Boosted Reptile Search Algorithm for Engineering and Optimization Problems

Abstract

Recently, various metaheuristic (MH) optimization algorithms have been presented and applied to solve complex engineering and optimization problems. One main category of MH algorithms is the naturally inspired swarm intelligence (SI) algorithms. SI methods have shown great performance on different problems. However, individual MH and SI methods face some shortcomings, such as trapping at local optima. To solve this issue, hybrid SI methods can perform better than individual ones. In this study, we developed a boosted version of the reptile search algorithm (RSA) to be employed for different complex problems, such as intrusion detection systems (IDSs) in cloud–IoT environments, as well as different optimization and engineering problems. This modification was performed by employing the operators of the red fox algorithm (RFO) and triangular mutation operator (TMO). The aim of using the RFO was to boost the exploration of the RSA, whereas the TMO was used for enhancing the exploitation stage of the RSA. To assess the developed approach, called RSRFT, a set of six constrained engineering benchmarks was used. The experimental results illustrated the ability of RSRFT to find the solution to those tested engineering problems. In addition, it outperformed the other well-known optimization techniques that have been used to handle these problems.

1. Introduction

In daily life, optimization is everywhere. Optimization is used widely in many fields, including system management, technical design, economics, and various engineering problems [1,2]. To optimize something means to make sure one or more goals of a particular situation are as ideal as possible. The goal of the optimization process is to investigate potential solutions and find the optimal one. Occasionally, many constraints are applied to solutions, making optimization more difficult [3].

In recent decades, the effectiveness of metaheuristic (MH) techniques in solving complicated problems with high dimensionality, multimodality, and non-differentiability has been proven by different studies in different applications [4,5,6]. As a result, the use of these algorithms has increased significantly, and there is a growing trend to suggest efficient upgrades and new metaheuristic algorithms. These algorithms primarily gain their inspiration from natural phenomena, physics laws, animals’ and birds’ behaviors, and so on. Earlier, some efficient optimization algorithms were developed, such as the genetic algorithm (GA) [7], artificial bee colony (ABC) [8], particle swarm optimization (PSO) [9], and the firefly algorithm [10]. Recently, there have been many newly developed metaheuristic optimization algorithms based on different inspirations, for example the sine–cosine algorithm (SCA) [11], the multi-verse optimizer (MVO) [12], Harris hawks optimization (HHO) [13], the marine predator algorithm [14], the Aquila optimizer [15], the arithmetic optimization algorithm [16], the Reptile search algorithm (RSA) [17], and many other optimization techniques.

Although those algorithms have shown good performance in different optimization and engineering algorithms, in some cases, they face critical limitations and shortcomings. According to the well-known no free lunch theorem, no one method has the ability of solving all problems. To this end, in recent years, researchers have used the hybridization concept to develop hybrid metaheuristic optimization algorithms to address different engineering and optimization issues.

For example, Houssein et al. [3] applied an improved equilibrium optimizer (EO) using a technique called the self-adaptive EO algorithm. They used four search techniques to enhance the search process of the EO optimization method. This method was employed to address different optimization engineering problems, and it showed better results compared to the original EO. Bo et al. [18] applied several search techniques, namely opposition-based learning and greedy search, to boost the chimp optimization algorithm (ChOA) search mechanism. The developed ChOA was employed to solve different constrained engineering problems and recorded significant outcomes. Shen et al. [19] introduced an enhanced whale optimization algorithm (WOA) to handle engineering design problems and global optimization issues. They used a technique called multi-population evolution to improve the original WOA. A modified SCA was developed by Yang et al. [20]. They suggested a multi-mechanism acting SCA method to overcome the drawbacks of the original SCA. This modified version of the SCA was utilized to address constrained engineering optimization problems. An improved SSA optimization technique was developed by Hong et al. [21] to solve engineering problems, using a new strategy called producer centralization. Furthermore, in [22], a modified improved moth–flame optimizer was suggested using two search mechanisms, namely Gaussian mutation and chaotic grouping. This modified method was evaluated with complex engineering and optimization problems. An enhanced FA was suggested by Peng et al. [23] to solve complex engineering optimization problems. They applied three search techniques to enhance the traditional FA. Zhang et al. [24] presented a modified AOA to be employed in various numerical and engineering optimization applications based on two techniques, called random high-speed jumping and multi-leader wandering around search strategy. Hybrid optimization algorithms have also been adopted in other research areas, such as a hybrid of AO and the seagull optimization algorithm for wind power forecasting [25] and a hybrid of AO with the SMA and SSO to predict CO₂ trapping in deep saline [26,27]. Another hybrid optimization method was developed by [28] using DE and the Cuckoo Search algorithm to optimize the ship pipe route. Moreover, a hybrid method of HHO and DE was suggested by Zhong et al. to solve the flight trajectory prediction problem [29].

In recent years, MH optimization techniques also have been investigated to address the problems of intrusion detection systems. Many optimization algorithms have been suggested as feature selection methods to improve the quality solutions of the IDS, such as PSO [30], the GA [31], AO [32], HHO [33], and many other approaches [1,2].

Paper Contribution

Inspired by the successful applications of the hybrid concept to overcome the drawbacks of the individual optimization algorithms, in this study, we present a new version of the reptile search algorithm using two techniques, namely the red fox optimization algorithm and the mutation operators. The RSA [17] is a recently developed optimization method emulating crocodiles’ hunting behaviors in nature. It has been adopted in various applications since it has high performance, such as in [34,35,36]. The red fox optimization algorithm [37] is also a nature-inspired technique based on the hunting behavior of the red fox. It has also been adopted in solving some optimization problems [38,39,40].

The main modification was performed using RFO to enhance the ability of the RSA to explore the search domain. Additionally, the triangular mutation operator (TMO) was applied to improve the exploitation of the RSA. This competition between the operators of the RSA, TMO, and RFO led to enhancing the rate of convergence toward the optimal solution.

In short, we can represent the contributions of this study as the following points:

• We propose a boosted version of the reptile search algorithm (RSA), called RSRFT, to address IDS problems in IoT and cloud environments, as well as complex and multidimensional engineering problems
• We employed the operators of the red fox optimization and triangular mutation operator to boost the performance of the RSA.
• We applied the RSRFT technique to solve different and complex engineering problems. We conducted a set of comparisons with other efficient techniques to verify the quality of RSRFT.

The rest of this study is organizes as follows: In Section 2, detailed background descriptions of the applied algorithms are presented. Section 3 introduces the descriptions of the RSRFT approach. In Section 4, we assess the quality of the modified RSA method using different datasets and with extensive comparisons to other methods, and in Section 5, we present the conclusion of this paper.

2. Background

2.1. The Reptile Search Algorithm

The conventional RSA, which simulates actual crocodile behavior, is presented in this section. This was accomplished using two steps: the local and global search stages.

