An Enhanced Symmetric Sand Cat Swarm Optimization with Multiple Strategies for Adaptive Infinite Impulse Response System Identification

Abstract

An infinite impulse response (IIR) system might comprise a multimodal error surface and accurately discovering the appropriate filter parameters for system modeling remains complicated. The swarm intelligence algorithms facilitate the IIR filter’s parameters by exploring parameter domains and exploiting acceptable filter sets. This paper presents an enhanced symmetric sand cat swarm optimization with multiple strategies (MSSCSO) to achieve adaptive IIR system identification. The principal objective is to recognize the most appropriate regulating coefficients and to minimize the mean square error (MSE) between an unidentified system’s input and the IIR filter’s output. The MSSCSO with symmetric cooperative swarms integrates the ranking-based mutation operator, elite opposition-based learning strategy, and simplex method to capture supplementary advantages, disrupt regional extreme solutions, and identify the finest potential solutions. The MSSCSO not only receives extensive exploration and exploitation to refrain from precocious convergence and foster computational efficiency; it also endures robustness and reliability to facilitate demographic variability and elevate estimation precision. The experimental results manifest that the practicality and feasibility of the MSSCSO are superior to those of other methods in terms of convergence speed, calculation precision, detection efficiency, regulating coefficients, and MSE fitness value.

1. Introduction

The IIR filter leverages the simplified scheme orders, fewer adjustable coefficients, and augmented spectrum allocation to accomplish high measurement precision and efficient convergence performance, which is frequently utilized in monitoring systems, wireless communication, signal manipulation, geographical exploratory, noise mitigation, and software-driven equalization. The error cost of the flexible IIR system identification is characterized by nonlinearity and non-differentiability. Numerous optimization techniques have recently been executed to construct the IIR filter system. The conventional design techniques utilize the Butterworth or Chebyshev Type I or Type II to design the IIR filter, which employs the bilinear transformation approach to convert the analog filter (s-domain) to the digital filter (z-domain). However, this technique has severe demerits in terms of the frequency-distorting effect, quantization errors, and nonlinear phase errors [1,2,3].

Later, gradient-based techniques such as Newton’s method, Quasi-Newton method, simplex technique, least mean square (LSM), linear programming, and least squares and its enhanced variants are used to resolve the IIR filter and minimize the error fitness value [4,5,6]. Most adaptive IIR filters are associated with nonlinear and multimodal fitness errors. These techniques are reliable and stable for the parameter estimation of a linear IIR system, and the unimodal objective function between an unidentified system’s input and the IIR filter’s output must be linear. However, these techniques have limitations: inefficient initial population quality, sluggish convergence acceleration, restricted evaluation precision, easy precocious convergence, slow detection efficiency, and combinatorial explosion. Meanwhile, these techniques exhibit some inherent demerits: the necessity of continuous and differentiable fitness, typically recognizing the regional extreme solution or retracing the sub-optimal solution, an inability to explore the enormous search domain, a constraint of the piecewise linear cost approximation (linear programming), and starting points becoming exceedingly sensitive as the solution variables increase [7,8]. They are uncertain and unstable when designing the higher-order systems of the IIR filter system because the poles may relocate the exterior of the unit circle.

Some swarm intelligence algorithms are applied to resolve the nonlinear, non-differentiable, multimodal, non-convex, and non-quadratic IIR system identification to overcome the above demerits. The swarm intelligence algorithms imitate the macro-intelligence characteristics and learning methods of groups of animals in nature, utilizing cooperation and information exchange to attain the most affordable and achievable solution. These algorithms exhibit parallel solid processing performance, global exploration ability, search efficiency, and strong adaptability to handle super-complex, large-scale, high-dimensional problems [9]. These algorithms establish single-objective and multi-objective mathematical models according to the different assessment environments and research objectives, which utilize different constraints and evaluation indicators to validate sustainability and feasibility and establish the most satisfactory value. These algorithms harmonize to the finest potential solution with faster detection efficiency and inhibit precocious convergence with superior escape ability. They ensure formidable discovery and extraction to recognize the most appropriate regulating coefficients and minimize the MSE.

The basic swarm intelligence algorithms are applied to resolve the IIR system identification and recognize the most appropriate regulating coefficients. Mahata et al. pursued a manta ray foraging approach to resolve the IIR system identification. The pole relating and initialization schedule ensured stability and reliability, which utilized chain, cyclone, and somersault scavenging to attain an attractive scavenging tactic and exemplary extraction efficiency to validate coherence and convergence. Still, it was impacted by an inappropriate selection of the fitness, robustness, resolution speed, and computational efficiency [10]. Izci et al. conducted a whale optimization approach to resolve IIR system identification, and this approach utilized circumnavigating prey, bubble-net assaulting, and pursuit for prey to stabilize localized exploitation and broad exploration to recognize the expansive accurate solution and the most appropriate regulation variables. The exploration and exploitation analysis of the proposed approach needed further research and discussion [11]. Singh et al. established a teaching-learning procedure to resolve the IIR system identification, and this approach utilized the average difference to maintain the diversity and robustness, advance extraction operation, and disrupted regional extreme solutions to identify the most satisfactory solution. The regulating coefficients, mean square error, convergence accuracy, and detection efficiency of the proposed approach needed to be further enhanced in addressing the higher-order IIR system identification [12]. Wang et al. explained an artificial fish swarm approach to resolve the IIR system identification, and this approach exhibited strong search ability and optimization efficiency to attain a lower ameliorated system error and greater estimated precision. However, the approach’s detection accuracy and convergence efficiency were associated with the appropriate selection of the control parameters, and the computational complexity was huge [13]. Ekinci et al. explored the atom search optimization to resolve IIR filter identification; this approach filtrated the finest potential solution, fostered selection probability, quickened the convergence procedure, and bolstered localized exploitation so that it maintained instructive sustainability and adaptability to prohibit search standstill and determine the finest global solution. The exploration and exploitation, diversity analysis, and hyperparameters tuning of the proposed algorithm needed further analysis and research [14].

However, the original algorithms may exhibit the demerits of slow convergence rate, low evaluation precision, and easy precocious convergence. To enhance convergence accuracy and detection efficiency, swarm intelligence algorithms with ensemble strategies are applied to resolve the identification of IIR systems. Zhang et al. devised a self-adaptation hybridized cross-mutation slime mold approach to resolve the IIR system identification; the Cauchy mutation operator augmented the mutation ability and evolutionary directions, the crossover rate equilibrium with differential vector information compensated for an individual-fitness relationship to expand the detection process, and the self-adaptation resume hybrid opposition learning elevated population diversity and alleviated search stagnation. This approach promoted and maintained instructive competitiveness and effectiveness to bolster resolution accuracy, a single objective approach with certain limitations in solving multi-objective issues [15]. Su et al. exploited the multi-objective particle swarm optimization to resolve the IIR system identification, and the approach incorporated the Pareto frontier efficiency and global detection ability to balance passband and stopband filter orders and qualitative concerns. The proposed approach significantly reduced the passband oscillation and maximized stopband suppression to ensure the predetermined indicators of the interrupted frequency and the IIR system order, which exhibited substantial sustainability and dependability in accomplishing the minimal fitness and facilitating regulation variables [16]. Rizk-Allah et al. developed sophisticated artificial rabbit optimization to solve the IIR filter identification. The adaptive local search mechanism reduced accuracy loss, enhanced local exploitation accuracy, and improved the optimization process, and the experience-based perturbed learning strategy promoted detection efficiency, increased population diversity, and avoided search stagnation. The sustainability and adaptability of the proposed approach are superior to those of other approaches in terms of the MSE fitness value, convergence accuracy, regulating coefficients, and evaluation efficiency [17]. Zhang et al. implemented an upgraded GJO to resolve the IIR filter identification, aiming to achieve the most appropriate regulating coefficients and minimize the mean square error. The elite opposition-based learning advanced demographic variety, promoted discovery efficiency, extended the detection domain, and avoided anticipation stagnation. The simplex approach accelerated the investigation of velocity, enhanced the extraction ability, fostered the estimation precision, and advanced the exploitation depth. The proposed approach advanced extraction operation and expedited evaluation efficiency to feature more enjoyable regulation variables and a more formidable convergence solution [18]. Niu et al. utilized an altered artificial ecosystem approach with a dynamical antagonistic learning tactic and the nonlinear adaptive weight coefficient to resolve IIR filter identification, and this approach facilitated population variety, fostered the extraction procedure, expedited evaluation efficiency, refrained precocious convergence, and bolstered broad exploration to maintain the finest potential solution [19].

The hybrid swarm intelligence algorithms are applied to bolster the exploration and exploitation and minimize the mean square error to resolve the IIR system identification. Kumar et al. incorporated a particle swarm optimization with a grasshopper optimization method to resolve IIR system identification; this approach exhibited remarkable intuitiveness and simplicity in identifying intelligent collaborative detection, accelerating exploitation efficiency, and facilitating solution accuracy, achieving a compensated estimation error. However, the proposed approach had limitations in the investigation research, parameter tuning process, computational complexity, and exploration and exploitation analysis [20]. Kaur et al. merged the chimp optimization approach with the cuckoo search approach to resolve IIR system identification; this approach combined exploration and exploitation to display sustainability and superiority in acquiring the most favorable evaluation solution and adjustment factors. However, the proposed approach’s convergence speed, estimation precision, and detection efficiency mainly depended on the appropriate selection of the regulating coefficients [21]. Ekinci et al. explored a cooperative technique with a gazelle optimization technique and simulated annealing method to resolve the IIR system identification; this approach not only achieved complementary advantages to avoid anticipation stagnation and enhance the detection efficiency, but also integrated prospecting and extracting to identify an extreme anticipation efficiency and identify precision.

Further research is needed to verify the stability and robustness of the control parameter adjustment and explore the adaptive technique of the proposed approach [22]. Ekinci et al. established a pattern search ameliorated arithmetic optimization technique to resolve the IIR system identification, and this approach utilized exploration and exploitation to fortify trustworthiness and dependability in accomplishing the superior regulation indicators and the highest-quality solution. The specific control parameters are associated with evaluation efficiency and estimation precision [23]. Liang et al. juxtaposed functional inequality-constrained optimization with a genetic algorithm to resolve IIR filter identification; this approach advanced initial population quality, escalated recognition depth, disrupted regional extreme solution, advanced extraction operation, and elevated estimation precision to improve convergence productivity and accomplished a more accurate solution [24].

Table 1. Brief literature review of existing methods for IIR system identification.

