MSAPO: A Multi-Strategy Fusion Artificial Protozoa Optimizer for Solving Real-World Problems

Abstract

Artificial protozoa optimizer (APO), as a newly proposed meta-heuristic algorithm, is inspired by the foraging, dormancy, and reproduction behaviors of protozoa in nature. Compared with traditional optimization algorithms, APO demonstrates strong competitive advantages; nevertheless, it is not without inherent limitations, such as slow convergence and a proclivity towards local optimization. In order to enhance the efficacy of the algorithm, this paper puts forth a multi-strategy fusion artificial protozoa optimizer, referred to as MSAPO. In the initialization stage, MSAPO employs the piecewise chaotic opposition-based learning strategy, which results in a uniform population distribution, circumvents initialization bias, and enhances the global exploration capability of the algorithm. Subsequently, cyclone foraging strategy is implemented during the heterotrophic foraging phase. enabling the algorithm to identify the optimal search direction with greater precision, guided by the globally optimal individuals. This reduces random wandering, significantly accelerating the optimization search and enhancing the ability to jump out of the local optimal solutions. Furthermore, the incorporation of hybrid mutation strategy in the reproduction stage enables the algorithm to adaptively transform the mutation patterns during the iteration process, facilitating a strategic balance between rapid escape from local optima in the initial stages and precise convergence in the subsequent stages. Ultimately, crisscross strategy is incorporated at the conclusion of the algorithm’s iteration. This not only enhances the algorithm’s global search capacity but also augments its capability to circumvent local optima through the integrated application of horizontal and vertical crossover techniques. This paper presents a comparative analysis of MSAPO with other prominent optimization algorithms on the three-dimensional CEC2017 and the highest-dimensional CEC2022 test sets, and the results of numerical experiments show that MSAPO outperforms the compared algorithms, and ranks first in the performance evaluation in a comprehensive way. In addition, in eight real-world engineering design problem experiments, MSAPO almost always achieves the theoretical optimal value, which fully confirms its high efficiency and applicability, thus verifying the great potential of MSAPO in solving complex optimization problems.

1. Introduction

Numerous real-life problems can often be reduced to the model building process of finding optimal solutions, a process that aims to achieve the best results through model solving [1]. The accelerated advancement of science and technology has led to the pervasion of real-world optimization challenges across a multitude of domains. These include image processing [2], feature selection [3], mechanical design [4], engineering problems [5,6], production scheduling and automation [7], and so on. These optimization problems are increasingly complex, and they are often characterized by high dimensionality [8], nonlinearity [9], discontinuities, and multiple constraints [10], thus putting traditional numerical optimization methods to a severe test and making it difficult to satisfy the growing demand for optimization [11].

Metaheuristic algorithms, by virtue of their intuitive and easy-to-understand principles, high randomness, high generality, and no need to rely on a specific problem background [12], have demonstrated extraordinary effectiveness in overcoming high-dimensional and large-scale optimization problems that are difficult to be mastered by traditional methods. This innovative methodology has not only broadened the horizon of optimization problems, but also attracted great attention and in-depth research in domestic and international academic circles [13]. Meta-heuristic algorithms can be classified into one of five categories: biological evolution-based, population intelligence-based, thinking cognition-based, physics and mathematics-based, and strategy enhancement-based [1].

The algorithms based on biological evolution are methods for conducting random search by emulating the genetic, mutation, and selection processes observed in biological evolution. The most classic algorithms are genetic algorithm (GA), proposed in 1975 [14], and differential evolution (DE), proposed in 1995 [15]. In 2013, Pinar proposed the backtracking search algorithm (BSA) [16]. In addition to this, other evolution-based algorithms proposed in recent years include spherical evolutionary algorithm (SEA) [17], geometric probabilistic evolutionary algorithm (GPEA) [18], etc.

Population intelligence-based algorithms implement stochastic search by simulating the instinctive behaviors of biological populations in nature (e.g., activities such as foraging, predation, migration, and courtship of different species). In 1995, Kennedy and Eberhart simulated the foraging behaviors of a flock of birds and proposed the classical particle swarm optimizer (PSO) [19]. In 2006, Dorigo et al. proposed ant colony optimizer (ACO) [20]. In 2014, Mirjalili proposed gray wolf optimizer (GWO) [21]. GWO has inspiration from the gray wolf population hierarchy mechanism and the collaborative strategy of wolf hunting. Subsequently, in 2016, he put forth whale optimization algorithm (WOA) [22]. Other notable algorithms are harris hawk optimization (HHO) [23], marine predator algorithm (MPA) [24], snake optimizer (SO) [25], nutcracker optimization algorithm (NOA) [26], genghis khan shark optimizer (GKSO) [27], and the recently proposed crested porcupine optimization (CPO) [28], secretary bird optimization algorithm (SBOA) [29], and so on.

Thinking cognition-based algorithms represent a methodology of random search whereby the thought-cognitive process of complex human behavioral patterns in nature and daily life is simulated. Rao proposed teaching-learning-based optimization (TLBO) [30] in 2011. Das proposed student psychology-based optimization algorithm (SPBO) [31] in 2020, inspired by the behavior of students who study hard to become the best in their class. There is also poor–rich optimization (PRO) [32], gaining–sharing knowledge-based algorithm (GSK) [33], human memory optimization algorithm (HMO) [34], and others.

Physics- and mathematics-based algorithms are employed to model physical phenomena and mathematical theorems. An illustrative example is sine cosine algorithm (SCA) [35]. Other algorithms include electrostatic discharge algorithm (ESDA) [36], archimedes optimization algorithm (AOA) [37], snow ablation optimizer (SAO) [38], fick’s law algorithm (FLA) [39], and others.

Policy-enhanced algorithms represent a novel category of algorithms that have been derived from the original heuristic algorithms. They effectively circumvent the inherent limitations of the original heuristics by incorporating innovative strategies or by drawing upon the superior attributes of other algorithms, thereby achieving enhanced performance and results in the optimization process. Examples include GSRPSO [40] and the collaboration-based hybrid GWO-SCA optimizer (cHGWOSCA) [41].

The “no free lunch” theorem [42] demonstrates that no universal optimization algorithm can achieve optimal performance on all problems. In view of this, many researchers will introduce improved strategies, algorithmic fusion, or innovative algorithms to solve complex optimization problems. Artificial protozoa optimizer (APO) is a novel metaheuristic algorithm proposed in this context.

Wang et al. [43] developed an artificial protozoan optimizer (APO) by simulating the foraging, dormant, and reproductive behaviors of protozoa. The APO employs two adaptively varying probability parameters to maintain equilibrium between the exploration and development phases of the optimization process, and introduces mapping vectors to vary the dimensions. The superiority and effectiveness of APO have been demonstrated in literature [43], which presents the results of experiments conducted with 32 intelligent optimization algorithms on the CEC2022 test set and five engineering applications. These experiments included statistical analyses and Wilcoxon rank sum tests. Nevertheless, the guided positional updates provided by unidentified individuals during the autotrophic phase may result in a slower convergence of the algorithm. Furthermore, the algorithm’s heterotrophic phase, which is influenced only by neighboring individuals, may result in a relatively straightforward descent into a local optimum.

To further enhance APO’s ability to find excellence, this paper puts forth a multi-strategy fusion artificial protozoa optimizer, or MSAPO, as a potential solution. Comparative experiments between MSAPO and established optimization algorithms are conducted on the CEC2017 and CEC2022 test sets, respectively. The results demonstrate the efficacy and competitive advantage of MSAPO. This paper makes the following contributions to the field:

(1)

First, we adopt the piecewise chaotic opposition-based learning strategy in the stochastic initialization process, which serves to augment the dispersion and exploration space of the initial solution set and improves the algorithm’s ability to search globally. Next, cyclone foraging strategy is implemented during the heterotrophic foraging phase. enabling the algorithm to identify the optimal search direction with greater precision, guided by the globally optimal individuals. In addition, hybrid mutation strategy is added in the reproduction phase to enhance the ability of the algorithm to quickly move away from the local optimum in the early phase and to converge accurately in the later phase. The incorporation of the crisscross strategy prior to the conclusion of the iteration not only enhances the algorithm’s capacity for efficient global domain search but also augments its capability to circumvent local optima.

(2)

The effectiveness of MSAPO is substantiated through its validation in three distinct dimensions of the CEC2017 test set and in the highest dimension of the CEC2022 test set. Furthermore, its performance is benchmarked against other state-of-the-art swarm intelligence optimization algorithms, and the experimental outcomes substantiate MSAPO’s superiority.

(3)

The efficacy of MSAPO in addressing practical engineering challenges has been substantiated through the examination of eight illustrative case studies.

The remainder of the paper is structured as illustrated in the following outline: a detailed description of the basic APO is given in Section 2. The four strategies introduced in this work and the proposed MSAPO algorithm are demonstrated in Section 3. Section 4 presents the findings of the experiments conducted on distinct test sets, offering a detailed analysis of the results. Section 5 examines the optimization outcomes of MSAPO in the context of eight engineering case studies. The final section, Section 6, presents the conclusion.

2. Artificial Protozoa Optimizer

Euglena is a representative unicellular protozoan that has some characteristics of both plants and animals, and can therefore obtain nutrients for survival by both autotrophic and heterotrophic means. Euglena reproduces asexually through a binary fission process [43]. Wang et al. developed an artificial protozoa optimizer (APO) by simulating the behaviors exhibited by Euglena, including foraging, dormant and reproductive behaviors. The mathematical model of the APO is described below, and the pseudo code is shown in Algorithm 1.

Algorithm 1: APO

Input: The population size N, the individual dimension D, Controlling parameters np, pfmax, and the maximum number of iterations T.

Output: The optimal individual ybest and its corresponding fitness fbest.

2.1. Population Initialization

The APO is founded upon the principles of traditional random sampling, which is employed to generate a population that is uniformly distributed at random. ${\mathit{y}}_i=\left[y_{i,1},y_{i,2},\dots ,y_{i,D}\right], i=\mathrm{1,2},\dots ,N$ represents the ith individual in a protozoan population of size $N$, each with dimension $D$. The fundamental population is produced through Equation (1),

$${\mathit{y}}_{\mathit{i}}=\mathbf{L}\mathbf{b}+\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{d}\times (\mathbf{U}\mathbf{b}-\mathbf{L}\mathbf{b}),$$

(1)

where $\mathbf{L}\mathbf{b}$ and $\mathbf{U}\mathbf{b}$ are the upper and lower search space bounds, $\mathbf{L}\mathbf{b}=[l{b}_1,l{b}_2,\dots ,l{b}_D]$, $\mathbf{U}\mathbf{b}=[u{b}_1,u{b}_2,\dots ,u{b}_D]$, where $l{b}_{di}$ and $u{b}_{di}$ are the dith component of $\mathbf{L}\mathbf{b}$ and $\mathbf{U}\mathbf{b}$, respectively. The vector $\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{d}$ is a random variable with elements distributed between 0 and 1, $\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{d}=[ran{d}_1,ran{d}_2,\dots ,ran{d}_D]$. In this section, the operator $\times$ denotes the Hadamard product.

At the beginning of each iteration, the current population of protozoa is sorted on the basis of the fitness value, as defined by the Equation (2). In the sorted population, a few individuals are randomly selected to enter the dormant or breeding state and the rest to enter the foraging state, based on the proportion fraction $pf$ in Equation (3).

$${\mathit{y}}_{\mathit{i}}=\mathrm{sort}\left({\mathit{y}}_i\right),$$

(2)

$$pf=p{f}_{max}\cdot rand$$

(3)

where the maximum proportional score $p{f}_{max}$ is set to 0.1 and $rand$ is a randomly generated number within the interval $\left[\mathrm{0,1}\right]$.

2.2. Foraging

Protozoa can take up the nutrients they need to survive in both autotrophic and heterotrophic ways.

2.2.1. Autotrophic Mode

In the right light, protozoa carry out plant-like photosynthesis, using chloroplasts to produce carbohydrates for energy, and protozoa are able to respond to light cues by moving closer or further away from the light to find a suitable habitat for survival. Protozoan populations will move towards individuals whose ambient light intensity is suitable for photosynthesis. The autotrophic model is mathematically represented by the following Equation (4),

$${\mathit{y}}_i^{new}(t+1)={\mathit{y}}_i(t)+f\cdot ({\mathit{y}}_j(t)-{\mathit{y}}_i(t)+{\displaystyle \frac{1}{np}}\cdot {\sum }_{k=1}^{np}{w}_a\cdot ({\mathit{y}}_{k-}(t)-{\mathit{y}}_{k+}(t)))\times {\mathbf{M}}_{\mathbf{f}},$$

(4)

$$f=rand\cdot (1+\mathit{cos}({\displaystyle \frac{t}{T}}\cdot \pi )),$$

(5)

$$n{p⌊{\displaystyle \frac{N-1}{2}}⌋}_{max}$$

(6)

$$w_a=e^{-\left|{\displaystyle \frac{f(y_{k-}(t))}{f(y_{k+}(t))+\mathrm{eps}}}\right|},$$

(7)

$${\mathbf{M}}_{\mathbf{f}}[di]=\left\{\begin{array}{cc}1,& \mathit{if} di \mathrm{is} \mathrm{in} \mathrm{randperm}(\mathrm{D},⌈D\cdot {\displaystyle \frac{i}{N}}⌉)\\ 0,& \mathit{otherwise}\end{array},\right.$$

(8)

where ${\mathit{y}}_i^{new}(t+1)$ is the position of the tth generation of protozoa ${\mathit{y}}_i(t)$ after renewal, the current and maximum number of iterations, respectively, are represented by the symbols $t$ and $T$. ${\mathit{y}}_{\mathit{j}}(t)$ is a randomly selected protozoan in generation $t$. The current populations have been sorted by fitness value from smallest to largest. ${\mathit{y}}_{k-}(t)$ denotes the kth left neighbour of the current individual ${\mathit{y}}_i(t)$, a protozoan with a ranking index lower than $i$, randomly selected from the population. ${\mathit{y}}_{k+}(t)$ denotes the kth right neighbour of ${\mathit{y}}_i(t)$, a random selection of protozoa, exhibiting a ranking index greater than $i$. $np$ denotes the number of selected pairs of neighbors. $f$ denotes the foraging factor. $w_a$ is the weight factor in the autotrophic mode, $f(•)$ calculates the fitness value of an individual. ${\mathbf{M}}_{\mathbf{f}}$ is a random mapping vector of size $D$ dimensions, which influences the mutation dimension in the foraging state.

2.2.2. Heterotrophic Mode

In conditions of low light, protozoa display behaviors that are characteristic of animals, including the absorption of organic matter from their surrounding environment. Assuming that food is abundant in the vicinity of ${\mathit{y}}_{near}(t)$, then the protozoa will move towards it. The is mathematical model of the heterotrophic model is as follows,

$${\mathit{y}}_{\mathit{i}}^{\mathit{n}\mathit{e}\mathit{w}}\left(t+1\right)={\mathit{y}}_i\left(t\right)+f\cdot \left({\mathit{y}}_{near}\left(t\right)-{\mathit{y}}_i\left(t\right)+{\displaystyle \frac{1}{np}}\cdot {\sum }_{k=1}^{np}{w}_h\cdot \left({\mathit{y}}_{i-k}\left(t\right)-{\mathit{y}}_{i+k}\left(t\right)\right)\right)\times {\mathbf{M}}_{\mathbf{f}},$$

(9)

$${\mathit{y}}_{\mathit{n}\mathit{e}\mathit{a}\mathit{r}}\left(t\right)=\left(1\pm \mathit{R}\mathit{a}\mathit{n}\mathit{d}\cdot \left(1-{\displaystyle \frac{t}{T}}\right)\right)\times {\mathit{y}}_i\left(t\right),$$

(10)

$$w_h=e^{-\left|{\displaystyle \frac{f\left(y_{i-k}\left(t\right)\right)}{f\left(y_{i+k}\left(t\right)\right)+\mathrm{eps}}}\right|},$$

(11)

where ${\mathit{y}}_{\mathit{n}\mathit{e}\mathit{a}\mathit{r}}\left(t\right)$ is the location near ${\mathit{y}}_{\mathit{i}}\left(t\right)$, “$\pm$” indicates that it is possible for ${\mathit{y}}_{near}\left(t\right)$ and ${\mathit{y}}_{\mathit{i}}\left(t\right)$ to be in different directions. A pair of neighbors of ${\mathit{y}}_{\mathit{i}}\left(t\right)$ denoted by ${\mathit{y}}_{\mathit{i}-\mathit{k}}\left(t\right)$ and ${\mathit{y}}_{\mathit{i}+\mathit{k}}\left(t\right)$. $w_h$ is the weight factor in heterotrophic mode.

In the foraging state, the autotrophic mode is concerned with a comprehensive search of the surrounding area, thereby enhancing the algorithm’s exploration capacity. Conversely, the heterotrophic mode is geared towards identifying regions with greater potential, thus bolstering the exploitation capability. The transition between autotrophic and heterotrophic modes is governed by Equation (12),

$${\mathit{y}}_i^{new}\left(t+1\right)=\left\{\begin{array}{cc}\mathrm{Equation} \left(4\right),& \mathrm{if} r \mathrm{a}\mathrm{n}\mathrm{d}<p_{ah}\\ \mathrm{Equation} \left(9\right),& \mathrm{otherwise}\end{array},\right.$$

(12)

$$p_{ah}={\displaystyle \frac{1}{2}}\cdot \left(1+\mathit{cos}\left( {\displaystyle \frac{t}{T}}\cdot \pi \right)\right),$$

(13)

The parameter $p_{ah}$, which represents the probability of a given protozoan exhibiting autotrophic or heterotrophic modes, is observed to decrease as the iteration proceeds. This shift in foraging tendency from autotrophic to heterotrophic modes is a consequence of the aforementioned decrease in $p_{ah}$.