2.1.1. Exploration Search

Following [17], the process of splitting the maximum number of generations into four parts will trigger the RSA to switch from the exploration to the exploitation stage. Moreover, the RSA exploration phase investigates the search domains using two primary search techniques in pursuit of a more reliable agent. There are two requirements for this search component. If the long walking plan is not adjusted by t ≤ $\frac{T}{4}$, the agents are modified by t ≤ 2$\frac{T}{4}$ and t > $\frac{T}{4}$. Equation (1) can be utilized to update the positions for the exploration stage:

$$x_{ij}(t+1)=\left\{\begin{array}{cc}Bes{t}_j\left(t\right)\times -{\eta }_{ij}\left(t\right)\times \beta -R_{ij}\left(t\right)\times rand,& t\le \frac{T}{4}\\ Bes{t}_j\left(t\right)\times x_{(r_1,j)}\times ES\left(t\right)\times rand,& t\le 2\frac{T}{4}andt>\frac{T}{4}\end{array}\right.$$

(1)

where $Bes{t}_j$ stands for the best solution at dimension j. $x_{(r_1,j)}$ denotes the value of a randomly selected solution at dimension j. $rand$ represents a random value, and $\beta =0.1$ as in [17]. T represents the total number of generations. N and ${\eta }_{ij}$ are the number of solutions and the hunting operator. The definition of ${\eta }_{ij}$ is given as

$${\eta }_{ij}=Bes{t}_j\left(t\right)\times P_{ij},$$

(2)

$$R_{ij}=\frac{Bes{t}_j\left(t\right)-x_{(r_2,j)}}{Bes{t}_j\left(t\right)+ϵ},$$

(3)

$$ES\left(t\right)=2\times r_3\times \left(1-\frac{1}{T}\right),$$

(4)

In Equation (4), $ϵ$ refers to a small value. $r_3\in [-1,1]$ is a random integer value, while $P_{ij}$ is defined as

$$P_{ij}=\alpha +\frac{x_{ij}-M\left(x_i\right)}{Bes{t}_j\left(t\right)\times (U{B}_{\left(j\right)}-L{B}_{\left(j\right)})+ϵ},$$

(5)

In Equation (5), $U{B}_{\left(j\right)}$ and $L{B}_{\left(j\right)}$ represent the boundaries of the search domain, and $\alpha =0.1$ [17]. M($x_i$) denotes the mean value of X, and it is given as

$$M\left(x_i\right)=\frac{1}{n}\sum _{j=1}^n{x}_{ij},$$

(6)

2.1.2. Exploitation Search

Using the hunting coordination plan, t ≤ 3$\frac{T}{4}$ and t > $2\frac{T}{4}$ adopt the agents in this stage. In contrast, if t ≤ T and t > 3$\frac{T}{4}$, the hunting cooperation plan is carried out. Equation (7) provides the updated value using the exploitation process.

$$x_{ij}(t+1)=\left\{\begin{array}{cc}Bes{t}_j\left(t\right)\times P_{ij}\left(t\right)\times rand,& t\le 3\frac{T}{4}andt>2\frac{T}{4}\\ Bes{t}_j\left(t\right)-{\eta }_{ij}\left(t\right)\times ϵ-R_{ij}\left(t\right)\times rand,& t\le Tandt>3\frac{T}{4}\end{array}\right.$$

(7)

2.2. Red Fox Algorithm

This optimizer begins by figuring out the parameters’ values and then creating the population of N foxes according to Equation (8):

$$X_{ij}=L{B}_j+r\times \left(U{B}_j-L{B}_j\right),i=1,\dots ,N,j=1,\dots ,D,r\in [0,1]$$

(8)

where $L{B}_j$ and $U{B}_j$ indicate the boundaries of the search space, while D represents the dimension of $X_{ij}$ foxes. Thereafter, the objective value of $X_i$ is evaluated then allocating the best solution $X_b$. The next process is to update $X_{ir}$ using the following formula:

$$X_{ir}=X_i+asign\left(X_b-X_i\right)$$

(9)

In Equation (9), $a\in [0,d_{ib}]$ stands for a scaling hyperparameter chosen randomly, whereas $d_{ib}$ is defined as

$$d_{ib}=\sqrt{\left|\right|X_i-X_b\left|\right|}$$

(10)

In case $X_{ir}$ has a fitness value better than its previous value, then we replace $X_i$ with $X_{ir}$. Otherwise, we preserved the $X_i$ value. The observation radius r is then adjusted by applying the formula Equation (11) in the case of the fox not being seen:

$$r=\left\{\begin{array}{c}a\frac{\sin \left({\varphi }_0\right)}{{\varphi }_0}if{\varphi }_0\ne 0\hfill \\ \theta otherwise\hfill \end{array}\right.$$

(11)

In Equation (11), $\theta \in [0,1]$ is a random value that is applied to control adverse weather conditions such as fog, rain, etc. After that, X is updated based on the following formula:

$$\left\{\begin{array}{c}\begin{array}{c}{x}_0^{new}=ar.\cos \left({\varphi }_1\right)+x_0^{act}\\ \begin{array}{c}{x}_1^{new}=ar.\sin \left({\varphi }_1\right)+ar.\cos \left({\varphi }_2\right)+x_1^{act}\\ x_2^{new}=ar.\sin \left({\varphi }_1\right)+ar.\sin \left({\varphi }_2\right)+ar.\cos \left({\varphi }_3\right)+x_2^{act}\end{array}\end{array}\\ ⋮\\ x_{n-1}^{new}=ar.\sin \left({\varphi }_1\right)+ar.\sin \left({\varphi }_2\right)+\cdots +ar.\sin \left({\varphi }_{n-1}\right)+x_{n-1}^{act}\end{array}\right.$$

(12)

where ${\varphi }_i\in [0,2\pi ],i=1,2,\dots ,n-1$ and each angular value represents a randomized one for each point.

This set of equations simulates a fox’s actions after it spots a target and tries to attack it. Then, the fitness value of $X_i$ is assessed and then sorted based on the values. Hunters may kill the worst solution, while Equation (13) is employed to update the X solution.

$$X=\left\{\begin{array}{c}\mathrm{nomadic}\mathrm{agent}\mathrm{if}>0.5\hfill \\ \mathrm{Reproduction}\mathrm{of}\mathrm{the}\mathrm{alpha}\mathrm{couple}\mathrm{otherwise}\hfill \end{array}\right.$$

(13)

where in the first branch of Equation (13); the new solutions wander outside the environment in search of a new space to repopulate their herd as nomadic agents. Outside of the habitat and within the search zone, the solution is chosen randomly. The center of the habitat ($C_H$) is calculated as

$$C_H=\frac{X_b+X_{\beta }}{2}$$

(14)

In Equation (14), $X_b$ and $X_{\beta }$ denote the first- and second-best solutions, respectively.

In addition, in the second case of Equation (13), the following formula is used to update X.

$$X=\frac{X_b+X_{\beta }}{2}$$

(15)

2.3. Triangular Mutation Operator

The mutation technique utilizes a integration vector, which is constructed using three randomly selected vectors along with three distinct vectors that have been selected during the competition (the best, worst, and better). This combination vector is then utilized to create the new mutant vector.

$$V_i(t+1)=M_1\times (X_b\left(t\right)-{\overline{X}}_{br}\left(t\right))+M_2\times (X_b\left(t\right)-X_w\left(t\right))+M_3\times (X_{br}\left(t\right)-X_w\left(t\right))+{\overline{X}}_c\left(t\right)$$

(16)

In Equation (16), $M_1,M_2,$ and $M_3$ refer to the mutation factors that are related to $x_i$ and they are produced according to the uniform distribution. In addition, three tournaments are given as $X_w\left(t\right)$, $X_{br}\left(t\right)$, $X_b\left(t\right)$. In addition, ${\overline{X}}_c\left(t\right)$ is the combination vector triangle at iteration t and formulated as:

$${\overline{X}}_c=w_1\times X_b+w_2\times X_{br}+w_3\times X_w,w_i⩾0,\sum _{i=1}^3{w}_i=1$$

(17)

where $w_i$ is a real weight and $w_i$ is formulated as $w_i=p_i/{\sum }_{i=1}^3{p}_i$.

