An Improved Reptile Search Algorithm Based on Lévy Flight and Interactive Crossover Strategy to Engineering Application

Abstract

In this paper, we propose a reptile search algorithm based on Lévy flight and interactive crossover strategy (LICRSA), and the improved algorithm is employed to improve the problems of poor convergence accuracy and slow iteration speed of the reptile search algorithm. First, the proposed algorithm increases the variety and flexibility of the people by introducing the Lévy flight strategy to prevent premature convergence and improve the robustness of the population. Secondly, an iteration-based interactive crossover strategy is proposed, inspired by the crossover operator and the difference operator. This strategy is applied to the reptile search algorithm (RSA), and the convergence accuracy of the algorithm is significantly improved. Finally, the improved algorithm is extensively tested using 2 test sets: 23 benchmark test functions and 10 CEC2020 functions, and 5 complex mechanical engineering optimization problems. The numerical results show that LICRSA outperforms RSA in 15 (65%) and 10 (100%) of the 2 test sets, respectively. In addition, LICRSA performs best in 10 (43%) and 4 (40%) among all algorithms. Meanwhile, the enhanced algorithm shows superiority and stability in handling engineering optimization.

1. Introduction

Almost all scientific and engineering fields can readily translate into optimization problems [1,2,3], given the different real-world requirements of various application domains [4]. These optimization problems often include many needs and requirements, such as uncertainty, multi-objective, and high dimensions, whereas finding a solution strategy for optimization problems is usually a specification [5]. In contrast, various complex issues emerge as the different domains are deepened. Therefore, a more robust and faster solution is needed to solve them. This trend suggests that scientists need to find an algorithm adaptable to various applications to meet the complex situations of difficult problems in cutting-edge engineering fields [6,7]. A wide variety of optimization algorithms are found in delving into the literature on the proposed optimization algorithms [8,9]. These evolved from traditional optimization techniques using planning techniques to nature-inspired metaheuristic algorithms, each with its highlights [10,11]. As a general-purpose optimization method, traditional algorithms often fail to provide solutions that can solve the problem when faced with optimization problems of high complexity and high gradient variability in high dimensions [12,13]. In contrast, the MH benefits from multi-group cooperation through shared experience to find the optimal solution accurately, bypassing the optimal local solution. It has the advantages of being easy to understand, converging quickly, and avoiding getting trapped in a local optimum [14,15,16].

In the latest decade, many MH algorithms have been proposed. In general, there is no standard/unique classification of metaheuristic algorithms. In this research, MH algorithms can be classified into four categories based on the source of inspiration: population intelligence (SI)-based, human-based algorithms, physics/chemistry-based algorithms, and evolutionary algorithms.

SI methods are usually inspired by the real-life and cooperative behaviors of different plants and animals in nature, and these methods search for the optimal solution adaptively by exchanging and sharing the information flow of multiple candidate solutions. Also, this approach is the most extended. Methods of this type include but are not limited to krill herd (KH) [17], GOA [18], moth flame optimization algorithm (MFO) [19], gray wolf optimizer (GWO) [20], red fox optimization algorithm (RFO) [21], slime mould algorithm (SMA) [22], aquila optimizer (AO) [23], Harris hawks optimization (HHO) [8], whale optimization algorithm (WOA) [24], sparrow search algorithm (SSA) [25], and butterfly optimization algorithm (BOA) [26].

Genetic laws often influence evolutionary algorithms, mimicking the crossover and variation behavior therein. Methods of this type include, but are not limited to, evolutionary strategy (ES) [27], genetic algorithm (GA) [28].

Physics-based and chemistry-based methods mimic physical theorems or chemical phenomena prevalent in the universe and life, usually based on universal rules to distinguish the interactions between candidate solutions. There are various methods in this class, such as atomic search optimization (ASO) [29], thermal exchange optimization (TEO) [30], multi-verse optimization algorithm (MVO) [31], chemical reaction optimization (CRO) [32], and gravity search algorithms (GSA) [33].

The final method classified in this research is a human-based algorithm, mainly motivated by social and natural behavior and guided by autonomous human thought. This method contains human mental search (HMS) [34], student psychology based-optimization algorithm (SPBO) [35], and TLBO [36].

SI-based method, as a subset of the MH algorithm, has some more significant advantages: (1) More frequent information exchange. (2) The structure of the algorithm is more straightforward. (3) It is less prone to fall into local optimal solutions. Therefore, SI-based methods have been applied to deal with complex sequential optimization issues such as feature selection [37] and surface shape parameter optimization [12,38]. However, many methods in SI are not powerful enough and often cannot handle optimization situations with a wide range of scope areas, which may be due to the low level of development and exploration and poor diversity of some methods [39]. However, the balance between exploration and development for an SI method is an area that requires too much attention. With the increasing demands of optimization problems, many studies have proposed search strategies to ensure that balance [40,41,42]. Recently, Abualigah et al. established a proposed search algorithm—reptile search algorithm—which is capable of selecting both local and global searches to solve complex optimization problems [7]. The method mainly simulates the encirclement mechanism in the exploration stage and the hunting mechanism in the exploitation stage of the crocodile.

However, some experimental results also demonstrate that RSA also has the problem of poor convergence and slow convergence when facing high-dimensional complex nonlinear optimization problems. Therefore, this paper uses some promising modification strategies to enhance the RSA algorithm capability. Firstly, Lévy flight has been considered an excellent strategy to help optimization algorithms improve their performance [43]. Zhang et al. applied the Lévy flight strategy to the backtracking search algorithm to solve the parameter estimation of the PV model [44]. OB-LF-ALO is an efficient algorithm optimized by Lévy flight and is employed to deal with some specific mechanical optimization in life [45]. Feng et al. proposed a Lévy flight-based gravitational search algorithm for solving ecological scheduling of step hydro power plants [46]. Further, Gao et al. suggested an enhanced chicken flock algorithm based on Lévy flight for the multi-objective optimization problem of integrated energy for smart communities, considering the utility of decision-makers [47]. Therefore, the Lévy flight strategy is introduced into the reptile search algorithm in this paper to improve the global search capability of RSA and guarantee the search to escape the local optimum. Moreover, influenced by the crossover operator [48] and the difference operator [49], an interactive crossover strategy is proposed in this paper. In this strategy, the candidate solutions are more influenced by the iterative process, making the difference between the preliminary search and the later development more obvious. Therefore, an improved reptile search algorithm (LICRSA) based on Lévy flight and interactive crossover is proposed in this study based on these two strategies.

The remainder of this paper is structured as follows: Section 2 presents a comprehensive description of the reptile search algorithm. Section 3 depicts the contents of both Lévy flight and crossover strategies. In Section 4, the algorithm is evaluated using 23 benchmark functions and CEC2020 and compared with different MH methods. Section 5 selects five commonly used mechanical engineering optimization problems the proposed algorithm solves. Section 6 summarizes the work.

2. Description of the Reptile Search Algorithm (RSA)

The reptile search algorithm mainly imitates the predation strategy and social behavior of crocodiles in nature [7]. The first is the encirclement mechanism in the exploration phase, and the second is the hunting mechanism in the exploitation phase. A standard MH algorithm tends to be first applied to a set of candidate solutions before the iteration starts. It is a random strategy generated.

$$Z=\left[\begin{array}{ccccc}{z}_{1,1}& \cdots & z_{1,j}& z_{1,d-1}& z_{1,d}\\ z_{2,1}& \cdots & z_{2,j}& \cdots & z_{2,d}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ z_{N-1,1}& \cdots & z_{N-1,j}& \cdots & z_{N-1,d}\\ z_{N,1}& \cdots & z_{N,j}& z_{N,d-1}& z_{N,d}\end{array}\right]$$

(1)

where z_i,j represents the jth dimension of the ith crocodile. N is all the crocodiles, and d denotes dimension. Z is N candidate solutions randomly generated by Equation (2).

$$Z_{i,j}=rand\times (up-low)+low,$$

(2)

where rand is a stochastic number; up and low are the maximum and minimum limits of the optimization problem.

Encircling behavior is the hallmark of the globalized search of RSA. The process is divided into two behaviors, aerial and abdominal walk. These two actions often do not allow crocodiles to get close to food. However, the crocodile will find the general area of the target food by chance after several search attempts because it is a global search of the whole solved spatial range. Meanwhile, ensure that the stage can be continuously adapted to the next developmental stage. The process is often limited to the first half of the total iteration. Equation (3) simulates the encircling behavior of the crocodile and is shown below.

$$z_{i,j}(t+1)=\left\{\begin{array}{c}bes{t}_j(t)\times (-{\eta }_{i,j}(t))\times \beta -R_{i,j}(t)\times r, t\le \frac{T_{Max}}{4}\\ bes{t}_j(t)\times z_{r_1,j}\times ES(t)\times r, \frac{T_{Max}}{4}\le t<\frac{2\times T_{Max}}{4}\end{array}\right.$$

(3)

where best_i,j (t) is the best-positioned crocodile at t iterations, and r is a stochastic number of 0 to 1. T_Max represents the maximum iteration. η_i,j is the operator of the ith crocodile at the jth dimension and is given by Equation (4). β is a sensitive parameter controlling the search accuracy and is given as 0.1 in the original text [7]. R_i,j is the reduce function, and the equation is used to decrease the explored area and is computed by Equation (5). r₁ is a stochastic integer between 1 and N, and z_r1,j is the jth dimension of the crocodile at position r₁. ES(t) is an arbitrary decreasing probability ratio between 2 and −2 and is given in Equation (6).

$${\eta }_{i,j}=bes{t}_{i,j}(t)\times P_{i,j},$$

(4)

$$R_{i,j}=\frac{bes{t}_j(t)-x_{r_2,j}}{bes{t}_j(t)+\epsilon },$$

(5)

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

(6)

where r₂ is a random integer between 1 and N, and r₃ is an integer number of −1 to 1. ε is a small value. P_i,j represents the percentage difference between the crocodile in the best position and crocodiles in the current position, updated as in Equation (7).

$$P_{i,j}=\alpha +\frac{z_{i,j}-M(z_i)}{bes{t}_j(t)\times (u{p}_j-lo{w}_j)+\epsilon },$$

(7)

where M (z_i) is the average position of the crocodile of z_i and is given in Equation (8), up_j and low_j are the maximum and minimum limits of the jth dimension. α is the same as β as a sensitive parameter to control the search accuracy. It was set to 0.1 in the original paper [7].

$$M(z_i)=\frac{1}{n}{\displaystyle \sum _{j=1}^n{z}_{i,j}},$$

(8)

Connected to the search process of RSA is the hunting process of local symbolic exploitation, which has two strategies in this part: coordination and cooperation. After the influence of the encirclement mechanism, the crocodiles almost lock the location of the target prey, and the hunting strategy of the crocodiles will make it easier for them to approach the target. The development phase will often find the near-optimal candidate solution after several iterations. The mathematical model of its simulated crocodile hunting behavior is presented in Equation (9). This development hunting strategy occurs in the second half of the iterations.

$$z_{i,j}(t+1)=\left\{\begin{array}{c}bes{t}_j(t)\times P_{i,j}(t)\times r, \frac{2\times T_{Max}}{4}\le t<\frac{3\times T_{Max}}{4} \\ bes{t}_j(t)-{\eta }_{i,j}(t)\times \epsilon -R_{i,j}(t)\times r,\frac{3\times T_{Max}}{4}\le t<\frac{4\times T_{Max}}{4}\end{array}\right.$$

(9)

where best_j is the optimal position of the crocodile, η_i,j is the operator of the ith crocodile at the jth dimension and is given by Equation (4). R_i,j is the reduce function, and the equation is used to decrease the explored area and is computed by Equation (5). Algorithm 1 is the pseudo-code for RSA.

Algorithm 1: The framework of the RSA

1: Input: The parameters of RSA including the sensitive parameter α, β, crocodile size (N), and the maximum generation TMax.

2: Initializing n crocodile, zi and calculate fi.

3: Determine the best crocodile bestj.

4: while (t ≤ tMax) do

5:  Update the ES by Equation (6).

6:  for i = 1 to N do

7:   for i = 1 to N do

8:    Calculate the η, R, P by Equations (4), (5) and (7).

9:    if t ≤ TMax/4 then

10:      zi,j (t + 1) = bestj (t) × −ηi,j × β − Ri,j (t) × r.

11:     else if TMax/4 ≤ t < 2 * TMax/4

12:      zi,j (t + 1) = bestj (t) × zi,j × ES (t) × r.

13:     else if 2*TMax/4 ≤ t < 3*TMax/4

14:      zi,j (t + 1) = bestj (t) × Pi,j (t) × r.

15:     else

16:      zi,j (t + 1) = bestj (t) −ηi,j × ε − Ri,j (t) × r.

17:     end if

18:    end for

19:   end for

20:   Find the best crocodile.

21:   t = t + 1.

22: end while

23: Output: The best crocodile.

3. The Proposed Hybrid RSA (LICRSA)

In this paper, the Lévy flight strategy and the interaction crossover strategy are chosen, respectively. The Lévy flight strategy helps candidate solutions leap out of local solutions and enhances the algorithm’s precision. An interaction crossover strategy mainly enhances the ability of algorithm development. The two methods are introduced primarily in the following.

3.1. Lévy Flight Strategy

The Lévy flight strategy is used in this paper to generate a random number with a replacement rand with the following characteristics: (1) The random numbers generated are occasionally large but mostly interspersed with small numbers in between. (2) The probability density function of the steps is heavy-tailed. This random number will help update the position to produce oscillations, perform a small search in the neighborhood by the fluctuations of the random number at one iteration, and help the candidate solution leap out of the local optimum.