2.3. Dormancy or Reproduction

2.3.1. Dormancy

Protozoa may enter a dormant state in unfavorable environments. In such cases, the population is maintained by the production of new individuals that replace those in the dormant state. The model of the dormant state, as it is represented mathematically, is as follows:

$${\mathit{y}}_i^{new}(t+1)=\mathbf{L}\mathbf{b}+\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{d}\times (\mathbf{U}\mathbf{b}-\mathbf{L}\mathbf{b}).$$

(14)

2.3.2. Reproduction

Protozoa may reproduce asexually by binary fission at appropriate ages and health conditions. The process of reproduction is modelled by the generation of replicate protozoa and the introduction of perturbations, expressed by Equation (15):

$${\mathit{y}}_{\mathit{i}}^{\mathit{n}\mathit{e}\mathit{w}}\left(t+1\right)={\mathit{y}}_i\left(t\right)\pm r \mathrm{a}\mathrm{n}\mathrm{d}\cdot \left(\mathbf{L}\mathbf{b}+\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{d}\times \left(\mathbf{U}\mathbf{b}-\mathbf{L}\mathbf{b}\right)\right)\times {\mathbf{M}}_{\mathbf{r}},$$

(15)

$${\mathbf{M}}_{\mathbf{r}}\left[di\right]=\left\{\begin{array}{cc}1,& \mathrm{if} di \mathrm{is} \mathrm{in} \mathit{randperm}\left(D,⌈D\cdot rand⌉\right)\\ 0,& \mathrm{otherwise}\end{array},\right.$$

(16)

In the case of the symbol “$\pm$”, this indicates that the mutation in question can be either forward or reverse. The symbol $\lceil •\rceil$ represents an upward rounding operation, while ${\mathbf{M}}_{\mathbf{r}}$ denotes a vector of random mappings that affect the dimensionality of the mutation during the process of reproduction.

The dormant state of the protozoa represents the exploratory phase of the algorithm, while the reproductive state corresponds to the developmental phase. The transition between these two states is governed by Equation (17).

$${\mathit{y}}_i^{new}(t+1)=\left\{\begin{array}{cc}\mathrm{Equation} (14),& \mathrm{if} r \mathrm{a}\mathrm{n}\mathrm{d}<p_{dr}\\ \mathrm{Equation} (15),& \mathrm{otherwise}\end{array},\right.$$

(17)

$$p_{dr}={\displaystyle \frac{1}{2}}\cdot (1+\mathit{cos}((1-{\displaystyle \frac{i}{N}})\cdot \pi )),$$

(18)

where $p_{dr}$ is the probability parameter between dormancy and reproduction.

At the conclusion of each iteration, APO determines the final population through a greedy selection process, with the fitness value, as calculated by the formula presented in Equation (19), serving as the determining factor.

$${\mathit{y}}_i(t+1)=\left\{\begin{array}{cc}{\mathit{y}}_i^{new}(t+1),& \mathrm{if} f({\mathit{y}}_i^{new}(t+1))<f({\mathit{y}}_i(t))\\ {\mathit{y}}_i(t),& \mathrm{otherwise}\end{array}\right..$$

(19)

3. The Proposed MSAPO

As evidenced in the literature [43], the basic APO has demonstrated effectiveness within the domain of optimization problems. However, the algorithm exhibits limitations when confronted with optimization challenges characterised by high complexity and multi-dimensionality. These include relatively slow convergence and a proclivity towards locally optimal solutions, which limit the effectiveness and applicability of the algorithm in solving complex optimization problems. In view of the aforementioned constraints, we put forward the suggestion of an enhanced APO, MSAPO. The objective is to address the shortcomings of the basic APO. The MSAPO flowchart is presented in Figure 1. The MSAPO integrates four additional strategies into the basic APO: As shown in Figure 1, MSAPO employs a segmented chaotic counter-learning strategy to enhance the diversity of the initial population, integrates cyclone foraging and hybrid mutation to prevent population premature convergence, and utilizes a crossover strategy to balance the population’s exploration and exploitation capabilities.

3.1. Piecewise Chaotic Opposition-Based Learning Strategy

The fundamental APO population initialization employs a random distribution, which is a straightforward approach but may result in an uneven spatial distribution of protozoan individuals. This, in turn, may prompt the algorithm to converge on a local optimum. To enhance the initial population quality of APO and circumvent the potential for local optimum convergence, piecewise chaotic opposition-based learning strategy is introduced in this section.

The fundamental premise of traditional OBL (opposition-based learning) is to derive the inverse solution to a feasible solution and select a superior candidate solution by evaluating the feasible solution and the inverse solution. The OBL formulation is as follows:

$${\mathit{y}}_{\mathit{i}\_\mathit{O}\mathit{B}\mathit{L}}=\mathbf{L}\mathbf{b}+\mathbf{U}\mathbf{b}-{\mathbf{y}}_i,$$

(20)

where ${\mathit{y}}_{\mathit{i}\_\mathit{O}\mathit{B}\mathit{L}}$ represents the inverse solution corresponding to the ith individual ${\mathit{y}}_{\mathit{i}}$ in the random initial population. The application of OBL serves to facilitate the investigation of hitherto unconsidered solution domains within the search field, thereby enhancing the diversity of the population.

In an effort to more fully address the issue of unequal initial population distribution, this paper introduces the use of Piecewise chaotic mapping in OBL. Piecewise chaotic mapping is a segmentally defined chaotic system that divides the entire definition domain into multiple subintervals and applies distinct nonlinear transformations on each subinterval. This type of mapping is typically characterised by more intricate mathematical expressions and a greater degree of chaotic behavior, making it well suited to scenarios that demand heightened complexity and flexibility. The aforementioned mathematical model of piecewise chaotic mapping can be expressed as the following Equation (21),

$$\mathbf{P}\mathbf{W}\mathbf{L}\mathbf{C}\mathbf{M}\left(k+1\right)=\left\{\begin{array}{c}\begin{array}{cc}{\displaystyle \frac{\mathit{P}\mathit{W}\mathit{L}\mathit{C}\mathit{M}\left(k\right)}{P}},& 0\le \mathit{P}\mathit{W}\mathit{L}\mathit{C}\mathit{M}\left(k\right)<P\\ {\displaystyle \frac{\mathit{P}\mathit{W}\mathit{L}\mathit{C}\mathit{M}\left(k\right)-P}{0.5-P}},& P\le \mathit{P}\mathit{W}\mathit{L}\mathit{C}\mathit{M}\left(k\right)<0.5\end{array}\\ \begin{array}{cc}{\displaystyle \frac{1-P-\mathit{P}\mathit{W}\mathit{L}\mathit{C}\mathit{M}\left(k\right)}{0.5-P}},& 0.5\le \mathit{P}\mathit{W}\mathit{L}\mathit{C}\mathit{M}\left(k\right)<1-P\\ {\displaystyle \frac{1-\mathit{P}\mathit{W}\mathit{L}\mathit{C}\mathit{M}\left(k\right)}{P}},& 1-P\le \mathit{P}\mathit{W}\mathit{L}\mathit{C}\mathit{M}\left(k\right)<1\end{array}\end{array}\right.,$$

(21)

where $\mathbf{P}\mathbf{W}\mathbf{L}\mathbf{C}\mathbf{M}(k)$ denotes the kth chaotic value of the Piecewise chaotic sequence $\mathbf{P}\mathbf{W}\mathbf{L}\mathbf{C}\mathbf{M}$, The preliminary value of the $\mathbf{P}\mathbf{W}\mathbf{L}\mathbf{C}\mathbf{M}$ sequence is established by the random number rand, which is drawn from the interval $\left[\mathrm{0,1}\right]$, and chaos parameter $P=0.1$.

The generation of chaotic sequences with a more balanced distribution is achieved through the application of Equation (21), which in turn is utilized to generate novel solutions, as illustrated in Equation (22),

$${\mathit{y}}_{\mathit{i}\_\mathit{C}\mathit{O}\mathit{B}\mathit{L}}=\mathbf{L}\mathbf{b}+\mathbf{U}\mathbf{b}-\mathbf{P}\mathbf{W}\mathbf{L}\mathbf{C}\mathbf{M}\times {\mathbf{y}}_i,$$

(22)

where ${\mathit{y}}_{\mathit{i}\_\mathit{C}\mathit{O}\mathit{B}\mathit{L}}$ represents the novel solution, which is analogous to the initial population ${\mathit{y}}_{\mathit{i}}$. $\mathbf{P}\mathbf{W}\mathbf{L}\mathbf{C}\mathbf{M}$ signifies the corresponding sequence of chaotic mappings. Furthermore, the operator $\times$ in this instance denotes the Hadamard product operation of two matrices.

Piecewise chaotic opposition-based learning strategy is utilized for the generation of a population of protozoa, which is then ranked and compared with the randomly generated population. The top $N$ superior individuals are subsequently selected as the initial population for subsequent operations. This strategy generates diverse populations through mirror flipping, retaining superior individuals as the initial population for iterative cycles. This approach enhances the diversity of the initial population and effectively prevents excessive clustering of the population near suboptimal positions. As this strategy operates solely during the initial population phase, it imposes minimal impact on the algorithm’s computational complexity.

3.2. Cyclone Foraging Strategy

In the autotrophic foraging phase, APO relies more on the guidance of random individuals to update the position. This process is exploratory but often accompanied by a high degree of blindness and randomness, resulting in a relatively slow speed of optimality search. In the heterotrophic foraging phase, the same characteristics will be observed. The algorithm is influenced by two types of individuals: those in nearby positions and the current individual’s front and rear neighbors, ranked according to the population’s fitness value. This can result in the algorithm reaching a local optimum, making it challenging to escape. To address this limitation, the cyclone foraging strategy [44] is integrated into the heterotrophic foraging phase of APO with the following mathematical model:

$${\mathit{y}}_{\mathit{i}}^{\mathit{n}\mathit{e}\mathit{w}}\left(t+1\right)=\left\{\begin{array}{cc}{\mathit{y}}_{best}\left(t\right)+r\cdot \left({\mathit{y}}_{best}\left(t\right)-{\mathit{y}}_i\left(t\right)\right)+\beta \cdot \left({\mathit{y}}_{best}\left(t\right)-{\mathit{y}}_i\left(t\right)\right),& i=1\\ {\mathit{y}}_{best}\left(t\right)+r\cdot \left({\mathit{y}}_{i-1}\left(t\right)-{\mathit{y}}_i\left(t\right)\right)+\beta \cdot \left({\mathit{y}}_{best}\left(t\right)-{\mathit{y}}_i\left(t\right)\right),& \mathrm{otherwise}\end{array}\right.$$

(23)

$$\beta =2\cdot e^{r_1\cdot {\displaystyle \frac{T-t+1}{T}}}\cdot \mathit{sin}\left(2\cdot \pi \cdot r_1\right),$$

(24)

where ${\mathit{y}}_i^{new}\left(t+1\right)$ is the position of the ith protozoa ${\mathit{y}}_i(t)$ after updating in generation $t$, ${\mathit{y}}_{best}(t)$ is the optimal individual in the tth generation of protozoa. The random numbers $r$ and $r_1$ are drawn from a uniform distribution on the interval $\left[\mathrm{0,1}\right]$, and $\beta$ represents the weight coefficient.

Figure 2 illustrates the behavioral pattern of APO subsequent to the introduction of cyclone foraging strategy. The red blob in Figure 2 represents the current optimal protozoan individual, and the black blob represents the candidate individual. As shown in Figure 2, during the update process, APO’s ${\mathit{y}}_{\mathit{i}}^{\mathit{n}\mathit{e}\mathit{w}}$ is simultaneously influenced by ${\mathit{y}}_{i-1}$ and ${\mathbf{y}}_{best}$, generating a combined force along the spiral trajectory that propels the population toward the optimal point at an accelerated pace. In this way, APO updates its candidate individual of the population through the application of Equations (23) and (24), thereby approaching the optimal individual with the trajectory of the spiral line. The guidance of the global optimal individual enables APO to locate the search direction with greater accuracy, reducing the necessity for random wandering and significantly improving the speed of locating the optimal solution. Through the population renewal mechanism of cyclone foraging, individuals can approach the global optimum via richer convergence pathways. This enhances diversity in population renewal while maintaining incremental optimality, thereby preventing local optima traps.

3.3. Hybrid Mutation Strategy

Figure 3 illustrates the variability of the density function curves of the standard Cauchy distribution (Cauchy (1,0)) and the standard Gaussian distribution (Gaussian (0,1)). In Figure 3, the Gaussian distribution and Cauchy distribution are continuous probability distributions that represent disturbance terms in nature. Their key difference lies in tail behavior: the Gaussian distribution exhibits exponential decay, resulting in rapid decreases in extreme outcomes, which characterizes it as a light-tailed distribution. In contrast, the Cauchy distribution has a power-law decay, which increases the likelihood of extreme outliers. In the Cauchy distribution, 1 denotes the scale parameter, which is half the width at half the maximum, while 0 denotes the position parameter, which is the peak of the distribution. In the Gaussian distribution, 0 is the mean and 1 is the standard deviation. With regard to the properties of probability distributions, the Cauchy distribution exhibits a greater probability of occurrence at both extremes, indicating a propensity for generating variance values that deviate from the origin. This makes it a more suitable choice for extensive searches in the initial stages of an iteration, whereas the Gaussian distribution displays a higher concentration near the origin, making it more appropriate for fine-tuning in the latter stages of an iteration.

The hybrid mutation strategy employs the mixed mutation mechanism of Gaussian and Cauchy distributions to adjust the position of individuals with the objective of achieving diversification of mutation methods. The specific mathematical model is as follows:

$${\mathit{y}}_i^{new}(t+1)={\mathit{y}}_i(t)+{\mathit{y}}_i(t)\times (w_1(t)\cdot \mathbf{C}\mathbf{a}\mathbf{u}\mathbf{c}\mathbf{h}\mathbf{y}(\mathrm{1,0})+w_2(t)\cdot \mathbf{G}\mathbf{a}\mathbf{u}\mathbf{s}\mathbf{s}\mathbf{i}\mathbf{a}\mathbf{n}(\mathrm{1,0})),$$

(25)

$$\left\{\begin{array}{c}{w}_1\left(t\right)=1-{\left({\displaystyle \frac{t}{T}}\right)}^3\\ w_2\left(t\right)={\left({\displaystyle \frac{t}{T}}\right)}^3\end{array}\right.,$$

(26)

where ${\mathit{y}}_i^{new}\left(t+1\right)$ is the position of the individual ${\mathit{y}}_i(t)$ after mutation, $\mathbf{C}\mathbf{a}\mathbf{u}\mathbf{c}\mathbf{h}\mathbf{y}(\mathrm{1,0})$ denotes a sequence of variants obeying Cauchy distribution, $\mathbf{G}\mathbf{a}\mathbf{u}\mathbf{s}\mathbf{s}\mathbf{i}\mathbf{a}\mathbf{n}(\mathrm{0,1})$ denotes a sequence of variants obeying Gaussian distribution. $w_1(t)$ and $w_2(t)$ denote the Cauchy and Gaussian variance factors, respectively, and the variations are shown in Figure 4. As shown in Figure 4, in the early stages of optimization, algorithms leverage outlier variance for exploration, and later transition to more minor errors for precise searches. Thus, parameters $w_1$ and $w_2$, which sum to 1, are used to balance these two aspects of the algorithm.

This paper introduces hybrid mutation strategy in the reproduction phase of the algorithm. By dynamically adjusting the weights of the Cauchy mutation factor and the Gaussian mutation factor, the algorithm is able to flexibly switch mutation modes during the iteration process, thereby achieving rapid escape from the local optimum in the early stage and accurate convergence in the later stage.

3.4. Crisscross Strategy

In order to circumvent the algorithm’s tendency to converge prematurely and to circumvent the potential pitfall of a local optimum, the APO introduces the Crisscross Strategy before the conclusion of the iteration [45]. The utilization of horizontal crossover in the search process can diminish the number of self-points encountered by the algorithm, thereby enhancing its global search capability. Vertical crossover, on the other hand, can facilitate the elimination of dimensions that have reached a point of convergence.

3.4.1. Horizontal Crossover

The horizontal crossover operation is analogous to the crossover operation in genetic algorithms, which facilitates the exchange of information between different populations in the same dimension. Initially, it is essential to randomly pair the individuals of the parents to perform the crossover operation in the dth dimension, as follows:

$$\left\{\begin{array}{c}C{y}_{i,d}\left(t\right)=r\cdot y_{i,d}\left(t\right)+\left(1-r\right)\cdot y_{j,d}\left(t\right)+c\cdot \left(y_{i,d}\left(t\right)-y_{j,d}\left(t\right)\right)\\ C{y}_{j,d}\left(t\right)=r\cdot y_{j,d}\left(t\right)+\left(1-r\right)\cdot y_{i,d}\left(t\right)+c\cdot \left(y_{j,d}\left(t\right)-y_{i,d}\left(t\right)\right)\end{array},\right.$$

(27)

where $C{y}_{i,d}(t)$ and $C{y}_{j,d}(t)$ denote the values of the dth dimensional components of the two individuals $\mathbf{C}{\mathbf{y}}_{\mathit{i}}(t)$ and $\mathbf{C}{\mathbf{y}}_{\mathit{j}}(t)$, respectively, produced by the ith protozoa ${\mathit{y}}_{\mathit{i}}\left(t\right)$ and the jth protozoa ${\mathit{y}}_{\mathit{j}}\left(t\right)$ in generation $t$ after a lateral crossover. $r$ is a random number between the interval [0, 1] and $c$ is a random number between the interval $[-1,1]$.

The solutions generated by the horizontal crossover must be evaluated in comparison with their parental counterparts, and the individuals exhibiting the highest degree of fitness are selected for retention. This approach guarantees the efficacy and precision of the algorithmic convergence and optimization.

3.4.2. Vertical Crossover

APO is prone to local optimality at a later stage, largely due to the fact that some individuals in the population exhibit local optimality in a specific dimension, which results in premature convergence of the entire system. To address this issue, MSAPO employs a strategy of horizontal and vertical crossover operations on newborn individuals, improving the algorithm’s capacity to circumvent local optima.

If we assume that newborn protozoan $\mathbf{C}{\mathbf{y}}_i(t+1)$ crosses vertically in the d₁th and d₂th dimensions, the calculation is as follows:

$$S{y}_{i,j}(t+1)=r\cdot C{y}_{i,d_1}(t+1)+(1-r)\cdot C{y}_{i,d_2}(t+1),$$

(28)

$${\mathit{y}}_i^{new}(t+1)=\left\{\begin{array}{cc}\mathit{S}{\mathit{y}}_i(t+1),& \mathit{if} f(\mathit{S}{\mathit{y}}_i(t+1))<f({\mathit{y}}_i^{new}(t+1))\\ {\mathit{y}}_i^{new}(t+1),& \mathrm{otherwise}\end{array}\right.,$$

(29)

where $\mathbf{S}{\mathbf{y}}_i(t+1)$ is an individual resulting from vertical crossover from individual $\mathbf{C}{\mathbf{y}}_i(t+1)$.

