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Article

Serval Optimization Algorithm: A New Bio-Inspired Approach for Solving Optimization Problems

by
Mohammad Dehghani
and
Pavel Trojovský
*
Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic
*
Author to whom correspondence should be addressed.
Biomimetics 2022, 7(4), 204; https://doi.org/10.3390/biomimetics7040204
Submission received: 30 October 2022 / Revised: 14 November 2022 / Accepted: 16 November 2022 / Published: 20 November 2022
(This article belongs to the Special Issue Bio-Inspired Design and Optimisation of Engineering Systems)

Abstract

:
This article introduces a new metaheuristic algorithm called the Serval Optimization Algorithm (SOA), which imitates the natural behavior of serval in nature. The fundamental inspiration of SOA is the serval’s hunting strategy, which attacks the selected prey and then hunts the prey in a chasing process. The steps of SOA implementation in two phases of exploration and exploitation are mathematically modeled. The capability of SOA in solving optimization problems is challenged in the optimization of thirty-nine standard benchmark functions from the CEC 2017 test suite and CEC 2019 test suite. The proposed SOA approach is compared with the performance of twelve well-known metaheuristic algorithms to evaluate further. The optimization results show that the proposed SOA approach, due to the appropriate balancing exploration and exploitation, is provided better solutions for most of the mentioned benchmark functions and has superior performance compared to competing algorithms. SOA implementation on the CEC 2011 test suite and four engineering design challenges shows the high efficiency of the proposed approach in handling real-world optimization applications.

1. Introduction

An optimization problem is a type of problem that has several feasible solutions. Optimization is the process of searching for the best solution among possible solutions for a problem [1]. Optimization is used in various science, engineering, technology, and real-world applications [2]. Finding the best optimum can achieve multiple benefits, such as reducing costs, maximizing profits, improving equipment efficiency, etc. For this reason, finding suitable and effective solutions for optimization applications is a fundamental challenge for scientists. From a mathematical point of view, an optimization problem is characterized by three main parts: decision variables, objectives, and constraints [3].
Problem-solving techniques in dealing with optimization tasks are classified into two groups: deterministic and stochastic approaches [4]. Deterministic methods in two categories, gradient-based and non-gradient-based, are effective in handling optimization problems that are linear, convex, differentiable, and have a continuous search space [5]. Among the deterministic methods of solving optimization problems are approaches such as dynamic programming, Newton methods, linear programming, gradient method, quadratic programming, and simplex methods, among the deterministic methods of solving optimization problems [6]. However, as optimization problems become more complex, the number of decision variables increases, and for many real-world applications, deterministic approaches lose their effectiveness. Features such as complexity, non-linearity, non-convexity, non-differentiability, discretization, and problems with high-dimensions objective functions or a discrete search space, etc., are the nature of many modern optimization problems and real-world applications. Such characteristics lead to disruption of the efficiency of deterministic approaches and the problem that they are getting stuck in local optima [7]. Such difficulties in deterministic approaches have led researchers to develop new methods called stochastic approaches to deal with optimization problems. Advantages such as simplicity of concepts, easy implementation, no dependence on the type of problem, no need for derivative information, efficiency in non-linear, non-convex, NP-hard, complex and high-dimensional problems, efficiency in non-linear, and unknown search spaces have made metaheuristic algorithms popular and widespread [8].
The process of solution finding in metaheuristic algorithms starts with the random generation of a number of feasible solutions in the search space. Then, these solutions are improved during iterations of the algorithm based on different steps of updating the metaheuristic algorithms. Finally, after the full implementation of the algorithm, the best feasible solution found during the iterations is presented as a solution to the problem [9]. Metaheuristic algorithms perform the search process in the problem-solving space at both global and local levels. Global search with the concept of exploration ability leads to scanning different regions of the problem-solving space and avoiding getting stuck in local optima. Local search with the concept of exploitation ability leads to finding better solutions in promising areas of search space. In addition to the appropriate quality in exploration and exploitation, the primary key to the success of metaheuristic algorithms in solving optimization problems is to create a balance between exploration and exploitation during algorithm iterations [10].
The nature of random search in metaheuristic algorithms leads to the fact that there is no guarantee that the solutions obtained from these methods are the best solution to the problem. However, these solutions are acceptable as quasi-optimal solutions. A metaheuristic algorithm that can provide better pseudo-optimal solutions closer to the global optimum has a superiority in competition with other metaheuristic algorithms. The desire of scientists to achieve better and more effective solutions for optimization applications has led to the introduction of numerous metaheuristic algorithms [11].
Due to the fact that numerous metaheuristic algorithms have been developed so far, the main research question is: Does the world still need to introduce newer metaheuristic algorithms? No Free Lunch (NFL) [12] theorem answers the question that the effective performance of a metaheuristic algorithm in solving a set of optimization problems does not guarantee the same performance of that algorithm in all optimization applications. According to the NFL theorem, there is no assumption about the efficiency or non-efficiency of an algorithm in handling an optimization problem. Therefore, it can only be claimed that a particular algorithm is the best optimizer for some optimization tasks. Instead, the NFL theorem encourages and motivates researchers to be able to provide more effective solutions for optimization tasks by designing new metaheuristic algorithms. This theorem has also inspired the authors of this article to develop a new metaheuristic algorithm to deal with optimization problems.
According to the concept of the NFL theorem, the randomness of the search process in metaheuristic algorithms, the failure to guarantee the achievement of the global optimal by metaheuristic algorithms, and failure of a metaheuristic algorithm to provide similar performance in all optimization applications, the world always needs to introduce newer metaheuristic algorithms to provide more effective solutions for optimization problems. In this regard, the goal of the paper is to introduce a new metaheuristic algorithm to provide an effective problem-solving tool for researchers to be able to achieve better solutions for optimization tasks. The proposed new metaheuristic algorithm is developed based on simulating the natural behavior of serval during hunting and chasing process. In the proposed method, by simulating the search process in two phases of (i) exploration with the aim of increasing the global search power of the algorithm in order to identify the main optimal area and prevent getting stuck in the local optimal and the (ii) exploitation with the aim of increasing the local search power in order to achieve better solutions, it is expected acquired more effective solutions that are closer to the global optimum in solving optimization problems.
This paper’s novelty and innovative aspects in designing a new optimizer called the Serval Optimization Algorithm (SOA) to deal with optimization tasks in different sciences. The main contributions of this paper are listed as follows:
  • SOA is a nature-inspired approach that simulates natural serval behaviors.
  • The essential inspiration of SOA is the serval strategy when hunting in three stages: selection, attack, and chase.
  • The mathematical model of SOA is presented in two phases: exploration and exploitation.
  • SOA capability is benchmarked in optimizing the CEC 2017 and CEC 2019 test suites.
  • The performance of SOA in handling real-world applications is evaluated on the CEC 2011 test suite and four engineering design challenges.
  • The performance of the proposed SOA approach is challenged in comparison with twelve well-known metaheuristic algorithms.
The article is organized as follows: a literature review is presented in Section 2. The proposed SOA approach is introduced and modeled in Section 3. Simulation studies and results are presented in Section 4. The effectiveness of SOA in handling real-world applications is challenged in Section 5. Conclusions and suggestions for future research are provided in Section 6.

2. Literature Review

Metaheuristic algorithms have been developed based on the simulation of various natural phenomena, natural behaviors of animals, birds, aquatic animals, insects, and other living creatures in the wild, physical laws and phenomena, biological sciences, genetics, human behaviors and interactions, rules of games, and other evolutionary phenomena. Therefore, based on the main idea used in the design, metaheuristic algorithms are classified into five groups: swarm-based, evolutionary-based, physics-based, game-based, and human-based.
Swarm-based algorithms are developed by being inspired by the swarming behavior of living organisms, such as, e.g., animals, birds, insects, and aquatics in nature. The most famous algorithms of this group can be mentioned: Particle Swarm Optimization (PSO) [13], Artificial Bee Colony (ABC) [14], and Ant Colony Optimization (ACO) [15]. PSO is developed based on the simulation of the movement of flocks of birds or fish that are searching for food. ABC is introduced inspired by the activities of a honey bee colony in obtaining food resources. ACO is designed based on modeling the ability of ants to find the optimal route between the nest and the food source. Searching for food resources and hunting strategy for providing food are natural behaviors among animals that are employed in the design of numerous metaheuristic algorithms such as the Coati Optimization Algorithm (COA) [16], Reptile Search Algorithm (RSA) [17], White Shark Optimizer (WSO) [18], Honey Badger Algorithm (HBA) [19], Golden Jackal Optimization (GJO) [20], African Vultures Optimization Algorithm (AVOA) [21], Grey Wolf Optimizer (GWO) [22], Whale Optimization Algorithm (WOA) [23], Marine Predator Algorithm (MPA) [24], and Tunicate Swarm Algorithm (TSA) [25].
Evolutionary-based algorithms are introduced with inspiration from biological and genetics sciences, random operators, concepts of natural selection, and survival of the fittest. Genetic Algorithm (GA) [26] and Differential Evolution (DE) [27] are among the most well-known and widely used metaheuristic algorithms that are designed based on reproduction simulation, Darwin’s theory of evolution, and stochastic operators such as selection, crossover, and mutation.
Physics-based algorithms are designed with inspiration from phenomena, concepts, and laws in physics. The Simulated Annealing (SA) [28] algorithm is one of the most famous physics-based approaches. The modeling of the metal annealing phenomenon in metallurgy has been the main idea in its design. Physical forces are the origin of the creation of algorithms such as the Spring Search Algorithm (SSA) [29] based on spring tensile force, the Gravitational Search Algorithm (GSA) [30] based on gravitational attraction force, and the Momentum Search Algorithm (MSA) [31] based on momentum force. The phenomenon of physical changes in water has been the main idea in Water Cycle Algorithm (WCA) design [32]. Concepts of cosmology have been the origin of Black Hole Algorithm (BHA) design [33]. Some of the most popular physics-based methods are: Equilibrium Optimizer (EO) [34], Electro-Magnetism Optimization (EMO) [35], Multi-Verse Optimizer (MVO) [36], Archimedes Optimization Algorithm (AOA) [37], Thermal Exchange Optimization (TEO) [38], and Lichtenberg Algorithm (LA) [39].
Game-based algorithms are developed with inspiration from various individual and group games, the behavior of players, coaches, referees, and other people influencing the game. Football Game Based Optimization (FGBO) [40] and Volleyball Premier League (VPL) [41] are two game-based approaches that are designed based on the modeling of holding league competitions. The common aspect of many games is the effort of players to earn points, which is the origin of the design of algorithms, including Darts Game Optimizer (DGO) [42], Puzzle Optimization Algorithm (POA) [43], Hide Object Game Optimizer (HOGO) [44], Archery Algorithm (AA) [8], and Tug of War Optimization (TWO) [45].
Human-based algorithms are introduced by taking inspiration from human behaviors, interactions, and thoughts. One of this group’s most widely used algorithms is Teaching-Learning Based Optimization (TLBO) [46], which is introduced based on the modeling of human behaviors between students and teachers in the classroom. Teammates’ efforts to achieve team goals have been the design idea of the Teamwork Optimization Algorithm (TOA) [47]. The therapeutic activities of doctors in treating patients have inspired the design of Doctor and Patient Optimization (DPO) [48]. Some of the other popular human-based methods are: Ali Baba and the Forty Thieves (AFT) [49], Coronavirus Herd Immunity Optimizer (CHIO) [50], War Strategy Optimization (WSO) [51], and Gaining Sharing Knowledge based Algorithm (GSK) [52].
Based on the best knowledge obtained from the literature review, no metaheuristic algorithm has been designed so far based on the simulation of natural behaviors of servals. At the same time, the serval’s strategy during hunting and capturing prey is an intelligent process with the potential to design an optimizer. In order to address this research gap, in this paper, the natural behavior of servals during hunting in nature is employed in the design of a new bio-inspired metaheuristic algorithm, which is introduced and modeled in the next section.

3. Serval Optimization Algorithm

This section is dedicated to the introduction and mathematical modeling of the proposed Serval Optimization Algorithm (SOA) approach.

3.1. Inspiration of SOA

Serval is a skilled predator that hunts its prey in three stages. First, using its strong sense of hearing, it identifies the position of the prey and observes it for up to 15 min without moving. Then, in the second stage, it moves towards the prey, jumps up to a height of 4 meters in the air with all four feet, and attacks this prey with its front paws. Finally, in the third stage, in a chasing process by running and jumping to catch the fleeing prey, the serval kills it and starts eating it [53].
Serval’s strategy during hunting is one of the most characteristic natural behaviors of this animal. This strategy is an intelligent process that can inspire the design of a new metaheuristic algorithm. Modeling the three-stage serval strategy during hunting is employed in SOA design, which is discussed below.