The definition of Lévy distribution is given by Equation (10) below.

$$Levy(\gamma )~u=t^{-1-\gamma }, 0<\gamma \le 2,$$

(10)

The step length of the Lévy flight can be calculated as in Equation (11).

$$s=\frac{U}{|V{|}^{1/\gamma }},$$

(11)

where, both U and V follow Gaussian distribution as shown in Equation (12).

$$U~N(0,{\sigma }_U^2), V~N(0,{\sigma }_V^2)$$

(12)

where σ_U and σ_V meet the following Equations (13) and (14).

$${\sigma }_U={\left(\frac{\mathsf{\Gamma }(1+\beta )·\sin (\pi ·\beta /2)}{\mathsf{\Gamma }((1+\beta /2)·\beta ·2^{(\beta -1)/2})}\right)}^{1/\beta },$$

(13)

$${\sigma }_V=1,$$

(14)

where $\mathsf{\Gamma }$ is a standard Gamma function and β is a parameter with a fixed 1.5.

Lévy flight strategy is deployed for the crocodile’s high and belly walking in the encircling mechanism to expand the surrounding search area effectively. Meanwhile, the Lévy flight strategy is also deployed for hunting coordination and cooperation in the hunting behavior phase to increase the flexibility of the optimal exploitation phase. The encircling mechanism based on Lévy flight is displayed in Equation (15). λ is a parameter with a fixed 0.1.

$$z_{i,j}(t+1)=\left\{\begin{array}{c}bes{t}_j(t)\times (-{\eta }_{i,j}(t))\times \beta -R_{i,j}(t)\times \lambda \times Levy(\gamma ), t\le \frac{T_{Max}}{4} \\ bes{t}_j(t)\times z_{r_1,j}\times ES(t)\times \lambda \times Levy(\gamma ), \frac{T_{Max}}{4}\le t<\frac{2\times T_{Max}}{4}\end{array}\right.$$

(15)

The hunting behavior (hunting coordination and cooperation) based on Lévy flight is displayed in Equation (16).

$$z_{i,j}(t+1)=\left\{\begin{array}{c}bes{t}_j(t)\times P_{i,j}(t)\times \lambda \times Levy(\gamma ), \frac{2\times T_{Max}}{4}\le t<\frac{3\times T_{Max}}{4} \\ bes{t}_j(t)-{\eta }_{i,j}(t)\times \epsilon -R_{i,j}(t)\times \lambda \times Levy(\gamma ),\frac{3\times T_{Max}}{4}\le t<\frac{4\times T_{Max}}{4}\end{array}\right.$$

(16)

3.2. Interaction Crossover Strategy

The interaction crossover strategy helps the candidate in the current position readjust by exchanging the information of the two candidate solutions and the candidate at the optimal position. The new position draws the information on the optimal solution and other candidate solutions to improve the searchability of the candidate solutions. First, a parameter controlling the activity of the crocodile population is defined, which has a linearly decreasing trend with the number of iterations. CF is defined as follows:

$$CF={\left(1-\frac{t}{T_{Max}}\right)}^{2\times \frac{t}{T_{Max}}},$$

(17)

where t represents the current iteration, T_Max stands for all iterations. The crocodile population is randomly divided into two parts with the same number of crocodiles. Two positions of crocodiles z_k1 and z_k2 are selected from these two parts, and then the positions are interacted with to update the two crocodiles. The updated strategy is the same as that provided by Equations (18) and (19).

$$z_{k1,j}(t+1)=z_{k1,j}(t)+CF\times (bes{t}_j(t)-z_{k1,j}(t))+c_1\times (z_{k1,j}(t)-z_{k2,j}(t)),$$

(18)

$$z_{k2,j}(t+1)=z_{k2,j}(t)+CF\times (bes{t}_j(t)-z_{k2,j}(t))+c_2\times (z_{k2,j}(t)-z_{k1,j}(t)),$$

(19)

where best_j is the best positioned crocodile, c₁ and c₂ are stochastic numbers in the interval 0 to 1, and z_k1 is the crocodile in position k₁. Figure 1 illustrates the introduction graph of the interaction crossover.

For positions of the updated crocodiles, this paper set up an elimination mechanism: retaining the crocodiles with better positions after the interactive crossover and eliminating the crocodiles with poorer abilities. The updates are as follows:

$$z_{i,j}(t+1)=\left\{\begin{array}{c}{z}_{i,j}(t), if f(z_{i,j}(t))<f(z_{i,j}(t+1)) \\ z_{i,j}(t+1), elseif f(z_{i,j}(t+1))<f(z_{i,j}(t))\end{array}\right.$$

(20)

3.3. Specific Implementation Steps of LICRSA

The Lévy flight strategy and the interactive crossover strategy are incorporated in RSA. These strategies effectively improve the accuracy and exploitation of the RSA algorithm while obtaining a better traversal of the cluster. The specific operation steps of LICRSA are displayed below.

Step1: Provide all the relevant parameters of LICRSA, such as the number of alligators N, the dimension of variables Dim, the upper limit up of all variables, the lower limit low of all variables, all iterations T_Max, and the sensitive parameter β.

Step2: Initialize N random populations and calculate the corresponding adaptation values. The best crocodile individuals are selected.

Step3: Update ES values by Equation (6) and calculate η, R, and P values, respectively.

Step4: While t < M_iter, the search phase of RSA: If t < T_Max/4, update the new situation of the crocodile by the first part of Equation (15), and similarly, if t is between T_Max/4 and 2T_Max/4, then update the crocodile position according to the second part of Equation (15).

Step5: The development phase of RSA: if t is between 2T_Max/4 and 3T_Max/4, the position of the crocodile is updated by the first part of Equation (16), and in the final stage of the iteration, when t > 3T_Max/4, the position of the crocodile is updated by the second part of Equation (16).

Step6: The fitness of the crocodile positions is re-evaluated, and the crocodiles with function values worse than the last iteration are eliminated, while the position of the optimal crocodile is updated.

Step7: Interactive crossover strategy: firstly, the crocodile population is divided into two parts, and then crocodiles are randomly selected from the two groups in turn for crossover, and their information is interacted by the positions of the two crocodiles updated by Equations (18) and (19), respectively.

Step8: According to the principle of Equation (20), crocodile individuals with poor predation ability are eliminated, and the optimal ones are renewed.

Step9: Determine if the iteration limit is exceeded and output the optimal value.

To show the hybrid RSA algorithm more clearly, in this paper, the pseudo-code of LICRSA is provided in Algorithm 2. Where line 10, line 12, line 14, and line 16 are the encirclement and predation phases of the Lévy flight strategy improvement. Lines 20–26 are the strategies for the interaction crossover. Figure 2 shows the flow chart of the LICRSA algorithm.

Algorithm 2: The framework of the LICRSA

1: Input: The parameters of LICRSA including the sensitive parameter α, β, crocodile size (N), the maximum generation TMax, and variable range ub, lb.

2: Initializing n crocodile, zi (i = 1, 2, …, N) and calculate the fitness fi.

3: Determine the best crocodile bestj.

4: while (t ≤ tMax) do

5:  Update the ES by Equation (6).

6:  for i = 1 to N do

7:   for i = 1 to N do

8:    Calculate the η, R, P by Equations (4), (5) and (7).

9:    if t ≤ TMax/4 then

10:      zi,j (t + 1) = bestj (t) × −ηi,j × β − Ri,j (t) × λ × levy(γ).

11:     else if TMax/4 ≤ t < 2 * TMax/4

12:      zi,j (t + 1) = bestj (t) × zi,j × ES (t) × λ × levy(γ).

13:     else if 2*TMax/4 ≤ t < 3*TMax/4

14:      zi,j (t + 1) = bestj (t) × Pi,j (t) × λ × levy(γ).

15:     else

16:      zi,j (t + 1) = bestj (t) −ηi,j × ε − Ri,j (t) × λ × levy(γ).

17:     end if

18:    end for

19:   end for

20:   Update the fitness and location of the crocodile.

21:   Replace the optimal location and optimal fitness.

22:   Group = permutate(n).

23:   for i = 1 to N/2 do

24:    let k1 = Group(2 × i − 1) and k2 = Group(2 × i).

25:    Update the CF by Equation (17).

26:    zk1,j (t + 1) = zk1,j (t) + CF × (bestj(t) − zk1,j (t)) + c1 × (zk1,j (t) − zk2,j (t))

27:    zk1,j (t + 1) = zk1,j (t) + CF × (bestj(t) − zk1,j (t)) + c1 × (zk1,j (t) − zk2,j (t))

28:   end for

29:   Find the best crocodile.

30:   t = t + 1.

31: end while

32: Output: The best crocodile.

3.4. Considering the Time Complexity of LICRSA

The estimation of LICRSA time complexity is mainly based on RSA with the addition of the interaction crossover part, and the Lévy strategy only improves the update strategy of RSA. The time complexity of RSA is:

O (T_Max × d × N) + O (T_Max × N)

(21)

The adaptation value evaluation mainly depends on the complexity of the problem, so it is not considered in this paper. In the crossover phase, the method is a two-part population operation process, so the time complexity is:

O (T_Max × d × N) + O (T_Max × N) + O (T_Max × d × N/2)

(22)

where d is the variables, N is all the crocodiles, and T_Max is the maximum generation.

4. Simulation Experiments

To demonstrate the effectiveness of this improvement plan, 23 standard benchmark functions and CEC2020 will be used to test the local exploitation capability of LICRSA, global search capability, and ability to step out of local optimization. The experimental parameters of this section are set: the crocodile size N = 30 and the maximum iterations T_Max = 1000. In addition, for each test function, all methods will be run 30 times independently in the same dimension, and the numerical results will be counted. The comparative MH algorithms in this section include reptile search algorithm (RSA) [7], aquila optimizer (AO) [23], arithmetic optimization algorithm (AOA) [4], differential evolution (DE) [49], crisscross optimization algorithm (CSO) [48], dwarf mongoose optimization (DMO) [6], white shark optimizer (WSO) [50], whale optimization algorithm (WOA) [24], wild horse optimizer (WHOA) [15], seagull optimization algorithm (SOA) [51], and gravitational search algorithm (GSA) [33] in 23 test functions. Additionally, reptile search algorithm (RSA) [7], aquila optimizer (AO) [23], arithmetic optimization algorithm (AOA) [4], differential evolution (DE) [49], dwarf mongoose optimization (DMO) [6], grasshopper optimization algorithm (GOA) [18], gravitational search algorithm (GSA) [33], honey badger algorithm (HBA) [52], HHO [8], SSA [11], and WOA [24] in CEC2020. It is worth noting that to guarantee the equity of the experiments, the parameter settings of all MH methods are the same as those provided in

Table 1. Parameters setting.

Algorithm	Parameters	Values
AOA [4]	µ	0.499
	a	a5
WSO [50]	fmin, fmax	0.07, 0.75
	τ, ao, a1, a2	4.125, 6.25, 100, 0.0005
GSA [33]	ElitistCheck, Rpower, Rnorm, alpha, G0	1, 1, 2, 20, 100
WOA [24]	A	Decreased from 2 to 0
	b	2
SSA [11]	Leader position update probability	c3 = 0.5
RSA [7]	α	0.1
	β	0.1
DE [49]	Scaling factor	0.5
	Crossover probability	0.5
WHO [15]	Crossover percentage	PC = 0.13
	Stallions percentage (number of groups)	PS = 0.2
	Crossover	Mean
HBA [52]	β (the ability of a honey badger to get food)	6
	C	2
SOA [51]	Control Parameter (A)	[2, 0]
	fc	2

and are consistent with the source paper.

4.1. Exploration–Exploitation Analysis

The two main parts of different MH algorithms are the exploration and exploitation phases. The role of exploration is that distant regions of the search domain can be explored, ensuring better search candidate solutions. Alternatively, exploitation is the convergence to the region where the optimal solution is promising and expected to be found through a local convergence strategy. Maintaining a balance between these two conflicting phases is critical for the optimization algorithm to locate the optimal candidate solution to the optimization problem [53]. Dimensional diversity in the search procedure can be expressed in Equations (23) and (24).

$$Di{v}_j=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}median(z_j)-z_{i,j},}$$

(23)

$$Div(t)=\frac{1}{d}{\displaystyle \sum _{j=1}^{d}Di{v}_j},t=1,2,\cdots T_{Max}$$

(24)

where z_i,j is the position of the ith crocodile in the jth dim, Div_j is the average diversity of dimension j, and median(z_j) is the median value of the jth dim of the candidate solution. N is all the crocodiles, d is the dim, and T_Max is the maximum iteration. The following equation calculates the percentage of exploration and exploitation.

$$Exploration\%=\frac{Div(t)}{\max (Div)}\times 100\%,$$

(25)

$$Exploitation\%=\frac{\left|Div(t)-\max (Div)\right|}{\max (Div)}\times 100\%,$$

(26)

where max(Div) is the maximum diversity of the iterative process.

The test set of CEC2020 is selected in this section, and Figure 3 demonstrates the exploration and exploitation behavior of LICRSA in the CEC2020 test set [5] during the search process. Analyzing Figure 3, LICRSA mainly starts from high exploration and low exploitation. Still, as the exploration proceeds, LICRSA quickly switches to high exploitation and converges to the region where the optimal solution is expected to be found. There is a partial addition of the exploration process during the iteration process, which is energetic in both exploration and utilization. Therefore, LICRSA can find better optimal solutions than other MH algorithms. In addition, Figure 4 shows the percentage of different functions, the rate of exploration is generally between 10% and 30%, which indicates that LICRSA provides enough exploration time. The rest of the effort is consumed in development. This result means that LICRSA maintains a sufficiently high exploration percentage. On the other hand, LICRSA development utilization is high.