The vertical crossover process generates offspring individuals with enhanced fitness by integrating information from disparate dimensions. Furthermore, a meritocracy mechanism, as illustrated by Equation (29), is employed to guarantee the efficacy of the crossover operation. The conjunction of horizontal and vertical crossover results in augmented global search abilities and an enhanced capacity to circumvent local optima. In the iterative process, once an individual has escaped from a local optimum through vertical crossover, its updated dimension information is rapidly disseminated throughout the population via horizontal crossover, thereby driving the population as a whole towards a superior solution.

The pseudo code for MSAPO is given in Algorithm 2.

Algorithm 2: MSAPO

Input: The population size N, the individual dimension D, Controlling parameters np, pfmax, and the maximum number of iterations T.

Output: The optimal individual ybest and its corresponding fitness fbest.

3.5. Computational Complexity

The basic APO performs sorting, dormancy and reproduction, autotrophic foraging and heterotrophic foraging, and fitness value calculations with a time complexity of $O(T\cdot N\cdot (\mathit{log}(N)+D+f(•)))$. Piecewise chaotic opposition-based learning strategy adds the generation of the inverse solution of the chaotic mapping and the comparison of the ordering with the original initial population, with an increased time complexity of $O(N\cdot D+2\cdot N\cdot \mathit{log}(2\cdot N))$. The time complexity of the heterotrophic phase introducing cyclone foraging strategy increases $O(0.5\cdot T\cdot (N-⌈N\cdot pf⌉)\cdot D)$. The time complexity of the hybrid mutation strategy is $O(T\cdot ⌈N\cdot pf⌉\cdot D)$. The crisscross strategy has a time complexity of $O(2\cdot T\cdot N\cdot D)$. To reduce the time complexity of the algorithm computation, the evaluation values are recorded once after the cross-sectional cross-computation, so the increased complexity of the computation of the fitness values that are retained optimally after the introduction of the strategy is $O(T\cdot N\cdot f(•))$. Therefore, the final time complexity of MSAPO is $O(N\cdot (D+2\cdot \mathit{log}(2\cdot N))+T\cdot N\cdot (3.5\cdot D+\mathit{log}(N)+2\cdot f(•)))$.

4. Numerical Experiments and Analysis

4.1. Baseline Algorithms and Benchmark Function Sets

In order to corroborate the enhanced performance of the MSAPO put forth in this paper, in this section, the CEC2017 and CEC2022 test sets are chosen for the purpose of conducting simulation experimentation and analyzing the resulting data. All functions of CEC2017 are rotated and shifted, increasing the difficulty of algorithmic optimization search. CEC2017 contains 29 test functions: unimodal functions (F1, F3), simple multimodal functions (F4–F10), hybrid functions (F11–F20), and composition functions (F21–F30). The CEC2022 test set includes 12 test functions. The unimodal function has only one global optimum, which is employed to assess the capability of the optimization algorithm to develop; the simple multimodal function contains multiple local optimums, which is utilized to evaluate the capacity of the algorithm to transcend the local optimum; and the hybrid function increases the complexity of the solution. Composition functions present more complex optimization problems and necessitate the utilization of algorithms with enhanced global exploration and local development capabilities. The efficacy of the optimization algorithm can be assessed in an objective manner through the utilization of test functions that encompass a diverse array of optimization problems.

The MSAPO algorithm is extensively compared with four classes of existing optimization algorithms: (1) The first class of algorithms: includes some well-known algorithms published in recent years, such as slime mold algorithm (SMA) [46], african vultures optimization algorithm (AVOA) [47], snake optimizer (SO) [25], artificial rabbits optimization (ARO) [48], nutcracker optimizer algorithm (NOA) [26] and PID-based search algorithm (PSA) [49]; (2) Algorithms in the second category: covering some widely cited and studied classical algorithms, including grey wolf optimizer (GWO) [21], whale optimization algorithm (WOA) [22] and salp swarm algorithm (SSA) [50]; (3) The third class of algorithms: includes variants of classical optimization algorithms PSO, such as XPSO [51], FVICLPSO [52] and SRPSO [53]; (4) Algorithms of the fourth category: including the high-performance optimizers LSHADE_cnEpSin [54] and LSHADE-SPACMA [55], which won the CEC2017 competition.

Initially, a comparison is made between MSAPO and the first three classes of algorithms under the highest dimension of the CEC2022 test set. To further validate the efficiency of MSAPO, experiments are conducted which include the fourth class of comparison algorithms under 30, 50 and 100 dimensions of the CEC2017 test set. In order to guarantee the comparability of the experimental results, the population size N of all algorithms was set to 100, and the maximum number of iterations $T_{max}$ set to 1000, corresponding to a maximum of 100,000 fitness evaluations. To eliminate the influence of chance factors, each algorithm was executed independently on 30 occasions.

In fact, we observed that certain algorithms employ strategies involving population reduction (such as LSHADE_cnEpSin and LSHADE_SPACMA). Some algorithms exhibit multiple evaluation behaviors (such as SO and NOA). For these algorithms, we recorded the fitness value corresponding to the global optimum after each evaluation. We then grouped these into 1000 sets of experimental data, each comprising 100 evaluations, thereby ensuring the fairness of the comparative trials.

4.2. Sensitivity Analysis of Parameter

In MSAPO, we introduce an additional chaotic parameter $P$. To investigate whether the value of parameter $P$ significantly impacts algorithm performance and to determine its approximate optimal value, we conducted sensitivity analysis experiments on this parameter.

According to Equation (21), the parameter $P$ takes values within the range $0<P<0.5$. Therefore, in this experiment, the candidate $P$ were set to 0.05, 0.1, …, 0.5. Table S1 presents the convergence curves of MSAPO under different parameter settings. According to Figure 5, the performance of MSAPO varies little across different parameters, indicating that the algorithm is not sensitive to the value of parameter $P$. However, in the scenario where $P=0.1$, the algorithm consistently achieves slightly better performance. Therefore, the default value of $P$ in this paper is set to 0.1.

Table S1 shows the average values and rankings obtained using different values of the chaotic parameter ($P$) on the 30-dimensional CEC2017 test set. As indicated by the ranking in the last row of Table S1, MSAPO achieves the best overall ranking on the test set when $P=0.1$. Therefore, the chaotic parameter $P=0.1$ is adopted for subsequent numerical experiments.

4.3. Ablation Experiment

MSAPO employs four strategies that collectively enhance its performance in complex engineering optimization. To validate their effectiveness, we conducted an experiment removing each strategy—named APO1, APO2, APO3, and APO4—representing the removal of 4 strategies, piecewise chaotic opposition-based learning strategy, cyclone foraging strategy, hybrid mutation strategy, and crisscross strategy. We performed ablation tests on a 30-dimensional CEC2017 dataset, comparing these algorithms with MSAPO.

Table 1. Statistical results of mean values in the melting experiment.

F	MSAPO	APO	APO1	APO2	APO3	APO4
F1	1.0010E+02	3.6670E+03	1.0020E+02	1.0010E+02	2.0805E+03	1.0122E+02
F3	3.0000E+02	3.0315E+04	3.0000E+02	3.0000E+02	4.5763E+03	3.2184E+02
F4	4.2279E+02	5.0889E+02	4.3407E+02	4.2423E+02	5.0893E+02	4.6988E+02
F5	5.3671E+02	5.3046E+02	5.3234E+02	5.3064E+02	5.2552E+02	5.4438E+02
F6	6.0000E+02	6.0000E+02	6.0000E+02	6.0000E+02	6.0000E+02	6.0000E+02
F7	7.6139E+02	7.5792E+02	7.5876E+02	7.6566E+02	7.5809E+02	7.6975E+02
F8	8.3273E+02	8.2728E+02	8.3413E+02	8.3562E+02	8.2829E+02	8.4597E+02
F9	9.0001E+02	9.0060E+02	9.0004E+02	9.0005E+02	9.0002E+02	9.0086E+02
F10	3.4372E+03	3.8150E+03	3.3170E+03	3.2895E+03	3.7963E+03	3.3472E+03
F11	1.1439E+03	1.1829E+03	1.1351E+03	1.1372E+03	1.1559E+03	1.1363E+03
F12	1.5912E+04	4.8696E+05	1.1148E+04	1.2251E+04	3.1425E+05	2.7184E+04
F13	4.1810E+03	1.0161E+04	1.5974E+03	1.5640E+03	1.2586E+04	1.8285E+03
F14	1.4485E+03	1.4410E+03	1.4459E+03	1.4415E+03	1.4383E+03	1.4470E+03
F15	1.5498E+03	1.5726E+03	1.5453E+03	1.5388E+03	1.5843E+03	1.5498E+03
F16	1.6966E+03	1.9763E+03	1.7532E+03	1.7935E+03	1.8927E+03	1.8121E+03
F17	1.7475E+03	1.7624E+03	1.7480E+03	1.7502E+03	1.7536E+03	1.7521E+03
F18	2.3738E+03	2.9820E+04	2.0872E+03	2.1971E+03	1.7743E+04	2.2266E+03
F19	1.9281E+03	1.9763E+03	1.9281E+03	1.9314E+03	2.7031E+03	1.9329E+03
F20	2.0782E+03	2.0780E+03	2.0622E+03	2.0786E+03	2.1098E+03	2.0933E+03
F21	2.3342E+03	2.3277E+03	2.3293E+03	2.3268E+03	2.3220E+03	2.3287E+03
F22	2.3000E+03	2.3005E+03	2.3000E+03	2.3000E+03	2.3000E+03	2.3002E+03
F23	2.6861E+03	2.6750E+03	2.6845E+03	2.6806E+03	2.6715E+03	2.6811E+03
F24	2.8520E+03	2.8461E+03	2.8546E+03	2.8550E+03	2.8404E+03	2.8545E+03
F25	2.8868E+03	2.8878E+03	2.8865E+03	2.8864E+03	2.8868E+03	2.8887E+03
F26	3.9985E+03	3.8484E+03	3.9364E+03	3.9543E+03	3.7921E+03	3.9921E+03
F27	3.2032E+03	3.2086E+03	3.2071E+03	3.2052E+03	3.2048E+03	3.2091E+03
F28	3.1103E+03	3.2287E+03	3.1517E+03	3.1103E+03	3.2041E+03	3.1527E+03
F29	3.3649E+03	3.4187E+03	3.3781E+03	3.3833E+03	3.3949E+03	3.3776E+03
F30	5.2553E+03	8.7990E+03	5.7758E+03	5.4629E+03	7.7271E+03	5.7277E+03
Rank	1	6	2	3	4	5

shows the average values of the optimal solutions for six algorithms, with partial convergence curves depicted in Figure 6. Results in Table 1 show the performance ranking: MSAPO > APO1 > APO2 > APO3 > APO4 > APO, indicating that removing any strategy reduces effectiveness. The crisscross strategy has the greatest impact on algorithm performance, while the chaotic opposition-based learning strategy has the least impact. The other two strategies also influence algorithm performance.

4.4. Algorithm Parameter Settings and Performance Indicators

The parameter settings of the algorithms employed in the experiment are presented in

Table 2. Parameter settings of the experimental algorithms.

Algorithm	Parameter	Value
MSAPO (Proposed)	neighbor pairs ($np$)	1
	proportion fraction maximum	0.1
	Chaotic parameter ($P$)	0.1
APO	neighbor pairs ($np$)	1
	proportion fraction maximum	0.1
SMA	Controlling parameter ($z$)	0.03
AVOA	Controlling parameters ($p_1, p_2, p_3, \alpha , \beta , \gamma$)	0.6, 0.4, 0.6, 0.8, 0.2, 2.5
SO	Threshold value	0.25
ARO	/	/
NOA	Controlling parameters ($P_{rb}, P_{a2}, \delta$)	0.2, 0.2, 0.05
PSA	Controlling parameters ($t, K_p, K_i, K_d$)	1, 1, 0.5, 1,2
GWO	Convergence constant $a$	Decreases Linearly from 2 to 0
WOA	Convergence constant $a$	Decreases Linearly from 2 to 0
	Spiral factor $b$	1
SSA	$c_1$	Decreases from 2 to 0
XPSO	acceleration constants ($cc$)	[1.0 0.5 0.5]
	weight of the search space	0.1
FVICLPSO	Controlling parameters ($\lambda ,w$)	0.3, 0
SRPSO	Controlling parameters ($w_{min}, w_{max}, c_1, c_2$)	0.5, 1.05, 1.49445, 1.49445
LSHADE_cnEpSin	change freq	0.5
	Controlling parameters ($p_b, p_s$)	0.4, 0.5
LSHADE_SPACMA	$L_{Rate}$	0.8
	Controlling parameter	10−8

. The paper employs five performance evaluation metrics to conduct a comprehensive analysis of the experimental results obtained from the MSAPO algorithm in comparison to other algorithms. These metrics encompass Mean, Best, standard deviation (Std), Mean Rank for Friedman’s test (MR), and Wilcoxon rank sum test for 30 runs on each test function.

Mean and Best values provide an intuitive reflection of the efficacy of the algorithmic solution, whereas Std offers insight into the concentration of the solution results, thereby indicating the stability of the algorithmic solution. The ranking of each algorithm on the current test function can be derived by ranking the mean rank (MR) of the Friedman test. The Wilcoxon rank sum test provides a means of ascertaining whether the fitness values of the two algorithms originate from distributions with an identical median, thereby indicating whether they are significantly disparate. In this study, a one-sided Wilcoxon rank sum test was conducted on MSAPO and other comparison algorithms, with a significance level of 0.05. The calculated p-value allows us to statistically determine the performance of MSAPO on the test set in relation to its comparison algorithms. In the context of the Wilcoxon rank sum test, “+” signifies that the MSAPO algorithm is demonstrably superior to the comparison algorithm, “−” signifies that the MSAPO algorithm is significantly inferior to the comparison algorithm, and “=” signifies that there is no significant distinction between the two. The integration of these indicators offers a comprehensive and detailed reference point for the analysis of the experimental results.

4.5. Analysis of Results Under CEC2022

Table 3. Comparison results of various algorithms under 20D CEC2022.

F	Index	MSAPO	APO	SMA	AVOA	SO	ARO	NOA	PSA	GWO	WOA	SSA	XPSO	FVICLPSO	SRPSO
F1	MR	1.0000	9.3667	5.2333	6.4333	10.0000	13.5333	7.2333	3.4667	11.4000	11.2667	2.0000	3.5667	13.4333	7.0667
	Rank	1	9	5	6	10	14	8	3	12	11	2	4	13	7
F2	MR	3.2333	7.8333	5.9333	8.2667	5.7000	10.4667	5.6333	6.0667	10.7667	12.3333	7.2667	10.8000	5.4667	5.2333
	Rank	1	9	6	10	5	11	4	7	12	14	8	13	3	2
F3	MR	1.0000	2.0667	8.5000	12.6000	7.0333	10.2000	5.2000	9.3667	9.8667	13.9333	12.4333	5.8000	2.9333	4.0667
	Rank	1	2	8	13	7	11	5	9	10	14	12	6	3	4
F4	MR	2.7667	1.1000	9.5667	10.4333	3.8667	12.4667	6.6000	9.0667	6.4667	12.5333	9.5667	3.7667	12.0667	4.7333
	Rank	2	1	9	11	4	13	7	8	6	14	10	3	12	5
F5	MR	1.0000	3.4000	8.6000	11.5333	3.9333	13.3333	2.5333	9.3333	7.7000	12.9333	8.9000	4.6667	11.6667	5.4667
	Rank	1	3	8	11	4	14	2	10	7	13	9	5	12	6
F6	MR	1.3667	4.1333	11.7667	7.1000	7.1667	8.9667	2.9333	6.5333	10.5667	10.8000	9.3000	6.4667	11.8333	6.0667
	Rank	1	3	13	7	8	9	2	6	11	12	10	5	14	4
F7	MR	1.7000	2.4000	6.9000	12.2000	6.0333	11.7333	6.1000	9.3000	8.2333	13.4000	11.4333	4.6000	5.0333	5.9333
	Rank	1	2	8	13	6	12	7	10	9	14	11	3	4	5
F8	MR	1.5333	3.2333	6.4333	10.8000	5.9000	10.6000	7.3667	5.6000	9.8667	12.4000	11.3333	5.0000	6.5333	8.4000
	Rank	1	2	6	12	5	11	8	4	10	14	13	3	7	9
F9	MR	1.0000	3.3667	8.2333	8.7667	5.3667	12.2333	5.4000	2.1000	12.5000	13.3000	10.4333	11.3667	6.6667	4.2667
	Rank	1	3	8	9	5	12	6	2	13	14	10	11	7	4
F10	MR	2.9667	4.7333	10.1000	10.1000	4.1667	10.5333	5.6667	9.4333	9.8667	12.3000	9.1667	6.9667	1.5667	7.4333
	Rank	2	4	11	12	3	13	5	9	10	14	8	6	1	7
F11	MR	1.6333	4.6000	10.2000	6.5000	7.2000	10.6333	7.3667	4.4333	13.6667	11.4000	8.5667	5.0667	9.0000	4.7333
	Rank	1	3	11	6	7	12	8	2	14	13	9	5	10	4
F12	MR	1.8000	2.3667	5.1000	9.4000	8.8667	12.7000	2.8667	9.9667	7.9333	13.0000	8.3000	11.5000	4.9000	6.3000
	Rank	1	2	5	10	9	13	3	11	7	14	8	12	4	6
+/=/−		compared	10/1/1	11/1/0	12/0/0	10/2/0	12/0/0	12/0/0	12/0/0	12/0/0	12/0/0	12/0/0	12/0/0	10/1/1	12/0/0
Average MR		1.7500	4.0500	8.0472	9.5111	6.2694	11.4500	5.4083	7.0556	9.9028	12.4667	9.0583	6.6306	7.5917	5.8083
Total Rank		1	2	9	11	5	13	3	7	12	14	10	6	8	4

and Table S2 present the statistical metrics of the 14 algorithms for the 12 test function optimization results under the 20-dimensional CEC2022 test set. The results include those of the Mean, Best, Std, MR, Rank and Wilcoxon rank-sum tests. Additionally, the table provides the average MR (Average MR) on the 12 test functions and the combined rank (Total Rank). As evidenced in Table 3 and Table S2, MSAPO exhibits the most optimal Mean performance across the eight test functions, F1, F3, F5–F9, and F12. It demonstrates the capacity to converge stably to its theoretical optimum on F1, F3, and F5, indicating a robust ability to navigate and overcome local optima. This makes MSAPO a versatile choice for diverse types of optimization problems. MSAPO achieved the smallest Std of all algorithms on functions F1, F3, F5, F6, F9, F10, and F12, showing the stability of its results. The results of the Wilcoxon rank sum test and the Rank metrics on the 12 tested functions indicate that MSAPO significantly outperforms the other compared algorithms on 10 functions and ranks first. Additionally, MSAPO achieves a second-place ranking on F4 and F10. With regard to function F4, APO exhibited a slight advantage over MSAPO, whereas FVICLPSO demonstrated superior performance on function F10. Collectively, MSAPO attained an average MR of 1.7500 across the 12 functions, ultimately securing the top ranking.