3.2. Algorithm Initialization

The proposed SOA approach is a population-based optimizer that is able to provide suitable solutions for optimization problems by using the search power of its search agents. Servals that look for prey in nature have a similar approach to the mechanism of search agents in identifying the optimal solution. For this reason, from a mathematical point of view, servals form the SOA population that seeks to achieve the optimal solution in the search space. Therefore, each serval is a candidate solution for the problem whose position in the search space determines the values of the decision variables. From a mathematical point of view, each serval is a vector, and their population together forms the SOA population matrix, which can be represented according to Equation (1). The initial position of servals in the search space at the beginning of the implementation of the algorithm is randomly generated using Equation (2).
X = [ X 1 X i X N ] N × d = [ x 1 , 1 x 1 , j x 1 , d x i , 1 x i , j x i , d x N , 1 x N , j x N , d ] N × d ,
x i , j = l b j + r i , j · ( u b j l b j ) ,   i = 1 , 2 ,   ,   N   and   j = 1 , 2 ,   , d ,
where X denotes the population matrix of serval locations, X i is the i th serval (candidate solution), x i , j is its j th dimension in search space (decision variable), N denotes the number of servals, d is the number of decision variables, r i , j are random numbers in the interval [ 0 , 1 ] , l b j , and u b j are the lower and upper bounds of the j th decision variable, respectively.
Since each serval is a candidate solution for the problem, the objective function of the problem can be evaluated based on the proposed values of each serval for the decision variables. Then, according to Equation (3), a vector can represent the values of the problem’s objective function.
F = [ F 1 F i F N ] N × 1 = [ F ( X 1 ) F ( X i ) F ( X N ) ] N × 1 ,
where F denotes the vector of objective function values and F i denote to the obtained objective function value from the i th serval.
Among the calculated values for the objective function, the best value is identified as the best candidate solution, and the member corresponding to it is determined as the best member of the population. Considering that in each SOA iteration, the positions of all population members are updated, of course, the best member should be updated in each iteration.

3.3. Mathematical Modelling of SOA

The process of updating SOA population members in the search space has two phases based on simulating the serval hunting strategy in nature. These phases are intended to model exploration in global search and exploitation in local search in SOA design.

3.3.1. Phase 1: Prey Selection and Attacking (Exploration)

The serval is an efficient predator that uses its strong sense of hearing to identify the location of its prey and then attack it. In the first phase of SOA, the positions of servals are updated based on the simulation of these two strategies. This update causes big changes in the position of servals and leads to a detailed scanning of the search space. The purpose of this phase of SOA is to increase the power of SOA exploration in global search and to identify the main optimal region.
In the SOA design, the position of the population’s best member is considered the prey position. First, the new position for the serval is calculated using Equation (4) to model the serval’s attack on the prey. Then, if this new position improves the value of the objective function, it replaces the previous serval position according to Equation (5).
x i , j P 1 = x i , j + r i , j · ( P j I i , j · x i , j ) ,   i = 1 , 2 ,   ,   N   and   j = 1 , 2 ,   , d ,
X i = { X i P 1 ,   F i P 1 < F i X i ,   e l s e
where X i P 1 denotes the new position of the i th serval based on the first phase of SOA, x i , j P 1 is its j th dimension, F i P 1 is its objective function value, r i , j are random numbers in interval [ 0 , 1 ] , P denotes the prey location, P j is its j th dimension, I i , j are numbers randomly selected from the set { 1 , 2 } , N is the total number of servals population, and d is the number of decision variables.

3.3.2. Phase 2: Chase Process (Exploitation)

After attacking the prey, the serval tries to stop the prey by leaping in a chase process, then kills it and feeds on it. In the second phase of SOA, this serval strategy is employed in updating the population position of SOA. The simulation of the chase process causes small changes in the positions of the servals in the search space. In fact, the purpose of this SOA phase is to increase the exploitation power of SOA in local search and find better solutions. In order to mathematically model the chasing process between the serval and the prey, a new random position near the serval is calculated using Equation (6). This new position, provided that it improves the value of the objective function, replaces the previous position of the corresponding serval according to Equation (7).
x i , j P 2 = x i , j + r i , j · ( u b j l b j ) t ,   i = 1 , 2 ,   ,   N ,   j = 1 , 2 ,   , d ,   and   t = 1 , 2 ,   ,   T ,
X i = { X i P 2 ,   F i P 2 < F i , X i ,   e l s e ,
where X i P 2 represents the new position of the i th serval based on second phase of SOA, x i , j P 2 is its j th dimension, F i P 2 denotes its objective function value, t is the iteration counter of the algorithm, and T represents to the total number of algorithm iterations.

3.4. Repetition Process, Pseudocode, and Flowchart of SOA

By updating all servals based on the first and second phases of SOA, the first iteration of the algorithm is completed. Then, based on the new positions of the servals and the new values obtained for the objective function, the algorithm enters the next iteration. The operation of updating the positions of servals is repeated until the last iteration of the algorithm based on Equations (4)–(7). After the complete implementation of SOA, the best candidate solution obtained during the algorithm’s execution is introduced as the solution to the problem. The SOA implementation process is presented in the form of a flowchart in Figure 1, and its pseudo code is presented in Algorithm 1.
Algorithm 1 Pseudocode of the SOA.
Start SOA.
1. Input problem information: variables, the objective function, and constraints.
2. Set the population size (N) and the total number of iterations (T)
3. Generate the initial population matrix at random.
4. Evaluate the objective function.
5. For t = 1 to N
6.   For i = 1 to N
7.   Phase 1: Prey selection and attacking (exploration)
8.     Update the best member of population as prey location.
9.     Calculate the new position of the ith SOA member based on attack simulation using Equation (4). x i , j P 1 x i , j + r i , j · ( P j I i , j · x i , j )
10.     Update the ith SOA member using Equation (5). X i { X i P 1 ,   F i P 1 < F i , X i ,   else ,
11.   Phase 2: Chase process (exploitation)
12.     Calculate new position of the ith SOA member based on simulation the chase using Equation (6).
x i , j P 2 x i , j + r i , j · ( u b j l b j ) t
13.     Update the ith SOA member using Equation (7). X i { X i P 2 ,   F i P 2 < F i , X i ,   else ,
14.   end
15.   Save the best candidate solution so far.
16. end.
17. Output the best quasi-optimal solution obtained with the SOA.
End SOA.

3.5. Computational Complexity of SOA

This subsection is dedicated to the computational complexity analysis of the proposed SOA approach. SOA initialization operation has a complexity equal to O ( N d ) , where N is the number of servals and d is the number of decision variables. The process of updating the SOA population and calculating the objective function in two phases has a complexity equal to O ( 2 N d T ) , where T is the maximum number of algorithm iterations. Therefore, the total computational complexity of SOA is O ( N d ( 1 + 2 T ) ) .

4. Simulation Studies and Results

This section is dedicated to evaluating the performance of SOA in solving optimization problems and achieving solutions for these problems. For this purpose, thirty-nine standard benchmark functions from the CEC 2017 test suite and CEC 2019 test suite have been employed. The CEC 2017 test suite has 30 benchmark functions, including 3 unimodal functions C17-F1 to C17-F3, 7 multimodal functions C17-F4 to C17-F10, 10 hybrid functions C17-F11 to C17-F20, and 10 composition functions C17-F21 to C17-F30. The C17-F2 function is not considered in the simulations due to its unstable behavior. The complete information of the CEC 2017 test suite is described in [54]. CEC 2019 test suite has 10 hard benchmark functions, the complete information of which is described in [55]. The quality of the SOA approach in optimization has been compared with the performance of twelve well-known metaheuristic algorithms. These algorithms include: (i) widely used and famous methods: GA, PSO, (ii) high cited methods: GSA, TLBO, MVO, GWO, WOA (iii) recently published methods MPA, TSA, RSA, AVOA, and WSO. The adjusted values for the parameters of competitor algorithms are listed in Table 1.
SOA and competitor algorithms are employed to optimize the 39 benchmark functions mentioned above. Simulation results are presented using six indicators: mean, best, worst, standard deviation (std), median, and rank.

4.1. Evaluation the CEC 2017 Test Suite

To analyze the quality of SOA and competitor algorithms in handling optimization problems, they have been implemented on the CEC 2017 test suite for dimensions d equal to 10, 30, 50, and 100. Table 2, Table 3, Table 4 and Table 5 present the results obtained from these implementations.
What can be concluded from the comparison of simulation results is that for the dimension d = 10 , SOA is the best optimizer in handling the functions C17-F3, C17-F6, C17-F7, C17-F10, C17-F19 to C17-F24, C17-F26 to C17-F30. For the dimension d = 30 , SOA is the best optimizer in handling the functions C17-F1, C17-F3 to C17-F5, C17-F7 to C17-F11, C17-F14 to C17-F16, C17-F20, C17-F21, C17-F23, C17-F26, C17-F27, C17-F29, and C17-F30. For the dimension d = 50 , SOA is the best optimizer in handling the functions C17-F1, C17-F3 to C17-F10, C17-F12 to C17-F20, C17-F22, C17-F23, and C17-F25, to C17-F30. For the dimension d = 100 , SOA is the best optimizer in handling the functions C17-F1, C17-F3 to C17-F5, C17-F7 to C17-F13, C17-F15, C17-F16, C17-F18, to C17-F22, C17-F24, to C17-F26, C17-F29, and C17-F30.
The optimization results show that the proposed SOA approach has provided superior performance compared to competing algorithms in the CEC 2017 test suite optimization by creating a suitable balance between exploration and exploitation. The performance of SOA and competitor algorithms in the optimization of the CEC 2017 test suite is drawn as a boxplot diagram in Figure 2, Figure 3, Figure 4 and Figure 5.

4.2. Evaluation the CEC 2019 Test Suite

This subsection tests the effectiveness of the proposed SOA approach and competing algorithms in solving the CEC 2019 test suite. The optimization results of C19-F1 to C19-F10 functions are published in Table 6.
What is evident from the comparison of the simulation results is that the proposed SOA approach is the first best optimizer C19-F1 to C19-F4, and C19-F6 to C19-F9 functions against competitor algorithms. The optimization results show that SOA’s proposed approach better handles the CEC 2019 test suite by winning the first rank compared to competitor algorithms. The performance of SOA and competitor algorithms in the optimization of the CEC 2019 test suite is drawn as a boxplot diagram in Figure 6.

4.3. Statistical Analysis

In this subsection, by providing a statistical analysis of the simulation results, it has been investigated how significant the superiority of the proposed SOA approach is against competitor algorithms from a statistical point of view. For this reason, the Wilcoxon rank sum test [56] is utilized, which is applicable to determine the significant difference between the average of two data samples. The results of applying the Wilcoxon rank sum test on the performance of the proposed SOA proposed approach and competitor algorithms are reported in Table 7. Based on the values obtained for the p-value index, in cases where the p-value is less than 0.05, the proposed SOA approach has a statistically significant superiority compared to the corresponding competitor algorithm.

5. SOA for Real-World Applications

This section is dedicated to analyzing the effectiveness of the proposed SOA approach in handling real-world applications. In this regard, SOA and competitor algorithms are employed to optimize the CEC 2011 test suite and four engineering design problems.

5.1. Evaluation the CEC 2011 Test Suite

This collection contains twenty-two real-world optimization problems (the C11-F3 function was excluded in the simulation studies). The CEC 2011 test suite details are described in [57]. The optimization results of the CEC 2011 test suite using SOA and competitor algorithms are published in Table 8.
The simulation results imply that the proposed SOA approach is the best optimizer for handling functions C11-F1, C11-F2, C11-F4 to C11-F6, C11-F8 to C11-F10, F12, F15, F18, and C11-F20 to C11-F22. A comparison of the simulation results indicates that the proposed SOA approach has an acceptable efficiency in dealing with real-world optimization problems against competitor algorithms. Additionally, the results of employing the Wilcoxon rank sum test on the performance of SOA and competitor algorithms on the CEC 2011 test suite show the statistically significant superiority of SOA in competition with the compared algorithms. The performance of SOA and competitor algorithms in dealing with the CEC 2011 test suite is plotted as a boxplot diagram in Figure 7.