4.2. Performance Evaluation for the 23 Classical Functions

The 23 benchmark functions include 7 unimodal, 6 multimodal, and 10 fixed-dimensional functions [5]. Among them, unimodal functions are used to explore MH algorithms, and multimodal functions are often used to understand the development of algorithms. The dimension function aims to know how well different exploration algorithms leap out of the local optimum and how accurately algorithms converge. It should be noted that the objective of all 23 functions is to minimize the problem. In addition, four criteria are used in this part of the experiment, including maximum, minimum, mean, and standard deviation (Std). In addition, the Wilcoxon signed-rank test (significance level = 0.05) is used to test for the numerical cases. Where “+” means that an algorithm is more accurate than LICRSA, “=” indicates that there is no significant difference between the two, and “−” indicates that an algorithm is less accurate than LICRSA. We also provide the p-values as reference data.

Table 2. Numerical results for best, worst, mean, std of the search algorithm in 12 of the 23 benchmark functions.

Function	Index	Algorithms
		RSA	AO	AOA	CSO	DE	DMO	WSO	WOA	WHOA	SOA	GSA	LICRSA
F1	Best	0	2.65 × 10−300	5.12 × 10−7	9.24 × 10−26	5.35 × 10−3	1.07 × 10−4	2.5012	3.21 × 10−166	6.71 × 10−105	0	4.60 × 10−17	0
	Worst	0	3.34 × 10−206	2.17 × 10−06	1.40 × 10−17	2.16 × 104	1.44 × 10−03	37.3501	9.22 × 10−146	8.44 × 10−92	0	2.55 × 10−16	0
	Mean	0	1.11 × 10−207	1.27 × 10−6	6.66 × 10−19	2.83 × 103	4.89 × 10−4	12.9127	3.07 × 10−147	5.18 × 10−93	0	1.27 × 10−16	0
	Std	0	0	4.26 × 10−7	2.63 × 10−18	5.08 × 103	3.35 × 10−4	9.1243	1.68 × 10−146	1.87 × 10−92	0	5.20 × 10−17	0
	Rank	1	4	9	7	12	10	11	5	6	1	8	1
F2	Best	0	1.52 × 10−153	1.34 × 10−9	2.68 × 10−16	7.29 × 10−3	3.88 × 10−4	0.2289	4.65 × 10−116	3.79 × 10−58	0	3.17 × 10−8	0
	Worst	0	1.71 × 10−99	2.57 × 10−3	3.98 × 10−13	42.4839	3.7448	1.1593	1.86 × 10−100	3.53 × 10−50	2.65 × 10−300	9.93 × 10−8	0
	Mean	0	6.53 × 10−101	3.55 × 10−4	1.72 × 10−14	7.9621	0.1367	0.5527	6.23 × 10−102	1.23 × 10−51	2.65 × 10−300	5.74 × 10−8	0
	Std	0	3.14 × 10−100	6.00 × 10−4	7.21 × 10−14	9.8024	0.6817	0.2489	3.40 × 10−101	6.44 × 10−51	0	1.58 × 10−8	0
	Rank	1	5	9	7	12	10	11	4	6	3	8	1
F3	Best	0	1.17 × 10−295	3.75 × 10−6	141.0483	2.54 × 103	2.49 × 104	164.7909	6.50 × 103	6.15 × 10−72	2.61 × 103	273.2429	0
	Worst	0	2.22 × 10−190	1.52 × 10−3	3.99 × 103	7.57 × 104	6.99 × 104	1.09 × 103	4.75 × 104	2.79 × 10−55	3.74 × 104	930.4876	0
	Mean	0	1.04 × 10−191	3.16 × 10−4	900.3036	2.08 × 104	5.19 × 104	575.8136	2.30 × 104	2.24 × 10−56	1.31 × 104	491.8570	0
	Std	0	0	3.36 × 10−4	777.9065	1.67 × 104	1.17 × 104	252.9466	1.03 × 104	6.90 × 10−56	8.92 × 103	150.6003	0
	Rank	1	3	5	8	10	12	7	11	4	9	6	1
F4	Best	0	6.29 × 10−154	0.0002	0.0075	45.5951	36.1441	5.3086	1.4451	4.89 × 10−41	3.77 × 10−128	1.52 × 10−8	0
	Worst	0	8.32 × 10−109	0.0402	0.0809	78.0134	62.1824	14.6068	87.4681	2.90 × 10−35	2.23 × 10−42	4.7297	0
	Mean	0	2.77 × 10−110	0.0084	0.0330	63.6120	45.8755	10.6183	43.7648	1.49 × 10−36	8.93 × 10−44	1.4748	0
	Std	0	1.52 × 10−109	0.0091	0.0205	8.1802	5.3785	2.1397	26.3941	5.37 × 10−36	4.10 × 10−43	1.5097	0
	Rank	1	3	6	7	12	11	9	10	5	4	8	1
F5	Best	6.72 × 10−29	6.83 × 10−6	26.6485	0.0055	869.9148	577.6902	293.4370	0.8391	24.4814	28.7246	26.0094	1.13 × 10−19
	Worst	28.9901	0.0088	28.0367	264.5424	6.96 × 107	7.73 × 103	2.18 × 103	28.7495	78.4426	28.8734	347.4929	10.8886
	Mean	7.7128	0.0018	27.4318	52.0036	8.42 × 106	2.67 × 103	812.9171	26.3925	29.1866	28.8012	49.0518	0.4101
	Std	13.0090	0.0024	0.3141	61.6943	1.60 × 107	2.12 × 103	470.7655	4.8696	13.3897	0.0403	65.9500	1.9898
	Rank	3	1	5	9	12	11	10	4	7	6	8	2
F6	Best	3.5635	1.69 × 10−8	1.2454	5.23 × 10−25	0.0228	8.03 × 10−5	5.5298	0.0073	5.27 × 10−12	0.5051	3.69 × 10−17	0.7079
	Worst	7.4998	3.30 × 10−4	2.0581	4.44 × 10−21	6.98 × 103	9.35 × 10−4	268.0956	0.3385	1.94 × 10−6	4.6282	2.80 × 10−16	4.7392
	Mean	6.4884	4.93 × 10−5	1.7396	5.00 × 10−22	710.2299	4.19 × 10−4	27.5978	0.0740	1.28 × 10−7	2.4897	1.06 × 10−16	1.4065
	Std	1.0043	7.83 × 10−5	0.1716	1.03 × 10−21	1.59 × 103	2.52 × 10−4	47.2090	0.0920	3.67 × 10−7	1.0757	5.66 × 10−17	0.9717
	Rank	10	4	8	1	12	5	11	6	3	9	2	7
F7	Best	4.67 × 10−6	7.17 × 10−8	7.42 × 10−7	0.0003	1.9479	0.0532	0.0275	2.91 × 10−5	1.78 × 10−5	1.38 × 10−6	0.0275	1.56 × 10−6
	Worst	2.92 × 10−4	1.78 × 10−4	9.71 × 10−5	0.0073	45.3077	0.1374	0.3394	0.0113	0.0015	1.99 × 10−4	0.1400	1.47 × 10−4
	Mean	4.39 × 10−5	5.82 × 10−5	2.64 × 10−5	0.0017	8.9688	0.0849	0.1140	0.0017	0.0005	6.14 × 10−5	0.0623	3.62 × 10−5
	Std	5.97 × 10−5	5.03 × 10−5	2.12 × 10−5	0.0014	9.0334	0.0240	0.0822	0.0026	0.0004	5.22 × 10−5	0.0247	3.62 × 10−5
	Rank	3	4	1	8	12	10	11	7	6	5	9	2
F8	Best	−5746.50	−12,569.29	−6195.38	−12,569.49	−8839.00	−4960.68	−8910.96	−12,569.48	−10,488.23	−12,569.49	−3688.28	−6842.38
	Worst	−4375.91	−4200.08	−4766.89	−12,451.05	−5365.73	−3794.91	−3456.50	−7192.25	−7284.72	−12,569.47	−1665.49	−4351.51
	Mean	−5443.79	−9725.52	−5544.32	−12,565.05	−7455.45	−4300.56	−7038.96	−11,053.13	−9089.71	−12,569.49	−2599.01	−5782.82
	Std	277.88	3560.82	365.06	21.61	827.49	322.21	1276.74	1870.29	602.23	0.00	453.89	598.89
	Rank	10	4	9	2	6	11	7	3	5	1	12	8
F9	Best	0	0	0	0	75.8378	146.1968	10.4790	0	0	0	12.9345	0
	Worst	0	0	6.47 × 10−7	6.25 × 10−13	261.5807	231.3507	33.3913	0	0	0	53.7277	0
	Mean	0	0	1.69 × 10−7	9.28 × 10−14	161.8547	203.2452	20.3445	0	0	0	24.9735	0
	Std	0	0	2.08 × 10−7	1.17 × 10−13	48.2843	19.8946	5.2764	0	0	0	9.4583	0
	Rank	1	1	8	7	11	12	9	1	1	1	10	1
F10	Best	8.88 × 10−16	8.88 × 10−16	2.23 × 10−6	1.68 × 10−13	19.9571	0.0039	2.3685	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16	4.84 × 10−9	8.88 × 10−16
	Worst	8.88 × 10−16	8.88 × 10−16	3.87 × 10−4	9.81 × 10−10	19.9753	0.0602	5.4315	7.99 × 10−15	7.99 × 10−15	8.88 × 10−16	1.05 × 10−8	8.88 × 10−16
	Mean	8.88 × 10−16	8.88 × 10−16	1.79 × 10−4	4.82 × 10−11	19.9648	0.0160	3.6012	4.20 × 10−15	2.90 × 10−15	8.88 × 10−16	7.90 × 10−9	8.88 × 10−16
	Std	0	0	1.11 × 10−4	1.84 × 10−10	0.0035	0.0109	0.6814	2.27 × 10−15	2.41 × 10−15	0	1.47 × 10−9	0
	Rank	1	1	9	7	12	10	11	6	5	1	8	1
F11	Best	0	0	3.50 × 10−6	0	0.0973	0.0017	1.0480	0	0	0	2.9419	0
	Worst	0	0	0.0074	0.7293	53.0911	0.3955	2.0351	0	0	0	13.8166	0
	Mean	0	0	0.0003	0.1491	9.1646	0.0460	1.1800	0	0	0	7.4863	0
	Std	0	0	0.0013	0.2365	12.3615	0.0845	0.1848	0	0	0	2.6727	0
	Rank	1	1	7	9	12	8	10	1	1	1	11	1
F12	Best	0.6944	1.33 × 10−8	0.5443	4.48 × 10−27	5.61 × 103	5.0890	0.2155	0.0008	1.61 × 10−13	0.0035	3.44 × 10−19	0.0201
	Worst	1.6297	8.11 × 10−6	0.6527	6.41 × 10−22	4.86 × 108	26.7847	8.2292	0.1443	0.1037	0.7420	1.4406	1.1004
	Mean	1.2466	1.16 × 10−6	0.5999	3.78 × 10−23	2.95 × 107	13.8952	1.5707	0.0112	0.0138	0.1820	0.1335	0.2364
	Std	0.3311	1.66 × 10−6	0.0291	1.35 × 10−22	9.56 × 107	5.2129	1.5644	0.0262	0.0358	0.1907	0.2965	0.2769
	Rank	9	2	8	1	12	11	10	3	4	6	5	7
F13	Best	6.28 × 10−32	1.50 × 10−8	2.8173	3.52 × 10−26	2.76 × 105	8.0132	3.6135	0.0262	8.46 × 10−11	0.1053	6.58 × 10−18	5.17 × 10−32
	Worst	2.9000	4.74 × 10−5	2.9661	3.14 × 10−20	4.22 × 108	44.9396	62.2468	0.8651	0.6225	1.9644	2.0793	5.25 × 10−31
	Mean	0.2133	8.29 × 10−6	2.9576	1.13 × 10−21	4.28 × 107	28.5879	16.4883	0.2789	0.0424	0.9593	0.1829	3.32 × 10−31
	Std	0.7186	1.11 × 10−5	0.0327	5.72 × 10−21	7.76 × 107	7.8756	11.8333	0.2312	0.1219	0.4784	0.4322	2.23 × 10−31
	Rank	6	3	9	2	12	11	10	7	4	8	5	1
F14	Best	1.0750	0.9980	0.9980	0.9980	0.9980	0.9980	0.9980	0.9980	0.9980	0.9980	0.9980	0.9980
	Worst	12.6705	12.6705	12.6705	12.6705	19.6152	0.9980	0.9980	10.7632	10.7632	12.6705	9.8172	10.7632
	Mean	3.4254	3.9024	10.5733	2.3695	5.1382	0.9980	0.9980	2.1452	1.9830	5.1199	3.1632	4.0650
	Std	3.0067	4.0939	3.9373	2.8235	5.0099	0	0	2.4762	2.1383	4.0803	2.1437	4.1863
	Rank	7	8	12	5	11	1	1	4	3	10	6	9
F15	Best	4.25 × 10−4	3.09 × 10−4	3.11 × 10−4	3.10 × 10−4	3.07 × 10−4	4.84 × 10−4	3.07 × 10−4	3.08 × 10−4	3.07 × 10−4	3.51 × 10−4	1.41 × 10−3	3.07 × 10−4
	Worst	2.60 × 10−3	6.55 × 10−4	1.09 × 10−1	2.63 × 10−2	2.26 × 10−2	1.10 × 10−3	3.07 × 10−4	1.51 × 10−3	2.04 × 10−2	5.58 × 10−3	1.12 × 10−2	1.59 × 10−3
	Mean	1.15 × 10−3	4.39 × 10−4	6.46 × 10−3	3.05 × 10−3	7.39 × 10−3	9.73 × 10−4	3.07 × 10−4	6.33 × 10−4	2.61 × 10−3	2.17 × 10−3	3.13 × 10−3	4.18 × 10−4
	Std	5.11 × 10−4	8.33 × 10−5	2.03 × 10−2	6.38 × 10−3	9.17 × 10−3	1.34 × 10−4	1.25 × 10−19	4.02 × 10−4	6.04 × 10−3	1.56 × 10−3	2.32 × 10−3	2.87 × 10−4
	Rank	6	3	11	9	12	5	1	4	8	7	10	2
F16	Best	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316
	Worst	−1.0250	−1.0307	−1.0316	−1.0043	−1.0303	−1.0316	−1.0316	−1.0316	−1.0316	−0.9721	−1.0316	−1.0316
	Mean	−1.0310	−1.0314	−1.0316	−1.0302	−1.0315	−1.0316	−1.0316	−1.0316	−1.0316	−1.0238	−1.0316	−1.0316
	Std	0.0012	0.0002	5.42 × 10−12	0.0056	0.0003	1.77 × 10−9	3.77 × 10−7	5.28 × 10−11	5.45 × 10−16	0.0161	5.45 × 10−16	6.65 × 10−16
	Rank	10	9	4	11	8	4	7	6	1	12	1	1
F17	Best	0.3982	0.3979	0.4015	0.3979	0.3979	0.3979	0.3979	0.3979	0.3979	0.3979	0.3979	0.3979
	Worst	0.4648	0.3981	5.3931	0.5369	0.6465	0.3979	0.3982	0.3979	0.3979	0.4050	0.3979	0.3979
	Mean	0.4084	0.3980	1.6508	0.4058	0.4148	0.3979	0.3979	0.3979	0.3979	0.3987	0.3979	0.3979
	Std	0.0156	6.38 × 10−5	1.1813	0.0292	0.0495	1.45 × 10−9	6.13 × 10−5	2.52 × 10−6	0	0.0015	0	0
	Rank	10	7	12	9	11	4	6	5	1	8	1	1
F18	Best	3.0000	3.0002	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000
	Worst	92.0344	3.0635	84.0000	30.0212	30.0000	3.0000	3.0000	3.0003	30.0000	34.1840	3.0000	3.0000
	Mean	7.7850	3.0131	12.0000	4.9489	4.8000	3.0000	3.0000	3.0000	4.8000	10.5840	3.0000	3.0000
	Std	17.3466	0.0153	17.8442	6.8260	6.8501	2.07 × 10−13	1.40 × 10−15	5.46 × 10−5	6.8501	12.8123	3.38 × 10−15	9.96 × 10−9
	Rank	10	6	12	9	8	3	1	5	7	11	2	4
F19	Best	−3.8625	−3.8626	−3.8628	−3.8628	−3.8628	−3.8628	−3.8628	−3.8628	−3.8628	−3.8626	−3.8628	−3.8628
	Worst	−3.7534	−3.8490	−3.8627	−3.7348	−3.8289	−3.8628	−3.8628	−3.8518	−3.8628	−3.0551	−3.8628	−3.8628
	Mean	−3.8285	−3.8591	−3.8628	−3.8380	−3.8608	−3.8628	−3.8628	−3.8601	−3.8628	−3.7390	−3.8628	−3.8628
	Std	0.0325	0.0029	1.68 × 10−5	0.0320	0.0065	2.12 × 10−15	2.71 × 10−15	0.0030	2.71 × 10−15	0.1959	2.39 × 10−15	7.73 × 10−7
	Rank	11	9	6	10	7	1	1	8	1	12	1	5
F20	Best	−3.0811	−3.2967	−3.3220	−3.3220	−3.3220	−3.3220	−3.3220	−3.3220	−3.3220	−3.1830	−3.3220	−3.3220
	Worst	−2.1505	−3.0812	−3.2031	−2.9599	−1.7061	−3.3003	−3.2031	−3.0777	−3.2031	−1.0816	−3.3220	−3.2007
	Mean	−2.7328	−3.1980	−3.2903	−3.2488	−3.1404	−3.3210	−3.3061	−3.2503	−3.2824	−2.7700	−3.3220	−3.2703
	Std	0.2495	0.0613	0.0535	0.0787	0.2896	0.0040	0.0411	0.0885	0.0570	0.4578	1.42 × 10−15	0.0600
	Rank	12	9	4	8	10	2	3	7	5	11	1	6
F21	Best	−5.0552	−10.1532	−10.1532	−10.1532	−10.1532	−10.1532	−10.1532	−10.1532	−10.1532	−9.4886	−10.1532	−10.1532
	Worst	−5.0552	−10.0970	−5.0552	−2.6829	−2.6305	−3.0705	−2.6829	−0.8820	−2.6305	−1.0152	−2.6305	−2.6305
	Mean	−5.0552	−10.1462	−9.1411	−9.0693	−5.9358	−9.0487	−9.9042	−8.1773	−7.8057	−3.6896	−5.6345	−7.1401
	Std	3.56 × 10−7	0.0124	2.0586	2.5136	3.1874	2.1675	1.3639	3.2052	3.2236	1.9158	3.5292	3.3801
	Rank	11	1	3	4	9	5	2	6	7	12	10	8
F22	Best	−6.0680	−10.4027	−10.4029	−10.4029	−10.4029	−10.4029	−10.4029	−10.4027	−10.4029	−10.3589	−10.4029	−10.4029
	Worst	−5.0877	−10.3763	−1.8376	−2.7519	−2.2436	−2.7519	−10.4029	−2.7520	−2.7519	−0.9701	−10.4029	−2.7519
	Mean	−5.1203	−10.3995	−8.1584	−8.9844	−6.2099	−9.4988	−10.4029	−9.0462	−8.8090	−3.3780	−10.4029	−6.3786
	Std	0.1790	0.0057	3.2929	2.9137	3.5547	1.8634	1.14 × 10−15	2.5524	2.9825	1.6987	1.36 × 10−15	3.2108
	Rank	11	3	8	6	10	4	1	5	7	12	1	9
F23	Best	−6.2744	−10.5363	−10.5364	−10.5364	−10.5364	−10.5364	−10.5364	−10.5363	−10.5364	−10.4861	−10.5364	−10.5364
	Worst	−5.1285	−10.4968	−2.4217	−3.8354	−2.2763	−3.0887	−3.8354	−1.6765	−2.4273	−1.2000	−10.5364	−1.6766
	Mean	−5.1667	−10.5293	−6.8474	−9.9110	−6.0285	−9.8067	−10.3130	−7.6976	−8.4699	−3.4333	−10.5364	−5.4245
	Std	0.2092	0.0093	3.6133	1.9191	3.6904	1.9174	1.2234	3.4119	3.2542	1.8170	2.58 × 10−15	3.5552
	Rank	11	2	8	4	9	5	3	7	6	12	1	10
Mean rank		6.3913	4.0435	7.5217	6.5217	10.5217	7.2174	6.6522	5.4348	4.4783	7.0435	5.8261	3.8696
Final ranking		6	2	11	7	12	10	8	4	3	9	5	1
+/−/=		1/11/11	6/6/11	0/6/17	5/1/17	1/6/16	3/6/14	7/4/12	3/7/13	7/8/8	1/6/16	5/6/12	1/11/11