Figure 7 and Figure 8 illustrate the mean convergence curves and box plots of the 14 algorithms under the 20-dimensional CEC2022 test set. As illustrated in Figure 7, for the unimodal function F1, APO is unable to achieve a high convergence accuracy for the algorithm due to its slow convergence speed. In contrast, MSAPO expedites the convergence speed and optimality-finding ability of the algorithm through the utilization of a high-quality initial population and a hybrid mutation strategy, thereby enabling it to reach the optimal solution in a more rapid manner. In the context of the simple multimodal function F4, MSAPO’s principal competitor is APO. MSAPO demonstrates a superior convergence rate in the initial phase of the iteration, while APO exhibits a greater capacity to identify the optimal solution in the subsequent phase. Despite MSAPO’s inability to retain its leading position in this domain, the performance gap between it and APO remains relatively narrow. In addressing the intricate optimization issues pertaining to other combinatorial and composite functions within the test set, MSAPO has demonstrated its capacity to identify superior fitness values with a reduced number of iterations, thereby attaining enhanced convergence precision. This outcome suggests that MSAPO enhances the algorithmic convergence precision and its capacity to circumvent local optima through the deployment of cyclone foraging strategy and crisscross strategy. As illustrated in Figure 8, MSAPO exhibits the narrowest and lowest box across the majority of test functions, thereby substantiating its superior stability and optimality-finding capabilities. Figure 9 presents radar charts of the comprehensive ranking of MSAPO in comparison with other algorithms. The size of the enclosed area in the chart is indicative of the algorithm’s performance, allowing for an intuitive understanding of MSAPO’s notable superiority.

4.6. Result Analysis on CEC2017

In order to circumvent the constraints of a singular test set, this section elects to utilize CEC2017, which exhibits an analogous function type to that of CEC2022 and a more expansive array of optimization functions, as the test set. Additionally, it introduces two triumphant algorithms from the CEC2017 test set competition, LSHADE_cnEpSin and LSHADE_SPACMA, as the comparison algorithms to substantiate the efficacy of the proposed MSAPO algorithm.

Table 4. Comparison results of 16 algorithms under 30-dimensional CEC2017.

F	Index	MSAPO	APO	SMA	AVOA	SO	ARO	NOA	PSA	GWO	WOA	SSA	XPSO	FVICLPSO	SRPSO	LSHADE_ cnEpSin	LSHADE_ SPACMA
F1	MR	2.8333	6.9000	10.2667	7.8333	10.2000	14.4333	7.9667	7.7667	15.9000	14.6667	7.7333	7.6333	9.6333	9.0667	2.1667	1.0000
	Rank	3	4	13	8	12	14	9	7	16	15	6	5	11	10	2	1
F3	MR	1.8000	11.0333	3.8333	8.5000	12.6000	14.5667	6.6000	5.5333	11.7667	15.5000	6.4667	7.1333	14.6000	9.6000	2.8000	3.6667
	Rank	1	11	4	9	13	14	7	5	12	16	6	8	15	10	2	3
F4	MR	1.4333	9.3000	7.1667	9.3667	6.8000	13.8667	6.3000	6.9000	13.2000	15.4000	9.6667	12.4000	9.7000	8.5000	3.7333	2.2667
	Rank	1	9	7	10	5	15	4	6	14	16	11	13	12	8	3	2
F5	MR	3.9000	3.3000	10.0333	14.3667	4.5000	13.5333	10.1667	10.4667	9.1333	15.8333	12.1667	5.6667	12.0000	6.3000	2.6000	2.0333
	Rank	4	3	9	15	5	14	10	11	8	16	13	6	12	7	2	1
F6	MR	1.0000	2.3667	10.2333	14.5667	8.6333	12.5000	6.6000	12.1000	11.2667	15.9667	14.3667	7.8667	5.4333	5.8333	4.1000	3.1667
	Rank	1	2	10	15	9	13	7	12	11	16	14	8	5	6	4	3
F7	MR	3.3667	3.4000	8.5667	15.2667	6.7000	13.6667	9.8333	12.1000	10.2000	15.6667	10.3667	5.4667	11.4333	6.1333	2.7333	1.1000
	Rank	3	4	8	15	7	14	9	13	10	16	11	5	12	6	2	1
F8	MR	5.2000	2.8667	9.8667	13.8000	4.2333	14.1333	9.8000	10.7667	8.6667	15.5000	12.2000	6.0000	13.0333	5.5667	2.5667	1.8000
	Rank	5	3	10	14	4	15	9	11	8	16	12	7	13	6	2	1
F9	MR	1.6000	3.7667	11.0333	13.9000	5.7000	13.6333	3.3667	11.3333	9.2333	15.7000	12.9333	6.9333	12.2333	6.8333	4.4333	3.3667
	Rank	1	4	10	15	6	14	2	11	9	16	13	8	12	7	5	3
F10	MR	3.9667	5.5667	7.4667	12.8000	9.8667	9.0000	13.9000	9.2000	7.8667	15.1333	9.4667	6.0667	13.1000	6.6333	3.1000	2.8667
	Rank	3	4	7	13	12	9	15	10	8	16	11	5	14	6	2	1
F11	MR	2.0000	4.8667	8.7000	9.0000	5.9000	15.8000	6.0000	9.3000	13.3333	15.0667	11.3333	7.4000	13.4667	7.0667	3.3000	3.4667
	Rank	1	4	9	10	5	16	6	11	13	15	12	8	14	7	2	3
F12	MR	2.5667	7.8000	10.4000	11.0333	6.3667	12.3333	4.4000	7.0667	14.4000	15.8667	12.9333	6.0667	13.5333	7.7000	2.0667	1.4667
	Rank	3	9	10	11	6	12	4	7	15	16	13	5	14	8	2	1
F13	MR	2.1667	6.6667	9.3667	12.1000	7.7333	11.2333	5.0333	7.0667	13.8667	15.0333	13.9667	6.6000	12.5333	7.5667	2.6667	2.4000
	Rank	1	6	10	12	9	11	4	7	14	16	15	5	13	8	3	2
F14	MR	2.2333	1.4333	11.6000	11.4333	8.9000	15.1333	3.9333	8.9000	11.2667	15.3667	9.4333	8.0667	12.5667	8.3333	4.0000	3.4000
	Rank	2	1	13	12	8	15	4	9	11	16	10	6	14	7	5	3
F15	MR	1.1667	2.5667	11.1667	11.6000	9.0333	11.2000	4.6333	8.2000	14.3333	15.2667	14.3667	8.0667	8.1333	8.7000	3.9667	3.6000
	Rank	1	2	11	13	10	12	5	8	14	16	15	6	7	9	4	3
F16	MR	1.7667	2.7667	8.6667	13.1000	6.3667	14.1667	7.9000	12.2000	9.5667	15.5667	12.1000	7.4000	8.8333	6.2333	3.7000	5.6667
	Rank	1	2	9	14	6	15	8	13	11	16	12	7	10	5	3	4
F17	MR	1.3333	2.4667	12.2000	13.1667	8.2333	14.7667	5.8000	12.1667	10.0000	14.9333	11.6000	7.0000	7.2000	7.4000	3.5667	4.1667
	Rank	1	2	13	14	9	15	5	12	10	16	11	6	7	8	3	4
F18	MR	2.6667	5.0667	12.0333	12.0000	9.5000	14.6333	3.7333	7.8667	11.4667	14.2333	10.2667	7.6333	11.1000	10.2000	1.7333	1.8667
	Rank	3	5	14	13	8	16	4	7	12	15	10	6	11	9	1	2
F19	MR	1.5667	2.7333	11.7667	10.1333	8.7000	11.2000	3.8000	9.1667	13.9333	15.7333	14.7000	9.0333	7.1000	9.0667	3.5000	3.8667
	Rank	1	2	13	11	7	12	4	10	14	16	15	8	6	9	3	5
F20	MR	2.1000	2.6000	11.5000	14.2667	6.7000	14.2667	6.1667	11.3000	10.5000	14.6000	12.1667	7.0667	7.1000	6.5333	3.8000	5.3333
	Rank	1	2	12	14	7	15	5	11	10	16	13	8	9	6	3	4
F21	MR	3.0667	2.5667	10.7000	14.3000	4.7667	13.5667	9.3667	10.7333	9.2000	15.8667	12.0000	6.1000	11.0667	6.4667	2.8000	3.4333
	Rank	3	1	10	15	5	14	9	11	8	16	13	6	12	7	2	4
F22	MR	2.6667	7.4667	12.9667	12.9000	11.2333	12.8667	6.6000	10.1333	11.5667	13.7667	8.9000	4.6833	10.2333	6.4333	2.5667	1.0167
	Rank	3	7	15	14	11	13	6	9	12	16	8	4	10	5	2	1
F23	MR	4.1000	2.2667	10.1000	14.5667	5.6667	14.0333	8.9333	11.5667	9.8667	15.6667	11.1667	6.1000	11.4667	5.7667	2.2667	2.4667
	Rank	4	1	10	15	5	14	8	13	9	16	11	7	12	6	2	3
F24	MR	4.1000	2.4333	10.2667	14.5667	5.7000	15.5333	9.0000	11.2667	9.3667	14.8000	9.5333	6.1667	12.8000	5.3000	2.7667	2.4000
	Rank	4	2	11	14	6	16	8	12	9	15	10	7	13	5	3	1
F25	MR	4.0000	6.1333	6.8667	8.2667	7.1333	14.4000	3.5333	9.5000	14.3667	15.4667	10.4000	11.0333	11.5667	5.6000	4.0000	3.7333
	Rank	3	6	7	9	8	15	1	10	14	16	11	12	13	5	4	2
F26	MR	5.6333	5.4333	11.0000	13.6333	8.5333	14.4667	5.7000	10.7000	9.8667	15.3000	8.6333	4.1667	8.5000	6.2667	3.9333	4.2333
	Rank	5	4	13	14	9	15	6	12	11	16	10	2	8	7	1	3
F27	MR	2.8667	2.4333	6.1000	12.9667	10.6333	14.1667	7.6000	11.4333	9.9667	15.4667	10.2000	12.0333	7.0000	5.0333	4.0667	4.0333
	Rank	2	1	6	14	11	15	8	12	9	16	10	13	7	5	4	3
F28	MR	2.1333	8.8667	9.5000	8.6667	9.9000	15.0000	4.4667	6.8667	14.7667	14.9000	9.7667	4.3667	12.9000	5.9667	5.4667	2.4667
	Rank	1	9	10	8	12	16	4	7	14	15	11	3	13	6	5	2
F29	MR	1.4667	3.1667	11.2667	13.4333	8.1000	13.3000	9.0667	10.7667	9.6667	15.9000	13.2000	6.7667	8.2667	4.7667	3.3000	3.5667
	Rank	1	2	12	15	7	14	9	11	10	16	13	6	8	5	3	4
F30	MR	2.0333	5.2000	8.6667	12.1333	5.8333	10.9000	9.6333	4.9000	15.0000	15.7333	14.2667	8.7333	11.9667	6.8667	2.2000	1.9333
	Rank	2	5	8	13	6	11	10	4	15	16	14	9	12	7	3	1
Average MR		2.6460	4.6011	9.7690	12.1954	7.7299	13.5276	6.8908	9.5609	11.5011	15.3069	11.2517	7.2293	10.6379	6.9563	3.2379	2.9580
Total Rank		1	4	10	14	8	15	5	9	13	16	12	7	11	6	3	2

,

Table 5. Comparison results of 16 algorithms under 50-dimensional CEC2017.

F	Index	MSAPO	APO	SMA	AVOA	SO	ARO	NOA	PSA	GWO	WOA	SSA	XPSO	FVICLPSO	SRPSO	LSHADE_ cnEpSin	LSHADE_ SPACMA
F1	MR	2.6000	10.0667	10.8000	5.6333	11.9333	15.8000	8.5667	5.3000	15.1667	14.0333	4.9000	5.1000	13.0000	6.0667	4.7333	2.3000
	Rank	2	10	11	7	12	16	9	6	15	14	4	5	13	8	3	1
F3	MR	1.2000	12.0333	3.4333	7.6333	12.4333	15.5000	5.5333	6.3333	10.2000	13.9000	7.1333	8.3333	15.2667	9.7333	2.8000	4.5333
	Rank	1	12	3	8	13	16	5	6	11	14	7	9	15	10	2	4
F4	MR	2.3333	8.5667	7.6667	7.3333	6.9000	15.6000	5.7333	4.6333	14.0667	15.0000	8.7333	11.5667	13.1667	6.8667	4.4667	3.3667
	Rank	1	10	9	8	7	16	5	4	14	15	11	12	13	6	3	2
F5	MR	3.9333	5.7000	9.5333	12.6000	2.4667	14.3333	11.4333	9.9333	8.7667	15.8000	11.1333	4.8333	14.4667	5.6333	3.6667	1.7667
	Rank	4	7	9	13	2	14	12	10	8	16	11	5	15	6	3	1
F6	MR	1.0000	2.0667	11.3000	14.4333	6.3667	12.1667	6.5000	11.9667	10.6000	16.0000	14.5333	7.9333	6.8333	5.3333	5.5667	3.4000
	Rank	1	2	11	14	6	13	7	12	10	16	15	9	8	4	5	3
F7	MR	2.7667	4.6667	8.5000	14.8333	5.7333	14.0333	9.7667	12.1000	9.0333	15.8333	10.3667	4.1333	12.1333	7.6667	3.2333	1.2000
	Rank	2	5	8	15	6	14	10	12	9	16	11	4	13	7	3	1
F8	MR	4.4000	5.3000	9.3000	13.0667	2.6333	13.8333	11.2667	10.1000	8.5000	15.4000	11.8000	5.0333	14.5667	5.7333	3.1667	1.9000
	Rank	4	6	9	13	2	14	11	10	8	16	12	5	15	7	3	1
F9	MR	1.6667	2.7333	11.4667	12.3333	6.4000	13.6000	7.1333	10.9000	9.1667	15.7000	11.7333	5.6000	15.0667	4.3333	4.6667	3.5000
	Rank	1	2	11	13	7	14	8	10	9	16	12	6	15	4	5	3
F10	MR	3.0667	8.3000	5.6667	10.3667	14.2333	6.9333	13.8000	5.5667	7.3333	13.7667	9.7333	4.7333	14.9333	5.9667	5.7667	5.8333
	Rank	1	10	4	12	15	8	14	3	9	13	11	2	16	7	5	6
F11	MR	1.1667	4.0667	8.6333	7.2333	9.7000	15.5667	7.0667	7.9333	14.4667	13.4000	11.2667	7.2000	14.5667	3.8667	5.3667	4.5000
	Rank	1	3	10	8	11	16	6	9	14	13	12	7	15	2	5	4
F12	MR	2.1333	7.2667	9.7333	10.3667	7.8667	14.3667	5.2000	6.7333	14.2667	15.1667	12.1667	5.7667	14.0000	6.9333	2.3000	1.7333
	Rank	2	8	10	11	9	15	4	6	14	16	12	5	13	7	3	1
F13	MR	2.2333	3.7000	9.8667	11.3000	7.9000	13.6000	8.2333	5.0667	15.6000	14.4000	11.7000	5.3000	14.3667	4.5667	4.5000	3.6667
	Rank	1	3	10	11	8	13	9	6	16	15	12	7	14	5	4	2
F14	MR	2.5333	5.2000	10.7333	11.0333	8.4667	15.7667	3.6000	8.1667	12.4000	14.4667	10.0000	7.0000	14.0667	8.6000	1.7667	2.2000
	Rank	3	5	11	12	8	16	4	7	13	15	10	6	14	9	1	2
F15	MR	6.2667	5.3000	10.9333	12.2333	7.7667	12.6333	4.8000	7.0333	15.1000	15.0667	13.4333	4.9000	10.7667	5.6333	2.1000	2.0333
	Rank	7	5	11	12	9	13	3	8	16	15	14	4	10	6	2	1
F16	MR	2.5000	4.0333	10.3333	13.3333	9.0000	13.5333	7.5000	11.1667	7.7000	15.8667	11.1000	6.0000	11.5000	3.9667	3.0667	5.4000
	Rank	1	4	10	14	9	15	7	12	8	16	11	6	13	3	2	5
F17	MR	1.3333	3.7000	10.4333	13.4667	7.0000	13.6333	6.4333	11.0667	8.3000	15.2667	11.9000	7.5000	10.2000	6.8000	2.9667	6.0000
	Rank	1	3	11	14	7	15	5	12	9	16	13	8	10	6	2	4
F18	MR	2.6000	5.4333	10.7333	9.8333	10.0333	14.6333	3.9333	7.6667	12.2333	15.2667	9.9000	7.6333	13.2333	9.4000	1.7333	1.7333
	Rank	3	5	12	9	11	15	4	7	13	16	10	6	14	8	1	2
F19	MR	1.6667	8.1000	7.1333	9.9333	8.8667	11.6667	4.3333	8.4333	14.2333	15.3000	15.3000	7.5333	11.2333	7.7333	2.6000	1.9333
	Rank	1	8	5	11	10	13	4	9	14	15	16	6	12	7	3	2
F20	MR	2.1000	3.2000	10.3667	13.5667	10.2333	12.9667	6.3667	11.2333	9.3000	14.9000	12.0667	6.5000	8.0333	4.9667	3.7000	6.5000
	Rank	1	2	11	15	10	14	5	12	9	16	13	6	8	4	3	7
F21	MR	2.0667	4.7000	9.4000	13.7667	2.9333	13.9667	10.5000	10.9000	8.9000	15.8000	10.7000	5.1333	13.9667	5.4000	3.7000	4.1667
	Rank	1	5	9	13	2	14	10	12	8	16	11	6	15	7	3	4
F22	MR	1.1667	8.8333	6.9333	10.5667	13.1000	8.4333	12.2667	8.3667	6.7667	14.4667	9.1333	5.2667	14.0333	5.5333	6.2000	4.9333
	Rank	1	10	7	12	14	9	13	8	6	16	11	3	15	4	5	2
F23	MR	1.9333	3.5000	9.4667	14.6333	4.9000	14.3667	10.5333	11.0333	9.2667	15.9667	9.9000	5.4000	12.7667	5.0000	3.3667	3.9667
	Rank	1	3	9	15	5	14	11	12	8	16	10	7	13	6	2	4
F24	MR	1.5333	3.6000	9.4333	14.2333	7.1667	15.7667	10.3333	10.9333	8.7667	15.0000	9.0667	5.0000	12.9667	5.1333	3.3667	3.7000
	Rank	1	3	10	14	7	16	11	12	8	15	9	5	13	6	2	4
F25	MR	6.7667	10.7000	4.8667	8.8000	6.0667	15.5667	7.5000	6.2333	14.4333	14.7667	5.5333	11.5667	13.1333	4.2667	2.0000	3.8000
	Rank	8	11	4	10	6	16	9	7	14	15	5	12	13	3	1	2
F26	MR	3.1000	6.0333	8.0000	13.8000	6.9333	14.3667	9.5000	12.2000	9.7000	15.9667	5.6000	4.0333	12.8333	5.4000	4.4333	4.1000
	Rank	1	7	9	14	8	15	10	12	11	16	6	2	13	5	4	3
F27	MR	2.1667	4.0000	7.1667	13.4000	9.3333	14.9333	9.9000	10.2333	10.1000	15.7667	9.6667	10.2000	8.7000	4.0000	4.1000	2.3333
	Rank	1	3	6	14	8	15	10	13	11	16	9	12	7	4	5	2
F28	MR	3.2000	11.2333	5.7667	7.5333	9.3000	15.3667	8.4333	6.8333	14.3667	14.5000	5.5000	10.7333	13.7667	4.0333	2.6000	2.8333
	Rank	3	12	6	8	10	16	9	7	14	15	5	11	13	4	1	2
F29	MR	1.6333	3.9000	9.8333	13.0333	4.9000	13.4667	9.2000	10.3667	10.1667	16.0000	13.9333	7.2667	10.3333	5.7333	3.5333	2.7000
	Rank	1	4	9	13	5	14	8	12	10	16	15	7	11	6	3	2
F30	MR	1.7333	4.8333	8.0333	9.8333	5.8000	12.5000	11.9667	4.8000	14.6000	15.9333	14.4667	10.5333	10.0000	3.2667	3.8000	3.9000
	Rank	1	6	8	9	7	13	12	5	15	16	14	11	10	2	3	4
Average MR		2.5103	5.8885	8.8092	11.3839	7.8057	13.7552	8.1839	8.7322	11.1552	15.1276	10.4276	6.8184	12.5483	5.7782	3.6299	3.4460
Total Rank		1	5	10	13	7	15	8	9	12	16	11	6	14	4	3	2