5.2. The SOA Testing on Engineering Optimization Problems

In this subsection, the performance of SOA in solving four engineering design problems from real-world applications is evaluated.

5.2.1. Pressure Vessel Design Problem

The pressure vessel design is a real-world challenge in engineering studies where the goal is to minimize the design cost. The schematic of this design is provided in Figure 8.
The mathematical model of pressure vessel design problem is as follows [58]:
Consider: X = [ x 1 ,   x 2 ,   x 3 ,   x 4 ] = [ T s ,   T h ,   R ,   L ] .
Minimize: f ( x ) = 0.6224 x 1 x 3 x 4 + 1.778 x 2 x 3 2 + 3.1661 x 1 2 x 4 + 19.84 x 1 2 x 3 .
Subject to:
g 1 ( x ) = x 1 + 0.0193 x 3     0 ,   g 2 ( x ) = x 2 + 0.00954 x 3   0 , g 3 ( x ) = π x 3 2 x 4 4 3 π x 3 3 + 1296000   0 ,   g 4 ( x ) = x 4 240     0 .
With
0 x 1 , x 2 100   and   10 x 3 , x 4 200 .
The optimization results of pressure vessel design using SOA and competitor algorithms are released in Table 9 and Table 10.
Based on the simulation results, the proposed SOA approach has provided the optimal solution with the values of the design variables equal to (0.778027, 0.384579, 40.31228, 200), and the value of the objective function equals to 5882.901. The analysis of the results shows that compared to competitor algorithms. Therefore, SOA has provided better performance in dealing with pressure vessel design. The convergence curve of SOA in achieving the solution for the pressure vessel design problem is drawn in Figure 9.

5.2.2. Speed Reducer Design Problem

The speed reducer design is an engineering subject aiming to minimize the speed reducer’s weight. The schematic of this design is provided in Figure 10.
The mathematical model of the speed reducer design problem is as follows [59,60]:
Consider: X = [ x 1 ,   x 2 ,   x 3 ,   x 4 ,   x 5   , x 6   , x 7 ] = [ b ,   m ,   p ,   l 1 ,   l 2 ,   d 1 ,   d 2 ] .
Minimize: f ( x ) = 0.7854 x 1 x 2 2 ( 3.3333 x 3 2 + 14.9334 x 3 43.0934 ) 1.508 x 1 ( x 6 2 + x 7 2 ) + 7.4777 ( x 6 3 + x 7 3 ) + 0.7854 ( x 4 x 6 2 + x 5 x 7 2 ) .
Subject to:
g 1 ( x ) = 27 x 1 x 2 2 x 3 1     0 ,   g 2 ( x ) = 397.5 x 1 x 2 2 x 3 1   0 , g 3 ( x ) = 1.93 x 4 3 x 2 x 3 x 6 4 1   0 ,   g 4 ( x ) = 1.93 x 5 3 x 2 x 3 x 7 4 1     0 , g 5 ( x ) = 1 110 x 6 3 ( 745 x 4 x 2 x 3 ) 2 + 16.9 · 10 6 1   0 , g 6 ( x ) = 1 85 x 7 3 ( 745 x 5 x 2 x 3 ) 2 + 157.5 · 10 6 1     0 , g 7 ( x ) = x 2 x 3 40 1     0 ,   g 8 ( x ) = 5 x 2 x 1 1     0 , g 9 ( x ) = x 1 12 x 2 1     0 ,   g 10 ( x ) = 1.5 x 6 + 1.9 x 4 1     0 , g 11 ( x ) = 1.1 x 7 + 1.9 x 5 1     0 .
With
2.6 x 1 3.6 ,   0.7 x 2 0.8 ,   17 x 3 28 ,   7.3 x 4 8.3 ,   7.8 x 5 8.3 ,   2.9 x 6 3.9 ,   and   5 x 7 5.5 .
The implementation results of the proposed SOA and competitor algorithms on the speed reducer design problem are released in Table 11 and Table 12.
Based on the simulation results, the proposed SOA approach has provided the optimal solution with the values of the design variables equal to (3.5, 0.7, 17, 7.3, 7.8, 3.350215, 5.286683) and the objective function equal to 2996.348. Analysis of the results shows that SOA has provided better performance in handling speed reducer design compared to competitor algorithms. The convergence curve of SOA while solving the speed reducer design problem is drawn in Figure 11.

5.2.3. Welded Beam Design

The welded beam design is a real-world application with the aim of minimizing the fabrication cost of the welded beam. The schematic of welded beam design problem is provided in Figure 12.
The mathematical model of welded beam design problem is as follows [23]:
Consider: X = [ x 1 ,   x 2 ,   x 3 ,   x 4 ] = [ h ,   l ,   t ,   b ] .
Minimize: f ( x ) = 1.10471 x 1 2 x 2 + 0.04811 x 3 x 4   ( 14.0 + x 2 ) .
Subject to:
g 1 ( x ) = τ ( x ) 13600     0 ,   g 2 ( x ) = σ ( x ) 30000     0 , g 3 ( x ) = x 1 x 4   0 ,   g 4 ( x ) = 0.10471 x 1 2 + 0.04811 x 3 x 4   ( 14 + x 2 ) 5.0     0 , g 5 ( x ) = 0.125 x 1   0 ,   g 6 ( x ) = δ   ( x ) 0.25     0 , g 7 ( x ) = 6000 p c   ( x )   0 .
Here
τ ( x ) = ( τ ) 2 + ( 2 τ τ ) x 2 2 R + ( τ ) 2   ,   τ = 6000 2 x 1 x 2 ,   τ = M R J , M = 6000 ( 14 + x 2 2 ) ,   R = x 2 2 4 + ( x 1 + x 3 2 ) 2 , J = 2 { x 1 x 2 2 [ x 2 2 12 + ( x 1 + x 3 2 ) 2 ] }   ,   σ ( x ) = 504000 x 4 x 3 2 ,
δ   ( x ) = 65856000 ( 30 · 10 6 ) x 4 x 3 3   ,   p c   ( x ) = 4.013 ( 30 · 10 6 ) x 3 2 x 4 6 36 196 ( 1 x 3 28 30 · 10 6 4 ( 12 · 10 6 ) ) .
With
0.1 x 1 ,   x 4 2   and   0.1 x 2 ,   x 3 10 .
The results of using the SOA and competing algorithms on the problem of welded beam design are released in Table 13 and Table 14.
Based on the simulation results, the proposed SOA approach has provided the optimal solution with the values of the design variables equal to (0.20573, 3.470489, 9.036624, 0.20573) and the objective function equal to 1.724852. Based on the statistical indicators, it is clear that SOA has provided a more effective capability in handling the welded beam design problem compared to competitor algorithms. The SOA con-vergence curve during welded beam design optimization is drawn in Figure 13.

5.2.4. Tension/Compression Spring Design

The tension/compression spring design is a real-world issue with the goal of minimizing the weight of tension/compression spring. The schematic of this design is provided in Figure 14.
The mathematical model of tension/compression spring design problem is as follows [23]:
Consider: X = [ x 1 ,   x 2 ,   x 3   ] = [ d ,   D ,   P ] .
Minimize: f ( x ) = ( x 3 + 2 ) x 2 x 1 2 .
Subject to:
g 1 ( x ) = 1 x 2 3 x 3 71785 x 1 4     0 ,   g 2 ( x ) = 4 x 2 2 x 1 x 2 12566 ( x 2 x 1 3 ) + 1 5108 x 1 2 1   0 ,
g 3 ( x ) = 1 140.45 x 1 x 2 2 x 3   0 ,   g 4 ( x ) = x 1 + x 2 1.5 1     0 .
With
0.05 x 1 2 ,   0.25 x 2 1.3   and   2   x 3 15 .
The simulation results of the tension/compression spring design problem using the SOA and competitor algorithms are released in Table 15 and Table 16.
Based on the simulation results, the proposed SOA approach has provided the optimal solution with the values of the design variables equal to (0.051689, 0.356718, 11.28897) and the objective function equal to 0.012665. Comparing the obtained results indicates the superiority of SOA in dealing with the tension/compression spring design problem compared to competing algorithms. The SOA convergence curve while achieving the optimal design for the tension/compression spring design problem is drawn in Figure 15.

6. Conclusions and Future Works

This paper introduced a new swarm-based metaheuristic algorithm named the Serval Optimization Algorithm (SOA) based on the simulation of serval behaviors in nature. The serval strategy during hunting in the three stages of prey selection, attack, and the chase is the fundamental inspiration of SOA. Different steps of SOA were stated and mathematically modeled in two phases of exploration and exploitation. The effectiveness of SOA in solving optimization problems was tested on thirty-nine benchmark functions from the CEC 2017 test suite and the CEC 2019 test suite. The SOA’s results were compared with the performance of the other twelve well-known metaheuristic algorithms. The optimization results showed that SOA had performed better by balancing exploration and exploitation and had superior performance compared to competitor algorithms. Employing the proposed approach in optimizing the CEC 2011 test suite and four engineering design challenges demonstrated SOA’s evident ability to address real-world applications.
The introduction of SOA enables several research tasks for future studies. Designing the multi-objective version of SOA and using it in multi-objective optimization problems, developing the binary version of SOA, and using it in applications that require binary algorithms, such as feature selection, are among the most special suggestions for future studies. The use of SOA in various optimization problems in science and real-world applications are among the other recommendations of this article for future research.