provides the numerical case of 12 MH algorithms in solving 23 benchmark problems, where the bold indicates the most available mean and standard deviation of all algorithms. From the table, LICRSA ranked first with an average of 3.8696, and LICRSA ranked first in 10 of these test problems and second or higher in 13. The LICRSA algorithm performs the best in single-peak, ranking second or higher in 6 out of 7 unimodal functions. Meanwhile, LICRSA also performed well in multimodal functions, obtaining optimal solutions in 4 of the 6 functions (F9, F10, F11, F13). In addition, AO ranked first in 5, outperforming other MH algorithms on F5, F11, and F21, second only to LICRSA. In addition, GSA showed dominance in the fixed dimension, achieving the best runs on F16, F17, F19, F20, F22, and F23. The results show that LICRSA has more robust solution accuracy than other existing algorithms in unimodal and multimodal functions when solving the 23 benchmark problems, indicating that LICRSA can obtain a greater chance of convergence success to the optimal value resulting in good development performance. It performs better than most search algorithms in the fixed dimension. Figure 5 provides a bar plot of the mean rank of different algorithms.

In this section, the convergence analysis of the search algorithm is performed on 23 benchmark test suites. Figure 6 provides the number of functions better than LICRSA, significantly the same as LICRSA, and worse than LICRSA. Figure 7 shows the results of the problems for the 23 test problems. Some of these functions have images for which we have taken logarithmic data processing. From the results in the figure, it can be noticed that LICRSA starts with a fast convergence and mutation behavior on most of the problems, which is more pronounced than the other tested algorithms. This result is due to the excellent exploration capability of the added interaction crossover strategy. Moreover, the results display that LICRSA has the fastest convergence on most functions than AO, AOA, RSA, CSO, DE, GSA, WOA, WHOA, DMO, SOA, and WSO. This result is due to the reduced search space and enhanced exploitation capability at iteration time by Lévy flight and crossover strategies. Also, from the convergence curves, LICRSA converges quickly to the global optimum for most unimodal and multimodal functions (F1, F3, F5, F9, F10, F11), indicating that LICRSA has better convergence capability. The results further verify the effectiveness of the Lévy flight and the interaction crossover strategy, which enhances the local search capability. In addition, Figure 8 provides the box plots of the 23 test functions, from which it is clear that the box of LICRSA is narrower, which also indicates the stability ability of LICRSA for multiple runs, and then LICRSA has some instability in the face certain fixed dimensional test functions.

Table 3. p-value and statistical results.

p-Value	RSA	AO	AOA	CSO	DE	DMO	WSO	WOA	WHOA	SOA	GSA
F1	NaN/=	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	NaN/=	1.21 × 10−12/−
F2	NaN/=	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.93 × 10−9/−	1.21 × 10−12/−
F3	NaN/=	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−
F4	NaN/=	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−
F5	1.95 × 10−3/−	0.0103/+	3.02 × 10−11/−	1.78 × 10−10/−	3.02 × 10−11/−	3.02 × 10−11/−	3.02 × 10−11/−	3.69 × 10−11/−	3.02 × 10−11/−	3.02 × 10−11/−	3.02 × 10−11/−
F6	3.69 × 10−11/−	3.02 × 10−11/+	1.09 × 10−5/−	3.02 × 10−11/+	3.52 × 10−7/−	3.02 × 10−11/+	3.02 × 10−11/−	3.02 × 10−11/+	3.02 × 10−11/+	4.98 × 10−4/−	3.02 × 10−11/+
F7	0.8073/=	0.0646/=	0.4733/=	3.02 × 10−11/−	3.02 × 10−11/−	3.02 × 10−11/−	3.02 × 10−11/−	8.09 × 10−10/−	1.41 × 10−9/−	0.0625/=	3.02 × 10−11/−
F8	2.89 × 10−3/−	0.0003/+	0.0436/−	4.11 × 10−12/+	2.92 × 10−9/+	2.36 × 10−10/−	2.32 × 10−6/+	3.02 × 10−11/+	3.02 × 10−11/+	2.27 × 10−11/+	3.02 × 10−11/−
F9	NaN/=	NaN/=	1.95 × 10−9/−	3.86 × 10−10/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	NaN/=	NaN/=	NaN/=	1.21 × 10−12/−
F10	NaN/=	NaN/=	1.21 × 10−12/−	1.21 × 10−12/−	1.13 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	3.06 × 10−9/−	2.64 × 10−5/−	NaN/=	1.21 × 10−12/−
F11	NaN/=	NaN/=	1.21 × 10−12/−	2.93 × 10−5/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	NaN/=	NaN/=	NaN/=	1.21 × 10−12/−
F12	1.45 × 10−10/−	3.02 × 10−11/+	3.83 × 10−6/−	3.02 × 10−11/+	3.02 × 10−11/−	3.02 × 10−11/−	1.31 × 10−8/−	2.61 × 10−10/+	5.94 × 10−9/+	0.5793/=	0.0012/+
F13	0.0232/+	3.00 × 10−11/−	3.00 × 10−11/−	3.00 × 10−11/−	3.00 × 10−11/−	3.00 × 10−11/−	3.00 × 10−11/−	3.00 × 10−11/−	2.98 × 10−11/−	3.00 × 10−11/−	2.98 × 10−11/−
F14	0.0619/=	0.0823/=	3.83 × 10−8/−	0.1257/=	0.0667/=	1.19 × 10−5/+	1.19 × 10−5/+	0.1738/+	0.0475/+	0.0064/−	0.1938/=
F15	3.04 × 10−9/−	5.23 × 10−5/−	0.0002/−	8.26 × 10−8/−	9.17 × 10−7/−	7.47 × 10−9/−	6.61 × 10−5/+	7.50 × 10−6/−	0.0628/−	1.00 × 10−9/−	2.20 × 10−11/−
F16	1.21 × 10−12/−	1.21 × 10−12/−	NaN/=	8.87 × 10−7/−	0.0419/−	0.0419/−	0.0815/=	NaN/=	NaN/=	1.21 × 10−12/−	NaN/=
F17	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	4.79 × 10−8/−	0.0003/−	0.1608/=	0.0815/=	1.70 × 10−8/−	NaN/=	1.21 × 10−12/−	NaN/=
F18	2.64 × 10−12/−	2.37 × 10−12/−	0.0261/−	2.73 × 10−12/−	0.4022/=	0.1608/=	0.1608/=	4.12 × 10−12/−	0.9591/=	1.13 × 10−11/−	0.1608/=
F19	1.72 × 10−12/−	1.72 × 10−12/−	4.10 × 10−11/−	1.93 × 10−12/−	0.0453/−	0.3337/=	0.3337/=	2.15 × 10−12/−	0.3337/=	1.72 × 10−12/−	0.3337/=
F20	3.00 × 10−11/−	2.31 × 10−6/−	0.6099/=	0.0061/−	0.1252/=	0.6734/=	1.90 × 10−7/+	0.0020/−	0.0002/+	3.00 × 10−11/−	1.93 × 10−10/+
F21	1.86 × 10−3/−	0.8180/=	0.8180/=	0.0012/+	0.2979/=	0.6830/=	6.50 × 10−6/+	0.2498/=	0.2160/=	5.54 × 10−5/−	0.3788/=
F22	0.0985/=	0.0267/+	0.3622/=	0.0729/=	0.4299/=	0.0694/=	1.62 × 10−8/+	0.2827/=	0.0016/+	3.93 × 10−5/−	1.62 × 10−8/+
F23	0.6094/=	0.0019/+	0.5291/=	0.0024/+	0.5080/=	0.0066/+	8.25 × 10−8/+	0.2572/=	0.0007/+	0.0480/−	1.66 × 10−8/+
+/−/=	1/11/11	6/6/11	0/6/17	5/1/17	1/6/16	3/6/14	7/4/12	3/7/13	7/8/8	1/6/16	5/6/12

provides the final results of p-values and statistics between each algorithm and LICRSA. 1/11/11, 6/6/11, 0/6/17, 5/1/17, 1/6/16, 3/6/14 for RSA, AO, AOA, CSO, DE, and DMO, respectively, and 7/4/12, 7/4/12, 7/7/13, 7/7/13, and 7/7/13 for WSO, WOA, WHOA, SOA, and GSA. 3/7/13, 7/8/8, 1/6/16, 5/6/12.