Moreover, the mutation is used to balance the exploitation propensity with the exploration capacity, with the primary mutation favoring the exploration phase. As a result, utilizing the presented mutation is twice as likely to succeed as using the fundamental principles. The differential evolution (DE) method is the foundation of the advanced mutation technique. Thus, using the fundamental mutation approach DE/rand/1/bin, the following new combination is created:

If$rand\le (2/3))$, then

$$V_i(t+1)=M_1\times (X_b\left(t\right)-X_{br}\left(t\right))+M_2\times (X_b\left(t\right)-X_w\left(t\right))+M_3\times (X_{br}\left(t\right)-X_w\left(t\right))+{\overline{X}}_c\left(t\right)$$

(18)

Else:

$$V_i(t+1)=X_{r_1}\left(t\right)+F\times (X_{r_1}\left(t\right)-X_{r_3}\left(t\right))$$

(19)

In Equation (18), F is a random value at $\left[(-1,0)\cup (0,1)\right]$ and $rand\in [0,1]$ a random value.

3. Proposed RSRFT Method

Within this section, we introduce the stages of the developed technique, which depends on modifying the performance of the RSA using TMO and RFO as given in Figure 1. The main objective of RFO is to facilitate the identification of a feasible solution within the feasible area, while TMO is applied to promote harmony between the exploitation and exploration. The initial population of RSRFT is constructed inside the search space’s confines. The best solution is then selected from the current population after the fitness function for each solution has been evaluated. The solutions are updated using the RSA, RFO, and TMO operators. The RFO technique will then be used to update a collection of individuals that have reached a local standstill. Until the terminal condition is satisfied, the updating of the population is conducted again. The next subsections offer a detailed explanation of the developed RSRFT approach.

3.1. Initial Phase

The RSRFT creates the N agent starting populations $\left(X_i\right)$ as

$$X_i=rand(1,D)\times (UB-LB)+LB,i=1,\dots ,N$$

(20)

In Equation (20), $rand\in [0,1]$ stands for a random vector with dimension D. N is the number of solutions.

3.2. Updating Phase

The current X is updated by using either the operators of the RSA, RFO, or TMO. This can be conducted through a set of steps. The first step is to calculate the fitness value for $X_i$. The smallest fitness value and its related solution are then identified. Then, according to the probability of the fitness value, either the operators of the RSA or RFO and TMO are used to update X. This probability is computed as

$$P{r}_i=\frac{Fi{t}_i}{{\sum }_{i=1}^{N}Fi{t}_i}$$

(21)

Then, the updating is performed using the following formula:

$$X_i(t+1)=\left\{\begin{array}{cc}{X}_i^{RSA}& ifP{r}_i>r_{pr}\\ X_i^{RT}& Otherwise\end{array}\right.,$$

(22)

where $r_{pr}$ is computed as

$$r_{pr}=min\left(Pr\right)+(max\left(Pr\right)-min\left(Pr\right))\times rand$$

(23)

In addition, the value of $X_i^{RT}$ is computed as

$$X_i(t+1)=\left\{\begin{array}{cc}{X}_i^{TMO}& if rand>0.5\\ X_i^{RFO}& Otherwise\end{array}\right.,$$

(24)

$$X_i^{TMO}=\left\{\begin{array}{cc}UseEquation\left(18\right)toupdate{X}_i& ifrand>0.5\\ UseEquation\left(19\right)toupdate{X}_i& Otherwise\end{array}\right.,$$

(25)

while the values of $X_i^{RFO}$ and $X_i^{RSA}$ refer to the updated value of $X_i$ using the operators of RFO and the RSA, respectively.

3.3. Terminal Phase

The stopping conditionsare examined during this phase, and in case they are satisfied, the steps of updating are halted and $X_b$ is returned. Otherwise, a new updating step is carried out.

The main steps of the proposed RSRFT are given in Algorithm 1.

Algorithm 1 The RSRFT method.

1: Initialize the parameters as N solutions.   2: Construct initial set of X solutions using Equation (20).   3: repeat   4:       Evaluate the fitness value of $X_i$, and find $X_b$.   5:       Compute $P{r}_i$ and $r_{pr}$ using Equations (21) and (23), respectively.   6:       for $i=1:N$ do   7:     if $P{r}_i>r_{pr}$ then   8:       Use the RSA to update $X_i$ as in Equations (1)–(7).   9:     else 10:       if $rand>0.5$ then Update $X_i$ using the operators of RFO as in Equations (8)–(15). 11:       else 12:        Use TMO to update $X_i$ as in Equation (25). 13:       end if 14:    end if 15:      end for 16:      $t=t+1$. 17: until Stopping conditions met 18: Return best solution $X_b$.

4. Experimental Results and Discussion

In this study, the performance of RSRFT was evaluated through two experimental series, which included engineering problems and improve the security in the IoT environment. The experiments illustrated in this section were conducted using MATLAB R2020b installed on machine configured with a 2.40 GHz Intel Core i5 CPU, 4.00 GB RAM, and a Windows 10 operating system.

4.1. Series of Analysis 1: Engineering Problems

The evaluation of the developed RSRFT to solve the constraint optimization engineering problem is presented in this section. Six engineering optimization problems—the design of welded beams, the design of tension/compression springs, and the design optimization challenge for pressure vessels—were the focus of this experiment. The next subsections address the problem definition and results.

4.1.1. Welded Beam Design Problem

Determining the welding beam’s settings that lower the fabrication costs are the major goal of solving welded beam design (WBD) problem (Figure 2). These parameters are the bar’s height (t), the length of an attached portion (l), the weld’s thickness (h), and the bar’s thickness (b).