Author Contributions

Conceptualization, P.T.; methodology, P.T.; software, M.D.; validation, P.T. and M.D.; formal analysis, M.D.; investigation, P.T.; resources, P.T.; data curation, P.T. and M.D.; writing—original draft preparation, P.T. and M.D.; writing—review and editing, P.T. and M.D.; visualization, P.T.; supervision, P.T.; project administration, M.D.; funding acquisition, P.T. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Project of Excellence Faculty of Science, University of Hradec Králové, grant number 2210/2022.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors thank University of Hradec Králové for support.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Flowchart of the proposed SOA.
Figure 1. Flowchart of the proposed SOA.
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Figure 2. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2017 test suite (for the dimension d = 10 ).
Figure 2. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2017 test suite (for the dimension d = 10 ).
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Figure 3. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2017 test suite (for the dimension d = 30 ).
Figure 3. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2017 test suite (for the dimension d = 30 ).
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Figure 4. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2017 test suite (for the dimension d = 50 ).
Figure 4. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2017 test suite (for the dimension d = 50 ).
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Figure 5. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2017 test suite (for the dimension d = 100 ).
Figure 5. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2017 test suite (for the dimension d = 100 ).
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Figure 6. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2019 test suite.
Figure 6. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2019 test suite.
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Figure 7. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2011 test suite.
Figure 7. Boxplot diagram of SOA and competitor algorithms performances on the CEC 2011 test suite.
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Figure 8. Schematic of the pressure vessel design.
Figure 8. Schematic of the pressure vessel design.
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Figure 9. SOA’s performance convergence curve on the pressure vessel design.
Figure 9. SOA’s performance convergence curve on the pressure vessel design.
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Figure 10. Schematic of speed reducer design.
Figure 10. Schematic of speed reducer design.
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Figure 11. SOA’s performance convergence curve on the speed reducer design.
Figure 11. SOA’s performance convergence curve on the speed reducer design.
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Figure 12. Schematic of the welded beam design.
Figure 12. Schematic of the welded beam design.
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Figure 13. SOA’s performance convergence curve on the welded beam design.
Figure 13. SOA’s performance convergence curve on the welded beam design.
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Figure 14. Schematic of the tension/compression spring design.
Figure 14. Schematic of the tension/compression spring design.
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Figure 15. SOA’s performance convergence curve on the tension/compression spring.
Figure 15. SOA’s performance convergence curve on the tension/compression spring.
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Table 1. Control parameters values.
Table 1. Control parameters values.
AlgorithmParameterValue
GA
TypeReal coded
SelectionRoulette wheel (Proportionate)
CrossoverWhole arithmetic ( Probability = 0.8 ,
α [ 0.5 ,   1.5 ] )
MutationGaussian ( Probability = 0.05 )
PSO
TopologyFully connected
Cognitive and social constant ( C 1 , C 2 ) = ( 2 ,   2 )
Inertia weightLinear reduction from 0.9 to 0.1
Velocity limit10% of dimension range
GSA
Alpha, G 0 , R n o r m , R p o w e r 20, 100, 2, 1
TLBO
T F : teaching factor T F =   round     [ ( 1 + r a n d ) ]
random numberrand is a random number between 0   and   1 .
GWO
Convergence parameter (a)a: Linear reduction from 2 to 0.
MVO
wormhole existence probability (WEP) Min ( WEP ) = 0.2 and Max ( WEP ) = 1 .
Exploitation accuracy over the iterations (p) p = 6 .
WOA
Convergence parameter (a)a: Linear reduction from 2 to 0.
r is a random vector in [ 0 ,   1 ] .
l is a random number from [ 1 ,   1 ] .
TSA
P m i n   and   P m a x   1, 4
c 1 , c 2 , c 3 random numbers lie in the interval [ 0 ,   1 ] .
MPA
Constant number P = 0.5
Random vector R is a vector of uniform random numbers in [ 0 ,   1 ] .
Fish Aggregating Devices (FADs) F A D s = 0.2
Binary vector U = 0 or 1
RSA
Sensitive parameter β = 0.01
Sensitive parameter α = 0.1
Evolutionary Sense (ES)ES: randomly decreasing values between 2 and −2
AVOA
L , L 2 0.8, 0.2
w 2.5
P 1 , P 2 , P 3 0.6, 0.4, 0.6
WSO
F m i n   and   F m a x 0.07, 0.75
τ , a 0 , a 1 , a 2 4.125, 6.25, 100, 0.0005
Table 2. Optimization results of the CEC 2017 test suite (for the dimension d = 10 ).
Table 2. Optimization results of the CEC 2017 test suite (for the dimension d = 10 ).
SOAWSOAVOARSAMPATSAWOAMVOGWOTLBOGSAPSOGA
C17-F1mean100.8273072.0474092.8641.09 × 10101006.28 × 1098734,8505565.47243,130,98592,363,844789.61763796.40816,141,320
best100.5137501.1341116.65819.41 × 10910014,208,1202734,0951329.94336,727.3975,427,798100.0205352.6583,264,472
worst100.98247395.23112,699.461.3 × 10101001.15 × 101021,710,54910,084.161.64 × 1081.31 × 1081902.6465631.97538,820,667
std0.2146763233.0235877.6131.61 × 1095.28 × 10−64.84× 1098,902,2804689.86780,613,75226,120,139779.92872493.30715,706,283
median100.90612195.9121777.6671.06 × 10101006.82× 1095,247,3785423.8934,281,71081,580,648577.90174600.49911,240,071
rank24613112871011359
C17-F3mean300464.5135302.019210,267.9530012,685.253260.516300.06352593.578858.810610,918.38315.580623,430.61
best300300.19273005527.2633008573.571669.5847300.0286329.8934607.09936863.28313.049815,808.63
worst300844.4021304.31913,744.5830016,868.147558.956300.11084362.2431095.09914,847.31319.445928,582.79
std1.89 × 10−13254.80572.3427493757.3154.04 × 10−113775.7472994.4780.0387871853.797199.54073293.0072.791395569.145
median300356.7296301.878910,899.9830012,649.642406.762300.05732841.088866.521910,981.47314.913424,665.52
rank16410212938711513
C17-F4mean403.0484407.2957405.07071414.847400452.2675452.3776403.3356417.5088429.1642404.8591405.1688413.6139
best402.5089406.6334401.3246874.8043400408.4697407.1862400.7995407.7759410.0639403.8009400.3294408.6659
worst403.9619407.9785406.96531943.823400484.0319516.3257405.041446.4927474.1362406.4845411.3318417.5843
std0.640450.6108812.657606456.31312.19 × 10−735.362551.197391.91161619.3226830.325811.2307355.5107454.388246
median402.8615407.2854405.99641420.38400458.2842442.9992403.751407.8834416.2283404.5755404.5069414.1026
rank27513111123910468
C17-F5mean520.0481517.9215547.3819578.3821524.1491555.48565.3388522.6386520.2654532.6729557.956531.8387531.6347
best506.9647514.9671528.8537562.5519510.9445536.7252552.0922512.9368516.9173527.4993552.7326515.9198525.2321
worst540.3655520.8979567.6569594.5714545.3806583.5143587.2286540.7974524.6775540.5986570.6416542.783537.0252
std14.406652.91607920.3851417.7594716.2450519.9516615.1903812.384413.3017515.6108368.57329512.18544.905863
median516.4311517.9106546.5084578.2025520.1356550.8402561.0172518.4101519.7334531.2967554.225534.326532.1407
rank21913510124381176
C17-F6mean600.0008600.8465618.7414644.0479623.8526629.3204637.3912601.3458600.1938606.3029618.6179602.4556607.4921
best600.0001600.0026617.6546640.5716614.9259618.7987627.5706600.5279600.1227604.6819603.1557600.4931605.6483
worst600.0024601.6626621.5024648.6505631.2704638.4976643.6766603.0893600.3623607.7053639.1099603.3747608.445
std0.0010590.9168381.8460133.6305476.8022049.1232537.556011.1792960.1138241.33577916.63341.3561571.252703
median600.0003600.8604617.9043643.4847624.607629.9926639.1589600.8831600.145606.4122616.1031602.9772607.9374
rank13913101112426857
C17-F7mean714.9232735.8553770.0335812.0262723.6362794.4765804.2404738.9214737.1898748.3656736.1961736.6172742.9681
best712.7689717.4734746.6222797.7337720.2506781.8441778.6661731.8761720.4588742.0222717.1787725.272732.2879
worst716.4969759.7501800.0727825.7699726.113805.1441842.8269754.0978752.0498753.5933757.0473753.4337750.6899
std1.56219417.994324.6058313.167482.81054711.9129628.1027510.3394612.944845.91365819.7252412.125038.93704
median715.2135733.0988766.7196812.3006724.0905795.4589797.7343734.8559738.1253748.9235735.2791733.8816744.4474
rank13101321112769458
C17-F8mean816.6276821.1931833.5798858.0199815.9157847.5389838.4987817.1652815.6853831.6274821.3916822.884820.143
best803.9798805.9698821.8891845.8319805.9698837.1798816.4064812.9367814.2627821.404812.9345817.9092811.2669
worst851.586853.9211850.7427863.7385827.5538854.4324852.3891821.8906819.3457842.065829.8487836.8133827.5287
std23.3197522.2132312.19588.25677211.063097.74107915.843293.6666642.454449.15377.1976929.2980617.809856
median805.4723812.4407830.8436861.2546815.0695849.2718842.5997816.9166814.5663831.5204821.3916818.4067820.8881
rank36101321211419785
C17-F9mean900.3851929.31311211.931514.1869001458.0391265.415900.2293965.985941.66249001084.117905.2287
best900.015900.0012958.1691410.468900990.26641008.524900.0018900.5441913.7529900900.9737901.4811
worst901.011971.59271724.0971663.0359002444.5941578.136900.91011155.325980.41599001338.977907.1148
std0.47198234.89239354.9109107.54172.05 × 10−8667.1913251.47610.453875126.269228.211680216.65762.557814
median900.2572922.82931082.7261491.629001198.6491237.501900.0026904.0356936.24059001048.258906.1594
rank46101321211387195
C17-F10mean1136.4711525.51833.32692.021398.2692196.0561831.8131696.6211618.1092333.7782369.9881799.8151931.023
best1015.3171118.7551517.4832508.2391269.371628.7271611.2711431.7211460.3992172.522069.6021548.1651789.372
worst1269.7612173.7552515.0463078.3441458.0362789.9581998.9592038.821803.2312471.1712484.0222102.9432091.463
std104.1524455.0288468.3656265.263287.62248516.1888161.4886255.5049164.4428130.9142200.6485289.059126.6268
median1130.4041404.7451650.3362590.7481432.8342182.7691858.511657.9721604.4022345.7112463.1631774.0751921.627
rank13813210754111269
C17-F11mean1110.2921124.0261151.9534189.3551101.742433.7661222.8521122.6661125.4311164.5991141.9811160.6281626.893
best1106.881117.9541118.2621484.1171100.6011175.7831164.0241105.3051113.1341157.0991121.0421117.921359.157
worst1113.471133.0781209.0266861.3241103.7875818.3161345.8341135.8271132.3541175.0831173.4961208.4511865.215
