203E+00	9.2443E−02	3.3860E+00	9.0316E−02
BSSA	4.6944E+00	6.0691E−02	3.7493E+00	1.4929E−01	3.4048E+00	6.7020E−02
BRUN	7.7956E+00	1.3153E−01	6.0882E+00	1.6609E−01	5.9267E+00	1.8925E−01

	Overall Rank
Rank	+/=/−	ARV
BRRWOA	8	∼	8
BWOA	3	0/0/15	3.4667
BMFO	4	0/0/15	3.7333
BGSA	2	0/0/15	3.2
BPSO	1	0/0/15	1.8667
BBA	6	0/0/15	4.6
BSSA	5	0/0/15	4.1333
BRUN	7	0/0/15	7

Table A14.

Comparison between the proposed BRRWOA classification method and the other six classification methods on average fitness values.

	Breastcancer		Exactly		HeartEW
	Avg	Std	Avg	Std	Avg	Std
BRRWOA	2.1913E−02	4.5553E−03	2.3077E−02	0.0000E+00	7.4316E−02	4.0325E−02
BACWOA	3.0651E−02	1.2209E−02	2.4615E−02	1.9861E−03	9.1852E−02	3.6663E−02
BCWOAI	3.4190E−02	9.3385E−03	2.4615E−02	1.9861E−03	8.5969E−02	4.9875E−02
BIWOA	4.0813E−02	1.7253E−02	4.7152E−02	1.8728E−02	1.4490E−01	3.4875E−02
BLWOA	3.4500E−02	1.3678E−02	2.9796E−02	4.1466E−03	1.0587E−01	2.3418E−02
BMWOA	2.8964E−02	1.2797E−02	2.6154E−02	2.4325E−03	1.0554E−01	3.0462E−02
BRDWOA	3.0611E−02	1.3441E−02	2.3462E−02	1.2163E−03	8.9430E−02	4.1881E−02
	M-of-n		Congress		Hepatitis
Avg	Std	Avg	Std	Avg	Std
BRRWOA	2.3077E−02	0.0000E+00	1.6534E−02	9.0144E−03	1.3684E−02	2.7179E−03
BACWOA	2.5950E−02	6.5799E−03	2.3737E−02	1.5043E−02	3.9805E−02	3.0781E−02
BCWOAI	2.4231E−02	1.8579E−03	2.6527E−02	1.1840E−02	2.7693E−02	1.5883E−02
BIWOA	5.1541E−02	1.7668E−02	4.8151E−02	1.7646E−02	7.9013E−02	5.2446E−02
BLWOA	2.9412E−02	4.2239E−03	3.1962E−02	1.6137E−02	4.4920E−02	3.1823E−02
BMWOA	2.5385E−02	2.6893E−03	2.6573E−02	1.4887E−02	2.6596E−02	1.7793E−02
BRDWOA	2.4231E−02	1.8579E−03	2.4652E−02	1.3464E−02	2.6860E−02	1.9700E−02
	USA		Wine		cmc
Avg	Std	Avg	Std	Avg	Std
BRRWOA	2.4096E−01	1.7129E−02	1.0000E−02	1.9861E−03	4.6824E−01	2.1952E−02
BACWOA	2.4765E−01	2.5854E−02	1.2692E−02	3.6488E−03	4.7356E−01	2.2384E−02
BCWOAI	2.4957E−01	1.0353E−02	1.4615E−02	3.5344E−03	4.7052E−01	1.3611E−02
BIWOA	2.8498E−01	1.2670E−02	2.4231E−02	4.4596E−03	4.8309E−01	2.0138E−02
BLWOA	2.5027E−01	1.6346E−02	1.8462E−02	3.0339E−03	4.7013E−01	1.7919E−02
BMWOA	2.5225E−01	1.8989E−02	1.3077E−02	4.1345E−03	4.7074E−01	2.9814E−02
BRDWOA	2.5031E−01	2.7539E−02	1.3077E−02	3.2434E−03	4.7027E−01	2.7125E−02
	Lymphography		Vehicle		Heart
Avg	Std	Avg	Std	Avg	Std
BRRWOA	1.4722E−02	6.1489E−03	1.9293E−01	2.5658E−02	6.5299E−02	3.0909E−02
BACWOA	3.3500E−02	4.2053E−02	2.1722E−01	4.1461E−02	8.1296E−02	1.8088E−02
BCWOAI	2.5778E−02	2.1426E−02	2.0278E−01	2.8582E−02	8.3989E−02	2.4866E−02
BIWOA	7.2341E−02	4.2882E−02	2.3187E−01	1.2417E−02	1.4726E−01	6.8206E−02
BLWOA	5.5757E−02	4.3895E−02	2.2707E−01	1.7500E−02	9.6524E−02	4.1873E−02
BMWOA	4.1748E−02	3.1327E−02	2.2923E−01	1.9081E−02	9.0641E−02	3.0948E−02
BRDWOA	2.5310E−02	1.7190E−02	2.1442E−01	2.0336E−02	7.9701E−02	3.4783E−02
	Glass		Vote		Zoo
Avg	Std	Avg	Std	Avg	Std
BRRWOA	1.0552E−01	5.0602E−02	1.8208E−02	1.7401E−02	1.0625E−02	1.6137E−03
BACWOA	1.2515E−01	5.3980E−02	2.6111E−02	2.0262E−02	1.1875E−02	2.8717E−03
BCWOAI	1.3536E−01	7.0241E−02	2.7042E−02	2.6617E−02	1.4375E−02	3.3593E−03
BIWOA	1.5243E−01	5.7766E−02	5.9222E−02	3.7555E−02	2.1562E−02	5.9748E−03
BLWOA	1.6165E−01	5.7317E−02	3.7613E−02	2.2606E−02	1.8125E−02	4.6117E−03
BMWOA	1.1409E−01	4.7424E−02	3.1051E−02	2.4758E−02	1.1562E−02	3.3105E−03
BRDWOA	1.4199E−01	4.1326E−02	3.0839E−02	3.4304E−02	1.1875E−02	3.8415E−03