The definition of WBD is formulated as

$$\begin{array}{cc}\hfill Consider& \overrightarrow{u}=\left[u_1{u}_2{u}_3{u}_4\right]=\left[hltb\right],\hfill \\ \hfill Minimize& f(\overrightarrow{u})=1.10471u_1^2{u}_2+0.04811u_3{u}_4(14.0+u_2),\hfill \\ \hfill Subjectto& g_1(\overrightarrow{u})=\tau (\overrightarrow{u})-{\tau }_{max}⩽0,\hfill \\ \hfill & g_2(\overrightarrow{u})=\delta (\overrightarrow{u})-{\delta }_{max}⩽0,\hfill \\ \hfill & g_3(\overrightarrow{u})=\sigma (\overrightarrow{u})-{\sigma }_{max}⩽0,\hfill \\ \hfill & g_4(\overrightarrow{u})=P-P_c(\overrightarrow{u})⩽0,\hfill \\ \hfill & g_5(\overrightarrow{u})=u_1-u_4⩽0,\hfill \\ \hfill & g_6(\overrightarrow{u})=1.10471u_1^2+0.04811u_3{u}_4(14.0+u_2)-5.0⩽0,\hfill \\ \hfill & g_7(\overrightarrow{u})=0.125-u_1⩽0\hfill \\ \hfill Variable{s}^{\prime }range& 0.1⩽u_1⩽2,\hfill \\ \hfill & 0.1⩽u_2⩽10,\hfill \\ \hfill & 0.1⩽u_3⩽10,\hfill \\ \hfill & 0.1⩽u_4⩽2\hfill \\ \hfill where& \tau (\overrightarrow{u})=\sqrt{{({\tau }^{{}^{\prime }})}^2+2{\tau }^{{}^{\prime }}{\tau }^{{}^{″}}\frac{u_2}{2R}+{({\tau }^{{}^{″}})}^2},\hfill \\ \hfill & {\tau }^{{}^{\prime }}=\frac{p}{\sqrt{2u_1{u}_2}},{\tau }^{{}^{″}}=\frac{MR}{J},\hfill \\ \hfill & M=P(L+\frac{u_2}{2}),\hfill \\ \hfill & R=\sqrt{\frac{u_2^2}{4}+{(\frac{u_1+u_3}{2})}^2},\hfill \\ \hfill & J=2\left\{\begin{array}{c}\sqrt{2u_1{u}_2}\left[\begin{array}{c}\frac{u_2^2}{4}+{(\frac{u_1+u_3}{2})}^2\end{array}\right]\end{array}\right\},\hfill \\ \hfill & \sigma (\overrightarrow{u})=\frac{6PL}{u_4{u}_3^2},\delta (\overrightarrow{u})=\frac{6P{L}^3}{E{u}_3^2{u}_4}\hfill \\ \hfill & P_c(\overrightarrow{u})=\frac{\sqrt[4.013E]{\frac{u_3^2{u}_4^6}{36}}}{L^2}(1-\frac{u_3}{2L}\sqrt{\frac{E}{4G}}),\hfill \\ \hfill & P=6000lb,L=14in.,{\delta }_{max}=0.25in.,\hfill \\ \hfill & E=30\times 1^{6}psi,G=12\times {10}^{6}psi,\hfill \\ \hfill & {\tau }_{max}=13600psi,{\sigma }_{max}=30000psi\hfill \end{array}$$

(26)

RSRFT was compared to several optimization techniques, such as the whale optimization algorithm (WOA) [41], MVO [12], evolutionary particle swarm optimization (CPSO) [42], LSHcEpS [43], co-evolutionary differential evolution (CSCA) [44], the simplex method (SIMPLEX) [45], Davidon–Fletcher–Powell (DAVID) [45], the gravitational search algorithm (GSA) [46], the GA [47], harmony search (HS) [48], and LSHADE_SPACMA (we renamed it as LSHSPCM) [49]. The developed RSRFT was conducted for 25 independent runs with $t_{max}=500$ and $N=25$.

Table 1. The value of the estimated parameters using RSRFT and other methods for solving the WBD problem.

Algorithm	H	L	t	b	Optimal Objective
RSRFT	0.20572	3.4704	9.0370	0.2057	1.72489
RFO	0.21846	3.51024	8.87254	0.22491	1.86612
LSHSPCM	$0.2057$	$3.4705$	$9.0366$	$0.2057$	$1.7249$
RSA	0.14468	3.514	8.9251	0.21162	1.6726
LSHcEpS	$0.2038$	$3.5148$	$9.0486$	$0.2057$	$1.7294$
OBLGOA [50]	$0.205769$	$3.471135$	$9.032728$	$0.2059072$	$1.7257$
RO [51]	$0.203687$	$3.528467$	$9.004233$	$0.207241$	$1.735344$
HS [48]	$0.2442$	$6.2231$	$8.2915$	$0.2443$	$2.3807$
DAVID [45]	$0.2434$	$6.2552$	$8.2915$	$0.2444$	$2.3841$
SIMPLEX [45]	$0.2792$	$5.6256$	$7.7512$	$0.2796$	$2.5307$
CPSO [42]	$0.202369$	$3.544214$	$9.04821$	$0.205723$	$1.72802$
MVO [12]	$0.205463$	$3.473193$	$9.044502$	$0.205695$	$1.72645$
GA [47]	$0.205986$	$3.471328$	$9.020224$	$0.20648$	$1.728226$
GSA [46]	$0.182129$	$3.856979$	10	$0.202376$	$1.87995$
CSCA [44]	$0.203137$	$3.542998$	$9.033498$	$0.206179$	$1.733461$
WOA [41]	$0.205396$	$3.484293$	$9.037426$	$0.206276$	$1.730499$

contains the results of RSRFT and the compared optimization techniques. The RSRFT technique had the lowest cost (as given in the column Optimal Objective), followed by LSHSPCM, while SIMPLEX had the highest cost. As a result, the design parameters obtained by RSRFT were more suitable for WBD.

4.1.2. Tension/Compression Spring Design Problem

Within this section, we evaluated the performance of RSRFT to allocate the parameters that are used to minimize spring weight during tension/compression spring design (TCSD), which is defined in Figure 3. Those parameters include the wire diameter (d), the mean coil diameter (D), and the number of active coils (N) of a spring. The formulation of TCSD is illustrated as

$$\begin{array}{cc}\hfill Consider& \overrightarrow{u}=\left[u_1{u}_2{u}_3\right]=\left[dDN\right],\hfill \\ \hfill Minimize& f(\overrightarrow{u})=(u_3+2)u_2{u}_1^2,\hfill \\ \hfill Subjectto& g_1(\overrightarrow{u})=1-\frac{u_2^3{u}_3}{71785u_1^4}⩽0,\hfill \\ \hfill & g_2(\overrightarrow{u})=\frac{4u_2^2-u_1{u}_2}{12566(u_2{u}_1^3-u_1^4)}+\frac{1}{5108u_1^2}⩽0,\hfill \\ \hfill & g_3(\overrightarrow{u})=1-\frac{140.45u_1}{u_2^2{u}_3}⩽0,\hfill \\ \hfill & g_4(\overrightarrow{u})=\frac{u_1+u_2}{1.5}-1⩽0,\hfill \\ \hfill Variable{s}^{\prime }range& 0.05⩽u_1⩽2\hfill \\ \hfill & 0.25⩽u_2⩽1.30\hfill \\ \hfill & 2.00⩽u_3⩽15\hfill \end{array}$$

(27)

In this experiment, RSRFT was compared with other approaches that have been used to solve the TCSD problem, for example CPSO [42], MVO [12], the GA [52], the GSA [41], the evolution strategy (ES) [53], the Ray–Saini method [54], the WOA [41], the CSCA [44], the method proposed by Belegundu and Arora [55], LSHSPCM, and LSHcEpS. The results of the comparison between RSRFT and the others are given in

Table 2. The value of the estimated parameters using RSRFT and others to solve the TCSD problem.