std2.8158986.4598239.950222416.6971.4177482258.97283.8207112.945478.8691578.6563622.3872842.52562221.4175
median1110.4081122.5361140.2634205.9891101.2861370.4831190.7751124.7671128.1171163.1061136.6921158.071641.6
rank24713112103596811
C17-F12mean1242.4222343.3341,181,9187.58 × 1081233.4411096,3814,964,945370,327756,766.53,736,0631,095,7795930.549913,806.8
best1207.991359.837382,2501.68 × 1081200.82238,588.441,889,17017,106.39235,233.1616,984.9509,5412404.891132,160.6
worst1330.824297.4992,143,4781.32 × 1091320.2892,832,54310,251,735674,113.11,939,1975,876,1921,853,20711,438.522,872,053
std59.085261323.889823,784.15.85 × 10858.102211,238,7413,677,451314,405.5801,178.82,473,613568,776.24251.0261,315,343
median1215.43918581,100,9717.69E × 1081206.327757,195.43,859,438395,044.3426,317.84,225,5371,010,1834939.391325,507
rank23101319125611847
C17-F13mean1314.6511329.42619,590.4136,919,8051306.1987713.77817,906.986218.48511,600.4410,709.3710,719.019477.39414,731.63
best1308.4061314.5372827.8353,065,0901301.3315119.7522690.1521643.8587514.5083941.495323.6062204.3658697.185
worst1318.0791363.91233,636.441.23 × 1081310.41411,977.7334,896.4918,348.4914,579.4217,269.7515,137.1316,209.7422,062.2
std4.42018123.1998315,927.2957,222,1404.140733006.08415,038.528095.6843088.6115648.284147.9545798.8455591.951
median1316.061319.62820,948.6911,023,4241306.5236878.81517,020.652440.79612,153.9110,813.1211,207.659747.73514,083.57
rank23121315114978610
C17-F14mean1414.721423.192068.495643.201404.483501.693400.441493.754803.041568.785878.195869.605357.57
best1412.451406.971699.954926.751401.001.55 × 1031648.111.4 × 10340501522.97484222642517.23
worst1419.771435.922935.307310.201406.975.45 × 1035862.66155151561652.018015.6972838985.62
std3.39064812.24421582.18131119.6122.502.21 × 1032049.5135510.960159.876241.49 × 1032.41 × 1032913.649
median1413.331424.931819.365167.921404.983.50 × 1033.05 × 1031.49 × 1035002.871550.065327.386965.794963.72
rank23611187495131210
C17-F15mean1512.831521.975583.0514,802.851500.967545.175337.481861.613291.731839.9625,559.974109.455985.44
best1511.371511.682115.432826.681500.281.68 × 1032420.261.54 × 10315751690.5911,95616223545.59
worst1514.981528.0813,461.8832,524.721502.002.09 × 1046967.84225577701979.2838,417.8068778873.14
std1.6547317.5369835293.32112967.070.799.04 × 1032099.343.67 × 1022997.628147.9881.26 × 1042.55 × 1032702.172
median1512.501524.073377.4411,930.011500.793.80 × 1035.98 × 1031.82 × 1031911.171844.9925,932.993968.995761.51
rank23912111856413710
C17-F16mean1611.841631.961825.202047.531601.431939.801958.001810.131816.181764.562108.531936.441791.70
best1607.701602.021645.361835.701600.831.80 × 1031845.631.61 × 10316681673.94197318421728.99
worst1615.651719.591950.572341.891601.902.06 × 1032123.07210719901899.692317.9520371848.57
std3.46090358.42219128.574213.78220.441.30 × 102120.10712.10 × 102140.496799.393181.57 × 10287.464.59923
median1612.001603.121852.442006.271601.491.95 × 1031.93 × 1031.76 × 1031803.641742.322071.581933.461794.62
rank23812110116741395
C17-F17mean1716.7031738.5181754.8131827.2741703.132030.6241805.6981769.8541764.0371782.1691857.8061788.9331757.909
best1713.7681710.8081736.9711809.0961701.4351769.611767.8311723.5491722.8621769.3321751.5481742.0541746.479
worst1719.1721753.3391802.1551837.1971704.7732407.0681821.4111790.0761797.9331799.2611993.651864.4241765.935
std2.32984718.9272931.6448412.489881.416707307.112225.4838531.1337138.4987815.20238123.464255.249928.187043
median1716.9371744.9631740.0621831.4021703.1571972.911816.7751782.8961767.6771780.0421843.0131774.6271759.61
rank23411113107681295
C17-F18mean1814.4051821.36212,583.126,109,1721801.27423,279.3512,052.731,257.2815,733.4532,735.8510,284.316716.70211,959.11
best1810.8091808.2275064.639302,305.51800.58214,032.496236.4087496.3332186.15413,630.196724.0694646.3733903.202
worst1819.1231830.20416,593.2117,734,8311801.95436,612.7420,512.1646,205.6135,629.0248,828.4612,582.8811,176.933,403.2
std3.56025410.396595168.7158,077,1120.6382219630.4846341.29716,873.7814,345.7915,253.632500.6313067.47914,314.21
median1813.8431823.50914,337.313,199,7771801.27921,236.0910,731.1235,663.5812,559.3134,242.3810,915.155521.7665265.028
rank23813110711912546
C17-F19mean1900.8291906.0017055.795754,594.91988.0967862.41911,722.221924.0213422.2682919.76443,205.6312,423.588045.107
best1900.1971902.8592196.81948,985.811941.3451936.7612247.9911904.8481931.0792082.11811,768.131992.9722511.267
worst1901.1321911.47714,063.291621,1732033.86914,054.7235,401.611943.4717740.0985097.51162,729.7132,949.6613,550.94
std0.4264174.0518265770.348709,246.537.971196845.28615,850.8215.985252879.0511456.03222,823.3314,370.325195.004
median1900.9941904.8345981.538674,110.51988.5857729.0984619.6391923.8822008.9472249.71449,162.357375.8478059.112
rank12713481036512119
C17-F20mean2011.4852042.5362182.8482239.1012108.712113.8282117.6622049.0312069.4772103.8122271.992148.1042046.167
best2003.982020.6222033.572176.3782029.5652032.5862074.1372023.772040.4892070.522201.3092086.3692040.594
worst2024.812065.2442315.7352298.5242143.2172186.9262165.2462088.0182141.2672190.8852371.8122206.5992054.028
std9.60406923.49206126.938660.1114853.0768971.1649938.7039830.4741848.0539158.2175482.9524549.25036.047881
median2008.5752042.1382191.0432240.752131.0292117.8992115.6322042.1682048.0752076.9222257.4192149.7232045.023
rank12111278945613103
C17-F21mean22002257.3782214.8152272.0212285.2412370.1292286.8412318.8562316.0592333.482380.5932298.4412248.355
best22002201.7762204.4292225.7042256.2722357.3892208.4732312.6282314.5552320.6832361.8412202.6412211.672
worst22002315.5022241.8592298.3562313.6862395.7482352.5722328.82317.3692339.952399.1852341.2262334.061
std4.36 × 10−663.3480818.0856932.1315530.3450417.7569674.574077.0573061.331898.67852115.6065764.8417857.45861
median22002256.1162206.4852282.0122285.5032363.692293.162316.9992316.1572336.6442380.6742324.9482223.844
rank14256127109111383
C17-F22mean2293.4522304.0532309.642961.2282300.4782852.9282310.942447.6592303.4512323.17623002590.6742315.89
best2249.6192302.8652304.6852736.9172300.2892325.932306.0262303.6552301.2292317.80623002301.372312.981
worst2309.5732306.7022311.9683125.8432300.6394289.0312315.3642876.0132308.4292330.14523003455.612322.759
std29.296821.781323.347251163.77410.148633958.35733.841076285.57123.345645.1281534.79 × 10−11576.62584.631892
median2307.3082303.3222310.9532991.0772300.4932398.3752311.1842305.4842302.0742322.37623002302.8582313.91
rank15613312710492118
C17-F23mean2610.5732629.1832645.2312708.1582612.2242726.8412653.0242616.8662616.4382639.3732806.2652640.9912667.811
best2607.6642617.3942632.6742676.8382606.932674.5862628.8322605.7352605.8982633.0392736.3312624.2482659.715
worst2612.0542640.7412664.3842751.8682615.9632799.2072684.8752626.6382621.9722644.2022955.1972656.692674.554
std1.99453910.5542614.913535.06474.48022556.7569423.832499.0400987.2706815.590693102.895214.705997.17143
median2611.2872629.2992641.9332701.9622613.0022716.7852649.1952617.5452618.9412640.1252766.7662641.5132668.487
rank15811212943613710
C17-F24mean25002628.4292777.9712866.4732569.1912809.6872721.2832747.6472747.382702.7152798.0852766.0892796.904
best25002500.0262749.5512831.73625002788.5152539.3682741.3912733.9652519.2282667.1042743.2022777.411
worst25002756.7282810.8392925.4452644.6842842.9162793.1272760.7862763.0432765.552910.0852813.422837.744
std0.000138148.116627.6391940.7419359.2631824.23185121.66618.94633413.17159122.332799.8077631.9576828.33329
median25002628.4812775.7472854.3562566.0412803.6592776.3182744.2062746.2562763.0412807.5752753.8662786.23
rank13913212576411810
C17-F25mean2906.5122922.2772912.0593302.4972897.7443106.262952.7322944.8962936.4072928.5642921.4982929.42955.349
best2899.6222897.9342899.1733227.9362897.7432996.2742950.0712943.6252913.9142903.2312899.5852899.5852949.724
worst2914.4722946.5852949.4683386.7782897.7463238.6622957.7372946.0572946.0692950.8152943.4262950.4812961.557
std7.98965327.156724.9416465.752380.001401109.19143.4366491.28087515.094324.715825.3025322.92634.8434
median2905.9762922.2942899.7973297.6372897.7433095.0532951.5612944.9512942.8232930.1052921.492933.7662955.059
rank25313112109864711
C17-F26mean2671.1952913.4232985.8643820.7882675.0013616.4993543.4852894.3193162.0493024.1963934.0652953.5752984.157
best2617.9322817.32628003472.5392600.0013138.7023155.6922801.2362817.0272962.221280029002918.692
worst2822.6543035.3473176.1484183.22429004076.9513938.7062975.7363958.7693149.0344458.1743043.1373058.781
std100.992890.24848214.6477306.4302149.9997384.0996344.427371.55693534.917884.63646768.331568.4864471.71749
median2622.0962900.5092983.6543813.6952600.0013625.1723539.7722900.1522936.1992992.7644239.0432935.5812979.577
rank14712211103981356
C17-F27mean3089.2493108.0493122.2993241.7883230.6173156.9313131.9953128.9783112.273107.9273236.3093104.6193158.594
best3088.9783097.473095.7433130.0533089.5183099.0043100.6393089.7353094.1813094.7353223.2313096.8873110.429
worst3089.5183119.4263187.8083448.333302.8443201.8413212.0343245.7643138.8243143.4783259.4863115.0153195.857
std0.3107169.90899443.79996141.007297.595351.2428953.5264377.8581320.8284723.7343616.133517.65784736.95242
median3089.253107.6513102.8233194.3853265.0533163.443107.6543090.2063108.0383096.7473231.2593103.2873164.045
rank14613119875312210
C17-F28mean31003206.3713298.813829.4843216.2213436.2143311.8273335.9373320.6773445.1663476.7063368.2713180.566
best31003101.233172.9053741.08331003216.3683185.8553164.4233227.5043224.7473462.4643239.913150.246
worst31003383.8833411.8223893.2953283.3383554.8633436.5613411.8253433.9533731.8133496.4993446.483250.807
std7.13 × 10−5125.184197.9149270.6491181.82053153.671130.6719115.1056103.9361210.510415.7708189.5077947.68498
median31003170.1863305.2563841.7793240.7733486.8123312.4453383.7493310.6263412.0523473.9323393.3483160.605
rank13513410687111292
C17-F29mean3138.6063179.8753296.0493393.5573289.5613381.1543360.1393232.253245.5843197.8413362.0243221.3453231.314
best3131.3593164.6053216.4763316.9593206.4223329.6783226.823135.3753159.2053187.3253241.1823171.93222.514
worst3148.3783205.5273382.8153465.9733346.8923413.5323430.3683320.0023367.4483209.2263673.0673259.4843241.71
std7.70776518.6575485.6326476.7812760.2381239.1509991.6635386.2856290.1438511.73545208.233337.864768.061759
median3137.3433174.6853292.4533395.6483302.4663390.7043391.6833236.8113227.8433197.4073266.9243226.9993230.516
rank12913812106731145
C17-F30mean3396.6487103.166315,325.23,935,4763593.41228,769.31854,82918,378.1725,404.81328,679.2837,788.8434,361.72254,186
best3395.2184030.14111,836.1885,863.33485.9189138.959207,480.46468.61410,065.8844,946.47644,043.45474.912282,133.6
worst3397.32415,735.06821,835.56,216,1633634.301458,333.25,681,03335,678.4134,710.86821,5111069,9501,705,0954,266,383
std0.9723095757.626338,593.42,231,80671.81478185,592.72,567,82512,749.4911,597.43352,907.1176,988.7847,168.31,783,824