4.3. Performance Evaluation for the CEC-2020 Test Functions

These 10 CEC2020 include a multi-peak function, 3 BASIC functions, 3 hybrid functions, and 3 composite functions [5]. The multimodal functions are usually employed to understand the development of the search algorithm, and hybrid functions can better combine the properties of sub-functions and maintain the continuity of global/local optima. The composite function combines various hybrid functions and can be employed to evaluate the convergence accuracy of the algorithm. Note that the objective of all 10 CEC2020 functions is also the minimization problem, and again, this section still uses the maximum, minimum, average, and standard deviation (Std) in the results of the numerical experiments. In addition, time metrics and p-values are also important factors in the evaluation system. In addition, the Wilcoxon signed-rank test (significance level = 0.05) is used to test the numerical cases; where “+” indicates that an algorithm is more accurate than LICRSA, “=” means that LICRSA is not different from MH, and “−” shows that a method is less accurate than LICRSA.

Table 4. Results of LICRSA and MH algorithms on CEC2020.

Function	Index	Algorithms
		RSA	AO	AOA	DMO	DE	GOA	GSA	HBA	HHO	SSA	WOA	LICRSA
Cec01	Best	5.25 × 109	2.31 × 105	1.17 × 1010	1.08 × 102	1.14 × 102	105.0520	100.1725	100.7955	1.53 × 105	100.0566	9.64 × 105	2.33 × 105
	Worst	1.95 × 1010	5.25 × 106	2.28 × 1010	2.32 × 104	1.94 × 109	1.74 × 109	2096.4092	1.27 × 104	1.97 × 106	1.03 × 104	5.97 × 107	4.91 × 107
	Mean	1.21 × 1010	1.07 × 106	1.80 × 1010	3.69 × 103	1.77 × 108	6.49 × 107	612.4027	4239.6784	5.64 × 105	2.54 × 103	7.75 × 106	1.24 × 107
	Std	3.67 × 109	9.43 × 105	2.92 × 109	5.14 × 103	5.02 × 108	3.16 × 108	569.1847	3604.0083	4.41 × 105	2.54 × 103	1.15 × 107	1.08 × 107
	Rank	11	6	12	3	10	9	1	4	5	2	7	8
Cec02	Best	2233.8285	1455.4387	1933.0870	2260.5966	1496.3762	1361.8998	1778.8631	1133.5292	1580.2253	1454.5649	1489.7424	1417.8513
	Worst	3010.5967	2375.2560	2592.4725	2853.3183	3513.4552	3086.1677	3144.7676	2862.7351	2601.7545	2553.6413	2789.9846	2055.7785
	Mean	2670.1586	2009.3093	2277.7283	2601.1095	2583.7915	2064.7889	2562.1481	1823.7617	2124.1770	1966.2834	2178.0227	1808.0986
	Std	196.3505	227.0938	180.5223	152.3035	484.9853	390.7645	337.9130	361.7304	258.7543	312.4417	345.7043	174.3287
	Rank	12	4	8	11	10	5	9	2	6	3	7	1
Cec03	Best	785.7836	723.2028	779.0469	729.1209	721.3701	720.3822	713.1061	717.6301	750.6805	720.5980	736.8136	724.4730
	Worst	829.3164	795.2918	811.5786	746.9329	836.8638	747.0424	720.3908	744.9706	832.0996	782.1344	821.4098	754.7121
	Mean	809.4439	751.4883	798.3304	736.7774	760.1927	730.7181	716.6472	731.2833	786.1963	738.3313	780.5855	745.0383
	Std	10.0994	14.9263	9.3729	5.1123	30.1127	7.4560	2.0621	7.6149	21.8597	13.4556	22.5280	7.0427
	Rank	12	7	11	4	8	2	1	3	10	5	9	6
Cec04	Best	1900	1900	1900	1901.2752	1900.9328	1900.4318	1900.4897	1900	1900	1900.7669	1900	1900
	Worst	1900	1900	1900	1902.6883	2004.6275	1903.5392	1901.3797	1900	1900	1903.7138	1900.6822	1900
	Mean	1900	1900	1900	1902.0151	1912.2234	1901.2387	1900.8736	1900	1900	1901.8996	1900.0665	1900
	Std	0	2.11 × 10−11	0	0.3909	22.9276	0.6453	0.2516	0	0	0.7755	0.1920	0
	Rank	1	6	1	11	12	9	8	1	1	10	7	1
Cec05	Best	3.66 × 105	3497.5707	2.62 × 105	3541.4806	2379.8512	2512.2289	4.84 × 104	2017.6515	4849.9726	2539.7317	6306.7836	2303.1418
	Worst	6.13 × 105	1.64 × 105	3.97 × 105	3.95 × 104	1.42 × 106	7.28 × 104	1.08 × 106	1.40 × 104	1.37 × 105	1.72 × 104	2.23 × 106	9158.3329
	Mean	5.12 × 105	3.35 × 104	3.30 × 105	1.64 × 104	1.33 × 105	1.32 × 104	5.42 × 105	5449.8243	5.28 × 104	5569.6814	2.91 × 105	4054.6215
	Std	7.09 × 104	4.16 × 104	4.48 × 104	9594.7660	2.95 × 105	1.67 × 104	2.14 × 105	2852.2526	4.78 × 104	3273.2674	4.81 × 105	2066.5253
	Rank	11	6	10	5	8	4	12	2	7	3	9	1
Cec06	Best	1837.8767	1631.6130	1875.0060	1602.0110	1719.7169	1617.9243	1852.2086	1601.5773	1614.0400	1602.7027	1730.8345	1601.5253
	Worst	2506.5128	2028.3591	2276.0362	1772.9683	2188.9717	2138.8684	2256.3544	2350.9541	2304.6563	1928.0415	1970.7864	1734.4909
	Mean	2121.9547	1726.3884	2085.6802	1641.7756	1922.3552	1866.8544	2046.9232	1792.6263	1850.0422	1737.6906	1825.6587	1652.9791
	Std	151.2529	79.0523	123.1877	46.4837	124.7613	127.5918	110.2315	162.2748	168.8979	96.7318	73.0895	50.6153
	Rank	12	3	11	1	9	8	10	5	7	4	6	2
Cec07	Best	5996.9070	2976.2827	2615.7983	2923.9581	2141.4152	2397.8704	1.70 × 104	2226.4011	2719.8088	2606.9487	6096.9952	2315.6893
	Worst	1.58 × 107	3.22 × 104	3066.1488	1.05 × 104	3.54 × 106	3.45 × 104	3.43 × 106	3.20 × 104	3.23 × 104	2.21 × 104	1.91 × 106	5017.1320
	Mean	1.18 × 106	8269.0073	2839.1191	4981.9422	1.31 × 105	8927.5566	7.72 × 105	3757.9868	1.07 × 104	7843.9504	2.06 × 105	2963.2802
	Std	3.06 × 106	6547.3923	112.1776	1970.2851	6.45 × 105	8898.3078	6.26 × 105	5355.4666	9058.4200	6026.2143	3.68 × 105	538.7066
	Rank	12	6	1	4	9	7	11	3	8	5	10	2
Cec08	Best	2677.9746	2304.9239	2780.6604	2293.9792	2300.2890	2300.9122	2300.0000	2300.3982	2306.2332	2241.5107	2247.2240	2239.7843
	Worst	3745.9831	2318.9154	4201.1369	2306.1535	3619.5033	3610.1215	2300.3975	2306.7348	2922.5468	2308.2670	3696.0385	2313.7217
	Mean	3104.4294	2310.3261	3810.0480	2302.3789	2388.0535	2460.4582	2300.0612	2302.0172	2334.2267	2301.3411	2398.8595	2304.0625
	Std	277.3221	3.1595	290.1835	2.3942	271.6648	406.0217	0.1259	1.3538	111.2365	11.4201	323.1133	13.7804
	Rank	11	6	12	4	8	10	1	3	7	2	9	5
Cec09	Best	2681.1103	2501.1045	2693.5585	2727.1328	2566.9071	2500.0001	2500.0000	2733.8946	2743.7557	2500.0001	2533.4263	2504.2400
	Worst	2983.5836	2808.1653	2977.6551	2771.0739	2807.6560	2836.4091	2844.9998	2781.0887	2988.9704	2768.0762	2842.1872	2775.6546
	Mean	2852.2613	2729.8452	2845.7016	2759.8759	2761.4579	2751.5740	2621.0530	2756.2384	2831.4282	2743.4119	2766.8724	2662.7931
	Std	48.3010	92.0422	87.2370	8.3941	40.7486	102.2275	151.4980	11.6448	54.7859	46.8448	65.3251	129.6056
	Rank	12	3	11	7	8	5	1	6	10	4	9	2
Cec10	Best	3200.6280	2625.7281	3333.3941	2897.9404	2899.1962	2800.0030	2897.9404	2897.7457	2898.2569	2897.7429	2899.5543	2898.4508
	Worst	3648.6327	2951.4412	4302.9415	2946.2567	3024.6391	3024.3401	2944.6881	2969.4344	3027.5770	2954.3007	3032.6286	2908.5549
	Mean	3377.4284	2924.9374	3811.5261	2922.2510	2946.3726	2928.6441	2934.6691	2933.7543	2927.2049	2934.2698	2950.6396	2901.3998
	Std	97.1819	60.2375	305.7569	20.9226	31.6026	38.4435	17.9829	22.3382	29.6627	21.9263	34.0961	2.4429
	Rank	11	3	12	2	9	5	8	6	4	7	10	1
Mean Rank		10.5	5	8.9	5.2	9.1	6.4	6.2	3.5	6.5	4.5	8.3	2.9
Final Ranking		12	4	10	5	11	7	6	2	8	3	9	1

provides a comparison between LICRSA and different MH algorithms. The table indicates that LICRSA has the best running results in four test functions, cec02, cec04, cec05, cec10, and the second ranking in cec06, cec07, and cec09, achieving the second-ranking. The overall average order of LICRSA of 2.9 is better than the other MH algorithms, indicating that LICRSA can handle this test function well. Figure 9 and Figure 10 provide the iterative and box-line plots for the 10 test functions of this test set, respectively, from which it is found that LICRSA can find the neighborhood of the optimal case in the earlier part of the iteration for most of the test functions, and has a faster convergence rate compared to the other comparative search algorithms. Additionally, in Figure 10 it can be found that LICRSA tends to have a narrower box. It has very stable results when run many times. Figure 11 provides the radar plots of the 12 algorithms, and the more concentrated shaded parts represent a smaller rank ranking and better performance in CEC2020. It can be seen from the figure that LICRSA has a more minor shaded part.

Table 5. p-value and statistical results on CEC2020.

p-Value	RSA	AO	AOA	DMO	DE	GOA	GSA	HBA	HHO	SSA	WOA
CEC01	3.02 × 10−11/−	9.06 × 10−8/+	3.02 × 10−11/−	3.02 × 10−11/+	8.66 × 10−5/−	8.48 × 10−9/−	3.02 × 10−11/+	3.02 × 10−11/+	1.55 × 10−8/+	3.02 × 10−11/+	0.0468/+
CEC02	3.02 × 10−11/−	0.0001/−	2.15 × 10−10/−	3.02 × 10−11/−	1.70 × 10−8/−	9.21 × 10−5/−	6.72 × 10−10/−	0.8073/=	5.86 × 10−6/−	0.0378/−	3.16 × 10−5/−
CEC03	3.02 × 10−11/−	0.0933/=	2.97 × 10−11/−	5.46 × 10−6/+	0.0594/=	1.85 × 10−8/+	3.02 × 10−11/+	4.31 × 10−8/+	2.15 × 10−10/−	0.0017/+	1.07 × 10−9/−
CEC04	NaN/=	NaN/=	NaN/=	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	1.21 × 10−12/−	NaN/=	NaN/=	1.21 × 10−12/−	0.0419/−
CEC05	3.02 × 10−11/−	1.31 × 10−8/−	3.02 × 10−11/−	3.50 × 10−9/−	4.31 × 10−8/−	0.0016/−	3.02 × 10−11/−	0.0163/−	4.18 × 10−9/−	0.0108/−	8.15 × 10−11/−
CEC06	3.02 × 10−11/−	8.29 × 10−6/−	3.02 × 10−11/−	0.4643/=	7.39 × 10−11/−	5.97 × 10−9/−	3.02 × 10−11/−	6.35 × 10−5/−	9.26 × 10−9/−	0.0002/−	3.69 × 10−11/−
CEC07	3.02 × 10−11/−	1.07 × 10−9/−	0.9000/=	3.82 × 10−9/−	0.8303/=	2.83 × 10−8/−	3.02 × 10−11/−	0.1715/=	5.46 × 10−9/−	7.60 × 10−7/−	3.02 × 10−11/−
CEC08	3.02 × 10−11/−	0.0002/−	3.02 × 10−11/−	1.39 × 10−6/+	0.2009/=	0.5793/=	2.64 × 10−9/+	5.53 × 10−8/+	3.81 × 10−7/−	6.74 × 10−6/+	7.09 × 10−8/−
CEC09	1.78 × 10−10/−	0.2838/=	5.60 × 10−7/−	0.4733/=	0.0108/−	2.88 × 10−6/−	0.1567/=	0.6520/=	2.60 × 10−8/−	0.5201/=	5.61 × 10−5/−
CEC10	3.02 × 10−11/−	4.74 × 10−6/−	3.02 × 10−11/−	0.0261/−	1.03 × 10−6/−	0.0040/−	1.33 × 10−5/−	0.0015/−	0.0224/−	0.0008/−	5.57 × 10−10/−
+/=/−	0/1/9	1/3/6	0/2/8	3/2/5	0/3/7	1/1/8	3/1/6	3/4/3	1/1/8	3/1/6	1/0/9

and

Table 6. Run time of LICRSA and other search algorithms on CEC2020.