Methods with Reference	Merits	Demerits	Gaps
Conventional design [1,2,3]	Bilinear transformation technique, quick and efficient implementation	Frequency warping effect, quantization errors, nonlinear phase errors	Lack of stability and robustness analysis
Gradient-based techniques [4,5,6,7,8]	Flexible development, low execution time, high efficiency and reliability in resolving the unimodal objective functions, simple implementation, low complexity	Inefficient initial population quality, sluggish convergence speed, restricted evaluation precision, easy precocious convergence, slow detection efficiency, combinatorial explosion, sensitivity to parameters	Lack of stability and robustness analysis
Basic swarm intelligence algorithms [10,11,12,13,14]	Easy to implement and understand, simple algorithm framework, few optimization parameters, convenient expansion, good parallelism and stability	Poor detection efficiency, slow convergence rate, and low estimation precision in resolving the large-scale, high-dimension, multi-objective and extremely complex optimization issues; the exploration and exploitation need to be enhanced	Lack of stability and robustness analysis analysis, ignoring the additive noise, lack of exploration and exploitation analysis, diversity analysis, hyperparameters tuning
Swarm intelligence algorithms with ensemble strategies [15,16,17,18,19]	Fosters selection probability, bolsters localized exploitation and broad exploration, increases population diversity, refrains from precocious convergence, disrupts regional extreme solution, efficiently balances exploration and exploitation, strong self-organization and robustness	The simulation result depends on the selection of the control parameters, the high computational complexity of the algorithm; further improvement needed in terms of detection efficiency, convergence rate, estimation precision, long execution time	Lack of stability and robustness analysis analysis, ignoring the additive noise, lack of exploration and exploitation analysis, diversity analysis, hyperparameters tuning
Hybrid swarm intelligence algorithms [20,21,22,23,24]	Realizes the complementary advantages between algorithms, intense exploration and exploitation, strong stability and robustness	The algorithm’s computational complexity is high, convergence rate and estimation precision depend on the selection of the control parameters, long execution time	Lack of different system identification structures, lack of stability and robustness analysis, ignoring the additive noise, lack of exploration and exploitation analysis, diversity analysis, failure to review some new approaches

summarizes the brief literature review of existing methods for IIR system identification.

The SCSO is motivated by the distinguishable properties of sand cats and mimics the surveillance and neutralization of prey to recognize the low-frequency noise, the prey location, and the most appropriate solution [25]. The basic SCSO exhibits deficiencies, such as inefficient initial population quality, sluggish convergence acceleration, and restricted evaluation precision. Therefore, an enhanced symmetric SCSO with multiple strategies (MSSCSO) is presented to accomplish the benchmark functions and adaptive IIR filter identification. The no-free-lunch (NFL) theorem stipulates that no swarm intelligence algorithm can resolve each optimization-related difficulty, which motivates us to establish an enhanced symmetric SCSO. The main contributions are summarized: (1) The ranking-based mutation operator, elite opposition-based learning strategy, and simplex method are introduced into the basic SCSO to accelerate convergence efficiency and advance estimation accuracy. (2) The ranking-based mutation operator filtrates the finest potential solution, fosters selection probability, quickens the convergence procedure, and bolsters localized exploitation. (3) The elite opposition-based learning strategy facilitates demographic variability, upgrades searched an area, expedites evaluation efficiency, refrains precocious convergence, and bolsters broad exploration. (4) The simplex method advances initial population quality, escalates recognition depth, disrupts regional extreme solutions, advances extraction operation, and elevates estimation precision. (5) The MSSCSO is evaluated using the CEC 2022 and three sets of the IIR filter identification compared to recently published approaches. (6) The MSSCSO exhibits strong stability and robustness to balance exploration and exploitation to determine a swifter convergent velocity, greater explored accuracy, lower standard deviation, and more accurate fitness.

The subsequent sections constitute this article. Section 2 reviews SCSO. Section 3 extends MSSCSO. Section 4 summarizes the simulation test and result analysis for benchmark functions. Section 5 conveys MSSCSO-based adaptive IIR system identification. Section 6 discusses the exploration and exploitation analysis and the diversity analysis. Section 7 outlines the conclusion and future research.

2. SCSO

The SCSO is motivated by the sand cat’s extraordinary features, imitating the surveillance and neutralization of prey to recognize expansive, accurate solutions by detecting low-frequency noise and the prey’s location. Figure 1 articulates the sand cat’s distinguishable properties: living, searching, and hunting.

2.1. Initial Population

The initialization matrix is constructed as follows:

$$Sand Ca{t}_i=\left[\begin{array}{cccc}{x}_{1,1}& x_{1,2}& \cdots & x_{1,d}\\ x_{2,1}& x_{2,2}& \cdots & x_{2,d}\\ ⋮& ⋮& ⋮& ⋮\\ x_{n,1}& x_{n,2}& \cdots & x_{n,d}\end{array}\right]$$

(1)

where $x_{i,j}$ signifies the jth dimension of the ith sand cat.

The calculation matrix of fitness is constructed as follows:

$$fitness=\left[\begin{array}{c}f(x_{1,1},x_{1,2},\cdots ,x_{1,d})\\ f(x_{2,1},x_{2,2},\cdots ,x_{2,d})\\ ⋮\\ f(x_{n,1},x_{n,2},\cdots ,x_{n,d})\end{array}\right]$$

(2)

2.2. Searching for Prey (Exploration)

The sand cat typically exploits low-frequency noise fabrication as an essential component of identifying and acquiring prey. The sensitivity $\overrightarrow{r_G}$ will mitigate continuously from 2 to 0, which encourages the sand cat to gradually approach and capture prey without scaring it away or forfeiting it. The $\overrightarrow{r_G}$ is constructed as follows:

$$\overrightarrow{r_G}=s_M-\left({\displaystyle \frac{S_M\times ite{r}_c}{ite{r}_{\max }}}\right)$$

(3)

where $S_M=2$ signifies the auditory traits, $ite{r}_c$ signifies the newest iteration, and $ite{r}_{\max }$ signifies the highest iteration.

The $\overrightarrow{R}$ and $\overrightarrow{r}$ are constructed as follows:

$$\overrightarrow{R}=2\times \overrightarrow{r_G}\times rand(0,1)-\overrightarrow{r_G}$$

(4)

$$\overrightarrow{r}=\overrightarrow{r_G}\times rand(0,1)$$

(5)

The position vector $\overrightarrow{Pos}$ is constructed as follows:

$$\overrightarrow{Pos}(t+1)=\overrightarrow{r}\cdot (\overrightarrow{Po{s}_{bc}}(t)-rand(0,1)\cdot \overrightarrow{Po{s}_c}(t))$$

(6)

where $\overrightarrow{Po{s}_{bc}}$ signifies the optimal position, $\overrightarrow{Po{s}_c}$ signifies the newest position.

2.3. Attacking on Prey (Exploitation)

Each sand cat migrates in an alternative circumferential direction of the exploitation region if the sensitivity region is circle-shaped. Figure 2 articulates the position refreshing procedure of the SCSO. The SCSO deploys the Roulette Wheel Section to identify an arbitrary angle $\theta$ to investigate the harvesting position. This quickens the convergence procedure, expedites evaluation efficiency, refrains precocious convergence, and elevates estimation precision. The positions are constructed as follows:

$$\overrightarrow{Po{s}_{rnd}}=\left|rand(0,1)\cdot \overrightarrow{Po{s}_b}(t)-\overrightarrow{Po{s}_c}(t)\right|$$

(7)

$$\overrightarrow{Pos}(t+1)=\overrightarrow{Po{s}_b}(t)-\overrightarrow{r}\cdot \overrightarrow{Po{s}_{rad}}\cdot \cos (\theta )$$

(8)

where $\overrightarrow{Po{s}_{rnd}}$ signifies the separation between the newest position $\overrightarrow{Po{s}_c}$ and the ideal position $\overrightarrow{Po{s}_b}$, and $\theta \in [0,360]$ signifies a random angle, $\cos \theta \in [0,1]$.

2.4. Exploration and Exploitation

The SCSO seamlessly switches between localized exploitation and broad exploration to expedite evaluation efficiency and attain the finest potential solution. When $\left|R\right|\le 1$, the SCSO instructs the sand cat to retrieve prey; it utilizes the Roulette Wheel Section to exploit and refrain from precocious convergence. When $\left|R\right|>1$, the SCSO instructs the sand cat to determine a new potential solution, which operates a prey-seeking procedure to explore and pursue prey. Figure 3 articulates the attacking prey (exploitation) versus searching for prey (exploration). The positions are constructed as follows:

$$\overrightarrow{X}(t+1)=\left\{\begin{array}{cc}\overrightarrow{Po{s}_b}(t)-\overrightarrow{Po{s}_{rnd}}\cdot \cos (\theta )\cdot \overrightarrow{r}\hfill & \hfill \left|R\right|\le 1;\mathrm{exploitation}\\ \overrightarrow{r}\cdot (\overrightarrow{Po{s}_{bc}}(t)-rand(0,1)\cdot \overrightarrow{Po{s}_c}(t))\hfill & \hfill \left|R\right|>1;\mathrm{exploitation}\end{array}\right.$$

(9)

Algorithm 1 emphasizes the pseudocode of SCSO.

Algorithm 1 SCSO

Begin Step 1. Randomly initialize the population, $r$, $r_G$ and $R$ Step 2. Ascertain the sand cat’s fitness and identify the finest prey $Po{s}_{bc}$ Step 3. while ($t<T$) do      for each sand cat do       Accomplish a randomized angle via Roulette Wheel Selection $(0^{°}\le \theta \le {360}^{°})$      if ($\left|R\right|\le 1$) then       Renew position via Equation (8)      else       Renew position via Equation (6)      end if     end for     $t=t+1$     end while     Restore the finest prey $Po{s}_{bc}$ End

3. MSSCSO

To advance initial population quality, convergence acceleration, evaluation precision, and solution efficiency, the expanded MSSCSO maintains instructive sustainability and adaptability to realize supplementary advantages and refrain from precocious convergence, and this integrates localized exploitation and broad exploration to expedite evaluation efficiency and elevates estimation precision.

3.1. Ranking-Based Mutation Operator

This strategy filtrates the finest potential solution, fosters selection probability, quickens convergence procedure, and bolsters localized exploitation. The most desirable prey is assigned the finest search agent, and the fitness is sorted from most significant to poorest [26,27,28]. The ranking of sand cat is constructed as follows:

$$R_i=N_p-i,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}i=1,2,\dots ,N_p$$

(10)

where $N_p$ signifies the population size. The selection probability $P_i$ is constructed as follows:

$$p_i={\displaystyle \frac{R_i}{N_p}},\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}i=1,2,\dots ,N_p$$

(11)

Algorithm 2 emphasizes the ranking-based mutation operator of “DE/rand/1”. Sand cats with the highest rating endure a more extensive probability of being established as the primordial vector or terminus vector, and the intention is to transmit the population’s genetic material to its descendants. A ranking-based mutation operator refrains from employing population sorting to ascertain the selection probability of the starting vector. The associated discovery procedure could dramatically diminish and facilitate voracious convergence if two differential variation vectors are extracted from the more prestigious vectors.