median3397.0244323.731163,814.64,319,9383626.71223,802.5765,402.115,682.8428,421.26224,129.6818,58113,438.712,234,113
rank13713261145810912
Sum rank4610621535687303272160182218265206223
Mean rank1.5862073.6551727.41379312.27586310.448289.379315.5172416.2758627.5172419.1379317.1034487.689655
Total rank13713212114581069
Table 3. Optimization results of the CEC 2017 test suite (for the dimension d = 30 ).
Table 3. Optimization results of the CEC 2017 test suite (for the dimension d = 30 ).
SOAWSOAVOARSAMPATSAWOAMVOGWOTLBOGSAPSOGA
C17-F1mean1575.1348.63 × 1093318.3444.38 × 101044,342.912.31 × 10102.19 × 109660,553.82.3 × 1096.17 × 10911,193,4712 × 1091.32 × 108
best291.57913.2 × 109301.98343.91 × 101023,422.061.8 × 10101.14 × 109489,540.51.48 × 1095.57 × 1092775.0875373.92684,091,185
worst2723.0031.55 × 10108149.685.39 × 101073,308.852.69 × 10104.06 × 1091,043,9853.25 × 1097.22 × 10939,077,3366.32 × 1092.15 × 108
std1312.3445.69 × 1093675.9556.83 × 10923,716.313.78 × 1091.36 × 109257,939.37.37 × 1087.68 × 10818,781,6362.98 × 10957,126,045
median1642.9767.91 × 1092410.8574.11 × 101040,320.372.38 × 10101.77 × 109554,344.82.24 × 1095.95 × 1092,846,8868.4 × 1081.15 × 108
rank11121331284910576
C17-F3mean781.875842,375.147,609.7878,405.91834.245448,537.76225,225.51031.67655,393.1436,906.86102,114.977,694.12134,031.2
best470.098531,992.725,840.4260,713.72466.356736,205.28165,937.7751.844542,537.228,114.7687,911.845616.7491,331.46
worst1159.52356,799.3961,573.9385,173.331251.79660,139.23339,833.41377.85569,178.0244,671.67112,455.1122,259.2171,056.6
std363.814911,011.0515,321.6811,832.89426.283211,184.0879,056.19259.085914,094.568186.67211,070.1352,140.1933,150.42
median748.940740,354.1651,512.3983,868.3809.414748,903.27197,565.4998.501954,928.6737,420.51104,046.491,450.26136,868.4
rank15610271338411912
C17-F4mean481.8263841.7389518.366110,440.38495.75282158.726841.9903495.544567.0971827.2028603.7402504.0106813.8432
best469.5735740.4671494.00596680.906482.2571388.877664.6617488.5097532.196630.1235582.046478.9838724.672
worst496.12711012.754537.729214,606.03517.08492948.4871220.665504.9644623.58821298.773628.6736536.8896894.6409
std12.21788127.725918.142313289.6515.86538664.9026255.08237.41856639.35261316.774320.3331724.0752671.3288
median480.8022806.8675520.864610237.29491.83462148.769741.317494.3509556.3021689.9572602.1205500.0845818.03
rank11051331211269748
C17-F5mean591.9605653.5151740.563909.9791604.5444853.4376764.4085616.2858623.7568746.731737.7933685.3203706.2255
best572.8715603.0483700.9927881.9704597.5869831.9647737.6831577.9794610.044696.6675716.8998620.4016695.0019
worst609.7335691.464803.4606946.4631612.4978888.9031787.6346671.5395645.0115779.7375765.6516768.6397714.7736
std17.4339839.3212446.3288830.809196.73485224.7077224.5529839.5036515.9855636.9587321.7221661.705439.752587
median592.6185659.7741728.8994905.7414604.0464846.4413766.1582607.8121619.9859755.2595734.3109676.12707.5632
rank15913212113410867
C17-F6mean606.6229644.2901649.6843688.3251603.2172667.3338677.7711623.4261610.7615641.7919660.113650.2099640.5046
best605.5854639.7766647.5717682.6749601.8096648.0678656.5877605.7405604.005636.0484659.3197643.6113632.6248
worst608.634653.2473652.9881695.4187605.8092682.7658699.2122638.1243616.1276646.9417661.167664.0501649.3386
std1.4355496.0860352.3409595.8858831.85994616.2059317.9127514.596945.1390164.4690850.8143019.3489127.046186
median606.1361642.0682649.0887687.6034602.6249669.2509677.6422624.9199611.4567642.0888659.9828646.5891640.0274
rank27813111124361095
C17-F7mean824.40191028.8911173.8371378.659871.16271261.6341237.611877.7531902.37271050.815987.1371961.4578998.7671
best808.2562937.00011050.1461364.064830.96121217.3491191.509830.7576869.29351023.079936.652940.4088990.4792
worst840.16611166.6371346.5711403.463963.27631286.6531300.094923.0444952.511093.221062.44991.39261006.751
std15.8714297.43161130.988717.5067262.0723330.4028153.1793938.7292340.1885333.7206755.1593824.886316.666849
median824.59271005.9641149.3151373.554845.20671271.2671229.42878.6053893.84361043.48974.7284957.015998.9192
rank18101321211349657
C17-F8mean876.44921.3526959.73971143.549905.34251109.5681022.72918.4966931.92041052.634972.3755928.34561000.158
best861.6931896.2341927.35421121.49882.72971050.063974.1633894.9188904.37271014.92946.2582899.4956989.1129
worst901.5509949.9888983.07331172.767930.59941173.2631072.325929.7411960.35941079.3621000.98953.23091019.463
std18.9608222.0425825.2629925.9778322.1431451.0428542.1531116.4174925.4017127.1542924.1039123.4977413.27682
median871.258919.5937964.26571139.97904.02041107.4731022.196924.6632931.47481058.127971.1318930.3279996.027
rank14713212103611859
C17-F9mean1195.5916940.9025082.06711,265.221229.2914,700.3312,240.593886.392149.4134773.8324288.6684170.3811390.886
best1005.4335371.933739.92210,986.111039.08212,687.958790.4392585.3851625.8754462.9573715.7191685.7361257.047
worst1413.6099175.4045799.19911,406.531446.19217,328.0716,223.676664.1752800.1985128.3255171.3296392.1861505.439
std174.84981601.414921.6272189.314172.36282274.1913233.821919.637510.5671310.8324641.20851950.664125.7143
median1181.6616608.1385394.57411,334.131215.94314,392.6511,974.1431482085.7894752.0234133.8114301.8021400.529
rank11091121312548763
C17-F10mean3816.3185075.7755669.4718287.8824220.8867602.7266780.7415112.8774492.3138516.9825023.9155002.4516061.83
best3461.6824293.9634863.1027401.5613924.0426749.6866256.5474322.2273596.228229.1344753.7174206.055451.612
worst4025.5256408.0486213.9658953.2034725.8568093.5577076.5226028.475426.1758698.7055457.7125740.7626971.01
std268.2071933.6489647.4401649.1055362.4291596.9208382.085785.8896962.8636207.2443323.6773627.4781645.3234
median3889.0334800.5455800.4088398.3824116.8237783.836894.9485050.4064473.4298570.0444942.1165031.4965912.35
rank16812211107313549
C17-F11mean1162.1331448.1161268.6969328.2141171.8135380.7916344.6941341.1461947.5391897.2763021.1191401.0995138.217
best1142.5851297.0051196.7047579.9071132.013668.834242.531265.4511440.9261744.8542320.741164.7481545.325
worst1182.4931551.2421337.2910,508.061215.9976958.5539436.311429.8852921.5482074.6053738.2181770.9839100.655
std18.852121.06858.609461341.60941.782691638.0692277.36882.79307687.1649140.7209668.1668268.4933518.582
median1161.7281472.1081270.3949612.4431169.6225447.895849.9691334.6241713.8411884.8223012.7581334.3324953.445
rank16313211124879510
C17-F12mean90,577.2585,200,12922,294,7661.17 × 101090,003.894.13 × 1091.9 × 10816,023,22930,541,4623.65 × 1082.13 × 108436,028.77,615,346
best53,334.116,278,4813,137,1391.04 × 101053,008.98.42 × 10854,166,3061969,7672402,6152.99 × 10841,143,265236,7735,611,464
worst186,847.71.92 × 10854,450,4581.47 × 1010185,636.89.43 × 1093.02 × 10827,669,71290,151,7804.98 × 1086.8 × 108912,580.89,655,726
std64,578.1677,647,07722,587,3072.04 × 10964,148.093.69 × 1091.08 × 10810,686,73640,183,99490,619,4763.12 × 108320,014.91,658,891
median61,063.671,256,86915,795,7341.08 × 101060,684.913.12 × 1092.03 × 10817,226,71914,805,7263.3 × 10865,228,755297,380.47,597,098
rank28613112957111034
C17-F13mean1698.70242,943.25159,812.11.13 × 10101687.0163.08 × 1091880,155115,014806,958.550,671,82738,902.0219,073,9958,631,128
best1666.69215,453.3988,435.665.92 × 1091655.8993.01 × 108874,962.743,649.190,835.1524,346,77531,538.4310,348.015,437,187
worst1739.00469,114.37252,818.61.38 × 10101727.8794.75 × 1093,971,612152,301.52,153,20079,648,60456,967.671,682,06911,994,455
std30.2576329,451.8768,264.233.62 × 10930.040961.98 × 1091,425,34448,532.06945,863.622,741,50512,177.8135,135,8042,736,987
median1694.55743,602.61148,997.11.27 × 10101682.1443.64 × 1091,337,022132,052.8491,899.749,345,96533,551.032,301,7818,546,436
rank24613112857113109
C17-F14mean1444.4011676.266289,413.42,344,3041496.974605,895.54,052,31531,475.3380,820.24125,817.11220,81140,538.981389,524
best1432.6191509.52840,389.431,178,2231482.99419,646.14500,090.91774.4843031.16256,550.37792,133.35098.10679,752
worst1454.9651968.988670,169.83,490,9541528.72,259,23810,915,28681,318.86304,629.3166,885.11843,312113,6322,018,125
std9.363157205.8048277,873.91,112,53621.288611,102,5184,662,12334,996.97149,234.951,967.8494,689.949,872.43884,978.4
median1445.0091613.274223,547.22,354,0201488.10172,349.172,396,94221,403.997810.246139,916.51,123,89921,712.911,730,110
rank13812291346710511
C17-F15mean1607.9634778.29639,852.036.38 × 1081689.00511,487,2332179,86142,062.955,302.76330,07917,042.722,342.611,660,817
best1597.1362049.81111,559.755.5 × 1081662.636510,798.941,148.4427,984.3629,781.281782,67512,076.249161.007162,991.5
worst1620.237380.57664,784.187.04 × 1081713.51230,936,7226697,05953,948.3395,876.7410,145,97723,149.4443,566.92,552,150
std11.792972216.52822,477.6275,373,69326.4320914,324,3583053,33010,716.8428,402.063805,8544629.29616,392.761,046,225
median1607.2434841.39841532.16.48 × 1081689.9357,250,705990,617.743,159.4547,776.396,695,83216,472.5718,321.271,964,064
rank13613212107811459
C17-F16mean1938.0242652.3213090.8935197.7582043.5153941.6473856.782718.0812844.0533652.1553800.7622579.7062881.999
best1875.0922467.9422618.6614355.1751960.6842696.1543503.4632471.4072350.8982855.6943606.062182.4952536.668
worst2022.8662873.3643646.4965943.3352140.4075949.2764712.0453124.713696.2244147.7423977.1243007.6923235.006
std63.16545180.599422.6948848.535776.106941411.044573.5086297.6959587.7307555.942165.3951348.332306.3028
median1927.0692633.9883049.2075246.2612036.4843560.5793605.8052638.1042664.5453802.5913809.9322564.3192878.16
rank14813212115691037
C17-F17mean1852.9022208.3092527.8653850.2651842.5583293.2112983.6952140.9442036.3122268.7152581.322297.2542189.114
best1788.7222004.3472371.8713442.1051779.7192060.0022607.1152039.7981924.9832092.2692474.2072111.5852010.935
worst1928.1032579.5232652.7074571.3631922.0015754.2393280.4962263.6742223.1792603.3482743.2772433.7512310.59
std71.09131262.7278122.841511.061971.752051677.199286.4468106.3038138.6211232.8375131.3936136.9696131.9916
median1847.3912124.6832543.443693.7961834.2562679.3023023.5842130.1531998.5422189.6222553.8982321.842217.466
rank26913112114371085
C17-F18mean1938.9373,105.142,825,58634,857,8721925.2621277,92312,144,452500,241.11900,9151,703,500549,137.3690,457.84,288,985
best1893.57935,316.23300,783.811,269,5021876.939562,332.55149,402138,664.6227,445.9539,938.8307,806.865,741.961,298,945
worst2006.212126,012.85,637,82168,482,1481993.8911,910,63431,414,794794,063.93,767,3762,823,7941,069,3021,843,3437,761,510
std55.1608643,259.82,501,28324,260,90255.11681631,242.312,853,387306,007.11,464,392954,837.7351,297.8816,443.52,854,320
median1927.96465,545.782,681,87029,839,9191915.111,319,3646,006,807534,1181,804,4191,725,134409,720.2426,373.24,047,743
rank23101317124985611
C17-F19mean1950.4487634.68272,080.611.04 × 1091934.53646,099,12316,697,045566,616.3776,641.66,505,03287,089.66155,5751,002,429
best1944.0484603.74115,265.27.54 × 1081925.9331968,4421190,037237,497.345,758.445,013,83447,176.612541.803178,602.2