Time	Algorithms
	RSA	LICRSA	AO	AOA	DMO	DE	GOA	GSA	HBA	HHO	SSA	WOA
cec01	26.1783	36.7426	9.8169	3.9698	37.9297	6.2376	50.7474	10.6257	5.9571	10.0409	3.2723	3.5769
cec02	26.5651	37.6956	11.1770	4.5525	38.8434	6.7670	51.5997	11.0695	6.3650	11.7236	3.7308	4.1962
cec03	27.0933	37.3185	10.8828	4.5389	39.1830	6.2591	51.1633	11.0190	6.4026	11.3222	3.6350	4.2131
cec04	26.3118	36.8044	10.4214	4.2247	37.9658	5.8460	50.2218	10.7967	6.3192	11.1092	3.4310	3.7895
cec05	26.6952	37.3432	10.7070	4.3864	38.8799	5.9315	51.8478	10.8394	6.2262	10.7683	3.5137	4.0030
cec06	26.5340	37.1979	10.6527	4.3489	38.7714	5.9239	52.6064	10.8391	6.5493	10.8512	3.5037	3.9887
cec07	26.5542	37.1876	10.6808	4.2912	38.9339	5.8890	53.1767	10.7757	5.9273	11.1000	3.4723	3.9353
cec08	28.1748	40.4331	14.0198	6.0621	42.2293	7.5555	55.7892	12.5570	7.5857	14.9357	5.0989	5.5953
cec09	28.6854	41.2043	14.7554	6.4715	49.3863	8.0228	52.8070	12.9327	8.1251	15.6190	5.4950	5.9161
cec10	28.1078	40.1146	13.5807	5.8813	43.0863	7.3519	52.2190	12.3228	7.5167	14.3918	4.9666	5.3896

provide the p-values of all the search algorithms run in the CEC2020 test set and the running time statistics, respectively. From the tables, it can be found that RSA, AO, AOA, DMO, DE, and GOA are 0/1/9, 1/3/6, 0/2/8, 3/2/5, 0/3/7, 1/1/8, while GSA, HBA, HHO, SSA, and WOA are 3/1/6, 3/4/3, 1/1/8, 3/1/6, 1/0/9. In Table 6, it can be found that LICRSA takes a longer running time, but the increase in running time is a typical situation due to the added improvement policy, while the running time of RSA itself is longer, and the increase in running time of LICRSA compared to RSA is not too much.

5. Case Studies of Real-World Applications

In this section, the performance of LICRSA is further evaluated through five engineering optimization examples. Also, in this experiment, the constraints are treated by linearly weighting the constraints into the objective function. This approach is commonly used [54]. Typically, the optimization equation with a minimization objective is defined as:

Minimize:

f(Z), Z = [z₁, z₂, … z_n]

(27)

Subject to:

$$\left\{\begin{array}{c}{g}_i(Z)\le 0, i=1,\cdots ,m.\\ h_j(Z)=0, j=1,\cdots ,l.\end{array}\right.$$

(28)

where m is the number of constraints and l is the number of equilibrium constraints. Z is a candidate solution that is in the feasible domain. Thus, the equation after linear weighting of the constraints to the objective function is described as:

$$f(Z)=f(Z)+\alpha {\displaystyle \sum _{i=1}^m\max \{g_i(Z), 0\}}+\beta {\displaystyle \sum _{j=1}^l\max \{h_j(Z), 0\}}.$$

(29)

where α is the weight of g(Z), and β is the weight of h(Z). In the practical case, to ensure that the solution is in the feasible domain, the candidate solution is severely penalized by larger weights on the objective function when the g(Z) and h(Z) constraints are exceeded.

5.1. Welded Beam Design Problem

The ultimate goal of the welded beam design optimization is to minimize the fabrication cost of the welded beam. Four decision variables exist for this design problem, namely: the thickness of the weld (h), length of the connected part of the reinforcement (l), the height of the reinforcement (t), and thickness of the reinforcement (b), with seven constraints [4]. The schematic diagram of the welded beam is provided in Figure 12. The mathematical expression of the welded beam optimization is as follows.

Minimize:

$$f(x)=1.10471x_1^2{x}_2+0.04811x_3{x}_4(14.0+x_2),$$

(30)

Subject to:

$$g_1(x)=\tau (x)-{\tau }_{\max }\le 0,$$

(31)

$$g_2(x)=\sigma (x)-{\sigma }_{\max }\le 0,$$

(32)

$$g_3(x)=\delta (x)-{\delta }_{\max }\le 0,$$

(33)

$$g_4(x)=x_1-x_4\le 0,$$

(34)

$$g_5(x)=P-P_C(x)\le 0,$$

(35)

$$g_6(x)=0.125-x_1\le 0,$$

(36)

$$g_7(x)=1.10471x_1^2+0.04811x_3{x}_4(14.0+x_2)-5.0\le 0,$$

(37)

where,

$$\tau (x)=\sqrt{{(\tau \prime )}^2+2\tau \prime \tau ″\frac{x_2}{R}+{(\tau ″)}^2},$$

(38)

$$\tau \prime =\frac{P}{\sqrt{2}{x}_1{x}_2},\tau ″=\frac{MR}{J},$$

(39)

$$M=P(L+\frac{x_2}{2}),$$

(40)

$$R=\sqrt{\frac{x_2^2}{4}+{(\frac{x_1+x_3}{2})}^2},$$

(41)

$$J=2\left\{\sqrt{2x_1{x}_2}\left[\frac{x_2^2}{4}+{\left(\frac{x_1+x_3}{2}\right)}^2\right]\right\},$$

(42)

$$\sigma (\overrightarrow{x})=\frac{6PL}{x_4{x}_3^2},\delta (\overrightarrow{x})=\frac{6P{L}^3}{E{x}_3^2{x}_4},$$

(43)

$$P_c(\overrightarrow{x})=\frac{\sqrt[4.013E]{\frac{x_3^2{x}_4^6}{36}}}{L^2}\left(1-\frac{x_3}{2L}\sqrt{\frac{E}{4G}}\right) ,$$

(44)

$$P=6000 lb,L=14 in,{\delta }_{\max }=0.25 in,E=30\times 1^{6}psi,$$

(45)

$$G=12\times {10}^{6}psi,{\tau }_{\max }=\mathrm{13,600}psi,{\sigma }_{\max }=\mathrm{30,000}psi$$

(46)

Variable range:

$$0.1\le x_1\le 2,0.1\le x_2\le 10,$$

(47)

$$0.1\le x_1\le 2,0.1\le x_2\le 10,$$

(48)

Eleven authoritative and most popular algorithms (AOA [4], WCA [55], EPO [56], LSA [40], GOA [18], ASO [29], GSA [33], HHO [8], SCA [42], WHOA [15], SSA [11]) are selected to ensure reliability during the experimental demonstration and compared with the results of LICRSA for solving and optimizing the welded beam design. The numerical results are presented in

Table 7. Numerical results for the welded beam design problem.

Algorithms	Variables				Fabrication Cost
	h	l	t	b
AOA	0.719368488	4.031933267	7.182900706	0.768872634	6.360027463
EPO	0.579158195	4.831560952	7.378944515	0.679429586	6.260178047
WCA	0.503927824	5.063922345	6.591757927	0.594335236	5.021144989
LSA	0.626915785	3.504310919	7.046422443	0.745973525	4.904396997
GOA	0.623784118	1.731105368	5.361811162	0.669418118	3.073084436
ASO	0.112461952	7.869657099	9.927606782	0.237246274	2.615188815
GSA	0.231152943	4.152861485	7.833468332	0.306215518	2.278234162
HHO	0.176535238	4.215064979	8.987259763	0.213592280	1.810408730
SCA	0.190963193	3.390767270	9.396083298	0.210010956	1.784850890
WHOA	0.240993638	2.747836964	8.472571962	0.241219438	1.783366573
SSA	0.164264918	4.298981953	9.063217616	0.205643980	1.759914368
LICRSA	0.201941354	3.086318875	9.022514058	0.206392365	1.668814555

, which displays the design variables and minimum expenditures for each algorithm optimization, and

Table 8. Statistics of 20 independent operations of welded beam.

Algorithms	Best	Worst	Mean	Std
AOA	2.690744810	11.478409822	6.360027463	2.220504135
EPO	2.141279184	18.277839444	6.260178047	3.535779646
WCA	1.984263283	18.512920304	5.021144989	4.069964644
LSA	3.017467298	7.603956337	4.904396997	1.306668793
GOA	1.900777191	4.661038249	3.073084436	0.726192820
ASO	1.814047463	5.031409356	2.615188815	0.818493036
GSA	1.949095983	2.814336686	2.278234162	0.244958370
HHO	1.668728085	2.092938322	1.810408730	0.133532471
SCA	1.711950739	1.837690043	1.784850890	0.032561529
WHOA	1.660343003	2.377960751	1.783366573	0.192005176
SSA	1.660791846	2.008564223	1.759914368	0.110515407
LICRSA	1.658808128	1.737698284	1.668814555	0.019963180

provides the statistics for 20 runs. In addition,

Table 9. Numerical comparison of the results obtained for the welded beam design problem in the literature.

Algorithms	Variables				Fabrication Cost
	h	l	t	b
AOA [4]	0.194475000	2.570920000	10.000000000	0.201827000	1.716400000
EPO [56]	0.205411000	3.472341000	9.035215000	0.201153000	1.723589000
WCA [55]	0.205728000	3.470522000	9.036620000	0.205729000	1.726427000
AHA [57]	0.205730000	3.470492000	9.036624000	0.205730000	1.724853000
DMO [6]	0.205570500	3.256772400	9.036177000	0.205769600	1.696400000
SO [58]	0.205700000	3.470500000	9.036600000	0.205700000	1.724851931
HHO [8]	0.204039000	3.531061000	9.027463000	0.206147000	1.731990570
SSA [11]	0.205700000	3.471400000	9.036600000	0.205700000	1.724910000
COOT [14]	0.198827100	3.337971000	9.191986000	0.198834500	1.670301154
HBA [51]	0.205700000	3.470400000	9.036600000	0.205700000	1.724510000
GJO [59]	0.205620000	3.471900000	9.039200000	0.205720000	1.725220000
LICRSA	0.201941354	3.086318875	9.022514058	0.206392365	1.668814555

provides the algorithms from the literature (AOA [4], WCA [55], EPO [56], AHA [57], DMO [6], SO [58], HHO [8], SSA [11], COOT [14], HBA [52]. GJO [59]) obtained. The table displays that LICRSA finds the optimal results and design variables for the welded beam design. The results show that LICRSA is one of the excellent methods to solve the problem.

5.2. Pressure Vessel Design Problem

The ultimate requirement for pressure vessel design is to minimize the cost of fabrication, welding, and materials for the pressure vessel [4]. Figure 13 prompts us that a hemispherical head capped the cylindrical vessel at both ends. Four relevant design variables need to be considered for optimization, including shell thickness T_s (x₁), head thickness T_h (x₂), internal diameter R (x₃), and vessel cylindrical cross-section length L (x₄). The mathematical optimization model for pressure vessel design is as follows:

Minimize:

$$f(\overrightarrow{x})=0.6224x_1{x}_3{x}_4+1.7781x_2{x}_3^2+3.1661x_1^2{x}_4+19.84x_1^2{x}_3,$$

(49)

Subject to:

$$g_1(\overrightarrow{x})=-x_1+0.0193x_3\le 0,$$

(50)

$$g_2(\overrightarrow{x})=-x_2+0.00954x_3\le 0,$$

(51)

$$g_3(\overrightarrow{x})=-\pi x_3^2{x}_4-\frac{4}{3}\pi x_3^3+1,296,000\le 0,$$

(52)

$$g_4(\overrightarrow{x})=x_4-240\le 0,$$

(53)

Variable range:

$$0\le x_1,x_2\le 99,$$

(54)

$$10\le x_3,x_4\le 200,$$

(55)

In this section, eleven different MH algorithms (AOA [4], CSO [48], HHO [8], BOA [26], GSA [33], GBO [13], SMA [22], SSA [11], COOT [14], HGS [39], AO [23]) are used to compare with the experimental results of LICRSA and provide the experimental results. Among them,

Table 10. Numerical results for the pressure vessel design problem.