,

Table 6. Comparison results of 16 algorithms under 100-dimensional CEC2017.

F	Index	MSAPO	APO	SMA	AVOA	SO	ARO	NOA	PSA	GWO	WOA	SSA	XPSO	FVICLPSO	SRPSO	LSHADE_ cnEpSin	LSHADE_ SPACMA
	MR	1.6667	12.0000	8.2333	5.8333	9.7667	16.0000	10.9667	6.8667	14.9667	14.0333	2.3667	2.3000	13.0000	8.8333	4.9333	4.2333
	Rank	1	12	8	6	10	16	11	7	15	14	3	2	13	9	5	4
	MR	1.0000	11.6667	3.8667	6.3000	9.4000	15.0667	4.2667	11.0667	7.1667	15.3000	7.6667	9.8667	14.4667	9.7000	3.4333	5.7667
	Rank	1	13	3	6	9	15	4	12	7	16	8	11	14	10	2	5
	MR	1.9000	10.7333	3.6667	9.2333	6.6000	16.0000	8.6000	7.1333	14.2667	14.7000	7.5333	11.4333	13.0333	4.7667	3.1000	3.3000
	Rank	1	11	4	10	6	16	9	7	14	15	8	12	13	5	2	3
	MR	2.0667	6.9333	9.2667	11.5667	3.4333	14.6000	12.5000	9.8000	8.5667	14.8333	11.2333	2.7000	15.4333	4.4667	5.1667	3.4333
	Rank	1	7	9	12	3	14	13	10	8	15	11	2	16	5	6	4
	MR	1.0000	2.7333	11.3000	14.3000	2.9333	12.2333	7.6333	12.0000	10.4667	16.0000	14.7000	7.0333	8.1000	6.1333	5.8000	3.6333
	Rank	1	2	11	14	3	13	8	12	10	16	15	7	9	6	5	4
	MR	1.5000	6.0333	8.1667	14.6000	4.1333	14.4333	9.0667	12.4000	10.0667	15.9667	10.7667	2.7000	12.0333	6.9000	4.9333	2.3000
	Rank	1	6	8	15	4	14	9	13	10	16	11	3	12	7	5	2
	MR	2.3333	6.9667	8.9333	12.5667	3.3333	14.6333	11.9333	10.2333	8.4000	15.5333	10.7667	2.5000	14.7000	4.7333	4.9000	3.5333
	Rank	1	7	9	13	3	14	12	10	8	16	11	2	15	5	6	4
	MR	1.2000	4.5667	11.5000	9.9333	3.7667	14.3333	8.4000	11.0333	10.4333	14.5667	11.0667	3.7333	16.0000	6.3333	5.6000	3.5333
	Rank	1	5	13	9	4	14	8	11	10	15	12	3	16	7	6	2
	MR	2.6000	10.0000	4.0667	6.8333	15.5000	6.2000	13.8000	4.8333	4.5333	12.9667	5.3667	4.1667	15.3000	9.2000	10.2000	10.4333
	Rank	1	10	2	8	16	7	14	5	4	13	6	3	15	9	11	12
	MR	1.0333	10.9333	2.6000	8.0667	12.6333	15.5667	9.1000	5.4333	12.5000	14.2000	7.4333	5.9333	15.0333	8.5333	3.4333	3.5667
	Rank	1	11	2	8	13	16	10	5	12	14	7	6	15	9	3	4
	MR	1.2000	8.1000	9.2333	9.0667	7.9333	15.9000	5.6000	4.6333	14.5333	13.2000	11.9333	9.4333	14.3667	5.8667	3.0000	2.0000
	Rank	1	8	10	9	7	16	5	4	15	13	12	11	14	6	3	2
	MR	1.5667	4.9667	11.5000	10.6000	9.4000	15.8667	3.0000	5.1667	15.0333	13.9667	10.5333	6.4000	13.0667	3.5667	6.5667	4.8000
	Rank	1	5	12	11	9	16	2	6	15	14	10	7	13	3	8	4
	MR	3.1667	9.7000	10.0333	8.3000	9.2333	15.6333	3.2667	7.6333	12.2667	13.9667	9.7000	5.7333	15.2667	8.5333	1.6333	1.9333
	Rank	3	10	12	7	9	16	4	6	13	14	11	5	15	8	1	2
	MR	1.7000	2.9333	10.7667	10.9333	8.3333	15.5000	5.7667	4.5000	15.3333	13.8000	11.5000	6.4000	13.1667	5.2667	6.3667	3.7333
	Rank	1	2	10	11	9	16	6	4	15	14	12	8	13	5	7	3
	MR	2.8667	5.4333	7.3333	9.8000	11.4667	12.9333	12.1667	7.7000	7.6333	15.9000	9.9667	5.2000	15.0333	2.1667	3.6333	6.7667
	Rank	2	5	7	10	12	14	13	9	8	16	11	4	15	1	3	6
	MR	1.6333	3.5667	9.2000	12.0000	10.8333	13.8667	7.9667	9.6667	5.9667	15.2000	10.2000	5.8000	14.8667	4.6000	3.3667	7.2667
	Rank	1	3	9	13	12	14	8	10	6	16	11	5	15	4	2	7
	MR	3.0000	7.1667	12.1333	8.2667	11.2000	14.7667	3.6333	7.2000	11.0667	12.5000	9.6333	5.8333	15.8667	10.3667	1.4333	1.9333
	Rank	3	6	13	8	12	15	4	7	11	14	9	5	16	10	1	2
	MR	3.7333	2.9000	9.7000	10.9333	8.1000	15.0000	4.6667	5.6333	14.7667	14.6000	13.4333	5.4333	12.1667	5.2000	5.0667	4.6667
	Rank	2	1	10	11	9	16	3	8	15	14	13	7	12	6	5	4
	MR	2.0333	4.3667	9.2333	11.6333	14.1000	11.0000	9.7667	8.4000	5.9667	14.1333	8.0667	4.2333	13.4333	6.7000	4.1667	8.7667
	Rank	1	4	10	13	15	12	11	8	5	16	7	3	14	6	2	9
	MR	1.1000	6.3333	9.0000	13.6333	3.3667	14.1000	11.0333	10.2333	8.9333	16.0000	10.7667	2.9333	14.2333	3.9333	5.3667	5.0333
	Rank	1	7	9	13	3	14	12	10	8	16	11	2	15	4	6	5
	MR	1.7333	10.5333	4.5000	8.0667	14.9667	6.7333	13.9000	4.2333	5.6667	13.4333	6.1667	4.3333	15.5333	6.7333	9.9000	9.5667
	Rank	1	12	4	9	15	7	14	2	5	13	6	3	16	8	11	10
	MR	1.0333	4.9333	7.5333	14.5000	2.0667	14.3667	9.0667	10.2333	9.9333	16.0000	11.2667	7.6667	13.1333	4.8333	5.3333	4.1000
	Rank	1	5	7	15	2	14	9	11	10	16	12	8	13	4	6	3
	MR	1.1000	4.4667	8.0333	14.3000	3.7000	14.7333	10.2000	11.1667	9.9333	15.9667	9.5333	7.2000	12.8667	2.4333	5.1333	5.2333
	Rank	1	4	8	14	3	15	11	12	10	16	9	7	13	2	5	6
	MR	2.2333	10.9333	4.0667	7.4333	7.9667	15.9667	8.1667	6.8667	14.4000	13.4000	8.4667	11.9000	14.2333	3.7000	3.7000	2.5667
	Rank	1	11	5	7	8	16	9	6	15	13	10	12	14	3	4	2
	MR	3.6000	5.0667	8.1000	14.3333	4.0667	14.6667	10.7000	11.7000	9.5000	16.0000	8.6000	3.5000	12.7000	3.1667	5.0000	5.3000
	Rank	3	6	8	14	4	15	11	12	10	16	9	2	13	1	5	7
	MR	2.2000	5.8333	5.0000	12.7667	6.0333	14.7667	9.6667	9.3333	11.6000	15.9000	9.3667	10.3000	13.3333	2.4333	4.3000	3.1667
	Rank	1	6	5	13	7	15	10	8	12	16	9	11	14	2	4	3
	MR	2.1333	11.5667	2.9000	7.1667	10.3333	15.2000	8.8333	6.7333	13.6333	13.3667	7.0333	11.0333	15.8000	3.7000	3.6667	2.9000
	Rank	1	12	2	8	10	15	9	6	14	13	7	11	16	5	4	3
	MR	1.8000	5.0000	7.2000	11.4333	3.4000	14.1667	11.1333	7.9000	10.6333	15.9667	12.9667	7.7000	14.0000	4.7667	3.7667	4.1667
	Rank	1	6	7	12	2	15	11	9	10	16	13	8	14	5	3	4
	MR	1.1000	6.9667	8.6667	11.3667	7.0333	14.5000	8.0667	3.0000	14.8000	15.3000	13.4000	9.5333	11.1333	4.9000	4.0667	2.1667
	Rank	1	6	9	12	7	14	8	3	15	16	13	10	11	5	4	2
Average MR		1.9046	7.0115	7.7839	10.5437	7.7598	14.0943	8.7195	8.0253	10.7920	14.7138	9.7046	6.3080	13.8379	5.6023	4.7230	4.4759
Total Rank		1	6	8	12	7	15	10	9	13	16	11	5	14	4	3	2

and Tables S3–S5 present the Mean, Best, Std, MR, Rank, and Average MR and Total Rank of the 16 algorithms under the three dimensions and on 29 functions of the CEC2017 test set.

Table 7. Wilcoxon rank sum test results of MSAPO and other algorithms under 30-dimensional CEC2017.

F	APO	SMA	AVOA	SO	ARO	NOA	PSA	GWO	WOA	SSA	XPSO	FVICLPSO	SRPSO	LSHADE_ cnEpSin	LSHADE_ SPACMA
F1	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	9.9998E−01 −	1.0000E+00 −
F3	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	9.9997E−01 −
F4	1.8449E−11 +	8.0661E−11 +	2.7470E−11 +	9.7839E−11 +	1.5099E−11 +	7.3215E−11 +	1.2193E−09 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	2.7863E−10 +	1.5099E−11 +	1.8449E−11 +	6.5555E−09 +	5.2033E−05 +
F5	1.4128E−01 =	1.6692E−11 +	1.5099E−11 +	7.7312E−01 =	1.5099E−11 +	1.5099E−11 +	1.6692E−11 +	1.6692E−11 +	1.5099E−11 +	1.5099E−11 +	9.2874E−04 +	1.5099E−11 +	7.4590E−07 +	9.9912E−01 −	9.9996E−01 −
F6	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +	1.4986E−11 +
F7	6.7350E−01 =	5.4683E−11 +	1.5099E−11 +	6.5555E−09 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.8449E−11 +	1.5099E−11 +	1.5099E−11 +	1.7962E−05 +	1.5099E−11 +	2.4005E−07 +	9.0490E−02 =	1.0000E+00 −
F8	1.0000E+00 −	2.2508E−11 +	1.5090E−11 +	9.9308E−01 −	1.5090E−11 +	1.5090E−11 +	1.5090E−11 +	4.8752E−10 +	1.5090E−11 +	1.5090E−11 +	2.1789E−02 +	1.5090E−11 +	1.5797E−01 =	1.0000E+00 −	1.0000E+00 −
F9	2.1536E−08 +	7.1337E−12 +	7.1337E−12 +	1.0808E−10 +	7.1337E−12 +	4.6311E−07 +	7.1337E−12 +	7.1337E−12 +	7.1337E−12 +	7.1337E−12 +	1.6221E−11 +	7.1337E−12 +	7.1337E−12 +	1.9564E−09 +	1.6715E−04 +
F10	1.2551E−02 +	2.6320E−04 +	2.2522E−11 +	2.6325E−05 +	5.1387E−07 +	1.5099E−11 +	1.0076E−08 +	9.3042E−07 +	1.5099E−11 +	4.1760E−08 +	4.9417E−03 +	1.5099E−11 +	7.5071E−03 +	4.0354E−01 =	9.9258E−02 =
F11	8.0311E−07 +	2.0998E−10 +	1.0772E−10 +	8.6470E−08 +	1.5099E−11 +	4.5316E−08 +	1.0772E−10 +	1.5099E−11 +	1.5099E−11 +	2.4876E−11 +	9.2837E−10 +	1.5099E−11 +	7.1471E−09 +	8.4066E−05 +	4.9417E−03 +
F12	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	9.7839E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	6.0116E−09 +	1.5099E−11 +	1.5099E−11 +	9.8966E−01 −	9.9998E−01 −
F13	2.3080E−10 +	1.6692E−11 +	1.5099E−11 +	2.7470E−11 +	1.5099E−11 +	2.4990E−09 +	1.0974E−08 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	6.2704E−08 +	1.5099E−11 +	2.0564E−07 +	2.7850E−03 +	5.1877E−02 =
F14	9.9916E−01 −	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.3610E−10 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.2457E−06 +	6.2385E−05 +
F15	3.7996E−07 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	2.2522E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.8449E−11 +	3.0605E−10 +	1.7486E−09 +
F16	1.5915E−03 +	8.8845E−11 +	1.5099E−11 +	6.5555E−09 +	1.5099E−11 +	6.6443E−11 +	1.6692E−11 +	6.6443E−11 +	1.5099E−11 +	1.5099E−11 +	1.2193E−09 +	1.6692E−11 +	2.4990E−09 +	1.3863E−05 +	2.7664E−08 +
F17	4.8959E−05 +	1.5099E−11 +	1.5099E−11 +	7.3215E−11 +	1.5099E−11 +	1.5099E−11 +	2.4876E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.0329E−11 +	1.5099E−11 +	1.5099E−11 +	2.7664E−08 +	1.6760E−08 +
F18	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	7.0334E−05 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	9.9971E−01 −	9.9957E−01 −
F19	5.1573E−03 +	1.5099E−11 +	1.5099E−11 +	2.4876E−11 +	1.5099E−11 +	2.4876E−11 +	2.0386E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	9.3655E−08 +	2.4990E−09 +
F20	1.5733E−02 +	7.3215E−11 +	1.5099E−11 +	8.6452E−07 +	1.5099E−11 +	7.3666E−08 +	3.3478E−11 +	6.0283E−11 +	1.5099E−11 +	1.5099E−11 +	7.7326E−10 +	1.7486E−09 +	7.7326E−10 +	9.4581E−05 +	2.2102E−06 +
F21	3.2553E−01 =	1.5099E−11 +	1.5099E−11 +	1.5915E−03 +	1.5099E−11 +	2.7863E−10 +	1.8449E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	5.9684E−07 +	1.5405E−08 +	2.4990E−09 +	5.2014E−01 =	7.8446E−01 =
F22	1.4299E−10 +	1.2145E−11 +	1.8184E−11 +	2.9970E−11 +	1.2145E−11 +	2.3037E−10 +	2.7131E−11 +	1.2145E−11 +	1.2145E−11 +	5.4197E−11 +	3.0356E−10 +	1.2145E−11 +	1.2989E−10 +	6.9649E−01 =	1.0000E+00 −
F23	9.9996E−01 −	1.5099E−11 +	1.5099E−11 +	1.3863E−05 +	1.6692E−11 +	4.4967E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.8294E−05 +	1.5099E−11 +	2.4909E−04 +	9.9996E−01 −	9.9972E−01 −
F24	9.9967E−01 −	1.5099E−11 +	1.5099E−11 +	2.0165E−03 +	1.5099E−11 +	1.6692E−11 +	1.5099E−11 +	1.9101E−10 +	1.5099E−11 +	1.5099E−11 +	3.4563E−04 +	1.5099E−11 +	1.5733E−02 +	9.9667E−01 −	9.9938E−01 −
F25	4.8959E−05 +	3.3681E−06 +	5.0177E−04 +	5.1387E−07 +	1.5099E−11 +	8.5000E−02 =	7.9820E−08 +	1.5099E−11 +	1.5099E−11 +	4.5316E−08 +	1.5099E−11 +	1.5099E−11 +	4.7341E−03 +	7.7312E−01 =	6.2040E−01 =
F26	9.9239E−02 =	1.8403E−11 +	2.5418E−08 +	2.4941E−09 +	1.5061E−11 +	8.9999E−01 =	1.6600E−06 +	3.0539E−10 +	1.5061E−11 +	3.9778E−03 +	3.7104E−01 =	8.8214E−03 +	2.3982E−01 =	9.9999E−01 −	9.9986E−01 −
F27	2.9727E−01 =	8.1753E−06 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	4.4967E−11 +	2.0998E−10 +	1.1857E−10 +	1.5099E−11 +	1.0772E−10 +	2.5362E−10 +	1.6692E−11 +	7.2061E−03 +	1.8354E−03 +	5.5546E−02 =
F28	1.5099E−11 +	1.6692E−11 +	2.4876E−11 +	1.5099E−11 +	1.5099E−11 +	1.3049E−10 +	2.7470E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.6937E−02 +	1.5099E−11 +	3.6945E−11 +	4.5316E−08 +	6.5671E−02 =
F29	1.9026E−07 +	1.5099E−11 +	1.5099E−11 +	2.2522E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	4.9593E−11 +	1.5099E−11 +	1.7486E−09 +	2.7310E−06 +	6.4302E−07 +
F30	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	2.3080E−10 +	1.5099E−11 +	1.5099E−11 +	9.7839E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	9.9258E−02 =	7.8446E−01 =
+/=/−	20/5/4	29/0/0	29/0/0	27/1/1	29/0/0	27/2/0	29/0/0	29/0/0	29/0/0	29/0/0	28/1/0	29/0/0	27/2/0	15/6/8	11/7/11

,

Table 8. Wilcoxon rank sum test results of MSAPO and other algorithms under 50-dimensional CEC2017.