worst1956.4759305.339160,7231.58 × 1091940.4461.4 × 10842,887,049962,525.92,874,1468,511,539117,244.7526,6262,377,684
std6.704242126.60363,361.953.68 × 1086.1949364,769,05518,312,639316,644.11,398,9641,468,16429,169.63249,691.1964,832.8
median1950.6358314.82456,167.149.2 × 1081935.88321,268,18511,355,547533,220.993,330.96,247,37891,968.6846,566.05726,714.7
rank23413112117810569
C17-F20mean2189.4422360.682678.2973018.9442202.082907.8192744.2252463.5742502.4582699.6373079.8152833.3662438.403
best2134.1962288.6322510.4612829.5582121.7562570.5042403.7582140.4612283.0512586.2272680.8862259.6382241.364
worst2259.8362498.5682933.8443140.2942261.9223136.6572907.1392802.382617.6042888.943587.3573572.4192528.431
std57.1465494.38145186.4711133.775760.93235261.575235.0817303.3727149.2588136.6338379.9235552.2857132.8408
median2181.8682327.7592634.4423052.9632212.3212962.0582833.0012455.7272554.5882661.693025.5092750.7032491.908
rank13712211956813104
C17-F21mean2334.2672451.5252451.1822708.0622372.3212623.7632600.992418.0652435.5492544.5322590.1712440.4052497.362
best2229.6082427.2412210.5632624.0322364.0182578.4732539.2612385.8972403.3542514.1632570.642423.642490.605
worst2384.4562472.6962619.7052807.8972387.9312707.1212653.4932460.0162517.6732556.9212628.3862466.8222506.684
std70.6076420.94728172.436779.9854510.6428458.0850749.4788331.6280855.0550820.4838625.9250219.614846.981597
median2361.5022453.0822487.232700.162368.6682604.7292605.6032413.1742410.5852553.5222580.832435.5792496.079
rank17613212113491058
C17-F22mean2332.5293959.4336059.498152.0752330.1637394.7037190.8626119.964511.8888205.8416631.6274407.8612804
best2302.5682653.9912403.9927043.9392303.4984181.5383233.914962.5842908.3583396.9884129.7752417.4592740.193
worst2373.4825235.8297475.8339265.8122405.0229991.8979030.2617463.7746446.61110,4547731.1397039.8982875.878
std35.417281461.3042441.084954.940449.932372954.1652683.6691031.3281696.0223239.011678.8392265.56662.5268
median2327.0343973.9577179.0678149.2752306.0677702.6898249.6386026.7414346.2919486.1897332.7964087.0442799.965
rank24712111108613953
C17-F23mean2691.4353077.8432949.2933294.9992826.5293,,206.5053089.6992776.372843.4332903.6343858.6282957.0342950.425
best2679.3782941.8132831.2333236.0122780.9053073.123021.8082743.7482731.3462892.1463741.5672853.2392899.501
worst2704.0983173.1983136.2763380.2922889.293316.1663188.2812813.7012939.2962918.3093974.4993093.2253010.503
std10.10131100.1635133.644462.7372746.7791101.864472.4288236.38104100.440511.04206123.4437101.786545.64687
median2691.1313098.1812914.8313281.8452817.963218.3683074.3542774.0142851.5462902.0413859.2232940.8372945.848
rank19612311102451387
C17-F24mean2896.2263408.4843199.1013463.6982877.1193391.2693316.3462915.6952958.9443090.0753408.4183091.6833257.36
best2888.0013305.5053048.8323366.8212870.9353343.0243234.9672877.5712894.1253059.4623368.1893016.8993199.87
worst2902.5893537.4173367.2713632.0062884.6333440.5123459.1072948.5543049.8843125.5283449.3453136.1573339.026
std6.73111396.59341139.9192122.52115.83924451.9282898.1274934.7635865.3634129.547535.8843754.991759.44844
median2897.1563395.5063190.1513427.9832876.4533390.773285.6562918.3282945.8843087.6553408.073106.8383245.273
rank21271311093451168
C17-F25mean2912.9423083.842910.4934702.9032906.9923544.9873087.4542893.7862978.1113041.5523004.3312912.1463085.075
best2909.213059.7732895.0964048.8892890.2343041.7583014.6442883.7572932.5712966.6972991.8112888.1223052.956
worst2917.0913116.9342952.855574.6172948.6724212.4673186.742913.8583018.1363093.0763017.9142942.5633105.734
std3.75901226.8268228.25826635.439827.9838489.968774.0592513.6199437.2470555.7841810.7790727.846823.07507
median2912.7343079.3252897.0134594.0532894.533462.8613074.2162888.7642980.8683053.2173003.82908.953090.805
rank59313212111687410
C17-F26mean2893.8117030.0267628.77610,262.032900.2747838.248165.0494768.3384910.8155876.5037785.8524194.9664581.948
best2807.9685403.736234.8549352.2322900.0766226.9547340.1144402.9594478.2583765.5726650.0132819.9414133.612
worst2927.2048050.9238445.90511,868.692900.4939673.838975.6545108.7175422.3427143.248377.2556371.4954920.061
std57.334271138.534973.17331181.0290.2078941474.61738.9123347.4211482.63341477.575805.99731521.981346.2337
median2920.0377332.7257917.1739913.6042900.2637726.0888172.2144780.8394871.3296298.68058.0693794.2154637.06
rank18913211125671034
C17-F27mean3215.33427.033367.6473811.953329.2723609.2123470.9913235.7643266.8483319.7385117.2723295.733464.964
best3201.2553331.0433273.5913507.8573290.5713282.2253306.3193224.663223.7553262.3734628.7293245.3823436.138
worst3224.2493639.1243449.7774124.0253421.1343869.4633676.9793250.0993311.7293362.0065473.5083341.9193489.332
std9.925705143.580292.80698264.142461.78087259.6228157.901211.0420638.6746347.4773413.434340.617522.19488
median3217.8473368.9773373.613807.9593302.6913642.5813450.3333234.1483265.9543327.2875183.4273297.813467.192
rank18712611102351349
C17-F28mean3310.3893608.0743279.5135904.2073265.2165091.3623573.3273237.1763491.1633607.883556.3463350.1733615.139
best3210.8943491.043245.9995568.3963206.0324122.4443506.1623218.463382.6823493.4433479.0243251.2033506.504
worst3398.3713834.8643314.3646255.9373373.7496052.2843629.1753255.9123763.1793765.0123718.623531.8043678.771
std76.84487160.547627.94958328.446975.21557889.44863.2023320.24324181.9531126.2162109.5052124.241975.08981
median3316.1463553.1963278.8445896.2473240.5425095.363578.9863237.1653409.3953586.5323513.8713308.8423637.641
rank41031321281697511
C17-F29mean3513.9113928.3154412.4895807.4923672.6684787.2715058.6044072.7093994.364518.1535204.7864232.1074550.496
best3438.9973649.8534031.9445122.1883586.2274310.3994512.6314002.8853773.1574421.9674908.074047.2144380.996
worst3601.8154058.0434644.4656750.7223787.3995070.8825647.8674125.8364324.0944661.4455483.8864365.6154714.488
std75.06742187.7454271.1545799.938787.08749350.1526515.240354.79906263.3728108.3907309.7752137.1358146.445
median3507.4154002.6834486.7745678.5293658.5234883.9015036.964081.0583940.0944494.5995213.5944257.7984553.251
rank13713210115481269
C17-F30mean8305.989137,769.61,529,4253.03 × 1099157.7466.15 × 10830,762,5563,049,9869,525,37825,150,2892,426,811104,576.7987,308.6
best6809.30759,525.81539,075.82.18 × 1097017.1667,706,7945,608,6861,699,1581,885,66119,520,5502,118,3898897.047568,046.1
worst9524.921183,375.52,708,6403.35 × 10910,822.161.23 × 10954,147,4885,439,81914,684,33535,666,5722,919,985321,178.61,356,063
std1134.90756,915.13907,077.75.7 × 1081704.626.99 × 10819,847,9121,654,3666,013,9877,204,436345,090145,516.1362,444.2
median8444.864154,088.41,434,9923.3 × 1099395.8266.1 × 10831,647,0262,530,48410,765,75922,707,0172,334,43544,115.671,012,563
rank14613212118910735
Sum rank4418319236657324307122167248245165219
Mean rank1.5172416.3103456.6206912.620691.96551711.1724110.586214.2068975.7586218.5517248.4482765.6896557.551724
Total rank16713212113510948
Table 4. Optimization results of the CEC 2017 test suite (for the dimension d = 50 ).
Table 4. Optimization results of the CEC 2017 test suite (for the dimension d = 50 ).
SOAWSOAVOARSAMPATSAWOAMVOGWOTLBOGSAPSOGA
C17-F1mean151,889.62.57 × 10109,830,1409.96 × 10105,506,8265.41 × 10107.14 × 1094,797,7539.99 × 1092.21 × 10101.82 × 10105.74 × 1098.1 × 109
best115,387.61.88 × 10101,170,3138.71 × 10103,290,1493.98 × 10105.01 × 1093,351,1155.9 × 1091.65 × 10101.45 × 10101.44 × 1096.57 × 109
worst247,412.73.39 × 101026,012,9401.09 × 10118,579,3386.44 × 10101.13 × 10105,622,4331.44 × 10103.09 × 10102.18 × 10101.05 × 10109.01 × 109
std63,842.316.64 × 10911,038,0469.48 × 1092,249,1331.16 × 10102.83 × 1091,007,7883.47 × 1096.74 × 1092.97 × 1094.87 × 1091.14 × 109
median122,379.12.51 × 10106,068,6531.01 × 10115,078,9095.6 × 10106.13 × 1095,108,7319.83 × 1092.05 × 10101.83 × 10105.54 × 1098.41 × 109
rank11141331262810957
C17-F3mean25,741.16109,166.7155,708.8167,728.727,077.85109,111239,463.355,133.09121,422.498,432.16189,072.7227,319.8289,533.1
best17,748.7587,516.26119,637152,152.918,689.8693,934.99149,929.135,708.23109,107.782,388.28170,742.6169,886.5206,512.9
worst31,049.77131,269.6189,453.3182,838.532,278.65143,029.3381,283.788,723.05137,403.710,7613.2213,623.9306,999.1361,157.1
std5842.38417,915.6231,532.2113,624.196102.04522,781.4599,541.924,999.812,061.2411,298.1320,728.7557,433.9367,162.39
median27,083.06108,940.5156,872.5167,961.628,671.4599,739.83213,320.248,050.53119,589.1101,863.6185,962.1216,196.8295,231.1
rank16892512374101113
C17-F4mean556.82094628.003711.759425,211.16603.19699115.471767.649592.4871175.3283505.1543225.817978.37871594.562
best537.43123095.048692.789116,625.85544.76057469.5911392.045533.3314951.14552035.1562692.306831.76011584.14
worst600.41236512.632744.129230,113.72642.974211,041.532186.232633.61511702.4925193.1953440.0771063.6831605.234
std29.388581751.26222.722176158.62943.340861468.831333.779248.89051353.30241632.081357.6048103.435510.48297
median544.724452.167705.059727,052.53612.52648975.381746.16601.50071023.8383396.1333385.4441009.0361594.437
rank11141331282610957
C17-F5mean677.7792825.0179879.25591166.95721.12151151.6161032.234716.9972732.73641062.166824.3543843.4191930.9574
best656.4368798.6558847.03491146.761700.28731006.991966.5428655.0431654.36511050.273768.6373757.5923874.5555
worst704.4044854.3394923.03721180.923741.8841243.5441058.35768.2161842.63291085.351863.1571935.7099986.5072
std23.35923.6723232.82515.6725916.98195101.259843.9096746.6917480.4385315.8456244.5027773.2548746.17203
median675.1377823.5383873.47571170.059721.15731177.9651052.021722.3649716.97371056.52832.8113840.1871931.3835
rank16813312102411579
C17-F6mean612.5444650.0934663.6411702.8348636.3891697.6476697.9072641.0802624.8983664.2849661.2626651.6176648.1166
best608.3547647.0002658.5488700.4264610.8922689.7925695.4772627.1604621.0722655.5853656.0718644.4713637.0315
worst618.0937652.6056669.415705.9343658.9156705.5216701.5624662.1438634.2509670.3405664.3465656.3579653.6417
std4.0655042.8094845.0200082.57830121.60586.4476682.76406115.498786.2841746.5367343.641125.4194927.510199
median611.8646650.3839663.3002702.4892637.8744697.6381697.2946637.5083622.1351665.6069662.3161652.8207650.8966
rank16913311124210875
C17-F7mean1020.0531634.9361719.6181959.3221061.0831830.7241790.7891104.0831082.3671452.4831449.51226.3911392.31
best984.97521274.2751644.2571873.4131012.3891488.4221657.4071042.23977.22251415.9131271.1181090.9991331.588
worst1069.8751936.7491789.9392069.0471096.422051.5251928.9881216.4581156.2521470.2531586.3731346.7371447.051
std36.24558272.864561.9324683.8888635.62031252.3578135.16477.1368875.4149224.69756142.0339110.346548.90072
median1012.681664.361722.1381947.4141067.7621891.4751788.381078.8221097.9961461.8841470.2551233.9141395.3
rank19101321211438756
C17-F8mean992.89441119.9351143.2491485.6841054.11501.0571300.0051093.3261037.121343.4921160.1731090.871239.259
best958.44091089.391094.351452.256984.20981375.1481199.3511031.358993.61261294.5751151.2181064.3591164.301