	Overall Rank
Rank	+/=/−	ARV
BRRWOA	1	∼	1
BACWOA	3	15/0/0	3.4667
BCWOAI	4	15/0/0	3.5333
BIWOA	7	15/0/0	6.9333
BLWOA	6	15/0/0	5.4667
BMWOA	5	15/0/0	4.0667
BRDWOA	2	15/0/0	2.8667

Table A15.

Comparison between the developed BRRWOA and the other six heuristics on the average KNN error values.

	Breastcancer		Exactly		HeartEW
	Avg	Std	Avg	Std	Avg	Std
BRRWOA	1.4286E−03	4.5175E−03	0.0000E+00	0.0000E+00	5.5556E−02	4.3649E−02
BACWOA	1.0042E−02	9.7320E−03	0.0000E+00	0.0000E+00	7.0370E−02	4.0759E−02
BCWOAI	1.1429E−02	1.1269E−02	0.0000E+00	0.0000E+00	6.2963E−02	5.2524E−02
BIWOA	1.4306E−02	1.5059E−02	1.6031E−02	1.7879E−02	1.1852E−01	3.8252E−02
BLWOA	1.0000E−02	1.5134E−02	1.0000E−03	3.1623E−03	8.1481E−02	2.3424E−02
BMWOA	1.0021E−02	1.3560E−02	0.0000E+00	0.0000E+00	8.5185E−02	3.5136E−02
BRDWOA	1.0000E−02	9.6421E−03	0.0000E+00	0.0000E+00	6.2963E−02	4.2945E−02
	M-of-n		Congress		Hepatitis
Avg	Std	Avg	Std	Avg	Std
BRRWOA	0.0000E+00	0.0000E+00	2.2727E−03	7.1870E−03	0.0000E+00	0.0000E+00
BACWOA	1.0000E−03	3.1623E−03	9.1966E−03	1.6161E−02	2.5833E−02	3.3380E−02
BCWOAI	0.0000E+00	0.0000E+00	6.8710E−03	1.1064E−02	5.8824E−03	1.8602E−02
BIWOA	1.9031E−02	1.5308E−02	1.8449E−02	1.8156E−02	5.7132E−02	5.5974E−02
BLWOA	1.0000E−03	3.1623E−03	9.3023E−03	1.6261E−02	1.9583E−02	3.1553E−02
BMWOA	0.0000E+00	0.0000E+00	1.1525E−02	1.2152E−02	6.6667E−03	2.1082E−02
BRDWOA	0.0000E+00	0.0000E+00	6.8710E−03	1.1064E−02	6.6667E−03	2.1082E−02
	USA		Wine		cmc
Avg	Std	Avg	Std	Avg	Std
BRRWOA	2.3416E−01	2.0603E−02	0.0000E+00	0.0000E+00	4.6365E−01	2.1739E−02
BACWOA	2.4016E−01	2.7729E−02	0.0000E+00	0.0000E+00	4.6982E−01	2.6158E−02
BCWOAI	2.4271E−01	1.1066E−02	0.0000E+00	0.0000E+00	4.6371E−01	1.5133E−02