F	APO	SMA	AVOA	SO	ARO	NOA	PSA	GWO	WOA	SSA	XPSO	FVICLPSO	SRPSO	LSHADE_ cnEpSin	LSHADE_ SPACMA
F1	1.5099E−11 +	1.5099E−11 +	1.7600E−07 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	7.4658E−05 +	1.5099E−11 +	1.5099E−11 +	4.7604E−04 +	2.9853E−05 +	1.5099E−11 +	1.6760E−08 +	1.2497E−03 +	7.2827E−01 =
F3	1.5099E−11 +	2.4876E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	8.0661E−11 +	2.4990E−09 +
F4	4.9593E−11 +	2.5362E−10 +	2.2863E−09 +	4.2424E−09 +	1.5099E−11 +	2.1553E−08 +	1.5786E−05 +	1.5099E−11 +	1.5099E−11 +	4.8778E−10 +	2.0386E−11 +	1.5099E−11 +	1.2193E−09 +	6.6248E−05 +	9.3341E−02 =
F5	2.9853E−05 +	1.5099E−11 +	1.5099E−11 +	9.9993E−01 −	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.6692E−11 +	1.5099E−11 +	1.5099E−11 +	1.2078E−02 +	1.5099E−11 +	1.2653E−04 +	3.4783E−01 =	1.0000E+00 −
F6	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +
F7	3.0052E−08 +	1.5099E−11 +	1.5099E−11 +	6.0116E−09 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.4563E−04 +	1.5099E−11 +	3.0605E−10 +	3.6322E−01 =	1.0000E+00 −
F8	7.2446E−02 =	1.5099E−11 +	1.5099E−11 +	9.9999E−01 −	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.8090E−01 =	1.5099E−11 +	1.6693E−03 +	9.9816E−01 −	1.0000E+00 −
F9	5.8736E−05 +	1.5099E−11 +	1.5099E−11 +	8.0661E−11 +	1.5099E−11 +	1.8449E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.0329E−11 +	1.5099E−11 +	7.7904E−09 +	2.7863E−10 +	1.1884E−07 +
F10	1.5984E−09 +	7.5890E−04 +	1.0772E−10 +	2.2522E−11 +	2.2220E−07 +	1.5099E−11 +	1.8219E−02 +	2.6325E−05 +	1.6692E−11 +	1.0974E−08 +	1.4603E−02 +	1.5099E−11 +	6.7869E−02 =	1.2164E−05 +	2.2102E−06 +
F11	1.8449E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	4.0764E−11 +	1.5099E−11 +	3.0605E−10 +	7.3215E−11 +	1.0169E−09 +
F12	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.5593E−09 +	1.5099E−11 +	1.8449E−11 +	8.1875E−01 =	9.7897E−01 −
F13	5.8688E−04 +	1.5099E−11 +	1.5099E−11 +	8.8845E−11 +	1.5099E−11 +	2.4876E−11 +	2.0420E−05 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	6.9263E−07 +	1.5099E−11 +	7.2116E−04 +	1.4395E−06 +	1.8852E−04 +
F14	2.5362E−10 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.1886E−03 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	5.0120E−02 =	3.7108E−01 =
F15	9.9336E−01 −	1.8229E−08 +	1.7371E−10 +	2.7071E−01 =	1.1857E−10 +	9.9833E−01 −	4.6427E−01 =	1.5099E−11 +	1.5099E−11 +	3.0329E−11 +	9.8594E−01 −	1.5099E−11 +	1.5367E−01 =	9.9992E−01 −	9.9992E−01 −
F16	6.5083E−04 +	2.0386E−11 +	1.5099E−11 +	5.3328E−08 +	1.5099E−11 +	8.8845E−11 +	6.6443E−11 +	4.8778E−10 +	1.5099E−11 +	7.3215E−11 +	3.3825E−05 +	1.5099E−11 +	1.0753E−02 +	3.0418E−01 =	1.1390E−05 +
F17	1.5984E−09 +	1.5099E−11 +	1.5099E−11 +	2.4990E−09 +	1.5099E−11 +	2.2522E−11 +	1.5099E−11 +	7.3215E−11 +	1.5099E−11 +	1.5099E−11 +	2.4876E−11 +	1.5099E−11 +	1.5794E−10 +	8.6452E−07 +	1.9101E−10 +
F18	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	5.0523E−09 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	9.9986E−01 −	9.9995E−01 −
F19	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	2.3080E−10 +	1.5099E−11 +	1.9101E−10 +	2.9369E−04 +	2.9727E−01 =
F20	2.6320E−04 +	2.7470E−11 +	1.5099E−11 +	9.2837E−10 +	1.6692E−11 +	3.8863E−09 +	2.2522E−11 +	8.0661E−11 +	1.5099E−11 +	1.5099E−11 +	1.3914E−07 +	1.6692E−11 +	5.0177E−04 +	9.4581E−05 +	3.6901E−10 +
F21	5.9684E−07 +	1.5099E−11 +	1.5099E−11 +	6.1809E−04 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.6692E−11 +	5.9684E−07 +	1.5099E−11 +	1.0979E−07 +	1.2164E−05 +	1.5786E−05 +
F22	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.6692E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.6692E−11 +	4.7570E−06 +	1.5099E−11 +	2.4876E−11 +	2.0386E−11 +	2.7863E−10 +
F23	1.1390E−05 +	1.5099E−11 +	1.5099E−11 +	1.1949E−08 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	8.6452E−07 +	1.5099E−11 +	3.2631E−07 +	6.6248E−05 +	4.4414E−06 +
F24	5.5386E−07 +	1.5099E−11 +	1.5099E−11 +	8.8845E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	4.9164E−08 +	1.5099E−11 +	7.7326E−10 +	6.2466E−06 +	1.3392E−06 +
F25	3.2639E−08 +	9.9667E−01 −	2.0165E−03 +	4.3764E−01 =	1.5099E−11 +	2.1156E−01 =	5.3951E−01 =	1.5099E−11 +	1.5099E−11 +	2.5805E−01 =	6.4352E−10 +	1.5099E−11 +	9.9902E−01 −	1.0000E+00 −	9.9997E−01 −
F26	6.6834E−06 +	1.7600E−07 +	6.6443E−11 +	2.4005E−07 +	2.0386E−11 +	4.2424E−09 +	1.5099E−11 +	2.4876E−11 +	1.5099E−11 +	3.9795E−03 +	9.3341E−02 =	1.5099E−11 +	7.2116E−04 +	5.3813E−03 +	6.7869E−02 =
F27	2.3195E−05 +	3.5593E−09 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	2.4876E−11 +	1.3049E−10 +	2.0386E−11 +	1.5099E−11 +	4.4967E−11 +	1.5099E−11 +	1.5099E−11 +	4.0600E−04 +	6.5083E−04 +	5.4933E−01 =
F28	1.5099E−11 +	1.1942E−04 +	1.0169E−09 +	2.2522E−11 +	1.5099E−11 +	4.0507E−10 +	8.4736E−10 +	1.5099E−11 +	1.5099E−11 +	7.4658E−05 +	1.7371E−10 +	1.5099E−11 +	2.7071E−01 =	1.2967E−01 =	1.3345E−01 =
F29	5.5386E−07 +	4.9593E−11 +	1.5099E−11 +	1.8229E−08 +	1.5099E−11 +	1.5099E−11 +	2.0386E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.3049E−10 +	1.5099E−11 +	1.4608E−09 +	2.7310E−06 +	7.9844E−04 +
F30	4.0507E−10 +	1.5099E−11 +	1.5099E−11 +	3.6945E−11 +	1.5099E−11 +	1.5099E−11 +	4.6301E−09 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	6.6834E−06 +	1.0783E−03 +	2.4713E−05 +
+/=/−	27/1/1	28/0/1	29/0/0	25/2/2	29/0/0	27/1/1	27/2/0	29/0/0	29/0/0	28/1/0	26/2/1	29/0/0	25/3/1	19/6/4	15/7/7

and

Table 9. Wilcoxon rank sum test results of MSAPO and other algorithms under 100-dimensional CEC2017.

F	APO	SMA	AVOA	SO	ARO	NOA	PSA	GWO	WOA	SSA	XPSO	FVICLPSO	SRPSO	LSHADE_ cnEpSin	LSHADE_ SPACMA
F1	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	5.3813E−03 +	2.1130E−03 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.6692E−11 +
F3	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.6945E−11 +
F4	1.5099E−11 +	2.2296E−04 +	1.5099E−11 +	3.3478E−11 +	1.5099E−11 +	1.6692E−11 +	1.0169E−09 +	1.5099E−11 +	1.5099E−11 +	1.0772E−10 +	4.4967E−11 +	1.5099E−11 +	1.2861E−07 +	2.7806E−04 +	1.5170E−03 +
F5	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.8486E−04 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	6.1001E−01 =	1.5099E−11 +	8.6470E−08 +	6.0283E−11 +	1.5159E−02 +
F6	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +
F7	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.3478E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	2.5456E−06 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	9.5562E−03 +
F8	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	9.9415E−03 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	8.7663E−01 =	1.5099E−11 +	2.7664E−08 +	2.4876E−11 +	1.5159E−02 +
F9	2.7863E−10 +	1.5099E−11 +	1.5099E−11 +	1.5794E−10 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	6.5555E−09 +	1.5099E−11 +	1.0974E−08 +	3.0329E−11 +	3.8863E−09 +
F10	1.5099E−11 +	4.5344E−03 +	4.1760E−08 +	1.5099E−11 +	3.2591E−09 +	1.5099E−11 +	6.6248E−05 +	2.9853E−05 +	1.5099E−11 +	1.9153E−05 +	9.1840E−03 +	1.5099E−11 +	2.7664E−08 +	1.5099E−11 +	1.5099E−11 +
F11	1.5099E−11 +	2.9837E−09 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.1857E−10 +	1.9101E−10 +
F12	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	9.7839E−11 +	1.5099E−11 +	1.5099E−11 +	8.8845E−11 +	1.5029E−04 +
F13	4.9593E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11V	1.5099E−11 +	5.2033E−05 +	6.4352E−10 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.7371E−10 +	1.5099E−11 +	3.1013E−04 +	1.5099E−11 +	3.0605E−10 +
F14	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	6.1001E−01 =	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.0000E+00 −	9.9999E−01 −
F15	1.5910E−04 +	1.5099E−11 +	1.5099E−11 +	1.6692E−11 +	1.5099E−11 +	5.0523E−09 +	4.4205E−07 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.2193E−09 +	1.5099E−11 +	7.4590E−07 +	1.4358E−10 +	2.8036E−05 +
F16	2.0564E−07 +	1.0974E−08 +	3.6945E−11 +	5.8687E−10 +	1.8449E−11 +	1.8449E−11 +	4.6301E−09 +	2.9837E−09 +	1.5099E−11 +	4.0764E−11 +	1.6840E−04 +	1.5099E−11 +	5.0120E−02 =	7.4827E−02 =	2.6320E−04 +
F17	5.0523E−09 +	1.6692E−11 +	1.5099E−11 +	4.0764E−11 +	1.5099E−11 +	1.8449E−11 +	4.0764E−11 +	3.0605E−10 +	1.5099E−11 +	8.0661E−11 +	3.5593E−09 +	1.5099E−11 +	1.6278E−07 +	1.3392E−06 +	2.0913E−09 +
F18	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.4563E−04 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	2.0386E−11 +	1.5099E−11 +	1.5099E−11 +	1.0000E+00 −	9.9992E−01 −
F19	3.6322E−01 =	3.0329E−11 +	1.5099E−11 +	3.2591E−09 +	1.5099E−11 +	1.4603E−02 +	1.5170E−03 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	4.5344E−03 +	1.5099E−11 +	1.1622E−02 +	1.8219E−02 +	2.1033E−02 +
F20	1.1884E−07 +	4.8778E−10 +	1.5099E−11 +	1.5099E−11 +	9.7839E−11 +	1.5099E−11 +	2.7863E−10 +	4.7570E−06 +	1.5099E−11 +	9.7839E−11 +	3.1013E−04 +	1.5099E−11 +	2.7310E−06 +	1.7962E−05 +	1.2861E−07 +
F21	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	4.4455E−10 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.3008E−08 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	8.0661E−11 +
F22	1.5099E−11 +	2.9293E−06 +	9.7839E−11 +	1.5099E−11 +	4.4455E−10 +	1.5099E−11 +	2.1765E−05 +	2.2220E−07 +	1.5099E−11 +	3.2639E−08 +	2.1088E−04 +	1.5099E−11 +	2.2220E−07 +	1.5099E−11 +	1.1857E−10 +
F23	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	7.0549E−10 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	2.7470E−11 +	1.5099E−11 +	1.5099E−11 +
F24	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	2.7470E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	4.5316E−08 +	3.0329E−11 +	1.5794E−10 +
F25	1.5099E−11 +	4.6056E−05 +	3.6945E−11 +	3.6945E−11 +	1.5099E−11 +	2.0386E−11 +	1.7371E−10 +	1.5099E−11 +	1.5099E−11 +	1.8449E−11 +	1.5099E−11 +	1.5099E−11 +	1.4457E−03 +	1.4603E−02 +	7.0617E−01 =
F26	1.0262E−03 +	4.5316E−08 +	1.5099E−11 +	9.9415E−03 +	1.5099E−11 +	2.0998E−10 +	6.0283E−11 +	5.3509E−10 +	1.5099E−11 +	4.1460E−06 +	3.6322E−01 =	1.5099E−11 +	6.6273E−01 =	7.2061E−03 +	1.5170E−03 +
F27	8.4736E−10 +	2.8000E−07 +	1.5099E−11 +	4.4455E−10 +	1.5099E−11 +	1.5099E−11 +	4.0764E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.4783E−01 =	1.2164E−05 +	2.1033E−02 +
F28	1.5099E−11 +	7.5071E−03 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	2.0386E−11 +	1.5099E−11 +	1.5099E−11 +	1.6692E−11 +	1.5099E−11 +	1.5099E−11 +	2.1088E−04 +	1.3863E−05 +	9.5562E−03 +
F29	2.0913E−09 +	3.6945E−11 +	1.5099E−11 +	5.5289E−05 +	1.5099E−11 +	1.5099E−11 +	3.3478E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	2.0386E−11 +	1.5099E−11 +	2.4005E−07 +	9.3042E−07 +	5.4534E−06 +
F30	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	3.6901E−10 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	1.5099E−11 +	6.7972E−08 +
+/=/−	28/1/0	29/0/0	29/0/0	29/0/0	29/0/0	28/1/0	29/0/0	29/0/0	29/0/0	29/0/0	26/3/0	29/0/0	26/3/0	26/1/2	26/1/2

show the statistical results of the Wilcoxon rank sum test with a significance level of 0.05.

The combined experimental results presented in Table 4, Table 5, Table 6 and Tables S3–S5 clearly illustrate that the MSAPO solution substantially exceeds the original APO for the 30-, 50-, and 100-dimensional unimodal functions F1 and F3 by orders of magnitude. In the 30- and 50-dimensional functions F1, MSAPO is ranked third, yet its Mean and Best metrics indicate that it identifies the theoretical optimum of the objective function. Furthermore, its overall performance is surpassed only by that of the two competition-winning algorithms. With a dimension of 100, MSAPO is the top-performing system in terms of F1. For function F3, MSAPO is the top-performing algorithm in all three dimensions, which indicates that the improvement strategy effectively enhances the exploration and development of the algorithm.

In the assessment of simple multimodal functions F4–F10, MSAPO achieved 3, 4 and 7 first places in 30, 50 and 100 dimensions, respectively. In 30 dimensions, MSAPO’s metrics on two functions exhibit slight inferiority compared to the original APO. However, in 50 and 100 dimensions, its overall performance is superior to the original APO, indicating that cyclone foraging strategy and crisscross strategy effectively enhance algorithm’s capacity to converge accurately and to circumvent local optima. As the dimension increases, MSAPO demonstrates a notable enhancement in its ability to solve problems, thereby indicating its efficacy in addressing complex, high-dimensional issues.