Algorithms	Variables				Minimize Total Cost
	x1	x2	x3	x4
AOA	27.530388	34.138394	63.069849	100.717010	26,631.517725
CSO	16.809529	8.928204	54.953414	75.344837	6697.878507
HHO	16.299333	8.018776	52.961474	82.816687	6454.926298
BOA	16.848615	8.220844	56.859497	60.135991	6407.727518
GSA	15.810543	7.963037	51.272371	90.229725	6384.404789
GBO	15.295085	7.579440	49.530667	110.866599	6294.769796
SMA	14.624979	7.242314	48.097655	132.372275	6196.180613
SSA	14.528154	7.419590	47.313095	127.549461	6184.961000
COOT	14.326820	7.240143	46.274177	136.306179	6129.945260
HGS	14.075195	7.035452	46.056238	147.375773	6063.502432
AO	13.630352	6.748551	45.315957	157.350850	6037.109755
LICRSA	13.586214	6.874278	45.168063	155.115594	6014.626610

provides the optimization results for the four design variables and the minimum overall cost.

Table 11. Statistics of 20 independent operations of pressure vessel.

Algorithms	Best	Worst	Mean	Std
AOA	5897.462372	46,501.56488	26,631.51772	13,276.53509
CSO	5628.027711	7674.402414	6697.878507	601.2257852
HHO	5627.760854	6960.328263	6454.926298	330.0697368
BOA	4136.939768	8959.042006	6407.727518	1197.933953
GSA	5978.230885	6917.020959	6384.404789	195.4921284
GBO	5654.370337	7332.841508	6294.769796	459.1550096
SMA	5654.370311	7332.851947	6196.180613	660.8464512
SSA	5653.718693	6820.410235	6184.961	333.793435
COOT	5654.370265	6820.410353	6129.94526	256.5232078
HGS	5654.370337	7339.600755	6063.502432	537.4203866
AO	5618.334168	6815.937061	6037.109755	383.0356993
LICRSA	5654.369001	5658.427987	5654.789764	1.222105494

provides the statistics for 20 independent runs. In addition,

Table 12. Numerical comparison of the results obtained for the pressure vessel in the literature.

Algorithms	Variables				Minimize Total Cost
	x1	x2	x3	x4
AOA [4]	0.830374	0.416206	42.751270	169.345400	6048.784400
HHO [8]	0.817584	0.407293	42.091746	176.719635	6000.462590
SMA [22]	0.793100	0.393200	40.671100	196.217800	5994.185700
AO [23]	1.054000	0.182806	59.621900	38.805000	5949.225800
SO [58]	0.781900	0.385700	40.575200	196.549900	5887.529768
GJO [59]	0.778296	0.384805	40.321870	200.000000	5887.071123
AVOA [60]	0.778954	0.385037	40.360312	199.434299	5886.676593
AHA [57]	0.778171	0.384653	40.319674	199.999262	5885.353690
COOT [14]	0.778170	0.384651	40.319618	200.000000	5885.348700
SMO [5]	0.778169	0.384649	40.319624	199.999928	5885.332950
CSA [61]	12.450698	6.154387	40.319619	200.000000	5885.332700
LICRSA	13.586214	6.874278	45.168063	155.115594	5654.789764

provides the results obtained by the algorithms (AOA [4], HHO [8], SMA [22], AO [23], SO [58], GJO [59], AVOA [60], AHA [57], COOT [14], SMO [5], CSA [61]) in the literature. According to the table, LICRSA has a minimum cost of 6014.62661, and the results also prove that LICRSA can handle the pressure vessel design problem very well.

5.3. Three-Bar Truss Design Problem

The ultimate goal of the three-bar truss design is to minimize the volume of a statically loaded three-bar truss constrained by the stresses (σ) in each bar. Two design variables exist for this problem as cross-sectional area A₁ = x₁ and cross-sectional area A₂ = x₂. A schematic design of the triple truss is given in Figure 14 [5]. The mathematical optimization model of the three-bar truss design is shown below.

Minimize:

$$f(\overrightarrow{x})=(2\sqrt{2}{x}_1+x_2)\ast l,$$

(56)

Subject to:

$$g_1(\overrightarrow{x})=\frac{\sqrt{2}{x}_1+x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}P-\sigma \le 0,$$

(57)

$$g_2(\overrightarrow{x})=\frac{x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}P-\sigma \le 0,$$

(58)

$$g_3(\overrightarrow{x})=\frac{1}{\sqrt{2}{x}_2+x_1}P-\sigma \le 0,$$

(59)

Variable range:

$$0\le x_1,x_2\le 1,$$

(60)

where,

$$\mathrm{l}=100 \mathrm{cm},P=2 \mathrm{KN}/{\mathrm{cm}}^2,\sigma =2 \mathrm{KN}/{\mathrm{cm}}^2.$$

(61)

The experiments of LICRSA and other well-known MH techniques (ASO [29], SCA [42], CSO [48], G-QPSO [62], AO [23], GSA [33], HHO [8], GWO [20], SSA [11], JS [16], GBO [13]) are provided in

Table 13. Results for the three-bar truss design.

Algorithms	Variables		Minimize Volume
	x1	x2
ASO	0.90514715	0.23210052	279.22432697
SCA	0.83127880	0.32661521	267.78267280
CSO	0.77483698	0.45542842	264.69983622
G-QPSO	0.79402041	0.39700591	264.28347770
AO	0.78613749	0.41791460	264.14472144
GSA	0.78190299	0.42840643	263.99620601
HHO	0.79160624	0.40082108	263.98216359
GWO	0.78910501	0.40707138	263.89982151
SSA	0.78800142	0.41017025	263.89748524
JS	0.78862473	0.40839151	263.89590834
GBO	0.78867699	0.40824306	263.89584379
LICRSA	0.78867514	0.40824829	263.89584338

. It is clear from the experimental results that LICRSA outperforms selected optimization methods. The statistical results of various optimizers, including the LICRSA, are shown in

Table 14. Statistics of 20 independent operations of three-bar truss design.

Algorithms	Best	Worst	Mean	Std
ASO	269.1574840	282.8427125	279.2243270	4.3189100
SCA	263.9019374	282.8427125	267.7826728	7.7260120
CSO	263.8959011	270.4633032	264.6998362	1.5273318
G-QPSO	263.9219460	265.0597391	264.2834777	0.3423240
AO	263.9245641	264.4579632	264.1447214	0.1553716
GSA	263.8974897	264.2263056	263.9962060	0.0817948
HHO	263.8968136	264.2487053	263.9821636	0.1092004
GWO	263.8968248	263.9127430	263.8998215	0.0038984
SSA	263.8958435	263.9092891	263.8974852	0.0030799
JS	263.8958468	263.8963071	263.8959083	0.0001060
GBO	263.8958434	263.8958446	263.8958438	3.90 × 10−07
LICRSA	263.8958434	263.8958434	263.8958434	2.92 × 10−14

. In addition,

Table 15. Numerical comparison of the results obtained for the three-bar truss design in the literature.

Algorithms	Variables		Minimize Volume
	x1	x2
CS [63]	0.78867000	0.40902000	263.97160000
AOA [4]	0.79369000	0.39426000	263.91540000
GBO [13]	0.78869300	0.40819700	263.89590000
GOA [18]	0.78889756	0.40761957	263.89588150
MVO [64]	0.78860276	0.40845307	263.89584990
GJO [59]	0.78865716	0.40829913	263.89584390
SSA [11]	0.78866541	0.40827578	263.89584340
HHO [8]	0.78866282	0.40828313	263.89584340
AVOA [60]	0.78868039	0.40823341	263.89584340
SMO [5]	0.78867694	0.40824317	263.89584338
CSA [61]	0.78867513	0.40824831	263.89584338
LICRSA	0.78867514	0.40824829	263.89584338

provides the results obtained by the algorithms (CS [63], AOA [4], GBO [13], GOA [18], MVO [64], GJO [59], SSA [11], HHO [8], AVOA [60], SMO [5], CSA [61]) in the literature. The numerical results in the table prove that LICRSA can find the best minimum volume.

5.4. Rolling Element Bearing Design

The ultimate goal of a rolling element bearing is to increase the dynamic load of the rolling bearing as much as possible. Ten design variables and nine nonlinear constraints exist for the design of rolling element bearings. These ten different design variables are ball diameter D_b, pitch diameter D_m, inner and outer raceway curvature coefficients f_i and f₀, the number of ball Z, K_Dmin, K_Dmax, $\epsilon$, e, and $\zeta$. In addition, five of the design parameters (K_Dmin, K_Dmax, $\epsilon$, e, and $\zeta$) exist only in the constraints and are not in the objective function, so they will indirectly radiate to the internal geometry. Let [x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀] = [D_b, D_m, f_i, f₀, Z, K_Dmin, K_Dmax, $\epsilon$, e, and $\zeta$]. A schematic diagram of the rolling element bearing structure is shown in Figure 15. The mathematical optimization of the rolling element bearing is formulated as follows.

Minimize:

$$f(\overline{x})=\left\{\begin{array}{c}{f}_c{Z}^{2/3}{D}_b^{1.8}, \mathrm{if} D_b\le 25.4\mathrm{mm}\\ 3.647f_c{Z}^{2/3}{D}_b^{1.4}, \mathrm{otherwise}\end{array}\right.$$

(62)

Subject to:

$$g_1(\overline{x})=Z-\frac{{\varphi }_0}{2{\sin }^{-1}(D_b/D_m)}-1\le 0,$$

(63)

$$g_2(\overline{x})=K_{\mathrm{Dmin}}(D-d)-2D_b\le 0,$$

(64)

$$g_3(\overline{x})=2D_b-K_{\mathrm{Dmax}}(D-d)\le 0,$$

(65)

$$g_4(\overline{x})=D_b-\zeta B_w\le 0,$$

(66)

$$g_5(\overline{x})=0.5(D+d)-D_m\le 0,$$

(67)

$$g_6(\overline{x})=D_m-(0.5+e)(D+d)\le 0,$$

(68)

$$g_7(\overline{x})=\epsilon D_b-0.5(D-D_m-D_b)\le 0,$$

(69)

$$g_8(\overline{x})=0.515-f_i\le 0,$$

(70)

$$g_9(\overline{x})=0.515-f_0\le 0,$$

(71)

where,

$$f_c=37.91{\left\{1+{\left\{1.04{\left(\frac{1-\gamma }{1+\gamma }\right)}^{1.72}{\left(\frac{f_i(2f_0-1)}{f_0(2f_i-1)}\right)}^{0.41}\right\}}^{10/3}\right\}}^{-0.3},$$

(72)

$$\gamma =\frac{D_b\cos (\alpha )}{D_m},f_i=\frac{r_i}{D_b},f_0=\frac{r_0}{D_b},$$

(73)

$${\varphi }_0=2\pi -2{\cos }^{-1}\left(\frac{{\{(D-d)/2-3(T/4)\}}^2+{\{D/2-(T/4)-D_b\}}^2-{\{d/2+(T/4)\}}^2}{2\{(D-d)/2-3(T/4)\}\{D/2-(T/4)-D_b\}}\right),$$

(74)

$$T-D-d-2D_b,D=160,d=90,B_w=30.$$

(75)

with bounds:

$$0.5(D+d)\le D_m\le 0.6(D+d),$$

(76)

$$0.15(D-d)\le D_b\le 0.45(D-d),$$

(77)

$$4\le Z\le 50,0.515\le f_0\le 0.6,$$

(78)

$$0.4\le K_{\mathrm{Dmin}}\le 0.5,0.6\le K_{\mathrm{Dmax}}\le 0.7,$$

(79)

$$0.3\le \epsilon \le 0.4,0.02\le e\le 0.1,0.6\le \zeta \le 0.85.$$

(80)

Table 16. Numerical results for the rolling element bearing problem.

Algorithms	Variables										Dynamic Load
	x1	x2	x3	x4	x5	x6	x7	x8	x9	x10
G-QPSO	125.00000	17.75037	5.10708	0.51500	0.51500	0.40000	0.60000	0.30000	0.02088	0.60000	35,135.13434
AOA	131.08999	18.55034	9.06984	0.56235	0.55575	0.44519	0.64439	0.34567	0.06421	0.69939	32,181.72474
SOA	130.83958	16.69825	6.05976	0.54163	0.54032	0.41555	0.63003	0.31008	0.03548	0.61423	23,408.13961
WOA	129.07495	18.00007	4.75565	0.59928	0.56702	0.43461	0.65271	0.31770	0.03425	0.60000	17,412.79383
SSA	127.86808	18.00000	4.93810	0.60000	0.58114	0.45416	0.65229	0.33863	0.05251	0.60000	17,224.02375
SCA	125.48131	18.05780	4.67540	0.59993	0.59815	0.46399	0.67166	0.34028	0.05208	0.60000	17,180.07707
AO	125.85880	18.01386	4.82810	0.60000	0.60000	0.46322	0.64500	0.34214	0.07292	0.60003	17,070.33342
CSO	126.28934	18.00008	4.55724	0.60000	0.60000	0.46434	0.63608	0.36713	0.09138	0.60000	17,038.34401
HGS	126.55000	18.00000	4.55671	0.60000	0.60000	0.40508	0.60611	0.31120	0.04294	0.60000	17,033.62576
HHO	128.51335	18.00076	5.04929	0.60000	0.60000	0.45586	0.67071	0.33525	0.07492	0.60000	17,003.61421
HBA	128.80000	18.00000	4.60738	0.60000	0.60000	0.45399	0.64505	0.32500	0.06208	0.60000	16,997.30994
LICRSA	129.55754	18.00458	4.90002	0.59998	0.59999	0.46100	0.67964	0.33308	0.08799	0.60000	16,994.75722

provides the design variables and optimal dynamic loads for LICRSA and eleven other MH algorithms (G-QPSO [62], AOA [4], SOA [51], WOA [24], SSA [11], SCA [42], AO [23], CSO [48], HGS [39], HHO [8], HBA [52]).