Algorithm 2 Ranking-based mutation operator of “DE/rand/1”

Begin Sort the population and distribute the selection probability $P_i$ Automatically identify $r_1\in \left\{1,N_p\right\}$ {base vector index} while $rand>p_{r_1}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_1==i$ Automatically identify $r_1\in \left\{1,N_p\right\}$ end Automatically identify $r_2\in \left\{1,N_p\right\}$ {terminal vector index} while $rand>p_{r_2}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_2==r_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_2==i$ Automatically identify $r_2\in \left\{1,N_p\right\}$ end Automatically identify $r_3\in \left\{1,N_p\right\}$ {starting vector index} while $r_3==r_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_3==r_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_3==i$ Automatically identify $r_3\in \left\{1,N_p\right\}$ end End

3.2. Elite Opposition-Based Learning Strategy

This strategy facilitates demographic variability, upgrades the search of an area, expedites evaluation efficiency, refrains from precocious convergence, and bolsters broad exploration. This approach specifies the most advantageous solution to serve as the search agent and executes the exploratory procedure by assessing the acceptable and the opposite solutions. A sand cat with the finest fitness is considered an elite solution $x_e=(x_{e,1},x_{e,2},\dots ,x_{e,D})$ [29,30,31]. The satisfactory solution $x_i=(x_{i,1},x_{i,2},\dots ,x_{i,D})$ and the opposite solution $x_i^{\prime }=(x_{i,1}^{\prime },x_{i,2}^{\prime },\dots ,x_{i,D}^{\prime })$ are constructed as follows:

$$x_{i,j}^{\prime }=k\cdot (d{a}_j+d{b}_j)-x_{e,j}, i=1,2,\dots ,n; j=1,2,\dots ,D$$

(12)

where $n$ signifies the population size, $D$ signifies the dimension, $k\in (0,1)$, and $d{a}_j$ and $d{b}_j$ signify the dynamic boundaries, which together are constructed as follows:

$$d{a}_j=\min (x_{i,j}), d{b}_j=\max (x_{i,j})$$

(13)

The dynamic boundary may alter the inverse solution’s discovery region and preserve the most outstanding solution. The individual $x_{i,j}^{\prime }$ is constructed as follows:

$$x_{i,j}^{\prime }=rand(d{a}_j,d{b}_j), if x_{i,j}^{\prime }<d{a}_j or x_{i,j}^{\prime }>d{b}_j$$

(14)

3.3. Simplex Method

This strategy advances initial population quality, escalates recognition depth, disrupts regional extreme solutions, advances extraction operation, and elevates estimation precision, which identifies the optimality of a solution by replacing the original solution with a feasible solution [32,33,34]. Figure 4 articulates the simplex method schematically.

• Step 1. Ascertain the fitness, filter out adequate solution $X_g$, inadequate solution $X_b$, and renewed solution $X_s$, and ascertain the fitnesses $f(X_g)$, $f(X_b)$, and $f(X_s)$.
• Step 2. Ascertain the midpoint $X_c$.

  $$X_c={\displaystyle \frac{X_g+X_b}{2}}$$

  (15)
• Step 3. Ascertain the refraction point $X_r$ and the fitness $f(X_r)$.

  $$X_r=X_c+\alpha (X_c-X_s)$$

  (16)

  where $\alpha =1$ signifies the reflectivity.
• Step 4. If $f(X_r)<f(X_g)$, the extension operation is computed as follows:

  $$X_e=X_c+\gamma (X_r-X_c)$$

  (17)

  where $\gamma =2$ signifies the extension factor. If $f(X_e)<f(X_g)$, extension point $X_e$ replaces $X_s$; otherwise, $X_r$ replaces $X_s$.
• Step 5. If $f(X_r)>f(X_s)$, the compression operation is constructed as follows:

  $$X_t=X_c+\beta (X_s-X_c)$$

  (18)

  where $\beta =0.5$ signifies the compression factor. If $f(X_t)<f(X_s)$, compression point $X_t$ replaces $X_s$; otherwise, $X_r$ replaces $X_s$.
• Step 6. If $f(X_g)<f(X_r)<f(X_s)$, the contraction point $X_w$ is computed as follows:

  $$X_w=X_c-\beta (X_s-X_c)$$

  (19)

  where $\beta =0.5$ signifies the contraction factor. If $f(X_w)<f(X_s)$, contraction point $X_w$ replaces $X_s$; otherwise, $X_r$ replaces $X_s$.

Symmetry is an indispensable characteristic in methodologies and applications, and this is essential in multiple distinctive disciplines. Symmetry in algorithms frequently emphasizes patterns within data, streamlines complicated difficulties, and advances methodology efficiency. In this paper, we derive motivation from symmetric cooperative swarms. The entire sand cat swarm is separated into superior and inferior individuals, encouraging each individual to carry out the multiple responsibilities and occupations and facilitating the investigation and utilization of the solution region.

Algorithm 3 emphasizes the pseudocode of MSSCSO.

Algorithm 3 MSSCSO

Begin Step 1. Randomly initialize the population, $r$, $r_G$ and $R$ Step 2. Ascertain the sand cat’s fitness and identify the finest prey $Po{s}_{bc}$ Step 3. while ($t<T$) do      for each sand cat do      Sort the population and distribute the selection probability $P_i$      /*ranking-based mutation stage*/      Automatically identify $r_1\in \left\{1,N_p\right\}$ {base vector index}      while $rand>p_{r_1}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_1==i$      Automatically identify $r_1\in \left\{1,N_p\right\}$      end      Automatically identify $r_2\in \left\{1,N_p\right\}$ {terminal vector index}      while $rand>p_{r_2}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_2==r_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_2==i$      Automatically identify $r_2\in \left\{1,N_p\right\}$      end      Automatically identify $r_3\in \left\{1,N_p\right\}$ {starting vector index}      while $r_3==r_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_3==r_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_3==i$      Automatically identify $r_3\in \left\{1,N_p\right\}$      end      /*end of ranking-based mutation stage*/        Accomplish a randomized angle via Roulette Wheel Selection $(0^{°}\le \theta \le {360}^{°})$      if ($\left|R\right|\le 1$) then          Renew position via Equation (8)      else          Renew position via Equation (6)      end if      The elite opposition-based learning strategy and simplex method are introduced into exploring and exploiting SCSO.      end for      $t=t+1$     end while     Restore the finest prey $Po{s}_{bc}$ End

4. Simulation Test and Result Analysis for Benchmark Functions

4.1. Experimental Setup

The computerized instruments are executed on a Windows 10 PC with an Intel Core i7-8750H 2.2 GHz CPU, a GTX1060, and 8 GB RAM. All approaches are written and operated in MATLAB R2018b.

4.2. CEC 2022 Benchmark Functions

To assess the practicality and feasibility, the MSSCSO is used to resolve the function optimization.

Table 2. Description of the CEC 2022 test suite.

Function Type	No.	Name	Dim	[lb,ub]	${\mathit{F}}_{\mathbf{min}}$
Unimodal function	$F_1$	Shifted and full rotated Zakharov function	10	[−100,100]	300
Basic functions	$F_2$	Shifted and full rotated Rosenbrock’s function	10	[−100,100]	400
	$F_3$	Shifted and full rotated expanded Schaffer’s $F_6$ function	10	[−100,100]	600
	$F_4$	Shifted and full rotated non-continuous Rastrigin’s function	10	[−100,100]	800
	$F_5$	Shifted and rotated Levy function	10	[−100,100]	900
Hybrid functions	$F_6$	Hybrid function 1 ($N=3$)	10	[−100,100]	1800
	$F_7$	Hybrid function 2 ($N=6$)	10	[−100,100]	2000
	$F_8$	Hybrid function 3 ($N=5$)	10	[−100,100]	2200
Composite functions	$F_9$	Composite function 1 ($N=5$)	10	[−100,100]	2300
	$F_{10}$	Composite function 2 ($N=4$)	10	[−100,100]	2400
	$F_{11}$	Composite function 3 ($N=5$)	10	[−100,100]	2600
	$F_{12}$	Composite function 4 ($N=6$)	10	[−100,100]	2700

summarizes the description of the CEC 2022 test suite [30,35].

4.3. Variables Setting

The MSSCSO is employed to resolve the CEC 2022, which is contrasted with bottlenose dolphin optimization (BDO) [36], Chornobyl disaster optimization (CDO) [37], golden jackal optimization (GJO) [38], Kepler optimization algorithm (KOA) [39], liver cancer algorithm (LCA) [40], spider wasp optimization (SWO) [41], rat swarm optimization (RSO) [42], hybrid leader-based optimization (HLBO) [43], waterwheel plant algorithm (WWPA) [44], sinh cosh optimization (SCHO) [45], and sand cat swarm optimization (SCSO) [25]. Each approach’s regulated variables are conventional speculative values extracted from the source papers.

BDO: accelerate factor $a_f=3.5$, randomized factor $S_r=0.8$, probability factor $p=0.5$, positive constant ${\theta }_{\min }=0.4$.

CDO: gamma speed $s_{\gamma }\in \left(1,300000\right)$, beta speed $s_{\beta }\in \left(1,270000\right)$, alpha speed $s_{\alpha }\in \left(1,16000\right)$, randomized value $r\in (0,1)$.

GJO: randomized value $rand\in [0,1]$, randomized value $r\in [0,1]$, positive constant $c_1=1.5$, randomized value $u\in (0,1)$, randomized value $v\in (0,1)$, positive constant $\beta =1.5$.

KOA: positive constant $T=3$, positive constant $\gamma =15$, positive constant ${\mu }_0=0.1$.

LCA: randomized value $r_d\in [0,1]$, tumor $width\in [0,1]$, tumor $depth\in [0,1]$, tumor $length\in [0,1]$, positive constant $f=1$, radius $r\in [0,1]$.

SWO: tradeoff rate $TR=0.3$, crossover rate $CR=0.2$, randomized value $l\in [-2,1]$, randomized value $p\in [0,1]$.

RSO: randomized value $R\in [1,5]$, randomized value $C\in [0,2]$.

HLBO: randomized value $r\in [0,1]$, integer value $I\in \{1,2\}$, positive constant $R=0.2$.

WWPA: randomized value $r_1\in [0,2]$, randomized value $r_2\in [0,1]$, randomized value $r_3\in [0,2]$, exponential value $K\in [0,1]$, randomized value $F\in [-5,5]$, randomized value $C\in [-5,5]$.

SCHO: positive constant $ct=3.6$, randomized value $r\in [0,1]$, sensitive factor $u=0.388$, sensitive factor $m=0.45$, tiny positive value $\epsilon =0.003$.

SCSO: sensitivity variable $r_G\in [0,2]$, phase control variable $R\in [-2r_G,2r_G]$.

MSSCSO: sensitivity variable $r_G\in [0,2]$, phase control variable $R\in [-2r_G,2r_G]$, randomized value $k\in (0,1)$, reflectivity $\alpha =1$, expansion coefficient $\gamma =1.5$, compression coefficient $\beta =0.5$, contraction coefficient $\beta =0.2$, scaling factor $F=0.7$.

4.4. Sensitivity Analysis

This section discusses the sensitivity investigation of the hyperparameter tuning of MSSCSO by varying the scaling factor $F$ and keeping other parameters fixed.

Table 3. Sensitivity results of MSSCSO for the different scaling factors $F$.