Algorithm	d	D	N	Optimal Objective
RSRFT	0.05147146	0.3515050	11.6013141	0.01266617
RFO	0.052667011	0.3806680	10.0213925	0.0126934
RSA [17]	0.057814	0.58478	4.0167	0.01176
LSHSPCM	$0.0535$	$0.4010$	$9.0962$	$0.012721349$
LSHcEpS	$0.0517$	$0.3567$	$11.2876$	$0.01266523$
OBLGOA [50]	$0.0530178$	$0.38953229$	$9.6001616$	$0.01270136$
Belegundu-Arora method [55]	$0.0500$	$0.3177$	$14.026$	$0.012730$
GA [52]	$0.05148$	$0.351661$	$11.632201$	$0.01270478$
WOA [41]	$0.051207$	$0.345215$	$12.004032$	$0.0126763$
CPSO [42]	$0.051728$	$0.357644$	$11.244543$	$0.0126747$
ES [53]	$0.051643$	$0.35536$	$11.397926$	$0.012698$
MVO [12]	$0.05251$	$0.37602$	$10.33513$	$0.012790$
GSA [41]	$0.050276$	$0.323680$	$13.525410$	$0.0127022$
Ray–Saini method [54]	$0.321532$	$0.050417$	$13.979915$	$0.013060$

. The RSRFT offered superior outcomes over the other methods.

4.1.3. Pressure Vessel Design Problem

This section presents the performance of developed RSRFT to solve another engineering problem, named the pressure vessel design (PVD) problem (see Figure 4), by determining the parameters that lead to minimizing the cost of the pressure cylinder. Those parameters are the thickness of the head $Th$, the length of the cylindrical section of the vessel L, the thickness $Ts$, and the inner radius R. The mathematical representation of PVD is given as

$$\begin{array}{cc}\hfill Consider& \overrightarrow{u}=\left[u_1{u}_2{u}_3{u}_4\right]=\left[T_s{T}_{h}RL\right],\hfill \\ \hfill Minimize& 0.6224u_1{u}_3{u}_4+1.7781u_2{u}_3^2\hfill \\ & +3.1661u_1^2{u}_4+19.84u_1^2{u}_3,\hfill \\ \hfill Subjectto& g_1(\overrightarrow{u})=-u_1+0.0193u_3⩽0,\hfill \\ \hfill & g_2(\overrightarrow{u})=-u_2+0.00954u_3⩽0,\hfill \\ \hfill & g_3(\overrightarrow{u})=-\pi u_3^2{u}_4-\frac{4}{3}\pi u_3^3+1296000⩽0,\hfill \\ \hfill & g_4(\overrightarrow{u})=u_4-240⩽0,\hfill \\ \hfill Variable{s}^{\prime }range& 0⩽u_1⩽99,\hfill \\ \hfill & 0⩽u_2⩽99,\hfill \\ \hfill & 10⩽u_3⩽200,\hfill \\ \hfill & 10⩽u_4⩽200\hfill \end{array}$$

(28)

The outcomes of the comparisons are listed in

Table 3. The value of the estimated parameters using RSRFT and other approaches to solve the PVD problem.

	Estimated Values for Parameters
Method	$\mathit{Th}$	$\mathit{Ts}$	$\mathit{R}$	$\mathit{L}$	Optimal Objective
RSRFT	0.81612257	0.403409949	42.2861349	174.325078	5953.4364
RFO	0.81425	0.44521	42.20231	176.62145	6113.3195
RSA	0.8400693	0.4189594	43.38117	161.5556	6034.7591
LSHSPCM	$0.9808$	$0.5251$	$50.1435$	$99.5532$	$6654.8902747$
OBLGOA [50]	$0.81622$	$0.40350$	$42.291138$	$174.811191$	$5966.67160$
PSO-DE [58]	$0.8125$	$0.4375$	$42.098446$	$176.6366$	$6059.71433$
HPSO [56]	$0.8125$	$0.4375$	$42.0984$	$176.6366$	$6059.7143$
ACO [57]	$0.8125$	$0.4375$	$42.098353$	$176.637751$	$6059.7258$
CDE [44]	$0.8125$	$0.4375$	$42.098411$	$176.63769$	$6059.734$
ES [53]	$0.8125$	$0.4375$	$42.098087$	$176.640518$	$6059.7456$
GA [52]	$0.8125$	$0.4375$	$42.097398$	$176.65405$	$6059.94634$
GSA [41]	$1.125$	$0.625$	$55.9886598$	$84.4542025$	$8538.8359$

. Those approaches are HPSO [56], ACO [57], the ES [53], the GA [52], PSO-DE [58], the GSA [41], CDE [44], and LSHSPCM [49]. From these results, we noticed that RSRFT was able to reach the minimum objective value (as given in the column Optimal Objective), which outperformed the other compared techniques.

4.1.4. Three-Bar Truss Design Problem Design

The major aim of Three-Bar Truss Design (TBTD) is to minimize the weight of the structure [59,60]. Figure 5 illustrates the geometry structure of TBTD, in which cross-sectional areas stand in for the design variables. The cross-section with $A_1(=x_1)$ and $A_2(=x_2)$ must be found according to [59] because of the system’s symmetry. Below is a diagram that represents the TBTD problem mathematically.

$$\begin{array}{cc}\hfill Consider& f(x)=(2\sqrt{2x_1}+x_2)\times l\hfill \\ \hfill Minimize& f(\overrightarrow{u})=(u_3+2)u_2{u}_1^2,\hfill \\ \hfill Subjectto& g_1\left(x\right)=\frac{\sqrt{x1}{x}_1+x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}P-\sigma \le 0,\hfill \\ \hfill & g_2\left(x\right)=\frac{x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}P-\sigma \le 0,\hfill \\ \hfill & g_3\left(x\right)=\frac{1}{\sqrt{2}{x}_2+x_1}P-\sigma \le 0\hfill \\ \hfill & g_4(\overrightarrow{u})=\frac{u_1+u_2}{1.5}-1⩽0,\hfill \\ \hfill Variable{s}^{\prime }range& l=100 \mathrm{cm},P=2 \mathrm{kN}/{\mathrm{cm}}^2,\sigma =2 \mathrm{kN}/{\mathrm{cm}}^2,\hfill \\ \hfill & (0\le x_1,x_2\le 1,)\hfill \end{array}$$

(29)

RSRFT was used to allocate the design variables. Furthermore, we compared RSRFT to other methods used to address the TBTD problem, which are shown in

Table 4. The value of the estimated parameters using RSRFT to solve the 3-bar truss design problem.

Algorithm	Estimated Values for Parameters		Optimal Weight
	${\mathit{x}}_1$	${\mathit{x}}_2$
RSRFT	0.78875052	0.4080351	263.89584
RFO	0.75356	0.55373	268.51195
RSA [17]	0.78873	0.40805	263.8928
DEDS [61]	0.78867513	0.40824828	263.89584
SSA [62]	0.78866541	0.408275784	263.89584
MBA [63]	0.7885650	0.4085597	263.89585
PSO-DE [58]	0.7886751	0.4082482	263.89584
Ray and Saini [54]	0.795	0.395	264.3
CS [64]	0.78867	0.40902	263.9716
AAA [65]	0.7887354	0.408078	263.895880
GOA [66]	0.78889755557	0.40761957011	263.89588149

. Since RSRFT obtained the smallest weight (as given under Optimal Weight), it is clear from the findings in this table that it offered the best solution. However, the outcomes of the traditional RSA were better than the competitive algorithms, including RSRFT.