In the context of hybrid functions F11–F20, MSAPO achieved a total of 7, 6, and 6 first-place rankings in 30, 50, and 100 dimensions, respectively. In the 30-dimensional experiment, MSAPO ranked third on function F12, with two competition winning algorithms as the main competitors. In regard to function F14, while MSAPO’s Mean and Std metrics are not as good as those of the original APO, its best metrics demonstrate superior performance, indicating an aptitude for identifying optimal outcomes. However, further enhancements are necessary to ensure stability. In regard to function F18, MSAPO’s performance is surpassed only by the two competition-winning algorithms. In the 50-dimensional experiments, MSAPO’s overall performance on function F12 is second only to LSHADE_SPACMA, but it excels in its Best metric. In terms of functions F14 and F18, MSAPO is ranked third, which is below the two competition-winning algorithms. In the ranking of function F15, APO, NOA, XPSO, LSHADE_cnEpSin and LSHADE_SPACMA demonstrated superior performance. In the 100-dimensional experiments, MSAPO ranked in the top three of all 10 hybrid functions, with individual functions not as good as the statistical metrics of the two competition-winning algorithms.

In the composition functions F21–F30, MSAPO obtained 2, 8 and 9 first places in 30, 50 and 100 dimensions, respectively. In 30 dimensions, APO demonstrates superior performance to MSAPO on five functions, indicating that APO is a more advantageous approach for addressing low-dimensional complex problems. In 50 dimensions, MSAPO demonstrates superior performance to the original APO across all metrics on 10 functions. However, it is ranked third behind the two competing algorithms on function F18 and exhibits suboptimal performance on function F25, where it is ranked eighth. In the 100-dimensional function F26, MSAPO is outperformed by XPSO and SRPSO. However, its Mean, Best, and Std exceed those of the other algorithms, indicating that MSAPO’s overall performance remains superior.

Table 7, Table 8 and Table 9 illustrate the Wilcoxon rank sum test statistics between MSAPO and the other 15 algorithms, evaluated in three dimensions. Table 7 illustrates that out of the 29 test functions of the 30-dimensional CEC2017, MSAPO performed better on 20 functions, comparable on 5 functions, and worse on 4 functions compared to the original APO. MSAPO’s principal competitors are LSHADE_cnEpSin and LSHADE_SPACMA, with MSAPO exhibiting inferior performance relative to LSHADE_cnEpSin on eight functions and to LSHADE_SPACMA on 11. In comparison to the other algorithms that were subjected to evaluation, MSAPO demonstrated superior performance across all 29 functions that were tested.

The 50-dimensional experimental results presented in Table 8 show that MSAPO’s performance is significantly improved compared to the 30-dimensional. Specifically, 27 functions demonstrate a performance that is superior to that of the original APO, while one function exhibits a performance that is comparable to that of the original APO, and one function exhibits a performance that is inferior to that of the original APO. However, MSAPO is significantly inferior to SMA and SRPSO on F25, to SO on F5, F8, and to NOA and XPSO on F15. For LSHADE_cnEpSin and LSHADE_SPACMA, MSAPO is inferior to LSHADE_cnEpSin on 4 functions, and to LSHADE_ SPACMA. Overall, there is still scope for enhancement in MSAPO’s optimised performance on F5, F7, F8, F15, F18 and F25.

The 100-dimensional test results presented in Table 9 demonstrate that MSAPO exhibits enhanced performance, significantly outperforming the original APO on 28 test functions. There is no discernible difference between MSAPO and APO on F19, and MSAPO identifies a superior Best. With the exception of F14 and F18, where the performance is not as optimal as that of LSHADE_cnEpSin and LSHADE_SPACMA, MSAPO demonstrates superior performance to the comparative algorithms across the remaining functions, thereby substantiating its robust capability in addressing high-dimensional optimization problems.

The Friedman average ranking data for the 29 functions in Table 4, Table 5, Table 6 and Tables S3–S5 were combined to produce the final rankings on the 30-, 50-, and 100-dimensions for MSAPO. These were found to be 2.6460, 2.5103, and 1.9046, respectively, which all rank first. In general, MSAPO demonstrates superior performance compared to the other 15 algorithms on all three dimensions of the CEC2017 test set. This illustrates its robust capability for optimization, particularly in complex high-dimensional optimization problems where it exhibits remarkable proficiency in optimization finding.

Furthermore, the convergence and stability of MSAPO are illustrated by the convergence curves and box plots of the 16 algorithms, and the convergence curves of the 16 algorithms on 29 test functions in 30, 50 and 100 dimensions are presented in Figure 10, Figure 11 and Figure 12, respectively. The convergence curves demonstrate that MSAPO exhibits a faster convergence rate than APO during the pre-iteration period. Furthermore, MSAPO identifies superior fitness values and attains higher convergence accuracy with a reduced number of iterations. In particular, the convergence performance of MSAPO is more pronounced in 100 dimensions than in 30 and 50 dimensions. The box plots for the three dimensions are provided in Figure 13, Figure 14, and Figure 15, respectively. The MSAPO exhibits the narrowest and lowest box in the case of the majority of the test functions, which indicates that it demonstrates superior stability and optimization-seeking ability.

4.7. Result Analysis on CEC2017

In this section, we conducted three independent numerical experiments using MSAPO. In the first experiment, we performed a sensitivity analysis on the newly added parameter P in MSAPO. The results indicate that changes in parameter P have little impact on the algorithm’s stability. When P is set to 0.1, the algorithm exhibits a slight advantage in convergence accuracy compared to other parameter values. In the second set of experiments, we sequentially removed each strategy component from MSAPO to validate the rationality of their combination. Results confirmed that every strategy component contributes positively to the algorithm, and the combined strategy set significantly outperforms the original algorithm. We compared the performance of MSAPO with optimized parameter P against dozens of benchmark algorithms. These benchmarks encompassed a comprehensive range of types, with standardized environments and parameter settings to ensure fairness and consistency across all evaluations. Experimental results demonstrate that MSAPO exhibits high convergence speed and accuracy, maintains exceptional stability across diverse test functions, and holds significant potential for tackling complex optimization problems in engineering.

5. Real-World Optimization Problems

This section employs MSAPO and 13 comparative algorithms (presented in Section 4.3) to address a set of 8 real-world constrained optimization problems [56]. The objective is to evaluate the efficacy of MSAPO in addressing engineering design optimization problems. The parameters of the given engineering problems are presented in

Table 10. Parameters of engineering problems.

Problems	Name	D	g	h	Theoretical Optimum
1	Process synthesis problem	7	9	0	2.9248305537
2	Weight minimization of a speed reducer	7	11	0	2994.4244658
3	Tension/compression spring design	3	3	0	0.0126652328
4	Welded beam design	4	5	0	1.6702177263
5	Three-bar truss design problem	2	3	0	263.89584338
6	Step-cone pulley problem	5	8	3	16.069868725
7	Gas transmission compressor design	4	1	0	2964895.4173
8	Himmelblau’s function	5	6	0	−30665.538672

. In order to evaluate the results yielded by each algorithm, four statistical indicators are employed: Best, Mean, Worst, and Std. The eight engineering constraint problems have different mathematical models and constraints, allowing for the effective testing of the algorithms’ ability and applicability in solving these problems. To facilitate a fair and consistent comparison, the population size, the number of independent runs of the algorithms, and the Maximum iteration times of the 14 algorithms are maintained at the same values throughout the process, which are set at 30, 30, and 500, respectively.

5.1. Process Synthesis Problem (PSP)

PSP can be defined as a mixed-integer nonlinear constrained optimization problem [56]. The problem comprises 7 free variables ($x_1~x_7$) and 9 nonlinear constraints ($g_1~g_9$). The mathematical model is as follows:

$$\begin{gathered} \mathbf{Minimize} f(\mathbf{x})=(1-x_1{)}^2+(2-x_2{)}^2+(3-x_3{)}^2+(1-x_4{)}^2+ \\ (1-x_5{)}^2+(1-x_6{)}^2-\mathit{ln}(1+x_7) \end{gathered}$$

subject to:

$$g_1\left(\mathit{x}\right)=x_1+x_2+x_3+x_4+x_5+x_6-5\le 0,$$

$$g_2\left(\mathit{x}\right)=x_1^2+x_2^2+x_3^2+x_6^2-5.5\le 0,$$

$$g_3\left(\mathit{x}\right)=x_1+x_4-1.2\le 0,$$

$$g_4\left(\mathit{x}\right)=x_2+x_5-1.8\le 0,$$

$$g_5\left(\mathit{x}\right)=x_3+x_6-2.5\le 0,$$

$$g_6\left(\mathit{x}\right)=x_1+x_7-1.2\le 0,$$

$$g_7\left(\mathit{x}\right)=x_2^2+x_5^2-1.64\le 0,$$

$$g_8\left(\mathit{x}\right)=x_3^2+x_6^2-4.25\le 0,$$

$$g_9(\mathbf{x})=x_3^2+x_5^2-4.64\le 0,$$

where the range of the variables are $0\le x_1,x_2,x_3\le 100$, $x_4,x_5,x_6,x_7\in \left\{\mathrm{0,1}\right\}$.

The numerical results obtained for the 14 algorithms employed to solve PSP are presented in

Table 11. Statistical results of PSP.

Algorithm	Best	Mean	Worst	Std
MSAPO	2.9248305537	2.9567965365	3.0817321271	0.0576094552
APO	2.9248399811	4.0124755032	10.7468528193	1.5855135136
SMA	6.9401199033	8.2048307039	11.4765727690	1.0678409572
AVOA	2.9248661295	3.2633425379	4.6900000052	0.6078710167
SO	2.9249114461	3.1079971435	4.2067341051	0.3686948968
ARO	2.9478917107	5.2080351840	12.5176062100	2.1988190414
NOA	2.9248305887	2.9382490990	3.0817321272	0.0397377470
PSA	2.9248428546	3.2819797321	4.7675006101	0.6159630085
GWO	2.9253957231	3.9055577651	7.1927710471	1.0964234765
WOA	2.9537565214	7.1137945439	13.3068528194	2.9114996820
SSA	2.9249923546	3.7996127327	4.9004992902	0.5862288981
XPSO	2.9249112624	3.7502193662	4.3239583112	0.5614549006
FVICLPSO	2.9248749385	3.3074364193	9.7542315171	1.2546785460
SRPSO	2.9248469916	3.4994773978	4.7884444002	0.6988961273

and Table S6. The optimal values are indicated by bold text in the tables, and the engineering problems that follow are similarly labelled. Table 11 illustrates that MSAPO’s Best and Worst are the smallest among all algorithms. Notably, MSAPO’s Best reaches the theoretical optimum of 2.9248305537 for PSP, which suggests that MSAPO has the capacity to resolve this specific issue in an exemplary manner. The inferior performance of MSAPO in comparison to NOA on Mean and Std indicates that MSAPO lacks the reliability and robustness necessary to effectively address PSP. The statistical indicators of MSAPO are superior to those of the original APO, which suggests that the incorporation of the four strategies has led to a notable enhancement in the algorithm’s performance. Table S6 presents the optimal design solution for PSP solved by MSAPO, which is (0.1983, 1.2806, 1.9547, 0.7564, −0.4323, 0.0864, 1.2050).

5.2. Weight Minimization of a Speed Reducer (WMSR)

The objective of WMSR is the design of a reducer for a small aircraft engine. In order to minimise the weight of the reducer, WMSR must satisfy 11 constraints, involving 7 variables. Figure 16 shows the schematic structure of the reducer and the specific mathematical model of the problem is as follows:

$$\begin{gathered} \mathbf{Minimize} f\left(\mathbf{x}\right)=0.7854x_1{x}_2^2\left(14.9334x_3+3.3333x_3^2-43.0934\right)+ \\ 0.7854(x_4{x}_6^2+x_5{x}_7^2)-1.508x_1(x_6^2+x_7^2)+7.447(x_6^3+x_7^3) \end{gathered}$$

subject to:

$$g_1\left(\mathbf{x}\right)=-x_1{x}_2^2{x}_3+27\le 0,$$

$$g_2\left(\mathbf{x}\right)=-x_1{x}_2^2{x}_3^2+397.5\le 0,$$

$$g_3\left(\mathbf{x}\right)=-x_2{x}_3{x}_4^{-3}{x}_6^4+1.93\le 0,$$

$$g_4\left(\mathbf{x}\right)=-x_2{x}_3{x}_5^{-3}{x}_7^4+1.93\le 0,$$

$$g_5\left(\mathbf{x}\right)=10x_6^{-3}\sqrt{16.91\times 10^6+(745x_2^{-1}{x}_3^{-1}{x}_4{)}^2}-1100\le 0,$$

$$g_6\left(\mathbf{x}\right)=10x_7^{-3}\sqrt{157.5\times 10^6+(745x_2^{-1}{x}_3^{-1}{x}_5{)}^2}-850\le 0,$$

$$g_7\left(\mathbf{x}\right)=x_2{x}_3-40\le 0,$$

$$g_8\left(\mathbf{x}\right)=-x_1{x}_2^{-1}+5\le 0,$$

$$g_9\left(\mathbf{x}\right)=x_1{x}_2^{-1}-12\le 0,$$

$$g_{10}\left(\mathbf{x}\right)=1.5x_6-x_4+1.9\le 0,$$

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

where the range of the variables are $2.6\le x_1\le 3.6$, $0.7\le x_2\le 0.8$, $17\le x_3\le 28$, $7.3\le x_4,x_5\le 8.3$, $2.9\le x_6\le 3.9$, $5\le x_7\le 5.5$.

The numerical results obtained for the 14 algorithms used to solve WMSR are presented in

Table 12. Statistical results of WMSR.

Algorithm	Best	Mean	Worst	Std
MSAPO	2994.4244658	2994.4244658	2994.4244658	8.40212E−13
APO	2994.4244689	2994.4244903	2994.4245585	1.98659E−05
SMA	2994.4263711	2994.4386356	2994.4941633	0.017359420
AVOA	2995.2005851	3003.6911932	3015.3627013	5.491776785
SO	2994.4244791	2994.8558129	3004.1586289	1.779878646
ARO	2996.0258442	3007.8257333	3062.6094724	1.45742E+01
NOA	2994.4244759	2994.4245298	2994.4246962	4.89070E−05
PSA	2994.4244658	2994.4269891	2994.4779000	0.010037723
GWO	3003.1078646	3010.5175094	3020.6678488	4.524933383
WOA	3009.4481980	3569.7259609	5590.4352755	7.76495E+02
SSA	3005.7441059	3036.9393368	3077.0139370	2.00322E+01
XPSO	3016.3628953	3021.7943767	3032.8365703	4.399323516
FVICLPSO	2994.4244658	2994.4244658	2994.4244658	1.33940E−10
SRPSO	2994.4244893	2994.4245727	2994.4247738	6.86375E−05

and Table S7. Table 12 demonstrates that the optimal value of 2994.4244658 for WMSR can be reached by Best, Mean and Worst of MSAPO. Furthermore, Std outperforms the other comparative algorithms, indicating that MSAPO is an effective and robust method for solving WMSR. It is also concerned that PSA and FVICLPSO can also solve for the theoretical optimum, suggesting that they converge with sufficient accuracy, but neither is as robust as MSAPO. Table S7 illustrates the optimal design solution as determined by MSAPO. The solution is (3.5, 0.7, 17, 7.3, 7.7153, 3.3505, 5.2867).

5.3. Tension/Compression Spring Design (T/CSD)

In the context of T/CSD, the objective is to minimise the weight of the tension/compression spring. The schematic structure is illustrated in Figure 17. The problem mainly consists of 3 continuous decision variables, and 4 constraints $g_1~g_4$ are to be satisfied, which are mathematically modelled as follows:

$$\mathbf{Minimize} f(\mathbf{x})=x_1^2{x}_2(2+x_3)$$

subject to:

$$g_1\left(\mathit{x}\right)=1-{\displaystyle \frac{x_2^3{x}_3}{71758x_1^4}}\le 0,$$

$$g_2\left(\mathit{x}\right)={\displaystyle \frac{1}{5108x_1^2}}+{\displaystyle \frac{4x_2^2-x_1{x}_2}{12566\left(x_1^3{x}_2-x_1^4\right)}}-1\le 0,$$

$$g_3\left(\mathit{x}\right)=1-{\displaystyle \frac{140.45x_1}{x_2^2{x}_3}}\le 0,$$

$$g_4(\mathbf{x})={\displaystyle \frac{x_1+x_2}{1.5}}-1\le 0,$$

where the range of the variables are $0.05\le x_1\le 2.00$, $0.25\le x_2\le 1.30$, $2.00\le x_3\le 15.0$.

The numerical results obtained for the 14 algorithms used to solve T/CSD are shown in

Table 13. Statistical Results of T/CSD.

Algorithm	Best	Mean	Worst	Std
MSAPO	0.0126652328	0.0126704094	0.0126948831	7.32811E−06
APO	0.0126652415	0.0126871934	0.0128011915	3.61245E−05
SMA	0.0126679717	0.0138679784	0.0167808421	0.001419395
AVOA	0.0126706339	0.0130264391	0.0146153522	0.000524042
SO	0.0126656203	0.0139378884	0.0177731593	0.001590096
ARO	0.0126797608	0.0139075607	0.0178426895	0.001733137
NOA	0.0126652731	0.0126676503	0.0126796726	3.20593E−06
PSA	0.0126661331	0.0135899226	0.0177565113	0.001410830
GWO	0.0126870740	0.0127901204	0.0134796661	0.000173615
WOA	0.0126654174	0.0136971554	0.0156213048	0.000951365
SSA	0.0126877585	0.0143562308	0.0253877782	0.002781414
XPSO	0.0127082064	0.0133515199	0.0152318504	0.000653020
FVICLPSO	0.0126705572	0.0130786819	0.0141212303	0.000435546
SRPSO	0.0127084467	0.0131377135	0.0142782804	0.000472955

and Table S8. Table 13 illustrates that Best of MSAPO is the smallest among all the algorithms and attaining the theoretical optimum of 0.0126652328 for T/CSD. However, the performance of MSAPO on Mean, Worst and Std is not as good as that of NOA, which indicates that the reliability and robustness of MSAPO for T/CSD are still insufficient. Table S8 indicates that the optimal design solution for the T/CSD solved by MSAPO is (0.0516875570, 0.3566815558, 11.2910874220).