Function	Result	Scaling Factor F
		0.1	0.3	0.5	0.7	0.9	0.11	0.13
F1	Best	300.3073	300.1744	300.1571	300.6584	301.1911	302.4874	301.6503
	Std	27.35221	20.69498	19.78377	20.28569	47.51898	44.99700	45.89157
	Mean	321.4812	323.6530	320.5828	318.9658	356.7509	352.9056	377.4065
	Median	309.1524	321.0634	318.1187	310.3300	344.9288	342.6918	386.6018
	Worst	394.7847	373.6334	378.0730	372.1049	498.4575	471.1267	485.6404
F2	Best	400.4242	400.1573	400.2118	400.0055	400.3607	400.4434	400.6102
	Std	26.43133	29.94478	37.69150	13.89718	3.372276	13.09777	14.30825
	Mean	424.7244	445.7873	439.7443	410.6535	407.3034	410.5346	411.9227
	Median	411.6288	447.6871	431.2803	407.4000	407.5555	407.6815	407.8502
	Worst	475.6888	514.4983	542.6178	470.8142	412.2323	472.5011	465.7700
F3	Best	600.6350	604.3034	601.6990	600.1112	600.2999	600.3239	600.3078
	Std	8.335732	8.254438	7.359391	2.594358	1.883909	2.479979	2.106617
	Mean	615.4231	616.4457	611.8143	602.2905	602.1334	602.6434	602.5331
	Median	615.9688	615.7775	610.2676	601.2250	601.8102	601.5358	601.9688
	Worst	635.8255	635.9863	632.3663	611.8587	610.7669	610.9289	609.6038
F4	Best	807.9609	813.9444	806.9650	804.9956	809.7937	805.5202	805.5820
	Std	6.923790	7.482125	6.157279	6.092363	4.890488	6.645209	7.098210
	Mean	823.8254	824.5767	820.9302	816.7228	817.9319	818.4581	818.4082
	Median	824.4116	824.9014	821.4310	815.7339	816.9648	817.2838	817.5602
	Worst	835.8218	838.8042	833.8812	831.8398	831.0936	830.9169	833.8749
F5	Best	900.0619	903.2530	901.8745	900.0761	900.0278	900.1004	900.0980
	Std	98.91538	119.9691	108.9970	10.84537	3.880727	9.692292	17.70980
	Mean	994.8915	999.9370	1009.578	907.1468	903.2252	906.7598	909.9744
	Median	969.2928	963.6642	981.3706	901.8173	902.3931	902.3058	901.7920
	Worst	1358.285	1372.578	1343.114	947.3566	920.6647	938.2756	986.2309
F6	Best	1862.773	1946.691	1905.818	1973.302	2169.275	1933.312	2469.662
	Std	1902.844	2033.640	1474.681	2141.748	2433.540	8388.875	2368.603
	Mean	4246.573	4175.993	3974.290	4515.021	5128.079	7043.900	6138.991
	Median	3773.378	3639.743	3740.014	4402.019	4151.588	5452.960	5734.492
	Worst	8123.173	8108.153	8175.531	8192.207	9377.553	49681.57	11236.24
F7	Best	2021.655	2016.735	2021.267	2001.167	2020.484	2020.895	2014.423
	Std	10.28591	12.45873	10.96718	11.57752	5.005222	5.379638	5.546232
	Mean	2038.748	2037.705	2039.983	2024.358	2025.220	2026.394	2025.945
	Median	2037.818	2035.994	2037.976	2024.058	2023.373	2024.669	2024.328
	Worst	2059.632	2065.161	2070.798	2059.712	2042.660	2042.665	2038.861
F8	Best	2213.617	2204.947	2209.794	2202.691	2202.819	2206.141	2203.812
	Std	3.287674	5.057958	4.132935	6.542852	5.688333	4.842115	5.877626
	Mean	2224.956	2224.397	2224.700	2221.238	2222.270	2223.126	2223.689
	Median	2224.863	2225.368	2225.915	2222.555	2223.998	2224.051	2225.141
	Worst	2232.605	2232.308	2229.803	2227.969	2227.876	2228.147	2230.420
F9	Best	2510.602	2518.721	2521.244	2508.868	2501.490	2512.881	2514.225
	Std	45.15194	36.26405	34.81383	10.02568	7.518856	4.555963	5.415469
	Mean	2582.450	2566.289	2567.374	2525.331	2524.904	2524.614	2523.913
	Median	2582.590	2566.398	2561.144	2524.991	2527.344	2524.959	2524.591
	Worst	2674.061	2650.742	2656.330	2564.274	2536.983	2531.966	2533.878
F10	Best	2500.213	2500.229	2500.186	2500.250	2500.285	2500.256	2500.339
	Std	57.37454	64.50411	62.01304	0.068716	0.097435	0.135486	0.086766
	Mean	2540.252	2559.726	2561.297	2500.379	2500.429	2500.471	2500.469
	Median	2500.527	2502.370	2554.732	2500.372	2500.418	2500.469	2500.453
	Worst	2628.760	2639.614	2636.280	2500.556	2500.683	2500.922	2500.668
F11	Best	2600.539	2600.913	2600.843	2600.851	2601.930	2601.776	2603.467
	Std	169.0856	66.55507	130.8655	63.13874	135.0487	112.4276	61.59830
	Mean	2678.347	2646.605	2652.849	2654.790	2710.581	2703.643	2689.650
	Median	2603.162	2604.155	2602.363	2606.769	2724.564	2732.358	2732.491
	Worst	3223.828	2788.254	3275.880	2752.298	3197.826	3191.905	2757.200
F12	Best	2851.128	2855.637	2850.101	2853.429	2853.414	2853.433	2856.232
	Std	14.27506	15.92806	8.744910	2.845476	2.391151	3.191990	2.704067
	Mean	2865.745	2867.139	2861.997	2860.225	2861.375	2861.021	2861.655
	Median	2861.552	2860.424	2860.280	2859.648	2861.668	2861.866	2861.316
	Worst	2901.384	2901.050	2900.071	2866.525	2866.180	2866.599	2866.945

summarizes the sensitivity results of MSSCSO for the different scaling factors $F$. For each approach, the average sample equals 50, the highest iteration equals 1000, and the detached operation equals 30. Best, Std, Mean, Median, and Worst signify the idealized value, standard deviation, average value, median value, and catastrophic value. To investigate the impact of the scaling factor $F$ on the convergence efficiency and estimation precision, the MSSCSO is carried out for several different values of the scaling factor $F$ taken as 0.1, 0.3, 0.5, 0.7, 0.9, 0.11, and 0.13, while keeping other parameters unchanged. The MSSCSO has a relatively good sensitivity to the scaling factor $F$. The difference between the results of different scaling factors $F$ is significant. When the scaling factor $F$ is set to 0.7, MSSCSO receives excellent reliability and robustness to avoid anticipation stagnation and incorporates exploration and exploitation to achieve a swifter convergent velocity, greater explored accuracy, and more accurate fitness.

4.5. Simulation Test and Result Analysis

The MSSCSO is utilized to resolve the CEC 2022 test suite, which contains unimodal, essential, hybrid, and composite functions. The ranking is based on the catastrophic value.

Table 4. Simulation results of different approaches for the CEC 2022.