BIWOA	2.7524E−01	1.2588E−02	0.0000E+00	0.0000E+00	4.7460E−01	1.9497E−02
BLWOA	2.3712E−01	1.9022E−02	0.0000E+00	0.0000E+00	4.6505E−01	1.9842E−02
BMWOA	2.4657E−01	2.1560E−02	0.0000E+00	0.0000E+00	4.6570E−01	3.2018E−02
BRDWOA	2.4191E−01	2.9912E−02	0.0000E+00	0.0000E+00	4.6227E−01	3.1448E−02
	Lymphography		Vehicle		Heart
Avg	Std	Avg	Std	Avg	Std
BRRWOA	0.0000E+00	0.0000E+00	1.7852E−01	2.4806E−02	4.4444E−02	3.8252E−02
BACWOA	1.3333E−02	4.2164E−02	1.9737E−01	4.3712E−02	5.9259E−02	1.9126E−02
BCWOAI	6.6667E−03	2.1082E−02	1.8099E−01	2.9318E−02	5.9259E−02	2.5897E−02
BIWOA	4.0476E−02	4.6940E−02	2.0811E−01	1.3743E−02	1.2222E−01	7.2093E−02
BLWOA	3.2083E−02	4.4879E−02	2.1036E−01	2.0085E−02	7.4074E−02	4.2767E−02
BMWOA	1.8799E−02	3.0326E−02	2.1030E−01	2.1041E−02	6.6667E−02	2.9215E−02
BRDWOA	5.8824E−03	1.8602E−02	1.9501E−01	1.8273E−02	5.5556E−02	4.0005E−02
	Glass		Vote		Zoo
Avg	Std	Avg	Std	Avg	Std
BRRWOA	8.8263E−02	5.3127E−02	6.6667E−03	1.4055E−02	0.0000E+00	0.0000E+00
BACWOA	1.0776E−01	5.6253E−02	1.3341E−02	1.7231E−02	0.0000E+00	0.0000E+00
BCWOAI	1.1733E−01	7.1903E−02	1.3333E−02	2.8109E−02	0.0000E+00	0.0000E+00
BIWOA	1.2946E−01	6.2308E−02	3.6681E−02	3.3414E−02	0.0000E+00	0.0000E+00
BLWOA	1.4501E−01	6.1054E−02	1.6567E−02	2.3164E−02	0.0000E+00	0.0000E+00
BMWOA	9.5533E−02	5.2074E−02	1.6567E−02	2.3164E−02	0.0000E+00	0.0000E+00
BRDWOA	1.2139E−01	4.3186E−02	1.6344E−02	3.5180E−02	0.0000E+00	0.0000E+00

	Overall Rank
Rank	+/=/−	ARV
BRRWOA	1	∼	1.0667
BACWOA	4	12/3/0	3.2667
BCWOAI	3	11/4/0	2.4667
BIWOA	7	13/2/0	6
BLWOA	6	13/2/0	4.4
BMWOA	5	11/4/0	3.6667
BRDWOA	2	10/4/1	2.2667

Table A16.

Comparison between the developed BRRWOA and the other six metaheuristics on the average value of the selected features.