4.1.5. Speed Reducer Problem

In order for an airplane’s propeller and engine to rotate at an efficient speed, a gearbox between them is needed (see Figure 6 [67]). The optimization strategy employed to address the speed reducer problem is intended for minimizing the design’s weight while still meeting the requirements for the bending stress of the gear teeth, surface stress, transverse shaft deflections, and stresses in the shafts. The best value for each of the seven design factors must be established to solve this problem.

These variables are the face width ($x_1$), the module of the teeth ($x_2$), the number of teeth on the pinion ($x_3$), the lengths of the first and second shafts between the bearings ($x_4$ and $x_5$, respectively), the first shaft’s diameter ($x_6$ and $x_7$, respectively), and the module of the teeth ($x_2$). Given that it contains more limitations than other issues, the speed reducer problem is defined as having a high degree of complexity. Below is a diagram that represents this issue mathematically.

$$\begin{array}{cc}\hfill Assume& f(x)=0.7854x_1{x}_2^2(3.3333x_3^2+14.9334x_3-43.0934)-\hfill \\ \hfill & 1.508x_1(x_6^2+x_7^2)+7.4777x_6^3+x_7^3+0.7854x_4{x}_6^2+x_5{x}_7^2\hfill \\ \hfill Minimize& f(\overrightarrow{u})=(u_3+2)u_2{u}_1^2,\hfill \\ \hfill Subjectto& g(1)=\frac{27}{x_1{x}_2^2{x}_3}-1\le 0,\hfill \\ \hfill & g(2)=\frac{397.5}{x_1{x}_2^2{x}_3^2}-1\le 0,\hfill \\ \hfill & g(3)=\frac{1.93x_4^3}{x_2{x}_3{x}_6^4}-1\le 0,\hfill \\ \hfill & g(4)=\frac{1.93x_5^3}{x_2{x}_3{x}_7^4}-1\le 0,\hfill \\ \hfill & g(5)=\frac{1}{110x_6^3}\sqrt{{(\frac{745x_4}{x_2{x}_3})}^2+16.9\times {10}^6}-1\le 0,\hfill \\ \hfill & g(6)=\frac{1}{85x_7^3}\sqrt{{(\frac{745x_5}{x_2{x}_3})}^2+157.5\times {10}^6}-1\le 0,\hfill \\ \hfill & g(7)=\frac{x_2{x}_3}{40}-1\le 0,\hfill \\ \hfill & g(8)=\frac{5x_2}{x_1}-1\le 0,\hfill \\ \hfill & g(9)=(x_1/12x_2)-1\le 0,\hfill \\ \hfill & g(10)=\frac{1.5x_6+1.9}{x_4}-1\le 0,\hfill \\ \hfill & g(11)=\frac{1.1x_7+1.9}{x_5}-1\le 0\hfill \\ \hfill Variable{s}^{\prime }range& (0.7\le x_2\le 0.8),(2.6\le x_1\le 3.6),(17\le x_3\le 28),(7.3\le x_4\le 8.3),\hfill \\ \hfill & (2.9\le x_6\le 3.9),(7.3\le x_5\le 8.3),(5\le x_7\le 5.5)\hfill \end{array}$$

(30)

Table 5. The value of the estimated parameters using RSRFT to solve the speed reducer design problem.

Method	Estimated Values for Parameters							Optimal Weight
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$
RSRFT	3.5000055	0.7	17	7.305888	8.004689	3.3502353	5.2868060	3000.97899
RFO	3.500001	0.7	17.00002	7.314497	8.0294718	3.350253	5.2867662	3001.5811
RSA [17]	3.50279	0.7	17	7.30812	7.74715	3.35067	5.28675	2996.5157
GA [69]	3.510253	0.7	17	8.35	7.8	3.362201	5.287723	3067.561
GSA [46]	3.600000	0.7	17	8.3	7.8	3.369658	5.289224	3051.120
HS [70]	3.520124	0.7	17	8.37	7.8	3.366970	5.288719	3029.002
SES [71]	3.506163	0.700831	17	7.460181	7.962143	3.362900	5.308949	3025.005127
MDA [72]	3.5	0.7	17	7.3	7.670396	3.542421	5.245814	3019.583365
SBSM [68]	3.506122	0.700006	17	7.549126	7.859330	3.365576	5.289773	3008.08
SCA [11]	3.508755	0.7	17	7.3	7.8	3.461020	5.289213	3030.563
CS [64]	3.5015	0.7000	17	7.6050	7.8181	3.3520	5.2875	3000.9810
PSO [73]	3.5001	0.7000	17.0002	7.5177	7.7832	3.3508	5.2867	3145.922
FA [74]	3.507495	0.7001	17	7.719674	8.080854	3.351512	5.287051	3010.137492
hHHO-SCA [75]	3.506119	0.7	17	7.3	7.99141	3.452569	5.286749	3029.873076

provides the comparative findings between RSRFT and the other MH approaches that have been given in the literature. It is clear that RSRFT surpassed the majority of the techniques that were examined (as given under Optimal Weight), and it was generally assigned the third rank. CS and SBSM were assigned the first and second ranks, respectively [64,68]. Since all three algorithms almost had the same ideal weight, there was little difference between them.

4.1.6. Multiple Disc Clutch Brake Problem

Finding the values of five design factors to reduce the mass of an MDCB is the primary goal of research into the multiple disc clutch brake (MDCB) problem, which was cited in [63]. The design parameters are shown in Figure 7 as the inner radius $x_1$, outer radius $x_2$, disc thickness $x_3$, actuation force $x_4$, and number of friction surfaces $x_5$. Below is a diagram that represents this issue mathematically.

$$\begin{array}{cc}\hfill Assume& f(x)=\Pi (r_o^2-r_i^2)t(Z+1)\rho \hfill \\ \hfill & 1.508x_1(x_6^2+x_7^2)+7.4777x_6^3+x_7^3+0.7854x_4{x}_6^2+x_5{x}_7^2\hfill \\ \hfill Minimize& f(\overrightarrow{u})=(u_3+2)u_2{u}_1^2,\hfill \\ \hfill Subjectto& g_1(x)=l_{max}-(Z+1)(t+\delta )\ge 0,g_2(x)=r_o-r_i-\Delta r\ge 0\hfill \\ \hfill & g_3(x)=P_{max}{\nu }_{srmax}-P_{rz}{\nu }_{sr}\ge 0g_4(x)=Pmax-Prz⩾0,\hfill \\ \hfill & g_5(x)={\nu }_{srmax}-{\nu }_{sr}\ge 0,g_6=T_{max}-T\ge 0\hfill \\ \hfill & g_7(x)=M_h-s{M}_s\ge 0,g_8(x)=T\ge 0\hfill \\ \hfill where& M_h=\frac{2}{3}\mu FZ\frac{r_o^3-r_i^2}{r_o^2-r_i^3},P_{rz}=\frac{F}{\Pi (r_o^2-r_i^2)},{\nu }_{rz}=\frac{2\Pi (r_o^3-r_i^3)}{90(r_o^2-r_i^2)},T=\frac{I_z\Pi n}{30(M_h+M_f)}\hfill \\ \hfill & \Delta r=20\mathrm{mm},I_z=55{\mathrm{kgmm}}^2,P_{max}=1\mathrm{MPa},F_{max}=1000\mathrm{N},\hfill \\ \hfill & {\nu }_{srmax}=10\mathrm{m}/\mathrm{s},l_{max}=30\mathrm{mm},r_{imin}=60,\hfill \\ \hfill & T_{max}=15\mathrm{s},\mu =0.5,s=1.5,M_s=40 \mathrm{Nm},M_f=3 \mathrm{Nm},n=250\mathrm{rpm},\hfill \\ \hfill & F_{max}=1000,Z_{min}=2,Z_{max}=9,r_{imax}=80,r_{omin}=90,\hfill \\ \hfill & r_{omax}=110,t_{min}=1.5,t_{max}=3,F_{min}=600\hfill \end{array}$$