Function	Result	BDO	CDO	GJO	KOA	LCA	SWO	RSO	HLBO	WWPA	SCHO	SCSO	MSSCSO
F1	Best	4925.789	8249.941	328.3754	16,253.06	8085.756	3867.746	1819.959	3932.907	8309.104	1833.181	322.7396	300.6584
	Std	1833.028	1.33$\times {10}^6$	2097.548	11,783.83	776.4972	4116.102	1448.814	3268.359	4852.778	61,039.22	1434.062	20.28569
	Mean	9656.029	270,488.7	2401.405	34,595.37	10,345.93	8855.867	2898.582	11,440.75	12,591.37	29,761.88	1309.624	318.9658
	Median	10,234.82	176,61.92	1772.794	31,105.16	10,767.56	8296.003	1856.988	11,725.69	10,863.60	7169.126	473.4388	310.3300
	Worst	13,376.90	7.30$\times {10}^6$	6500.795	56,400.90	11,022.00	17,229.55	6223.824	19,334.03	30,465.83	262,160.3	4667.163	372.1049
	Rank	5	12	6	10	2	8	4	7	9	11	3	1
F2	Best	829.4427	806.7506	407.6932	702.1547	624.9459	449.6727	569.4683	846.0378	1026.749	801.1575	400.1207	400.0055
	Std	240.5120	15.21908	24.26073	987.3036	1019.987	50.27431	206.8468	220.4127	1045.765	1000.582	33.18334	13.89718
	Mean	1165.991	849.1894	448.5079	2466.162	2250.694	530.7590	790.3062	1184.146	3036.182	1781.587	429.6792	410.6535
	Median	1091.252	852.3551	455.0632	2459.777	2046.559	519.9205	720.5938	1173.474	3225.143	1227.342	408.9474	407.4000
	Worst	1679.642	870.1469	487.0120	4940.152	5749.846	641.2478	1305.500	1664.289	5004.719	4342.539	502.0246	470.8142
	Rank	8	2	3	9	11	5	6	7	12	10	4	1
F3	Best	636.7920	628.4212	600.1967	675.9502	623.1902	616.4829	627.8814	650.0869	668.1951	627.6416	600.5163	600.1112
	Std	6.541113	4.020216	3.235669	8.538123	12.84364	8.493622	7.418231	6.405414	6.479185	17.92734	7.573123	2.594358
	Mean	656.2125	635.2412	604.6347	692.7154	665.5881	634.3778	643.7557	662.9752	681.1905	654.1414	610.8692	602.2905
	Median	656.3028	634.8934	604.0315	693.2727	668.6956	632.9724	643.3323	662.6562	679.9632	654.4357	609.4264	601.2250
	Worst	670.9830	644.6496	612.8190	708.2447	683.1953	653.7690	661.1006	675.1539	691.8179	700.069	628.9012	611.8587
	Rank	6	3	2	10	11	9	7	4	5	12	8	1
F4	Best	835.3624	832.5326	813.5885	876.4839	853.2148	837.0745	825.9955	837.5426	862.9625	838.5356	817.4304	804.9956
	Std	7.870513	6.262158	10.10826	11.46466	10.06061	9.519617	7.130501	10.09995	8.393425	11.08817	6.988211	6.092363
	Mean	853.8336	846.3934	827.8400	898.2084	875.2959	852.8306	839.3213	855.2635	881.4053	858.4868	828.9365	816.7228
	Median	854.6623	845.5254	825.0855	904.2125	876.8686	853.5633	839.3571	855.1270	882.8744	859.3410	829.1559	815.7339
	Worst	868.3677	860.5760	853.0835	909.6414	893.7319	875.2241	854.7475	875.2821	893.5234	877.6968	842.3118	831.8398
	Rank	5	2	10	12	8	7	4	9	6	11	3	1
F5	Best	1188.073	1275.414	900.9022	2983.761	1508.611	977.5962	1065.345	1247.959	1746.702	1206.901	900.7257	900.0761
	Std	170.4931	96.00600	91.90270	302.2772	279.2597	258.8724	114.0445	188.7156	153.9177	440.5012	129.9267	10.84537
	Mean	1562.193	1404.611	972.2127	3426.435	1940.832	1338.877	1296.001	1630.661	2208.417	1598.969	1031.926	907.1468
	Median	1564.401	1391.264	951.5516	3485.005	1941.104	1342.861	1299.212	1611.167	2236.375	1532.964	995.7267	901.8173
	Worst	1863.978	1671.877	1347.808	3733.018	2441.794	1915.348	1591.674	2086.508	2559.262	3771.114	1496.126	947.3566
	Rank	7	3	2	11	10	9	4	8	6	12	5	1
F6	Best	2.05$\times {10}^6$	4.64$\times {10}^7$	3448.482	2.16$\times {10}^8$	1.12$\times {10}^7$	9038.476	4897.245	2255.491	1.54$\times {10}^7$	7362.507	1938.987	1973.302
	Std	6.79$\times {10}^7$	2.92$\times {10}^8$	1972.504	7.68$\times {10}^8$	7.66$\times {10}^7$	2.07$\times {10}^6$	1.67$\times {10}^7$	1.56$\times {10}^7$	6.85$\times {10}^7$	8.11$\times {10}^8$	2250.499	2141.748
	Mean	9.98$\times {10}^7$	2.10$\times {10}^8$	7969.726	1.20$\times {10}^9$	1.10$\times {10}^8$	1.11$\times {10}^6$	6.97$\times {10}^6$	3.05$\times {10}^6$	1.17$\times {10}^8$	5.02$\times {10}^8$	4249.465	4515.021
	Median	8.59$\times {10}^7$	6.46$\times {10}^7$	8294.423	1.07$\times {10}^9$	1.07$\times {10}^8$	2.36$\times {10}^5$	2.69$\times {10}^4$	6.65$\times {10}^4$	1.14$\times {10}^8$	1.00$\times {10}^8$	3778.622	4402.019
	Worst	2.46$\times {10}^8$	9.53$\times {10}^8$	1.47$\times {10}^4$	3.06$\times {10}^9$	2.16$\times {10}^8$	9.07$\times {10}^6$	5.66$\times {10}^7$	8.55$\times {10}^7$	2.21$\times {10}^8$	3.54$\times {10}^9$	8186.026	8192.207
	Rank	7	10	1	11	9	4	6	5	8	12	3	2
F7	Best	2071.142	2115.190	2021.747	2111.493	2039.149	2054.527	2060.302	2097.086	2092.817	2115.381	2013.144	2001.167
	Std	19.66678	7.595220	14.31978	58.45546	45.75851	22.22771	28.52728	28.35122	30.46134	52.95661	27.02902	11.57752
	Mean	2111.562	2136.684	2045.262	2241.436	2151.434	2086.697	2086.549	2159.502	2191.014	2192.986	2045.393	2024.358
	Median	2110.646	2138.762	2042.527	2239.525	2148.629	2085.686	2083.649	2164.046	2198.537	2190.078	2039.688	2024.058
	Worst	2153.631	2146.214	2079.277	2346.168	2239.668	2142.546	2191.996	2215.862	2240.532	2331.348	2129.024	2059.712
	Rank	7	10	1	11	9	4	6	5	8	12	3	2
F8	Best	2244.850	2218.107	2221.919	2320.820	2236.046	2227.601	2228.669	2224.842	2250.125	2224.984	2209.145	2202.691
	Std	20.80617	4.158169	3.159479	1226.407	79.27812	12.27367	28.47745	28.38429	52.30765	141.8536	4.753517	6.542852
	Mean	2282.770	2229.619	2225.664	2893.113	2331.207	2243.272	2256.533	2262.251	2326.146	2371.111	2227.030	2221.238
	Median	2282.577	2230.040	2225.233	2569.242	2312.812	2242.107	2248.092	2262.338	2306.144	2299.889	2226.427	2222.555
	Worst	2349.706	2238.458	2232.937	9113.386	2502.489	2290.662	2361.180	2348.539	2471.151	2695.023	2234.457	2227.969
	Rank	6	2	1	12	10	5	8	7	9	11	3	4
F9	Best	2689.438	2656.222	2529.412	2763.846	2720.123	2501.430	2563.344	2684.120	2712.392	2653.578	2529.287	2508.868
	Std	27.03878	1.458474	35.86557	187.3811	85.30585	41.90462	42.79407	13.26892	40.87431	37.37313	41.92706	10.02568
	Mean	2737.129	2658.258	2589.261	3045.117	2851.621	2577.667	2659.585	2707.853	2818.665	2698.775	2579.727	2525.331
	Median	2732.001	2658.254	2583.562	3006.600	2841.078	2581.562	2667.191	2706.252	2813.997	2706.640	2582.722	2524.991
	Worst	2796.155	2662.427	2660.870	3655.114	3021.756	2682.051	2759.172	2752.530	2907.409	2772.768	2683.415	2564.274
	Rank	4	1	5	12	11	8	10	3	7	6	9	2
F10	Best	2524.034	2617.031	2500.378	2541.004	2530.898	2501.528	2508.385	2539.240	2559.044	2563.400	2500.369	2500.250
	Std	31.56800	136.8292	61.09714	240.9939	663.2109	79.14043	30.89074	204.1968	514.8208	338.6159	64.43356	0.068716
	Mean	2565.077	2729.747	2549.847	2698.455	3215.576	2581.953	2515.619	2871.122	3398.702	2974.808	2559.468	2500.379
	Median	2559.226	2698.023	2500.609	2611.429	2948.854	2542.406	2509.145	2827.971	3328.672	2915.630	2500.696	2500.372
	Worst	2652.408	3348.214	2644.563	3796.995	4994.002	2739.043	2678.422	3311.689	4131.091	4028.904	2645.483	2500.556
	Rank	3	7	4	9	12	6	2	8	11	10	5	1
F11	Best	2961.518	3333.839	2729.998	5291.300	3296.152	2811.470	2899.327	2989.809	3807.391	3337.346	2600.910	2600.851
	Std	373.4564	4.709603	209.2536	2.7$\times {10}^{-12}$	500.8876	422.8975	310.2735	1717.658	314.3944	593.7176	155.5155	63.13874
	Mean	3812.036	3342.067	2948.272	5291.300	4686.232	3479.935	3315.277	5161.047	4668.855	3990.278	2803.975	2654.790
	Median	3740.752	3342.149	2871.473	5291.300	4909.681	3561.035	3282.278	4963.890	4763.186	3880.141	2742.402	2606.769
	Worst	4393.412	3351.079	3471.845	5291.300	5113.701	4659.780	4090.669	11367.20	5066.628	5273.214	3131.513	2752.298
	Rank	8	2	5	1	10	9	6	12	7	11	4	3
F12	Best	2872.613	2987.326	2859.822	2891.037	2995.417	2874.042	2871.779	2884.172	2892.118	3008.190	2862.568	2853.429
	Std	7.917417	29.73760	9.735571	47.77224	115.5592	4.739680	36.53797	2.911470	1.439513	141.8999	15.69188	2.845476
	Mean	2892.342	3065.753	2868.682	2964.658	3194.247	2899.137	2901.327	2899.390	2899.740	3212.078	2870.729	2860.225
	Median	2892.118	3063.466	2865.081	2964.306	3190.925	2900.002	2885.013	2900.002	2900.002	3184.156	2865.424	2859.648
	Worst	2912.796	3114.900	2897.656	3059.869	3436.487	2900.002	3016.087	2900.002	2900.002	3513.019	2938.428	2866.525
	Rank	5	8	6	10	11	4	9	3	1	12	7	2

summarizes the simulation results of different approaches for the CEC 2022. To alleviate deficiencies of SCSO, such as inefficient initial population quality, sluggish convergence acceleration, and restricted evaluation precision, the ranking-based mutation operator, elite opposition-based learning strategy, and simplex method are incorporated into SCSO. For F1, F2, F3, F4, F5, and F10, the idealized values, standard deviations, average values, median values, and catastrophic values of the MSSCSO have been substantially augmented; the MSSCSO exhibits instructive sustainability and adaptability to refrain from precocious convergence and recognize the finest potential solution. The idealized values, standard deviations, average values, median values, and catastrophic values of MSSCSO are superior to those of BDO, CDO, GJO, KOA, LCA, SWO, RSO, HLBO, WWPA, SCHO, and SCSO. The MSSCSO utilizes the ranking-based mutation operator to filtrate the finest potential solution, foster selection probability, quicken the convergence procedure, and bolster localized exploitation. The MSSCSO utilizes broad discovery and localized extraction to enhance convergence acceleration and obtain better assessment metrics. The MSSCSO exhibits the most minor standard deviations and the best ranking, and it has strong stability and reliability. For F6 and F8, the standard deviations of the MSSCSO are worse than those of the SCSO. Still, the idealized values, standard deviations, average, median, and catastrophic values of the MSSCSO are more accurate than those of the BDO, KOA, LCA, SWO, RSO, HLBO, WWPA, and SCHO. The MSSCSO utilizes the elite opposition-based learning strategy to facilitate demographic variability, upgrade searched areas, expedite evaluation efficiency, refrain from precocious convergence, and bolster broad exploration. For F7, F9, F11, and F12, the MSSCSO has strong stability and reliability to enhance the idealized values, standard deviations, average, median, and catastrophic values. The assessment metrics and convergence efficiency of the MSSCSO have been substantially augmented. The idealized values, standard deviations, average, median, and catastrophic values of the MSSCSO are superior to those of the BDO, CDO, LCA, SWO, RSO, HLBO, SCHO, and SCSO. The MSSCSO utilizes the simplex method to advance initial population quality, escalate recognition depth, disrupt regional extreme solutions, advance extraction operation, and elevate estimation precision. The MSSCSO with symmetric cooperative swarms not only maintains instructive sustainability and adaptability to capture supplementary advantages and prohibit search standstill but also integrates prospecting and extracting to strengthen convergence productivity and accomplish the most satisfactory solution.

Figure 5 articulates the convergence trajectories of each approach. The convergence curve is a spontaneous technique used to visualize each approach’s utilization efficiency and resolution precision. When the comparison algorithms stabilize after a series of iterations, the approach exhibits superior evaluation precision and swifter utilization efficiency, which has instructive sustainability and adjustability to discover the most satisfactory solution. The MSSCSO endures an exceptional detection efficiency to mitigate anticipation stagnation and achieve the global optimal solution, which swiftly converges early and reliably re-explores later. The MSSCSO exhibits strong reliability and stability to incorporate discovery and extraction, superior convergence speed, calculation precision, and detection efficiency. For unimodal functions, essential functions, hybrid functions, and composite functions, the overall convergence efficiency and estimation precision of MSSCSO are superior to those of BDO, CDO, GJO, KOA, LCA, SWO, RSO, HLBO, WWPA, SCHO, and SCSO. The MSSCSO exhibits simple architectures, excessive optimization efficiency, marvelous parallelism, outstanding robustness, and stability, and it encounters excellent superiority and predictability in refraining from combinatorial explosion and precocious convergence. Figure 6 articulates the ANOVA tests of each approach. The standard deviation is a spontaneous technique to visualize each approach’s stability and reliability. The approach exhibits superior durability and dependability to foster optimization efficiency, advance extraction operation, and elevate estimation precision. The standard deviations of MSSCSO are superior to those of BDO, CDO, GJO, KOA, LCA, SWO, RSO, HLBO, WWPA, SCHO, and SCSO. The MSSCSO exhibits the formidable sustainability and dependability to disrupt the stagnation of exploration and recognize the most desirable solution. The MSSCSO incorporates the utilization and investigation to attain a swifter convergence velocity, greater estimation precision, superior regulation variables, lower standard deviation, and more accurate fitness.

Table 5. The execution time of different approaches for the CEC 2022 test functions.

Function	BDO	CDO	GJO	KOA	LCA	SWO	RSO	HLBO	WWPA	SCHO	SCSO	MSSCSO
F1–12	189.2856	51.8795	67.0120	76.1346	48.2005	43.2389	38.7741	103.4885	65.2331	96.4285	52.2834	79.8849

summarizes the execution time of different approaches for the CEC 2022 test functions. For each approach, BDO is 189.2856 s, CDO is 51.8795 s, GJO is 67.0120 s, KOA is 76.1346 s, LCA is 48.2005 s, SWO is 43.2389 s, RSO is 38.7741 s, HLBO is 103.4885 s, WWPA is 65.2331 s, SCHO is 96.4285 s, SCSO is 52.2834 s, MSSCSO is 79.8849 s. The execution time of the MSSCSO is better than those of the BDO, HLBO, and SCHO, but worse than those of the CDO, GJO, KOA, LCA, SWO, RSO, WWPA, and SCSO. The ranking-based mutation operator, elite opposition-based learning strategy, and simplex method are introduced into the SCSO, which increases the execution time needed to identify the most satisfactory solution.