	Breastcancer		Exactly		HeartEW
	Avg	Std	Avg	Std	Avg	Std
BRRWOA	3.7	0.67495	6	0	5.6	1.3499
BACWOA	3.8	1.1353	6.4	0.5164	6.5	1.6499
BCWOAI	4.2	1.0328	6.4	0.5164	6.8	1.0328
BIWOA	4.9	1.7288	8.3	0.48305	8.4	1.6465
BLWOA	4.5	0.84984	7.5	0.52705	7.4	1.075
BMWOA	3.5	1.354	6.8	0.63246	6.4	1.4298
BRDWOA	3.8	1.1353	6.1	0.31623	7.7	1.3375
	M-of-n		Congress		Hepatitis
Avg	Std	Avg	Std	Avg	Std
BRRWOA	6	0	4.6	1.7127	5.2	1.0328
BACWOA	6.5	0.97183	4.8	1.5492	5.8	1.4757
BCWOAI	6.3	0.48305	6.4	2.2211	8.4	2.7162
BIWOA	8.7	0.94868	9.8	2.2509	9.4	2.1705
BLWOA	7.4	0.5164	7.4	1.2649	10	1.0541
BMWOA	6.6	0.69921	5	2.0548	7.7	1.4944
BRDWOA	6.3	0.48305	5.8	2.4855	7.8	1.5492
	USA		Wine		cmc
Avg	Std	Avg	Std	Avg	Std
BRRWOA	3.7	0.94868	2.6	0.5164	5	0.94281
BACWOA	3.9	0.99443	3.3	0.94868	4.9	0.99443
BCWOAI	3.8	0.63246	3.8	0.91894	5.4	0.69921
BIWOA	4.7	1.0593	6.3	1.1595	5.8	0.91894
BLWOA	5	1.0541	4.8	0.78881	5.1	0.8756
BMWOA	3.6	0.96609	3.4	1.075	5.1	0.99443
BRDWOA	4.1	0.8756	3.4	0.84327	5.6	1.1738
	Lymphography		Vehicle		Heart
Avg	Std	Avg	Std	Avg	Std
BRRWOA	5.3	2.2136	8.4	1.8974	6	1.9437
BACWOA	7.5	2.4608	10.7	1.3375	6.5	1.354
BCWOAI	7	1.5635	11.1	1.4491	7.2	1.1353
BIWOA	12.2	1.3984	12.3	1.4181	8.1	2.0248
BLWOA	9.1	1.5239	9.8	1.3984	6.8	0.91894
BMWOA	8.6	2.3664	10.6	2.9515	7.1	1.4491
BRDWOA	7.1	2.6854	10.5	2.3214	7	1.5635
	Glass		Vote		Zoo
Avg	Std	Avg	Std	Avg	Std
BRRWOA	3.9	0.8756	3.8	1.6865	3.4	0.5164
BACWOA	4.1	0.99443	4.3	2.0575	3.8	0.91894
BCWOAI	4.3	0.94868	4.6	2.2211	4.6	1.075
BIWOA	5.3	1.0593	7.8	2.5298	6.9	1.912
BLWOA	4.3	0.67495	7	1.6997	5.8	1.4757
BMWOA	4.2	0.91894	4.9	3.1073	3.7	1.0593
BRDWOA	4.8	0.78881	4.9	1.7288	3.8	1.2293

	Overall Rank
Rank	+/=/−	ARV	ARV
BRRWOA	1	∼	1.2	1.2
BACWOA	2	14/0/1	2.8	2.8
BCWOAI	5	15/0/0	4.2	4.2
BIWOA	7	15/0/0	6.8667	6.8667
BLWOA	6	15/0/0	5.2667	5.2667
BMWOA	3	13/0/2	3.2667	3.2667
BRDWOA	4	15/0/0	3.8667	3.8667

Table A17.

Comparison between the developed BRRWOA and the other six metaheuristics on average running time.