(31)

Table 6. The value of the estimated parameters using RSRFT to solve the multiple disc clutch brake problem.

Method	Estimated Values for Parameters					Optimal Weight
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$
RSRFT	69.003908	89.003914	1	789.52330	2.965888	0.307109
RFO	72	93	762	2	1	0.25359
RSA [17]	70.0347	90.0349	1.0000	801.7285	2.9740	0.31176
TLBO [76]	70	90	1	810	3	0.313656611
NSGA-II [77]	70	90	1.5	1000	3	0.470400
WCA [78]	70	90	1	910	3	0.313656
MVO [79]	70	90	1	910	3	0.313656
CMVO [79]	70	90	1	910	3	0.313656
MFO [80]	70	90	1	910	3	0.313656

lists the outcomes of RSRFT and other MH approaches collected from the literature. From this table, it is clear that RSRFT obtained the ideal cost (0.31176) (as given under Optimal Weight), which was then followed by CMVO, MFO, MVO, and the WCA, which had the same efficiency.

4.2. Series of Analysis 2: RSRFT for Security in the IoT

Within this section, we evaluated the applicability of RSRFT to improve the detection of IDSs in the IoT environment [81]. In general, the problem definition of IDSs in the Internet of Things (IoT) is the need to detect and prevent unauthorized access, data breaches, and malicious activities in IoT devices and networks. With the rapid growth of the IoT, there has been an increase in the number and complexity of connected devices, which has also led to an increase in the number of potential attack vectors and vulnerabilities. As a result, there is a growing need for effective IDSs that can identify and respond to these threats in real-time [82].

The challenge of developing IDSs for the IoT is the complexity of the networks and the diversity of the devices. The systems need to be designed to detect and respond to a wide range of attack types, including malware, denial-of-service attacks, and insider threats. Furthermore, the systems need to be able to identify anomalous behaviors and patterns that may indicate an attack, while also avoiding false positives [83].

To address this problem, we present an alternative technique to detect the intrusions in the IoT. In this experiment, RSRFT was used as a feature selection technique to enhance the process of selecting the important features of the collected data in the IoT environment. This can be achieved through by the Boolean version of RSRFT by converting the real value of each solution into a binary value.

The proposed RSRFT starts by generating the value of the initial population. This can be defined as

$$X=LB+rand\times (UB-LB)$$

(32)

where $UB$ and $LB$ are the upper and lower boundaries, respectively. In this experiment, the values of $UB=1$ and $LB=0$, while $rand\in [0,1]$. This step is followed by obtaining the binary of $X_i$ using the following formula:

$$B{X}_i=\left\{\begin{array}{cc}1& if{X}_i>0.5\\ 0& otherwise\end{array}\right.$$

(33)

The next process is to assess the quality of the selected features, which correspond to the ones in $X_i$, using the following formula.

$$Fi{t}_i=\eta \times {\gamma }_i+(1-\eta )\times (\frac{|B{X}_i|}{D})$$

(34)

where ${\gamma }_i$ denotes the classification error of the KNN classifier, whereas the term $(\frac{|B{X}_i|}{D})$ refers to the ratio of features selected using $X_i$. The value of $\eta \in [0,1]$ is applied as the weight of the two terms of Equation (34).

Thereafter, according to the fitness values, we determined the best of them and their corresponding solution $X_b$. Then, we used this solution and the operators of RSRFT that are discussed in Section 3.2 and Section 3.3 to update X. The next step after reaching the stop conditions is to compute the efficiency of $X_b$ using the testing set of IoT data.

In the following sections, we introduce the datasets used in this experiment to assess RSRFT as an FS method. The comparison was conducted with moth–flame optimization (MFO) [60], the bat (BAT) algorithm [84], transient search optimization (TSO) [83], and the RSA. The parameter of each approach was assigned based on the original implementation.

4.2.1. Dataset Description

The proposed algorithm was assessed using the KDDCup-99, NSL-KDD, BoT-IoT, and CICIDS-2017 datasets. The KDD-99 dataset, created by the Defense Advanced Research Project Agency (DARPA) in 1988, contains network traffic data. Another kind of dataset, called NSL-KDD, was also used, which is a derived dataset with no duplicate network traffic records and grouped into four types: Probe, U2R, R2L, and DoS, in addition to normal data. The CICIDS-2017 dataset [85] was obtained from 25 clients at the Canadian Institute for Cybersecurity (CIC) using the CICFlowMeter tool to simulate realistic network traffic obtained using protocols such as HTTP, SSH, HTTPS, FTP, and email. Additionally, the Bot-IoT dataset [86] includes 3.5 million records produced from different IoT devices and features botnet attacks.The description of all of these datasets is given in

Table 7. Description of the datasets.

Target Class	KDDCup-99		NSL-KDD		Target Class	Bot-IoT		Target Class	CICIDS-2017
	Train	Test	Train	Test		Train	Test		Train	Test
Normal	97,278	60,593	67,343	9710	Normal	370	107	Benign	727,397	163,572
DoS	391,458	229,853	45,927	7458	DoS	1,320,148	385,309	DDoS	112,901	25,388
Probe	4107	4166	11,656	2422	DDoS	1,541,315	330,112	FTP-Patator SSH-Patator	6997 5201	1574 1169
R2L	1126	16,189	995	2887	Reconnaissance	72,919	18,163	PortScan Brute Force	140,043 1329	31,492 299
U2R	52	228	52	67	Theft	65	14	SQL Injection XSS	19 575	4 129

.