Table 6. Results of the p-value Wilcoxon rank-sum test on the benchmark functions.

Function	CGJO vs
	BDO	CDO	GJO	KOA	LCA	SWO	RSO	HLBO	WWPA	SCHO	SCSO
F1	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	8.99$\times {10}^{-11}$	2.93$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	1.61$\times {10}^{-10}$
F2	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	1.85$\times {10}^{-8}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	4.08$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	5.55$\times {10}^{-5}$
F3	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	2.05$\times {10}^{-3}$	3.00$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	7.09$\times {10}^{-8}$
F4	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	8.88$\times {10}^{-6}$	2.69$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	4.98$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	6.01$\times {10}^{-8}$
F5	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	2.20$\times {10}^{-7}$	2.55$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	4.57$\times {10}^{-9}$
F6	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	2.83$\times {10}^{-8}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	1.55$\times {10}^{-9}$	3.52$\times {10}^{-7}$	3.02$\times {10}^{-11}$	9.92$\times {10}^{-11}$	3.40$\times {10}^{-2}$
F7	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	1.73$\times {10}^{-7}$	3.02$\times {10}^{-11}$	4.08$\times {10}^{-11}$	3.34$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	1.17$\times {10}^{-4}$
F8	3.02$\times {10}^{-11}$	1.43$\times {10}^{-8}$	1.00$\times {10}^{-3}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.69$\times {10}^{-11}$	3.02$\times {10}^{-11}$	9.92$\times {10}^{-11}$	3.02$\times {10}^{-11}$	1.21$\times {10}^{-10}$	4.12$\times {10}^{-6}$
F9	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	1.09$\times {10}^{-10}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	9.83$\times {10}^{-8}$	3.34$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	6.72$\times {10}^{-10}$
F10	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	1.10$\times {10}^{-8}$	2.98$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	2.23$\times {10}^{-9}$
F11	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	8.15$\times {10}^{-11}$	1.21$\times {10}^{-12}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	2.60$\times {10}^{-5}$
F12	2.39$\times {10}^{-11}$	3.02$\times {10}^{-11}$	1.70$\times {10}^{-8}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	5.00$\times {10}^{-9}$

summarizes the results of the p-value Wilcoxon rank-sum test on the benchmark functions. Wilcoxon’s rank-sum test is administered to ascertain if MSSCSO differs noticeably from other approaches [46,47,48]. $p<0.05$ shows the distinguished discrepancy. $p\ge 0.05$ shows no distinguished discrepancy. The experimental results demonstrate a distinguished discrepancy between MSSCSO and different approaches, and the data endure an appropriate degree of authenticity and dependability that was not established by accident.

5. MSSCSO-Based Adaptive IIR System Identification

5.1. Adaptive IIR System Identification

The MSSCSO is implemented to resolve the IIR system identification, and the principal objective is to recognize the most accurate regulating coefficients, minimize MSE between an unidentified system’s input and the IIR filter’s output, and match the unknown system’s transfer function. The adaptive IIR system exhibits some drawbacks: (1) It is unstable due to the inappropriate selection of denominator coefficients. The proper selection of search space overcomes this problem. (2) It cannot provide an exact linear phase response. The ranking-based mutation operator, elite opposition-based learning strategy, and simplex method are added to the SCSO to overcome drawbacks and improve the convergence speed and calculation accuracy. The MSSCSO maintains instructive stability and adaptability to capture supplementary advantages, prohibit discovery stagnation, and exhibit excellent global exploration and local exploitation to achieve the most satisfactory solution. The MSSCSO has strong robustness and reliability to minimize the MSE according to the stability of the IIR system, and the minimum solution of the MSE is regarded as the global optimal solution. System identification is advantageous for establishing a mathematical representation by investigating the unidentified structure’s input and the IIR filter’s output. Figure 7 articulates the adaptive IIR system identification via MSSCSO.

The input $x(n)$ and output $y(n)$ of the IIR filter is constructed as follows:

$${\displaystyle \sum _{l=0}^L{a}_{l}y}(n-l)={\displaystyle \sum _{k=0}^K{b}_{k}x(n-k})$$

(20)

where $L$ signifies the feedforward order, $K$ signifies the feedback, $a_l$ signifies the pole factor, and $b_k$ signifies the zero factor. The $H(z)$ is constructed as follows:

$$H(z)={\displaystyle \frac{{\displaystyle {\sum }_{k=0}^K{b}_k{z}^{-k}}}{1+{\displaystyle {\sum }_{l=0}^L{a}_l{z}^{-l}}}}$$

(21)

The discrepancy between the unidentified system’s input and the IIR filter’s output is $e(n)=y_0(n)-y(n)$. The MSE is constructed as follows:

$$MSE=J(w)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=1}^N{e}^2(n)}$$

(22)

where $N$ signifies the input sample size and $w$ signifies the coefficient factor, $w={(a_1,a_2,\dots ,a_L,b_0,b_1,\dots ,b_K)}^T$.

The simple steps of the proposed model are summarized as follows:

(1)

Give the input variable $x(n)$ and output variable $y(n)$ of the adaptive IIR system identification.

(2)

Establish the mathematical expression for the IIR filter’s input and output; the formula is ${\displaystyle \sum _{l=0}^L{a}_{l}y}(n-l)={\displaystyle \sum _{k=0}^K{b}_{k}x(n-k})$.

(3)

Introduce noise disturbance $v(n)$ and an adaptive optimization algorithm (MSSCSO).

(4)

Utilize the unknown IIR system and adaptive IIR systems to construct the transfer function of the IIR filter $H(z)={\displaystyle \frac{{\displaystyle {\sum }_{k=0}^K{b}_k{z}^{-k}}}{1+{\displaystyle {\sum }_{l=0}^L{a}_l{z}^{-l}}}}$.

(5)

Establish the fitness evaluation function of the mean square error (MSE) between an unidentified system’s input and the IIR filter’s output. The MSE is $MSE=J(w)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=1}^N{e}^2(n)}$.

5.2. MSSCSO-Based Adaptive IIR System Identification

Algorithm 4 emphasizes the MSSCSO-based adaptive IIR system identification. Figure 8 articulates the flowchart of MSSCSO for adaptive IIR system identification.

Algorithm 4 MSSCSO-based adaptive IIR system identification

Begin Step 1. Randomly initialize the population,$r$, $r_G$ and $R$ Step 2. Ascertain the error fitness via Equation (22) and identify the finest prey $Po{s}_{bc}$ Step 3. while ($t<T$) do       for each sand cat do       Sort the population and distribute the selection probability $P_i$       /*ranking-based mutation stage*/       Automatically identify $r_1\in \left\{1,N_p\right\}$ {base vector index}       while $rand>p_{r_1}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_1==i$       Automatically identify $r_1\in \left\{1,N_p\right\}$       end       Automatically identify $r_2\in \left\{1,N_p\right\}$ {terminal vector index}       while $rand>p_{r_2}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_2==r_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_2==i$       Automatically identify $r_2\in \left\{1,N_p\right\}$       end       Automatically identify $r_3\in \left\{1,N_p\right\}$ {starting vector index}       while $r_3==r_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_3==r_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}or\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}_3==i$       Automatically identify $r_3\in \left\{1,N_p\right\}$       end       /*end of ranking-based mutation stage*/        Accomplish a randomized angle via Roulette Wheel Selection $(0^{°}\le \theta \le {360}^{°})$       if ($\left|R\right|\le 1$) then     Renew position via Equation (8)       else     Renew position via Equation (6)       end if       The elite opposition-based learning strategy and simplex method are introduced into the exploration and exploitation of SCSO.       Ascertain the error fitness via Equation (22)       end for       $t=t+1$   end while   Restore the finest prey $Po{s}_{bc}$ End

5.3. Computational Complexity of MSSCSO

Time complexity: This measures each algorithm’s computational time and resource consumption to resolve the optimization issues. The big-O notation efficiently verifies the stability and reliability. The time complexity mainly relies on the population size, the maximum iteration, and the issue dimension. The ranking-based mutation operator, elite opposition-based learning strategy, and simplex method are added to the SCSO to promote exploration and exploitation. The MSSCSO contains three main parts: population initialization, searching for prey (exploration) and attacking prey (exploitation), updating the sand cat’s location, and halting judgment. In MSSCSO, the population size is $N$, the maximum iteration is $T$, and the issue dimension is $D$. The population initialization is $O(N\times D)$. The searching for prey (exploration) is $O(T\times N\times D)$. The attacking on prey (exploitation) is $O(T\times N\times D)$. The updating of the sand cat’s location is $O(T\times N\times D)$. The updating of the sand cat’s location with the ranking-based mutation operator, elite opposition-based learning strategy, and simplex method is $O(T\times N\times D)$. The halting judgment is $O(1)$. Therefore, the total time complexity of MSSCSO is $O(N\times D+T\times N\times D)$. Removing lower-order terms, the overall time complexity of the MSSCSO is $O(T\times N\times D)$.

Space complexity: This is measured by an algorithm’s additional storage. In MSSCSO, the population size is $N$ and the issue dimension is $D$. The MSSCSO with symmetric cooperative swarms maintains instructive sustainability and adaptability to capture supplementary advantages, prohibit search standstill, and identify the most satisfactory solution. The overall space complexity of the MSSCSO is $O(N\times D)$. The space efficiency of the MSSCSO is effective and stable.

5.4. Variables Setting

The MSSCSO is implemented to accomplish the IIR filter, which is contrasted with the coati optimization algorithm (COA) [49], GOOSE (GOOSE) [50], horned lizard defense tactics (HLOA) [51], Chornobyl disaster optimization (CDO) [37], liver cancer algorithm (LCA) [40], sinh cosh optimization (SCHO) [45], spider wasp optimization (SWO) [41], and sand cat swarm optimization (SCSO) [25]. Each approach’s regulated variables are conventional speculative values extracted from the source papers.

COA: randomized value $r\in [0,1]$, integer value $I\in \{1,2\}$.

GOOSE: weight of stone $randi\in [5,25]$, randomized value $rnd\in [0,1]$, randomized value $pro\in [0,1]$, randomized value $coe\in [0,1]$.

HLOA: hue angle $h\in (0,2\pi )$, positive constant $\partial =2$, randomized value $Ligh{t}_{1,2}\in \left[0,0.4046661\right]$, randomized value $Dar{k}_{1,2}\in \left[0.544051,1\right]$, randomized value $walk\in [-1,1]$, randomized value $\sigma \in [0,1]$.

CDO: gamma speed $s_{\gamma }\in \left(1,300000\right)$, beta speed $s_{\beta }\in \left(1,270000\right)$, alpha speed $s_{\alpha }\in \left(1,16000\right)$, randomized value $r\in (0,1)$.

LCA: randomized value $r_d\in [0,1]$, tumor $width\in [0,1]$, tumor $depth\in [0,1]$, tumor $length\in [0,1]$, positive constant $f=1$, radius $r\in [0,1]$.

SCHO: positive constant $ct=3.6$, randomized value $r\in [0,1]$, sensitive factor $u=0.388$, sensitive factor $m=0.45$, tiny positive value $\epsilon =0.003$.

SWO: tradeoff rate $TR=0.3$, crossover rate $CR=0.2$, randomized value $l\in [-2,1]$, randomized value $p\in [0,1]$.

SCSO: sensitivity variable $r_G\in [0,2]$, phase control variable $R\in [-2r_G,2r_G]$.