	Breastcancer		Exactly		HeartEW
	Avg	Std	Avg	Std	Avg	Std
BRRWOA	1.1129E+03	1.1710E+02	3.5346E+02	4.0515E+01	2.1275E+02	2.8438E+01
BACWOA	1.3227E+01	4.3954E−01	2.8217E+00	2.1377E−01	1.5739E+00	1.5203E−01
BCWOAI	1.3149E+01	3.4866E−01	2.8226E+00	2.5309E−01	1.6020E+00	1.4531E−01
BIWOA	2.4491E+00	1.1551E−01	4.6994E−01	6.3931E−02	2.5870E−01	2.1269E−02
BLWOA	2.6627E+01	7.1513E−01	5.3085E+00	4.1770E−01	3.1590E+00	2.5877E−01
BMWOA	1.2995E+01	5.1829E−01	2.6011E+00	2.1679E−01	1.5790E+00	1.4249E−01
BRDWOA	2.6045E+01	6.3087E−01	5.1657E+00	3.7665E−01	3.0856E+00	2.6156E−01
	M-of-n		Congress		Hepatitis
Avg	Std	Avg	Std	Avg	Std
BRRWOA	3.7361E+02	2.6047E+01	2.8124E+02	3.5578E+01	2.5021E+02	3.3931E+01
BACWOA	2.6261E+00	2.0858E−01	1.8054E+00	1.3806E−01	1.4337E+00	1.2472E−01
BCWOAI	2.6470E+00	2.4616E−01	1.7074E+00	1.6702E−01	1.4279E+00	1.3874E−01
BIWOA	4.4828E−01	2.7106E−02	2.7727E−01	2.1480E−02	2.3845E−01	2.1171E−02
BLWOA	5.3787E+00	3.9964E−01	3.3804E+00	2.5015E−01	2.7958E+00	2.0243E−01
BMWOA	2.5935E+00	2.4007E−01	1.7312E+00	1.6763E−01	1.4380E+00	1.3838E−01
BRDWOA	5.1772E+00	3.5826E−01	3.3332E+00	2.5102E−01	2.8316E+00	2.8026E−01
	USA		Wine		cmc
Avg	Std	Avg	Std	Avg	Std
BRRWOA	5.1325E+02	5.6713E+01	2.2343E+02	2.3687E+01	9.2809E+02	1.0393E+02
BACWOA	4.3911E+00	3.4431E−01	1.5256E+00	1.5247E−01	7.3527E+00	1.7375E−01
BCWOAI	4.4998E+00	4.3889E−01	1.5326E+00	1.4211E−01	7.5257E+00	3.7262E−01
BIWOA	7.4624E−01	1.0743E−01	2.7504E−01	2.0800E−02	1.2933E+00	7.5342E−02
BLWOA	9.2590E+00	1.0203E+00	3.0221E+00	2.7645E−01	1.5190E+01	3.7789E−01
BMWOA	4.4857E+00	3.8324E−01	1.5088E+00	1.4295E−01	7.3627E+00	2.9984E−01
BRDWOA	9.1145E+00	9.3548E−01	3.0219E+00	2.4343E−01	1.7413E+01	3.6184E+00
	Lymphography		Vehicle		Heart
Avg	Std	Avg	Std	Avg	Std
BRRWOA	2.3862E+02	3.6493E+01	3.6044E+02	4.3340E+01	2.1485E+02	3.5486E+01
BACWOA	1.5152E+00	1.5439E−01	2.4548E+00	2.0626E−01	1.6958E+00	1.6736E−01
BCWOAI	1.5138E+00	1.3668E−01	2.4032E+00	1.9320E−01	1.6998E+00	1.7965E−01
BIWOA	2.5218E−01	3.1938E−02	3.8854E−01	4.5252E−02	2.8063E−01	3.6472E−02
BLWOA	3.0224E+00	2.9980E−01	4.9157E+00	3.7110E−01	3.3533E+00	3.3559E−01
BMWOA	1.5514E+00	1.6079E−01	2.4388E+00	1.7041E−01	1.6905E+00	1.4060E−01
BRDWOA	3.0168E+00	2.7915E−01	4.7644E+00	3.4224E−01	3.3282E+00	3.4401E−01
	Glass		Vote		Zoo
Avg	Std	Avg	Std	Avg	Std
BRRWOA	1.9806E+02	1.9847E+01	2.5311E+02	2.7839E+01	2.7837E+02	3.8312E+01
BACWOA	1.7639E+00	1.5968E−01	1.5864E+00	1.3360E−01	1.5375E+00	1.6094E−01
BCWOAI	1.7539E+00	1.5451E−01	1.5538E+00	1.3643E−01	1.5472E+00	1.5711E−01
BIWOA	2.9723E−01	3.4202E−02	2.6028E−01	2.4319E−02	2.3999E−01	1.8983E−02
BLWOA	3.6098E+00	3.3084E−01	3.0982E+00	3.0341E−01	3.1513E+00	2.9678E−01
BMWOA	1.7604E+00	1.7277E−01	1.5724E+00	1.3577E−01	1.5866E+00	1.4268E−01
BRDWOA	3.4577E+00	3.7564E−01	3.1558E+00	3.1858E−01	3.1061E+00	2.8832E−01

	Overall Rank
Rank	+/=/−	ARV
BRRWOA	7	∼	7
BACWOA	4	0/0/15	2.9333
BCWOAI	2	0/0/15	2.8667
BIWOA	1	0/0/15	1
BLWOA	6	0/0/15	5.6667
BMWOA	2	0/0/15	2.8667
BRDWOA	5	0/0/15	5.2

Data availability

Data will be made available on request.