4.2.2. Evaluation Criteria and Experimental Setup

In this study, various performance and evaluation criteria were applied to evaluate the effectiveness of RSRFT, including the average accuracy $\left(A{V}_{Acc}\right)$, the average recall $\left(A{V}_{Sens}\right)$, and the average precision $\left(A{V}_{Prec}\right)$:

• $A{V}_{Acc}$: This stands for the rate of the correct detection of intrusions in the IoT environment. $A{V}_{Acc}$ can be formulated as

  $$A{V}_{Acc}=\frac{1}{N_r}\sum _{k=1}^{N_r}Ac{c}_{Best}^k,Ac{c}_{Best}=\frac{TP+TN}{TP+FN+FP+TN}$$

  (35)

  where $N_r=25$ is the total number of runs. FN, TN, TP, and FP refer to false negative, true negative, true positive, and false positive, respectively.
• $A{V}_{Sens}$: This can also be referred to as the true positive rate, and it describes the percentage of correctly predicted positive intrusions. $A{V}_{Sens}$ is computed as

  $$A{V}_{Sens}=\frac{1}{N_r}\sum _{k=1}^{N_r}Sen{s}_{Best}^k,Sen{s}_{Best}=\frac{TP}{TP+FN}$$

  (36)
• $A{V}_{Prec}$: This describes the ratio of true detections among all correct detection samples, and it is formulated as

  $$A{V}_{Prec}=\frac{1}{N_r}\sum _{k=1}^{N_r}Pre{c}_{Best}^k,Pre{c}_{Best}=\frac{TP}{FP+TP}$$

  (37)

4.2.3. Results and Discussion

The performance of the enhanced RSA based on TMO and RFO as the feature selection technique to improve the detection of IDSs in the IoT environment is discussed in this section.

Table 8. Results of RSRFT for improving the detection of IDSs in the IoT environment.

		Train				Test
		Accuracy	Precision	F1-Measure	Recall	Accuracy	Precision	F1-Measure	Recall
KDD99	RSRFT	99.946	99.483	99.943	99.923	93.615	92.649	90.380	93.495
	RFO	92.275	92.414	97.304	93.126	84.375	82.501	87.351	85.225
	BAT	98.007	94.847	97.337	98.247	90.347	89.134	90.093	90.587
	TSO	95.439	91.027	97.437	94.919	87.536	80.791	87.479	87.016
	MFO	96.073	97.631	98.371	97.123	88.175	87.763	88.420	89.225
	RSA	99.910	99.909	99.906	99.910	92.040	89.684	89.985	92.040
NSL-KDD	RSRFT	99.382	99.545	99.548	99.301	76.407	82.371	72.731	77.107
	RFO	91.947	92.080	96.968	92.797	67.951	71.131	68.907	68.801
	BAT	97.669	94.501	96.989	97.909	73.671	73.501	68.905	73.911
	TSO	95.078	90.657	97.067	94.558	71.330	71.298	69.697	70.810
	MFO	95.745	97.297	98.035	96.795	71.626	76.122	69.844	72.676
	RSA	99.201	99.158	99.148	99.201	76.107	82.171	71.731	76.107
BIoT	RSRFT	99.568	99.568	99.568	99.568	99.512	99.420	99.080	99.064
	RFO	99.472	99.472	99.472	99.472	98.956	98.957	99.005	98.964
	BAT	99.475	99.475	99.474	99.475	99.019	98.987	99.012	99.021
	TSO	99.460	99.459	99.459	99.460	98.986	98.941	99.005	98.981
	MFO	99.480	99.480	99.480	99.480	98.998	99.013	99.020	99.009
	RSA	98.829	98.829	98.829	98.829	99.020	99.098	99.070	99.038
CIC2017	RSRFT	99.941	99.920	99.918	99.931	99.931	99.947	99.983	99.931
	RFO	99.690	99.490	99.450	99.690	99.430	99.240	99.190	99.430
	BAT	99.490	99.630	99.440	99.640	99.230	99.360	99.180	99.380
	TSO	99.680	99.750	99.680	99.710	99.420	99.480	99.420	99.450
	MFO	99.360	99.370	99.480	99.430	99.100	99.120	99.220	99.170
	RSA	99.911	99.910	99.889	99.911	99.911	99.907	99.888	99.911

shows the average results of the developed RSRFT to improve the detection of IDSs in the IoT environment using the training and testing sets. From these results, it can be noticed that RSRFT had a high ability to improve the detection of the IDSs in terms of accuracy. In addition, it can be observed that the behavior of RFO alone was the worst one among the tested algorithms using the training and testing sets. According to the results of the precision, recall, and F1-measure, it can be noticed that the developed algorithm was the first-ranked algorithm. This indicated that the ability of the developed RSRFT was better than exploring and exploiting the search space.

Figure 8 and Figure 9 show the average accuracy, precision, F1-measure, and recall using the training and testing sets, respectively. From these figures, we can see that the average of each measure (i.e., accuracy, precision, F1-measure, and recall) of RSRFT was better than that of the other methods using the training and testing sets. This was followed by the RSA, which provided better results than the other methods.

Moreover, we used the Friedman test as a non-parametric test to assess if there was a significant difference between the developed RSRFT and the others.

Table 9. The mean rank using the Friedman test.

		p-Value	RSRFT	RFO	BAT	TSO	MFO	RSA
Training	Accuracy	0.046	6	2.25	3.5	2.25	3	4
	Precision	0.0543	5.75	2.25	3.25	2	3.5	4.25
	F1-Measure	0.0208	6	1.75	2.25	3	4	4
	Recall	0.0435	6	2	3.5	2.5	3	4
Testing	Accuracy	0.0064	6	1.75	3.5	2.25	2.5	5
	Precision	0.0064	6	1.75	3.25	2	3	5
	F1-Measure	0.0101	6	1.625	2.5	2.625	3.5	4.75
	Recall	0.0054	6	1.5	3.5	2.5	2.5	5

shows the mean rank obtained using each approach over the training and testing sets and among the performance measures. From these results, it can be noticed that the developed RSRFT had the highest mean rank, and the difference between it and the others was significant since the p-value was less than 0.05, except for the precision value using the training set.

From the previous results of the two experimental series, it can be observed that the developed RSRFT had high efficiency to determine the solution of different engineering problems, whereas we noticed the high ability of RSRFT to enhance the process of improving the detection of IDSs in the IoT environment by reducing the number of selected features. This resulted from boosting the operators of the RSA, RFO, and TMO to improve the rate of convergence towards the optimal solution. However, there are some limitations in RSRFT, for example the initial population can influence the performance of the developed method. In addition, the diversity of the solutions needs more improvement.

5. Conclusions

In this paper, the use of an enhanced version of the reptile search algorithm (RSA) to solve engineering problems was shown to be a highly effective approach. By combining the strengths of the RSA with those of the red fox algorithm (RFO) and triangular mutation operator (TMO), this modification was able to effectively explore a wider range of possible solutions and find improved solutions more quickly. The results of this study demonstrated the effectiveness of the developed approach, named RSRFT, in solving a variety of engineering problems and showed that this algorithm can provide reliable and accurate solutions in a variety of contexts. Overall, the use of RSRFT represents a promising advancement in the field of engineering and has the potential to greatly improve the design and optimization of various systems and technologies. In the second experiment series, we evaluated RSRFT as a feature selection method to improve the classification of intrusion detection systems (IDSs) in the IoT and cloud environments. Four IDS datasets were utilized to test the performance of RSRFT, namely, KDD99, NSL-KDD, BIoT, and CIC2017. The results also confirmed the superiority of RSRFT.

In future works, the presented RSRFT can be extended to other areas, including task scheduling in cloud and fog environments, medical applications, agriculture, and others. Additionally, it may be tested on multi-objective optimization problems and solar cell parameter estimation.