MSSCSO: sensitivity variable $r_G\in [0,2]$, phase control variable $R\in [-2r_G,2r_G]$, randomized value $k\in (0,1)$, reflectivity $\alpha =1$, expansion coefficient $\gamma =1.5$, compression coefficient $\beta =0.5$, contraction coefficient $\beta =0.2$, scaling factor $F=0.7$.

5.5. Simulation Test and Result Analysis

The MSSCSO maintains instructive sustainability and adaptability to capture supplementary advantages and prohibit search standstill, which integrates investigation and extraction to strengthen utilization efficiency and accomplish the most satisfactory solution.

For case 1, each approach facilitates a first-order IIR filter to recognize a second-order procedure; the unidentified procedure $H_P(z)$ and IIR filter $H_M(z)$ are constructed as follows:

$$H_P(z)={\displaystyle \frac{0.05-0.4z^{-1}}{1-1.1314z^{-1}+0.25z^{-2}}}$$

(23)

$$H_M(z)={\displaystyle \frac{b}{1-a{z}^{-1}}}$$

(24)

For case 2, each approach facilitates a second-order IIR filter to recognize a second-order procedure; the unidentified system $H_P(z)$ and IIR filter $H_M(z)$ are constructed as follows:

$$H_P(z)={\displaystyle \frac{1}{1-1.4z^{-1}+0.49z^{-2}}}$$

(25)

$$H_M(z)={\displaystyle \frac{b}{1+a_1{z}^{-1}+a_2{z}^{-2}}}$$

(26)

For case 3, each approach facilitates a higher-order IIR filter to recognize a higher-order procedure; the unidentified system $H_P(z)$ and IIR filter $H_M(z)$ are constructed as follows:

$$H_P(z)={\displaystyle \frac{1-0.4z^{-2}-0.65z^{-4}+0.26z^{-6}}{1-0.77z^{-2}-0.8498z^{-4}+0.6486z^{-6}}}$$

(27)

$$H_M(z)={\displaystyle \frac{b_0+b_1{z}^{-1}+b_2{z}^{-2}+b_3{z}^{-3}+b_4{z}^{-4}}{1+a_1{z}^{-1}+a_2{z}^{-2}+a_3{z}^{-3}+a_4{z}^{-4}}}$$

(28)

The MSSCSO is executed to eliminate the adaptive IIR system identification, and the principal objective is to establish the most appropriate regulation variables and the minimum achievable solution, which diminishes the estimation error between an unidentified system’s input and the IIR filter’s output.

Table 7. Experimental results (MSE) of each approach for case 1.

Result	COA	GOOSE	HLOA	CDO	LCA	SCHO	SWO	SCSO	MSSCSO
Best	0.010425	0.009364	0.009782	0.013410	0.012880	0.010675	0.013887	0.009837	0.008475
Worst	0.018360	0.018838	0.019014	0.018875	0.021729	0.019553	0.026223	0.018100	0.016172
Mean	0.014099	0.012353	0.012995	0.016588	0.018085	0.013451	0.018294	0.013034	0.011493
Std	0.002573	0.002330	0.002372	0.001860	0.002278	0.002525	0.003629	0.002657	0.001287
Rank	7	4	5	2	3	6	9	8	1

summarizes each approach’s experimental results (MSE) for case 1.

Table 8. The finest evaluation variables of each approach for case 1.

Result	COA	GOOSE	HLOA	CDO	LCA	SCHO	SWO	SCSO	MSSCSO
$a$	0.919218	0.896197	0.921217	0.103887	0.136351	0.912120	0.942938	0.953202	−0.10047
$b$	−0.23761	−0.32082	−0.29290	0.162300	0.135762	−0.24297	−0.27375	−0.22891	0.100398

summarizes the finest evaluation variables of each approach for case 1.

Table 9. The average evaluation variables of each approach for case 1.

Result	COA	GOOSE	HLOA	CDO	LCA	SCHO	SWO	SCSO	MSSCSO
$a$	0.505677	0.633612	0.572077	0.236824	0.250614	0.505595	0.742000	0.605683	0.804566
$b$	−0.11108	−0.13753	−0.19323	0.064053	0.027292	−0.14041	−0.25005	−0.15625	−0.23740

summarizes the average evaluation variables of each approach for case 1.

Table 10. Experimental results (MSE) of each approach for case 2.

Result	COA	GOOSE	HLOA	CDO	LCA	SCHO	SWO	SCSO	MSSCSO
Best	6.96$\times {10}^{-5}$	3.31$\times {10}^{-9}$	3.68$\times {10}^{-5}$	0.123240	0.015756	2.81$\times {10}^{-6}$	0.004408	1.10$\times {10}^{-8}$	3.14$\times {10}^{-9}$
Worst	0.228098	0.244192	0.236883	0.264669	0.279143	0.260364	0.377817	0.244137	1.17$\times {10}^{-6}$
Mean	0.068231	0.048700	0.081601	0.212765	0.201706	0.128699	0.112613	0.057494	1.81$\times {10}^{-7}$
Std	0.082517	0.088644	0.072197	0.033496	0.067422	0.105720	0.101871	0.089363	2.58$\times {10}^{-7}$
Rank	5	6	4	2	3	9	8	7	1

summarizes each approach’s experimental results (MSE) for case 2.

Table 11. The finest evaluation variables of each approach for case 2.

Result	COA	GOOSE	HLOA	CDO	LCA	SCHO	SWO	SCSO	MSSCSO
$a_1$	−1.39981	−1.39999	−1.41159	−1.06573	−1.40094	−1.40155	−1.31698	−1.39996	−1.40000
$a_2$	0.492175	0.489979	0.500264	0.137023	0.481619	0.491472	0.412350	0.489950	0.489998
$b$	0.998340	1.000000	0.983396	0.954633	0.821923	0.996758	1.053119	0.999953	1.000000

summarizes the finest evaluation variables of each approach for case 2.

Table 12. The average evaluation variables of each approach for case 2.

Result	COA	GOOSE	HLOA	CDO	LCA	SCHO	SWO	SCSO	MSSCSO
$a_1$	−0.99638	−0.97431	−1.00668	0.062561	−0.26140	−0.61345	−1.00668	−0.98987	−1.40011
$a_2$	0.313780	0.444394	0.179891	0.21005	0.143142	0.104067	0.201868	0.287284	0.490102
$b$	0.691226	0.824775	0.705912	0.22144	0.236695	0.533497	1.052941	0.768647	0.999844

summarizes the average evaluation variables of each approach for case 2.

Table 13. Experimental results (MSE) of each approach for case 3.

Result	COA	GOOSE	HLOA	CDO	LCA	SCHO	SWO	SCSO	MSSCSO
Best	0.017946	0.000297	0.007608	0.012796	0.020442	0.006856	0.018455	0.000546	0.000221
Worst	0.031682	0.024122	0.080126	0.027749	0.033494	0.075749	0.156516	0.056702	0.015203
Mean	0.025111	0.004569	0.048196	0.018297	0.027239	0.042278	0.080582	0.008727	0.004805
Std	0.003385	0.005982	0.019007	0.003816	0.003466	0.024086	0.037226	0.010906	0.004760
Rank	1	5	7	3	2	8	9	6	4

summarizes each approach’s experimental results (MSE) for case 3.

Table 14. The finest evaluation variables of each approach for case 3.

Result	COA	GOOSE	HLOA	CDO	LCA	SCHO	SWO	SCSO	MSSCSO
$a_1$	0.953586	0.007352	−0.20786	0.878369	0.905309	−0.27987	0.022008	−0.00541	0.008618
$a_2$	0.953586	−0.04654	−0.13216	0.265057	0.905486	−0.16811	−0.04303	−0.00298	−0.02547
$a_3$	0.953586	0.005118	−0.17535	0.539142	0.905668	−0.06786	−0.42077	0.027448	−0.00666
$a_4$	0.953586	−0.80073	−0.40395	0.177061	0.903126	−0.36939	−0.26702	−0.83054	−0.83264
$b_0$	0.953586	0.996689	0.914360	0.972802	0.902231	0.997788	0.911890	0.946202	0.997731
$b_1$	0.953586	0.001516	−0.25320	0.638042	0.906205	−0.19897	0.196905	−0.04245	−0.01344
$b_2$	0.953586	0.326476	0.077720	0.332544	0.902735	0.187419	0.082871	0.372305	0.341360
$b_3$	0.953586	0.005216	−0.09017	0.667846	0.904501	−0.27117	−0.32481	0.013260	−0.03256
$b_4$	0.953586	−0.32506	−0.11180	0.274141	0.905339	−0.04838	−0.06055	−0.27418	−0.33272

summarizes the finest evaluation variables of each approach for case 3.

Table 15. The average evaluation variables of each approach for case 3.

Result	COA	GOOSE	HLOA	CDO	LCA	SCHO	SWO	SCSO	MSSCSO
$a_1$	0.689101	0.026186	−0.08864	0.367087	0.88964	0.046238	−0.01561	−0.01891	−0.04763
$a_2$	0.686175	−0.12576	0.029259	0.140799	0.889428	0.021665	−0.07093	−0.19670	−0.24465
$a_3$	0.657135	0.128047	0.049946	0.347353	0.890935	0.032133	−0.07362	0.019701	0.005844
$a_4$	0.589617	−0.46737	−0.00219	0.104341	0.887833	−0.01196	−0.03842	−0.32093	−0.41933
$b_0$	0.760431	0.962152	0.352217	0.830797	0.892001	0.494711	0.466569	0.868908	0.928592
$b_1$	0.654353	0.013291	0.000101	0.222736	0.889528	0.034416	−0.02076	−0.02577	−0.05120
$b_2$	0.729208	0.240603	0.144326	0.226274	0.891394	0.076728	0.059102	0.094151	0.095193
$b_3$	0.631146	0.118033	0.094043	0.257169	0.891455	−0.02049	−0.04550	−0.01461	−0.00774
$b_4$	0.671518	−0.12822	0.136392	0.242747	0.888826	0.061440	0.121246	0.034869	−0.01609

summarizes the average evaluation variables of each approach for case 3. For each approach, the average sample equals 30, the highest iteration equals 500, and the detached operation equals 30. Best, Worst, Mean, and Std signify the idealized value, catastrophic value, average value, and standard deviation, which comprehensively confirm the sustainability and profitability of MSSCSO. The ranking is established by measuring the standard deviation. To alleviate the deficiencies of the inefficient initial population quality, sluggish convergence acceleration, and restricted evaluation precision, the expanded MSSCSO is implemented to accomplish the IIR filter. For case 1, the assessment metrics of the MSSCSO are more accurate than those of the COA, GOOSE, HLOA, CDO, LCA, SCHO, SWO, and SCSO, and the MSSCSO encounters courageous investigation and extraction to refrain from sluggish searching and accomplish the most satisfactory solution. The finest or average evaluation variables of the MSSCSO are superior to those of the COA, GOOSE, HLOA, CDO, LCA, SCHO, SWO, and SCSO, and the MSSCSO receives fantastic consistency and endurance to ascertain the most effective evaluation variables. For case 2, the evaluation precision and solution efficiency of the MSSCSO have been substantially advanced compared to the SCSO. The assessment metrics and the finest or average evaluation variables of the MSSCSO are superior to those of the COA, GOOSE, HLOA, CDO, LCA, SCHO, SWO, and SCSO. The MSSCSO maintains discovery and extraction to attain a swifter convergence velocity, greater estimation precision, superior evaluation variables, lower standard deviation, and more accurate fitness. For case 3, the Best, Worst, and Mean of the MSSCSO are more precise than those of the COA, GOOSE, HLOA, CDO, LCA, SCHO, SWO, and SCSO, and the MSSCSO maintains instructive sustainability and adaptability to capture supplementary advantages and prohibit search standstill. The Std and ranking of the MSSCSO are superior to those of the GOOSE, HLOA, SCHO, SWO, and SCSO, but worse than those of the COA, LCA, and CDO, and the MSSCSO exhibits relative stability and robustness. The experimental results indicate that MSSCSO integrates prospecting and extracting to strengthen convergence productivity and accomplish the most satisfactory solution.