Table 17. Statistics of 20 independent operations of rolling element bearing.

Algorithms	Best	Worst	Mean	Std
GQPSO	34,869.42553	39,975.38601	35,135.13434	1139.29012
AOA	20,268.61547	54,937.54545	32,181.72474	10,755.46822
SOA	16,176.43125	36,056.93710	23,408.13961	5031.47835
WOA	16,982.83369	17,848.50849	17,412.79383	304.91575
SSA	16,972.94537	17,915.57397	17,224.02375	254.19596
SCA	16,925.01445	17,571.79443	17,180.07707	186.87150
AO	17,023.73892	17,114.12475	17,070.33342	25.72117
CSO	16,977.92561	17,062.90482	17,038.34401	28.02089
HGS	16,958.20229	17,058.76692	17,033.62576	44.67701
HHO	16,958.25317	17,058.77361	17,003.61421	36.71393
HBA	16,958.20229	17,058.76692	16,997.30994	44.91810
LICRSA	16,958.20229	17,058.76692	16,994.75722	37.14945

provides the statistics for 20 independent runs. In addition,

Table 18. Numerical comparison of the results obtained for the rolling element bearing in the literature.

Algorithms	Variables										Dynamic Load
	x1	x2	x3	x4	x5	x6	x7	x8	x9	x10
AHA [57]	125.71841	21.42535	10.52798	0.51500	0.51516	0.47022	0.64082	0.30001	0.09512	0.68224	85,547.49822
AVOA [60]	125.72272	21.42329	11.00116	0.51500	0.51500	0.40443	0.61868	0.30000	0.06913	0.60247	85,539.15785
HBO [65]	125.71939	21.42322	11.00000	0.51500	0.51500	0.48806	0.67937	0.30001	0.06902	0.60832	85,532.57000
GBO [13]	125.00000	21.87500	11.28817	0.51500	0.51500	0.41484	0.62866	0.30000	0.02033	0.67206	85,245.06110
CSA [61]	125.00000	21.41800	11.35600	0.51500	0.51500	0.40000	0.70000	0.30000	0.02000	0.61200	85,201.64100
TSA [66]	125.00000	21.41750	10.94100	0.51000	0.51500	0.40000	0.70000	0.30000	0.02000	0.60000	85,070.08000
SOA [50]	125.00000	21.41892	10.94123	0.51500	0.51500	0.40000	0.70000	0.30000	0.02000	0.60000	85,068.05200
EPO [56]	125.00000	21.41890	10.94113	0.51500	0.51500	0.40000	0.70000	0.30000	0.02000	0.60000	85,067.98300
MVO [64]	125.60020	21.32250	10.97338	0.51500	0.51500	0.50000	0.68782	0.30135	0.03617	0.61061	84,491.26600
COOT [14]	125.00000	21.87500	10.77700	0.51500	0.51500	0.43190	0.65290	0.30000	0.02000	0.60000	83,918.49200
HHO [8]	125.00000	21.00000	11.09207	0.51500	0.51500	0.40000	0.60000	0.30000	0.05047	0.60000	83,011.88329
LICRSA	129.55754	18.00458	4.90002	0.59998	0.59999	0.46100	0.67964	0.33308	0.08799	0.60000	16,994.75722

provides the results obtained by the algorithms (AHA [57], AVOA [60], HBO [65], GBO [13], CSA [61], TSA [66], SOA [51], EPO [56], MVO [64], COOT [14], HHO [8]) in the literature. The LICRSA can find the optimal design variables and dynamic loading. Thus, LICRSA proves to be a great advantage in dealing with the problem.

5.5. Speed Reducer Design

Reducers are one of the most critical components of a gearbox system. The ultimate requirement for the gearbox is weight minimization while satisfying 11 constraints. In addition, seven design variables exist for this design. x₁ is the face width, x₂ is the tooth mode, x₃ represents the number of teeth in the pinion, x₄ represents the length of the first shaft between the bearings, the length of the second shaft between the bearings is x₅, and x₆ and x₇ represent the diameters of the first shaft and the second shaft. Figure 16 provides a schematic diagram of the reducer [5]. Also, the mathematical design equation for the reducer design is as follows.

Minimize:

$$\begin{array}{c}f(\overrightarrow{x})=0.7854x_1{x}_2^2(3.3333x_3^2+14.9334x_3-43.0934)\\ -1.508x_1(x_6^2+x_7^2)+7.4777(x_6^3+x_7^3),\end{array}$$

(81)

Subject to:

$$g_1(\overrightarrow{x})=\frac{27}{x_1{x}_2^2{x}_3}-1\le 0,$$

(82)

$$g_2(\overrightarrow{x})=\frac{397.5}{x_1{x}_2^2{x}_3^2}-1\le 0,$$

(83)

$$g_3(\overrightarrow{x})=\frac{1.93x_4^3}{x_2{x}_3{x}_6^4}-1\le 0,$$

(84)

$$g_4(\overrightarrow{x})=\frac{1.93x_5^3}{x_2{x}_3{x}_7^4}-1\le 0,$$

(85)

$$g_5(\overrightarrow{x})=\frac{\sqrt{{(\frac{745x_4}{x_2{x}_3})}^2+16.9\times {10}^6}}{110.0x_6^3}-1\le 0,$$

(86)

$$g_6(\overrightarrow{x})=\frac{\sqrt{{(\frac{745x_4}{x_2{x}_3})}^2+157.5\times {10}^6}}{85.0x_6^3}-1\le 0,$$

(87)

$$g_7(\overrightarrow{x})=\frac{x_2{x}_3}{40}-1\le 0,$$

(88)

$$g_8(\overrightarrow{x})=\frac{5x_2}{x_1}-1\le 0,$$

(89)

$$g_9(\overrightarrow{x})=\frac{x_1}{12x_2}-1\le 0,$$

(90)

$$g_{10}(\overrightarrow{x})=\frac{1.5x_6+1.9}{x_4}-1\le 0,$$

(91)

$$g_{11}(\overrightarrow{x})=\frac{1.1x_7+1.9}{x_5}-1\le 0,$$

(92)

Variable range:

$$2.6\le x_1\le 3.6,0.7\le x_2\le 0.8,$$

(93)

$$17\le x_3\le 28,7.3\le x_4\le 8.3,$$

(94)

$$7.8\le x_5\le 8.3,2.9\le x_6\le 3.9,5\le x_7\le 5.5.$$

(95)

Table 19. Numerical results for the speed reducer design.

Algorithms	Variables							Minimum Weight
	x1	x2	x3	x4	x5	x6	x7
LSO	3.551135803	0.736923623	25.12309012	8.20716774	8.222940068	3.810971145	5.46787447	5477.462985
BAT	3.368142883	0.70000847	22.64438282	7.872881603	8.035242714	3.672563578	5.342158123	4213.055197
BOA	3.523249019	0.707558217	21.39373073	7.870916877	7.948465392	3.629370446	5.331842865	4095.416921
DMO	3.598259019	0.708842963	17	7.639422189	8.041575889	3.407858186	5.305442136	3111.977751
SCA	3.582632258	0.7	17.00413795	7.680864826	8.100725845	3.446360885	5.342929824	3101.694019
WOA	3.522023781	0.7	17.03522238	7.695659172	8.040955524	3.489935053	5.327565943	3086.729202
HHO	3.550532222	0.7	17.02298238	7.507127823	7.957178853	3.381293938	5.293488313	3037.864187
SSA	3.506903016	0.7	17.00000006	7.693799023	8.037812331	3.444231088	5.286766517	3033.402389
CSO	3.525023295	0.705664363	17	7.3	7.8	3.351257141	5.286754733	3032.813598
GWO	3.502456111	0.700011724	17.00198205	7.573843774	7.881657163	3.358079603	5.28817256	3004.756902
SCSO	3.500778985	0.700004857	17.00005987	7.723582413	7.945917609	3.353286754	5.286988024	3004.480179
LICRSA	3.50009504	0.70001495	17.00039814	7.344740534	7.826768272	3.350802205	5.286723295	2997.550054

provides the optimal design variables and minimum weights for LICRSA and 11 other MH algorithms (LSO [67], BAT [9], BOA [26], DMO [6], SCA [41], WOA [24], HHO [8], SSA [11], CSO [48], GWO [20], SCSO [41]) for the reducer problem.

Table 20. Statistics of 20 independent operations of speed reducer design.

Algorithms	Best	Worst	Mean	Std
LSO	3375.49065	6783.32679	5477.46298	1160.36894
BAT	3134.25557	5489.06816	4213.05520	760.69313
BOA	3049.42307	5671.13463	4095.41692	900.43458
DMO	3057.84155	3260.93923	3111.97775	54.51718
SCA	3046.18056	3198.09847	3101.69402	40.56807
WOA	2996.84150	3192.65667	3086.72920	61.34899
HHO	3005.69276	3079.26997	3037.86419	22.22538
SSA	2999.04829	3106.79787	3033.40239	31.04791
CSO	2996.30435	3195.02510	3032.81360	53.56356
GWO	2999.76380	3009.91957	3004.75690	3.18261
SCSO	2997.27696	3013.60422	3004.48018	4.96548
LICRSA	2996.30156	3007.99931	2997.55005	3.49117

provides the statistics for 20 independent runs. In addition,

Table 21. Numerical comparison of the results obtained for the speed reducer design in the literature.

Algorithms	Variables							Minimum Weight
	x1	x2	x3	x4	x5	x6	x7
GOA	3.5126	0.7033	17.2246	7.9131	7.9627	3.6567	5.2784	3169.32
GSA	3.6	0.7	17	8.3	7.8	3.369658	5.289224	3051.12
SCA	3.508755	0.7	17	7.3	7.8	3.46102	5.289213	3030.563
EA	3.506163	0.700831	17	7.46018	7.962143	3.3629	5.309	3025.005
FA	3.507495	0.7001	17	7.719674	8.080854	3.351512	5.287051	3010.137492
AO	3.5021	0.7	17	7.3099	7.7476	3.3641	5.2994	3007.7328
MVO	3.508502	0.7	17	7.392843	7.816034	3.358073	5.286777	3002.928
GWO	3.50669	0.7	17	7.380933	7.815726	3.357847	5.286768	3001.288
CS	3.5015	0.7	17	7.605	7.8181	3.352	5.2875	3000.981
MFO	3.4976	0.7	17	7.3	7.8	3.3501	5.2857	2998.54
AOA	3.50384	0.7	17	7.3	7.72933	3.35649	5.2867	2997.9157
LICRSA	3.50009504	0.70001495	17.00039814	7.344740534	7.826768272	3.350802205	5.286723295	2997.550054

provides the results obtained by the algorithms (GOA [18], GSA [33], SCA [42], EA [68], FA [69], AO [23], MVO [64], GWO [20], CS [63], MFO [19], AOA [4]) in the literature. The results show that LICRSA achieves the best result among the search algorithms for both minimum weights. Therefore, LICRSA can solve the reducer problem well.

6. Discussion

According to the numerical situation of this study, the proposed LICRSA algorithm shows excellent performance compared with other optimization algorithms. The validation experiments are divided into three main parts. The first part compares the LICRSA algorithm with the original RSA and different optimization algorithms on 23 test functions. The LICRSA algorithm ranks first in 10 functions, accounting for 43% of all functions. In addition, the LICRSA algorithm outperforms RSA in 15 functions, implying that introducing the Lévy flight and interactive crossover strategy enhances the exploration and development capabilities. The second part compares the LICRSA algorithm with other optimization algorithms in the CEC2020 test set. After analyzing the comparative metrics such as best, mean, worst, and std of solving the benchmark function, the LICRSA also solves the test function with higher accuracy, better stability, and faster convergence. The final section will discuss the capability of LICRSA with different algorithms in solving five complex engineering examples. The simulation results verify the usefulness and dependability of the LICRSA algorithm for real optimization problems. Of course, there is room for improved algorithm stability when solving engineering optimization problems.

7. Conclusions

This paper proposes an improved reptile search algorithm based on Lévy flight and interactive crossover strategies. To improve the performance of the original RSA algorithm, two main approaches are employed to address its shortcomings. As a general modification technique to improve its optimization capability, the Lévy flight strategy enhances the global exploration of the algorithm and jumps out of local solutions. Meanwhile, the interactive crossover strategy, as a variant of the crossover operator, inherits the local exploitation ability and enhances the fit with iterations. Twenty-three benchmark test functions, the IEEE Conference Evolutionary Computation (CEC2020) benchmark, and five engineering design problems are used to test the proposed LICRSA algorithm and verify the effectiveness of LICRSA. The experimental results show that LICRSA obtains more suitable optimization results and significantly outperforms competing algorithms, including DMO, WOA, HHO, DE, AOA, and WSO. The LICRSA has a more significant global search capability and higher accuracy solutions. In addition, the results of the tested mechanical engineering problems prove that LICRSA outperforms other optimization methods in total cost evaluation criteria. LICRSA has excellent performance and can be utilized by other researchers to tackle complex optimization problems.

Similarly, the applicability of RSA will be investigated. For example, to further improve the spatial search capability, we try to integrate RSA with other general intelligent group algorithms to improve the global performance of RSA. We use RSA to provide optimal parameters for machine learning models in privacy-preserving and social networks in practical applications. Furthermore, the proposed LICRSA algorithm may also have excellent application potency in solving other complex optimization problems, such as shop scheduling problems [70], optimal degree reduction [71], image segmentation [72], shape optimization [73], and feature selection [74], etc.