5.4. Welded Beam Design (WBD)

The objective of WBD [57] is to find the lowest solution for the manufacturing cost by adjusting the decision variables. The configuration of WBD is depicted in Figure 18, which contains 4 design variables and 5 constraints and is mathematically modelled as follows:

$$\mathbf{Minimize} f(\mathit{x})=1.10471x_1^2{x}_2+0.04811x_3{x}_4(x_2+14)$$

subject to:

$$g_1\left(\mathit{x}\right)=x_1-x_4\le 0,$$

$$g_2\left(\mathit{x}\right)=\delta \left(\mathit{x}\right)-{\delta }_{max},$$

$$g_3\left(\mathit{x}\right)=P-P_c\left(\mathit{x}\right)\le 0,$$

$$g_4\left(\mathit{x}\right)=\tau \left(\mathit{x}\right)-{\tau }_{max},$$

$$g_5\left(\mathit{x}\right)=\sigma \left(\mathit{x}\right)-{\sigma }_{max},$$

where:

$$\tau =\sqrt{{\tau }^{\prime 2}+{\tau }^{″2}+2{\tau }^{\prime }{\tau }^{″}{\displaystyle \frac{x_2}{2R}}},{\tau }^{″}={\displaystyle \frac{RM}{J}},{\tau }^{\prime }={\displaystyle \frac{P}{\sqrt{2}{x}_1{x}_2}},$$

$$R=\sqrt{{\displaystyle \frac{x_2^2}{4}}+{\left({\displaystyle \frac{x_1+x_3}{2}}\right)}^2},J=2\left({\displaystyle \frac{x_2^2}{4}}+{\left({\displaystyle \frac{x_1+x_3}{2}}\right)}^2\right)\sqrt{2}{x}_1{x}_2,$$

$$\sigma \left(\mathit{x}\right)={\displaystyle \frac{6PL}{x_3^2{x}_4}},\delta \left(\mathit{x}\right)={\displaystyle \frac{6P{L}^3}{E{x}_3^2{x}_4}},P_c\left(\mathit{x}\right)={\displaystyle \frac{4.013E{x}_3{x}_4^3}{6L^2}}\left(1-{\displaystyle \frac{x_3}{2L}}\sqrt{{\displaystyle \frac{E}{4G}}}\right),$$

$$L=14\mathrm{in},P=6000\mathrm{lb},E=30\times {10}^6\mathrm{psi},{\sigma }_{max}=30000psi,$$

$${\tau }_{max}=13600psi,G=12\times {10}^6\mathrm{psi},{\delta }_{max}=0.25in$$

with bounds: $0.125\le x_1\le 2$, $0.1\le x_2,x_3\le 10$, $0.1\le x_4\le 2$.

The numerical results obtained for the 14 algorithms employed to solve WBD are presented in

Table 14. Statistical results of WBD.

Algorithm	Best	Mean	Worst	Std
MSAPO	1.6702177263	1.6702177263	1.6702177263	1.29848E−13
APO	1.6702183328	1.6778701701	1.7842171850	0.023168762
SMA	1.6702487613	1.6751435731	1.6933727705	0.005433919
AVOA	1.6713845802	1.7633507602	1.8167103299	0.055203758
SO	1.6702280567	1.7231724113	2.1226856454	0.118830458
ARO	1.6975457985	2.5877041799	3.8347423812	0.640174267
NOA	1.6702235402	1.6702423297	1.6702993293	1.73743E−05
PSA	1.6703761548	1.9560517976	3.3272873272	0.411996244
GWO	1.6724873061	1.6754632067	1.6795218097	0.002139746
WOA	1.7399909610	2.6743731903	4.8117067328	0.798276810
SSA	1.6931472325	1.8481224290	2.1500098911	0.117680402
XPSO	1.6702177263	1.6702277075	1.6704642030	4.48547E−05
FVICLPSO	1.7020056012	2.1013504928	2.8391906018	0.306116774
SRPSO	1.6702196010	1.6705078375	1.6743392358	0.000944901

and Table S9. Table 14 shows that the MSAPO Best, Mean and Worst values reach the theoretical optimum of 1.6702177263 for WBD, and that its standard deviation is also significantly better than that of the other algorithms under comparison. Although XPSO’s Best can also reach the theoretical optimum, it is less stable. Table S9 indicates that the best design solution for WBD solved by MSAPO is (0.1988323072, 3.3373652986, 9.1920243225, 0.1988323072).

5.5. Three-Bar Truss Design Problem (TBTD)

In the context of TBTD, the primary objective is to minimise the volume of the three-rod truss, as illustrated schematically in Figure 19. Given that the cross-sectional areas of rods $x_1$ and $x_3$ are identical in the three-rod truss, only rods $x_1$ and $x_2$ are selected as the optimization variables, and the variable $\sigma$ represents the stress that the three-rod truss is subjected to at each truss member. The particular mathematical model is as follows:

$$\mathbf{Minimize} f(\mathbf{x})=(2\sqrt{2}{x}_1+x_2)\times l$$

subject to:

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

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

(30)

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

(31)

where

$$l=100, P=2, \mathrm{and} \sigma =2.$$

with bounds: $0\le x_1,x_2\le 1$.

The numerical results obtained for the 14 algorithms used to solve TBTD are presented in

Table 15. Statistical results of TBTD.

Algorithm	Best	Mean	Worst	Std
MSAPO	263.89584338	263.89584337646	263.89584337646	1.73446E−13
APO	263.89584338	263.89584337658	263.89584337805	3.22794E−10
SMA	265.95665827	270.29189062052	275.61967619448	2.355049747
AVOA	263.89600718	263.92932771908	264.13246725461	0.049148698
SO	263.89584735	263.89696737254	263.90656067239	0.002099651
ARO	263.89584349	264.10592832336	265.44290385159	0.392013217
NOA	263.89584338	263.89584337649	263.89584337672	5.44970E−11
PSA	263.89593168	263.91828913156	264.02871756670	0.036145621
GWO	263.89687204	263.90453317790	263.93006692201	0.007237521
WOA	263.90872632	265.47952244582	276.93666249896	2.597531606
SSA	263.89584427	263.89866520994	263.91408335124	0.004029022
XPSO	263.89584473	263.89604262610	263.89732158037	0.000360897
FVICLPSO	263.89584516	265.19163306555	276.52973206872	2.696934891
SRPSO	263.89584824	263.89631291786	263.89841919628	0.000590811

and Table S10. Table 15 illustrates that the three algorithms, MSAPO, APO and NOA, collectively achieved the theoretical optimum value of 263.89584338 for TBTD. Nevertheless, MSAPO displays superior performance in terms of Mean, Worst and Std metrics in comparison to the other algorithms. Table S10 indicates that the optimal design solution for TBTD solved by MSAPO is (0.7886751377, 0.4082482817).

5.6. Step-Cone Pulley Problem (SCP)

The main objective of SCP [58] is to minimise the weight of a 4th order conical pulley using 5 variables. This problem contains 11 nonlinear constraints. The structure of step-cone pulley is shown schematically in Figure 20. The formula is as follows:

$$\mathbf{Minimize} f(\mathbf{x})=\rho \omega [d_1^2\{1+({\displaystyle \frac{N_1}{N}}{)}^2\}+d_2^2\{1+({\displaystyle \frac{N_2}{N}}{)}^2\}+d_3^2\{1+({\displaystyle \frac{N_3}{N}}{)}^2\}+d_4^2\{1+({\displaystyle \frac{N_4}{N}}{)}^2\}]$$

subject to:

$$h_1\left(\mathit{x}\right)=C_1-C_2=0,$$

$$h_2\left(\mathit{x}\right)=C_1-C_3=0,$$

$$h_3\left(\mathit{x}\right)=C_1-C_4=0,$$

$$g_{\mathrm{1,2},\mathrm{3,4}}\left(\mathit{x}\right)=R_i\ge 2,$$

$$g_{\mathrm{5,6},\mathrm{7,8}}(\mathbf{x})=P_i\ge (0.75\ast 745.6998),$$

where $\rho =7200 \mathrm{kg}/{\mathrm{m}}^3,a=3 \mathrm{m},\mu =0.35,s=1.75 \mathrm{MPa},t=8\mathrm{mm},$ and $C_i$ is the belt length to obtain speed $N_i$, $R_i$ and $P_i$ respectively represents the tension ratio and the power transmitted at each step, the formulas are as follows

$$C_i={\displaystyle \frac{\pi d_i}{2}}\left(1+{\displaystyle \frac{N_i}{N}}\right)+{\displaystyle \frac{(\frac{N_i}{N}-1{)}^2}{4a}}+2a,$$

$$P_i=st\omega [1-\mathit{exp}[-\mu \{\pi -2{\mathit{sin}}^{-1}\{({\displaystyle \frac{N_i}{N}}-1){\displaystyle \frac{d_i}{2a}}\}\}]]{\displaystyle \frac{\pi d_i{N}_i}{60}},$$

$$Ri=\mathit{exp}[\mu \{\pi -2{\mathit{sin}}^{-1}\{({\displaystyle \frac{N_i}{N}}-1){\displaystyle \frac{d_i}{2a}}\}\}],\hspace{0.33em}(i=\mathrm{1,2},\mathrm{3,4}).$$

The numerical results obtained for the 14 algorithms used to solve SCP are given in

Table 16. Statistical results of SCP.

Algorithm	Best	Mean	Worst	Std
MSAPO	16.090274300	16.426945548	17.093777443	0.375730967
APO	16.839771316	17.061646619	17.168089491	0.079676469
SMA	16.331295978	17.617562900	18.372309194	0.597217204
AVOA	16.728701723	17.540694286	18.238688014	0.417091341
SO	16.095500756	17.033309112	18.255985273	0.610190959
ARO	2.71328E+03	9.67806E+06	8.48363E+07	1.83876E+07
NOA	16.574080056	16.949591089	18.272980440	0.340123629
PSA	16.090380768	16.434240825	17.039689943	0.278482514
GWO	1.42761E+06	2.71227E+07	6.07531E+07	1.75319E+07
WOA	20.455741626	3.82896E+10	3.51545E+11	9.98215E+10
SSA	16.245060480	17.338372133	18.243798196	0.448903199
XPSO	16.754399927	7.57199E+06	5.39672E+07	1.48767E+07
FVICLPSO	16.232505496	1.33555E+02	1.63368E+03	3.36975E+02
SRPSO	16.280795257	16.757981083	17.167246649	0.305472989

and Table S11. As illustrated in Table 16, the MSAPO exhibits superior performance in terms of the Best and Mean metrics when compared to the other algorithms. However, according to the Std metric, it displays slightly reduced stability. The optimal design solution solved by MSAPO in Table S11 is (38.41396, 52.85864, 70.47270, 84.49572, 90).

5.7. Gas Transmission Compressor Design (GTCD)

GTCD [59] has 4 decision variables and 1 constraint. The specific mathematical model is as follows:

$$\begin{gathered} \mathbf{Minimize} f(\mathbf{x})=7.72\times 10^8{x}_1^{-1}{x}_2^{0.219}-765.43\times 10^6{x}_1^{-1}+ \\ 8.61\times 10^5{x}_1^{1/2}{x}_2{x}_3^{-2/3}{x}_4^{-1/2}+3.69\times 10^4{x}_3 \end{gathered}$$

subject to:

$$g_1(\mathbf{x})=x_2^{-2}+x_2^{-2}{x}_4-1\le 0,$$

with bounds: $20\le x_1,x_3\le 50$, $1\le x_2\le 10$, $0.1\le x_4\le 60$.

The numerical results obtained for the 14 algorithms employed to address GTCD are presented in

Table 17. Statistical results of GTCD.

Algorithm	Best	Mean	Worst	Std
MSAPO	2964895.4159	2964895.4159	2964895.4159	1.27673E−09
APO	2964895.4159	2964895.4180	2964895.4685	9.59826E−03
SMA	2964895.7984	2970576.9272	2986613.5619	6.07937E+03
AVOA	2965125.9575	2988025.3479	3050307.0127	2.42679E+04
SO	2964895.9219	2965144.1092	2967033.7792	4.48052E+02
ARO	2965405.3971	4846531.7268	9269316.8937	1.86248E+06
NOA	2964895.4159	2964895.4161	2964895.4190	5.88091E−04
PSA	2964909.4536	2967324.6361	2973467.1197	2.52014E+03
GWO	2965013.9619	2965785.7088	2967150.4799	6.17094E+02
WOA	2965320.6740	3026321.8463	3198106.6990	5.59910E+04
SSA	2969856.0018	3153184.2561	3491860.7554	1.65074E+05
XPSO	2964909.5186	2965984.0353	2969341.8160	1.23956E+03
FVICLPSO	2965548.9840	3284138.6178	6057429.3534	6.61504E+05
SRPSO	2964913.0328	2967587.7629	2982479.5468	4.01481E+03

and Table S12. Table 17 illustrates that Best metric of three algorithms, MSAPO, APO and NOA, is the smallest among the compared algorithms. Meanwhile, Mean, Worst and Std metrics of MSAPO demonstrate a notable superiority over those of the other algorithms. Table S12 denotes that the optimal design solution for GTCD solved by MSAPO is (50, 1.178283951, 24.592590288, 0.388353071).

5.8. Himmelblau’s Function (HF)

Himmelblau’s function is a general benchmark for the analysis of nonlinear constrained optimization algorithms [56], comprising 6 nonlinear constraints and 5 variables. The specific modelling of the problem is as follows:

$$\mathbf{Minimize} f(\mathbf{x})=37.293239x_1+5.3578547x_3^2+0.8356891x_1{x}_5-40792.141$$

subject to:

$$g_1\left(\mathbf{x}\right)=-G_1\le 0,$$

$$g_2\left(\mathbf{x}\right)=G_1-92\le 0,$$

$$g_3\left(\mathbf{x}\right)=90-G_2\le 0,$$

$$g_4\left(\mathbf{x}\right)=G_2-110\le 0,$$

$$g_5\left(\mathbf{x}\right)=20-G_3\le 0,$$

$$g_6(\mathbf{x})=G_3-25\le 0,$$

where

$$G_1=0.0006262x_1{x}_4+0.0056858x_2{x}_5-0.0022053x_3{x}_5+85.334407,$$

$$G_2=0.0029955x_1{x}_2+0.0021813x_3^2+0.0071317x_2{x}_5+80.51249,$$

$$G_3=0.00125447x_1{x}_3+0.0019085x_3{x}_4+0.0047026x_3{x}_5+9.300961,$$

with bounds: $78\le x_1\le 102$, $33\le x_2\le 45$, $27\le x_3,x_4,x_5\le 45$.

Table 18. Statistical results of Himmelblau’s function.

Algorithm	Best	Mean	Worst	Std
MSAPO	−30665.538672	−30665.538672	−30665.538672	1.02453E−11
APO	−30665.538333	−30665.510505	−30665.087633	0.082171306
SMA	−30665.538492	−30665.516100	−30665.389277	0.032493939
AVOA	−30665.538466	−30586.627024	−30211.002626	1.50529E+02
SO	−30665.538671	−30665.070138	−30652.087564	2.454381948
ARO	−30574.908116	−29982.671102	−29177.285241	3.42334E+02
NOA	−30665.538294	−30665.533662	−30665.524086	0.003413955
PSA	−30665.538670	−30662.245165	−30581.025254	1.54469E+01
GWO	−30663.897917	−30657.336829	−30643.939903	4.045396571
WOA	−30585.962162	−29725.582260	−28958.010459	4.10427E+02
SSA	−30637.811103	−30495.116447	−30150.498669	1.27269E+02
XPSO	−30644.802248	−30610.381679	−30476.647281	3.21344E+01
FVICLPSO	−30659.626134	−30501.241730	−29955.150983	1.74227E+02
SRPSO	−30665.537638	−30664.535049	−30636.737930	5.250368207

and Table S13 present the results of 14 algorithms to solve Himmelblau’s function. Table 18 demonstrates that MSAPO’s Best, Mean, and Worst values align with the Himmelblau’s function theoretical optimal value of −30665.538672. Additionally, MSAPO’s Std value exhibits a notable superiority over those of other algorithms, indicating that MSAPO possesses exceptional capabilities in solving Himmelblau’s function. From Table S13, the optimal design solution for Himmelblau’s function solved by MSAPO is (78, 33, 29.99525603, 45, 36.77581291).

5.9. Conclusion on Engineering Optimization Problems

In the section addressing practical engineering optimization problems, we conducted eight independent experiments, each corresponding to a distinct engineering optimization problem. Across these experiments, MSAPO demonstrated superior overall performance in most problems, ranking highly across the majority of evaluation metrics. Benefiting from the enhancement provided by the cross-strategy approach, MSAPO exhibits the ability to avoid local optima. Furthermore, its cyclonic search pattern accelerates convergence speed in high-dimensional problems. However, while the algorithm achieved competitive optimality in a few optimization problems, it exhibited weaker stability compared to algorithms like NOA in certain low-dimensional complex constraint problems (e.g., process synthesis, stepped cone pulleys). Consequently, there remains room for improvement in this algorithm.

6. Conclusions and Future Work

This paper proposes an enhanced APO algorithm, designated MSAPO, which incorporates piecewise chaotic opposition-based learning strategy, cyclone foraging strategy, hybrid mutation strategy, and crisscross strategy, alongside the APO. The combination of these four strategies enhances the initial population diversity, augment the local search capability of the algorithm, and diminish the probability of the algorithm attaining a local optimum. The comparison of the results with other excellent optimization algorithms on the CEC2017 and CEC2022 test functions reveals that the proposed MSAPO has significantly enhanced its computational accuracy and convergence speed. Notably, MSAPO demonstrates a remarkable capacity to identify optimal solutions, particularly in high-dimensional problems, which substantiates its competitiveness in optimization problems. The experimental results of the engineering examples demonstrate that MSAPO is capable of identifying superior optimization solutions when confronted with genuine problems. Nevertheless, MSAPO is not without its own set of limitations. The incorporation of the vertical and horizontal crossover strategy towards the conclusion of the algorithmic process has the effect of increasing its overall complexity, thereby rendering it less optimal for achieving rapid convergence when confronted with specific functional equations. In the future, the algorithm’s problem-solving ability will be enhanced by incorporating a variety of strategies to address diverse types of problems. Furthermore, the potential of the proposed MSAPO to optimize intricate optimization problems in diverse fields represents a promising avenue for future research.