Figure 9 articulates the convergence trajectories of each approach for case 1. Figure 10 articulates the ANOVA tests of each approach for case 1. Figure 11 articulates the convergence trajectories of each approach for case 2. Figure 12 articulates the ANOVA tests of each approach for case 2. Figure 13 articulates the convergence trajectories of each approach for case 3. Figure 14 articulates the ANOVA tests of each approach for case 3. The MSSCSO is impacted by the scaling factor $F$ and the distinguishable properties of sand cats, which makes MSSCSO converge toward the optimal solution faster than other comparison approaches and subsequently re-exploit the evaluation accuracy. The overall convergence velocity and estimation precision of the MSSCSO for cases 1, 2, and 3 are superior to those of the COA, GOOSE, HLOA, CDO, LCA, SCHO, SWO, and SCSO; the MSSCSO not only maintains an equilibrium between localized extraction and nationwide investigation to restrict explore stagnation and strengthen convergence acceleration but also fosters selection probability and facilitates population variety to elevate estimation precision and expedite evaluation efficiency. For cases 1 and 2, the standard deviations of the MSSCSO are superior to those of the COA, GOOSE, HLOA, CDO, LCA, SCHO, SWO, and SCSO. The MSSCSO portrays noteworthy sustainability and superiority in recognizing the best solution and evaluation variables. For case 3, the standard deviation of the MSSCSO is superior to those of the GOOSE, HLOA, SCHO, SWO, and SCSO, but is worse than those of the COA, LCA, and CDO. The MSSCSO encounters relative stability and reliability to disrupt regional extreme solutions, foster convergence acceleration, bolster evaluation precision, and advance solution efficiency. The MSSCSO exhibits instructive sustainability and adaptability to foster selection probability, quicken the convergence procedure, bolster localized exploitation and broad exploration, facilitate population variety, broaden the searched area, refrain from precocious convergence, advance initial population quality, escalate recognition depth, and advance extraction operation. The experimental results manifest that MSSCSO maintains discovery and extraction to attain a swifter convergence velocity, greater estimation precision, superior regulation variables, lower standard deviation, and more accurate fitness.

Table 16. The execution time of different approaches for cases 1, 2, and 3.

Case	COA	GOOSE	HLOA	CDO	LCA	SCHO	SWO	SCSO	MSSCSO
Case 1	9.6983	11.2469	12.3444	14.5551	5.3123	4.8513	3.6918	5.2933	8.9495
Case 2	10.4756	12.6700	12.8912	15.1089	5.5421	5.0947	3.6831	5.9431	9.8702
Case 3	11.2005	13.9707	13.7214	17.3076	5.6333	5.3919	3.5543	7.2435	10.6880

summarizes the execution time of different approaches for cases 1, 2, and 3. For MSSCSO, the execution time of case 1 is 8.9495 s, the execution time of case 2 is 9.8702 s, and the execution time of case 3 is 10.6880 s. The overall execution time of the MSSCSO is better than that of the COA, GOOSE, HLOA, and CDO, but it is worse than that of the LCA, SCHO, SWO, and SCSO. The MSSCSO exhibits strong stability and robustness to balance exploration and exploitation and achieve a more accurate solution.

Table 17. Results of the p-value Wilcoxon rank-sum test.

Result	COA	GOOSE	HLOA	CDO	LCA	SCHO	SWO	SCSO
Case 1	1.11$\times {10}^{-4}$	4.29$\times {10}^{-2}$	9.07$\times {10}^{-3}$	8.15$\times {10}^{-11}$	6.07$\times {10}^{-11}$	3.37$\times {10}^{-4}$	8.99$\times {10}^{-11}$	5.94$\times {10}^{-4}$
Case 2	3.02$\times {10}^{-11}$	6.63$\times {10}^{-3}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	3.02$\times {10}^{-11}$	1.91$\times {10}^{-2}$
Case 3	3.02$\times {10}^{-11}$	3.87$\times {10}^{-2}$	6.07$\times {10}^{-11}$	1.61$\times {10}^{-10}$	3.02$\times {10}^{-11}$	1.78$\times {10}^{-10}$	3.02$\times {10}^{-11}$	7.48$\times {10}^{-5}$

summarizes the results of the p-value Wilcoxon rank-sum test. The experimental results demonstrate a distinguished discrepancy between MSSCSO and other approaches, and the data endure an appropriate degree of authenticity and dependability that was not established by accident.

6. Exploration and Exploitation Analysis, Diversity Analysis

Exploration and exploitation are two essential search properties that inhibit precocious convergence, facilitate detection quality, and achieve the most satisfactory solution. There is a robust relationship between the exploration and exploitation of the unique search mechanism and the convergence accuracy. MSSCSO employs exploration to create low-frequency noise fabrication, search for prey, and capture prey, which is conducive to establishing the most appropriate solution in the search region and reducing the convergence efficiency of the optimization approach. The MSSCSO not only utilizes the exploitation to assault and retrieve prey but also utilizes the Roulette Wheel Section to refrain from precocious convergence, which is conducive to accelerating the convergence speed and advancing the possibility of regional extreme solutions. The appropriate ratio between exploration and exploitation is necessary to ensure the MSSCSO’s reasonable performance. The MSSCSO has strong overall searchability, stability, and robustness.

The MSSCSO is employed to accomplish the CEC 2022 and the adaptive IIR system identification, which utilizes a set of better candidate solutions to achieve exploration and exploitation, recognize the most satisfactory solution, and minimize the estimation error between an unidentified system’s input and the IIR filter’s output. Generally, the sand cat with the superior solution can guide the entire search process to achieve global exploration and local exploitation. When the distance between the sand cats gradually increases, the exploration’s impact will become progressively more apparent, and the exploitation’s impact will gradually weaken. A diversity measurement evaluates the increase and decrease in distance between the sand cats. The population diversity is constructed as follows:

$$Di{v}_j={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\left|median\left(x^j\right)-x_i^j\right|}$$

(29)

$$Div={\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=1}^{m}Di{v}_j}$$

(30)

where $median\left(x^j\right)$ signifies the median of the jth dimension, $x_i^j$ signifies the jth dimension of ith sand cat, $N$ signifies the population size, and $m$ signifies the design variables. The mathematical expression to measure the percentage of exploration and exploitation is constructed as follows:

$$\mathrm{Exploration}\%=\left({\displaystyle \frac{Div}{Di{v}_{\max }}}\right)\times 100$$

(31)

$$\mathrm{Exploration}\%=\left({\displaystyle \frac{\left|Div-Di{v}_{\max }\right|}{Di{v}_{\max }}}\right)\times 100$$

(32)

where $Di{v}_{\max }$ signifies the maximum diversity. The two elements of exploration and exploitation are both mutually conflicting and complementary. In evaluating the balance impact, the median value utilizes the reference element to avoid optimization discrepancies, and the balance impact is affected by $Di{v}_{\max }$ in the entire optimization process. The $Di{v}_{\max }$ is employed as a reference to evaluate the rate of exploration and exploitation. The experimental results indicate that MSSCSO integrates exploration and exploitation to exhibit a swifter convergent velocity, greater explored accuracy, superior regulation variables, lower standard deviation, and more accurate fitness.

7. Conclusions and Future Research

The SCSO is motivated by the distinguishable properties of sand cats and mimics the surveillance and neutralization of prey to recognize the low-frequency noise, the prey location, and the most appropriate solution. The ranking-based mutation operator, elite opposition-based learning strategy, and simplex method are incorporated into the SCSO to enrich the population quality, convergence acceleration, evaluation precision, and solution efficiency. The expanded MSSCSO with symmetric cooperative swarms is implemented to accomplish the benchmark functions and adaptive IIR filter identification, and the principal objective is to recognize the most appropriate regulating variables and minimize the estimation error between an unidentified system’s input and the IIR filter’s output. The ranking-based mutation operator filtrates the finest potential solution, fosters selection probability, quickens the convergence procedure, and bolsters localized exploitation. The elite opposition-based learning strategy facilitates demographic variability, upgrades the search of an area, expedites evaluation efficiency, refrains from precocious convergence, and bolsters broad exploration. The simplex method advances initial population quality, escalates recognition depth, disrupts regional extreme solution, advances extraction operation, and elevates estimation precision. The MSSCSO exhibits simple architectures, excessive optimization efficiency, marvelous parallelism, outstanding robustness, and stability. The MSSCSO encounters excellent superiority and predictability in refraining from combinatorial explosion and precocious convergence. The experimental results demonstrate that MSSCSO incorporates the utilization and investigation to attain a swifter convergent velocity, greater explored accuracy, superior regulation variables, lower standard deviation, and more accurate fitness.

The limitations of the MSSCSO are summarized as follows: (1) The simulation experiments are primarily focused on the adaptive IIR system identification, and further investigations are needed to explore the optimization efficiency and convergence accuracy in a broader range of application fields. (2) The hyperparameters tuning process may require careful consideration to achieve the most satisfactory solution for different problem domains. (3) The proposed algorithm consumes more execution time to recognize the most appropriate regulating coefficients and minimize the MSE. (4) The computational complexity, mathematical logic reasoning, algorithm convergence proof, and dynamic parameter selection will affect the convergence speed and calculation accuracy. (5) The convergence efficiency and estimation accuracy of MSSCSO need to be further strengthened by introducing advanced search strategies, alternative encoding formats, and hybrid algorithmic structures to capture supplementary advantages and inhibit precocious convergence. (6) The MSSCSO is utilized to resolve super-complex, large-scale, high-dimensional, and multi-objective optimization problems, which may not be able to effectively balance exploration and exploitation to determine the finest potential solution.

Our future research will focus on the following aspects: (1) We will utilize more different IIR filter examples or large-scale engineering problems to evaluate and validate the effectiveness and stability of the MSSCSO. The MSSCSO will be compared with state-of-the-art methods to show its competitive performance. We will use statistical analysis to illustrate the discrepancies between the two data sets. (2) We will verify the algorithm convergence and computational complexity from theoretical and practical perspectives. We will exploit dynamic control parameters to enhance the stability and robustness. (3) We will execute advanced search strategies, alternative encoding formats, and hybrid algorithmic structures to capture supplementary advantages and avoid precocious convergence. The renewed SCSO will integrate localized exploitation and broad exploration to strengthen population quality, convergence acceleration, evaluation precision, and solution efficiency. (4) The renewed SCSO will be executed to improve characteristic Dabie Mountain forestry and crops, incorporating Huoshan Dendrobium, Lu’an Guapian, Chinese herbal medicine, and Shucheng Camellia oleifera. The quality of agricultural equipment and intelligent operation will be strengthened by intelligent detection, intelligent control, and specialized agricultural equipment.