worst1018.1611182.2891193.2031509.2751144.5241561.0621386.41211.0781054.8291379.6241175.0971135.1921276.922
std25.3841343.5437956.2037623.9655667.852987.4129878.0344881.1476529.1216238.0902210.6230731.9293553.04527
median997.48791104.0311142.7211490.6031043.8321534.0091307.1331065.4331050.021349.8851157.1881081.9651257.907
rank16712313105211849
C17-F9mean3379.96831,574.5413,994.5838,384.663535.57243,997.338,999.5512,924.5310,955.8224,490.3311,230.4613,220.9911,936.19
best2720.79827,546.8813,334.9636,059.862795.90934,228.7927,771.854379.0227695.71317,011.1510,226.788972.3439776.976
worst4482.20535,368.0114,911.5140,280.94766.80651,238.857,115.1920,429.4913,132.4927,535.9612,132.6421,544.0514,035.6
std766.33664276.902681.87041998.844858.0767407.68912,869.398072.6392466.0825024.465792.22215669.6122264.86
median3158.43531,691.6413,865.9338,598.943289.78645,260.835,555.5813,444.8111,497.5426,707.1111,281.2211,183.7911,966.1
rank11081121312639475
C17-F10mean6174.0727403.7048579.60114,818.966667.02113,370.0312,530.297041.6127036.96814,639.168868.7418635.02511,353.66
best5757.8956703.8288018.60214,572.315919.30812,643.4411,659.826312.6516849.6513,468.427973.3887336.09710,137.65
worst6627.517941.4789205.32415,177.467192.28913,961.8713,907.527530.8547297.50215,497.989980.3479821.61111,876.02
std461.6323515.452503.2166286.0404585.7901662.72661020.996557.1729220.1985848.7871837.3571033.431825.7338
median6155.4427484.7568547.23914,763.036778.24513,437.4112,276.927161.4727000.3614,795.118760.6148691.19611,700.49
rank15613211104312879
C17-F11mean1334.964491.2991635.40222,214.141309.93310,609.273565.6281464.3138372.9545000.77115,005.371892.93115,726.03
best1281.1713159.7291507.53819,751.851305.5626841.7223031.1321362.164716.4724243.98314,069.521462.89511,246.91
worst1372.226882.5621798.58124,081.651315.87715,049.353780.7661592.31712,480.165708.21917,016.792863.25519,576.5
std38.835751664.8134.24521809.2334.4003854211.171358.468112.54753546.225655.68991355.471655.24593430.853
median1343.2243961.4531617.74522,511.531309.14510,2733725.3071451.3878147.5935025.44114,467.581622.78716,040.36
rank27413110639811512
C17-F12mean11,251,7315.27 × 10978,049,6217.57 × 101012,693,0271.72 × 10101.61 × 10998,207,7471.57 × 1093.95 × 1092.31 × 1091.16 × 1092.07 × 108
best10,038,7991.65 × 10933,060,1305.52 × 10109,956,2127.73 × 1098.92 × 10863,359,9963.8 × 1082.82 × 1097.6 × 1082.74 × 10896,807,585
worst12,129,5449.38 × 1091.21 × 1081.04 × 101118,145,1982.85 × 10102.96 × 1091.62 × 1082.93 × 1095.13 × 1094.15 × 1091.74 × 1093.39 × 108
std1,025,4913.48 × 10946,894,5922.24 × 10103,729,8908.96 × 1099.5 × 10846,032,9521.37 × 1099.8 × 1081.4 × 1096.93 × 1081.02 × 108
median11,419,2905.02 × 10979,267,8897.19 × 101011,335,3491.62 × 10101.3 × 10983,612,3251.48 × 1093.93 × 1092.16 × 1091.32 × 1091.95 × 108
rank11131321284710965
C17-F13mean25,194.617.45 × 108158,4434.58 × 101029,207.771.23 × 10101.55 × 108169,862.12.14 × 1086.76 × 10819,712,0462.11 × 10826,307,383
best14,947.6739,849,03436,363.72.32 × 101014,829.625.77 × 10972,249,48489,730.531.19 × 1084.77 × 10833,125.9560217.586,804,282
worst35,197.792.32 × 109349,245.46.59 × 101049,867.221.62 × 10102.35 × 108250,596.33.63 × 1087.89 × 10866,446,1655.9 × 10843,575,192
std10,557.91.08 × 109133,793.41.79 × 101016,453.734.55 × 10967,992,27865,749.651.14 × 1081.41 × 10831,690,7142.77 × 10815,084,746
median25,316.493.12 × 108124,081.54.71 × 101026,067.121.36 × 10101.56 × 108169,560.91.88 × 1087.19 × 1086184,4471.28 × 10827,425,029
rank11131321274910586
C17-F14mean1561.629954,429.11301,71451,521,2401644.84411,042,5716537,253338,3622,186,3411,214,91316,125,876177,015.611,328,374
best1551.5740,522.92403,134.715,801,5841610.492744,015.31234,192193,899.1117,686.71,015,5563,656,07754,705.128,041,233
worst1569.0112,245,9633,100,5861.04 × 1081665.85938,689,46718,106,774480,695.44,353,6881,446,28926,477,284410,064.815,650,354
std7.48164929,949.31226,14837,650,67124.5515718,475,0347,877,558121,825.32,228,784193,566.310,346,08315,8781.13165,277
median1562.968765,615.1851,568.742,984,4091651.5142,368,4023,404,022339,426.62,136,9961,198,90417,185,072121,646.310,810,955
rank15713210948612311
C17-F15mean2088.20284,644.1340,352.974.46 × 1092168.0642.15 × 10945,457,765106,61623,648,13750,130,6252.1 × 10820,135.5413,007,299
best2001.11619,632.7524,871.693.48 × 1092081.998733200312,695,48761,228.51114,896.224,391,48620291.992938.6251,429,335
worst2202.268176,250.974,3945.28 × 1092262.0886.23 × 10986,925,294163,102.242,481,43077,779,1358.16 × 10840,169.4125,370,774
std84.6148866,115.4422,955.177.97 × 10877.265392.81 × 10935,216,05942,226.9918,140,24125,125,6374.04 × 10817,100.379,825,680
median2074.71271,346.4231,073.094.53 × 1092164.0861.19 × 10941,105,139101,066.625,998,11049,175,94012,713,83318,717.0612,614,543
rank15413212968101137
C17-F16mean2737.2313413.8094503.4247943.4252910.2625681.856366.8383202.223082.3435298.3074079.1093392.6543966.724
best2549.1622716.4724180.0525899.3122532.0984684.455729.7842828.4372728.3695156.6853701.7512883.7763447.318
worst2910.8183865.4384935.66711,948.513329.9826106.0826736.1053693.0783311.2695434.4874494.8333690.9714613.582
std148.0198530.6386363.54252755.255326.6354668.1979465.9646390.8859248.6798114.3378375.0628374.9784542.6677
median2744.4713536.6634448.9886962.9412889.4855968.4336500.7313143.6833144.8675301.0284059.9253497.9353902.998
rank16913211124310857
C17-F17mean2573.1953002.3783658.25811,639.652700.3965430.2564790.5623274.992870.2343934.0823937.1823256.4843396.164
best2411.3632677.7163193.2848466.862459.0433655.9513430.3442935.2772516.3543671.3763461.2193075.8953257.524
worst2687.1873690.4934212.58315,174.642833.2427076.825593.0943534.4173082.954167.4824255.0793551.1073543.127
std115.9525464.2714492.96182763.849165.21911442.127967.311265.5602264.1476206.5056348.598224.8963117.1755
median2597.1152820.6523613.58311,458.552754.655494.1265069.4053315.1332940.8153948.7354016.2153199.4673392.002
rank14813212116391057
C17-F18mean14,578.012,805,3002,568,7231.2 × 10815,143.9845,107,78261,393,8532,423,53421,017,6558731,2718,959,584832,165.916,730,316
best4589.064379,650.7332,655.253,792,8214782.9915,112,94912,472,6751,316,4873,654,6882,959,6804,236,144500,424.37,543,771
worst30,709.15,279,2704,705,0671.66 × 10832,192.731.12 × 1081.43 × 1083,480,08464,837,31221,724,95216,743,5561,208,42121,978,246
std11,718.822,146,2472,215,52055,161,29912,301.3247,374,61161,960,9051,015,36229,414,1788,763,2975,701,618292,071.86,671,844
median11,506.932,781,1412,618,5851.29 × 10811,800.0931,796,85444,854,4582,448,7827,789,3115,120,2267,429,317809,909.118,699,624
rank16513211124107839
C17-F19mean2108.057159,032.8276,582.54.09 × 1092212.4731.16 × 1093,880,3475,795,75111,522,36469,314,944481,438.727,395.04852,195.6
best2041.29636,866.4996,991.52.76 × 1092118.38910389370424,737.41,799,2282,465,41829,753,913276,793.46758.912657,534.7
worst2184.699419,582.6570,566.35.06 × 1092328.2462.36 × 1097,483,3049,042,44732,549,6181.43 × 1081,055,17578,011.971,075,172
std77.04148178,299205,522.61.02 × 109108.41811.32 × 1093,181,5283,059,23014,306,74952,264,278382,629.333,896.96171,296
median2103.11789,841.03219,386.14.27 × 1092201.6281.14 × 1093,806,6736,170,6645,537,21052,379,219296,893.212,404.65838,037.7
rank14513212891011637
C17-F20mean2547.2093273.2893337.7484223.7312931.5383847.9534145.3573157.8493220.1283846.3774165.343170.4073108.672
best2498.0652844.2332688.2353927.4782786.5663412.0713572.7222814.4512952.3233618.0133846.162963.6483017.065
worst2648.1263919.7583912.4574391.2133205.9824148.5264368.7313500.3783597.9634078.0284468.5863305.2493201.274
std68.60929507.0025524.2096204.2398187.4237312.5293382.4528282.9635276.6289252.1705255.2723155.278792.03002
median2521.3223164.5823375.154288.1172866.8023915.6084319.9893158.2833165.1133844.7344173.3073206.3643108.174
rank17813210114691253
C17-F21mean2461.0152750.032785.9553080.6952442.5582978.1553074.5982538.4012541.0132858.8892878.6042691.4352775.299
best2454.0622655.452655.9122968.6242438.7882961.7822916.8372465.5812500.982816.4162802.9472602.6442699.723
worst2467.752890.7392985.8393173.8472445.9373018.4983252.4482618.4922568.7592900.4662921.0822738.3882838.696
std5.753449100.6273142.212797.295913.78576426.98508139.203175.6941628.6669135.3846653.1080160.4859763.22481
median2461.1232726.9662751.0353090.1542442.7532966.173064.5532534.7672547.1572859.3372895.1942712.3542781.388
rank26813111123491057
C17-F22mean5068.17910,601.6311,695.7317,156.725289.64914,744.4114,773.489918.288391.52716,328.3712,006.9511,379.8611,852.76
best2317.6599792.0139254.4816,976.742386.83713,539.0312,269.299328.1016397.78315,366.0311,736.649262.035080.869
worst8010.13111,189.4113,600.2917,419.318618.20915,424.9416,718.8410,360.0110,211.5917,081.3812,207.0513,321.1315,030.32
std3175.556587.46191976.242202.75373338.394827.8011899.776440.27491597.655721.302208.20341660.844566.475
median4972.46410,712.5511,964.0817,115.425076.77615,006.8315052.99992.5048478.36716,433.0412,042.0511,468.1413,649.93
rank15713210114312968
C17-F23mean2879.8643742.8033333.2823984.373013.4844203.963625.6262987.9783030.1493353.5194909.233393.5383411.219
best2819.0953528.5643242.0243931.8582911.8233785.873524.3372952.6452998.0213281.3724698.9383325.5793287.366
worst2929.6663941.543420.5344030.283057.884675.0383725.8343017.0483058.1983472.3515093.6023455.8193511.652
std45.73851215.6684.2772341.0496168.2742415.883187.1708627.5636626.8374882.51679161.893853.58173114.0835
median2885.3473750.5533335.2863987.673042.1164177.4663626.1672991.113032.1893330.1774922.1893396.3763422.929
rank11051131292461378
C17-F24mean3083.5044142.6763560.7584611.7993081.9814037.3583901.953096.8463208.0723525.0454499.53660.7663832.45
best3015.9653876.673440.3674085.6273000.5113762.7863823.2383056.2883106.1453473.4434461.4013604.2823679.231
worst3132.9784662.2063761.4385903.3163136.0034367.0773988.1853126.9443377.683567.2114558.0363744.374008.357
std51.40778355.0229139.0862869.70758.68838279.975168.729529.97032117.540646.1085644.9247764.59743155.6822
median3092.5364015.9143520.6144229.1273095.7054009.7853898.1883102.0763174.2323529.7634489.283647.2063821.106
rank21161311093451278
C17-F25mean3068.7334544.5063193.31812,627.243087.8086295.7354867.6623090.153831.6964514.7774376.1083145.6394421.074
best3046.8553802.5613163.26510,093.243054.9965817.7694493.1663057.2953605.513911.354001.1373096.164179.04
worst3077.5215346.1913243.54114,183.953131.8496738.6065247.9773113.0894055.2534952.4885087.8893185.5494659.918
std14.64504632.252134.790921922.17433.4426392.6728344.007923.71233.4356519.0428511.070739.00329196.3794
median3075.2794514.6353183.23213,115.893082.1946313.2844864.7523095.1083833.0114597.6344207.7033150.4224422.67
rank11051321211369748
C17-F26mean4455.59410,887.8911,546.2815,965.144869.19414,453.4615,596.316182.3737275.8799760.46212,162.557406.2027234.428
best3501.586702.7110,993.215,295.973470.60413,342.3714,726.995832.2056155.4139467.02911,786.483655.9936898.45
worst6191.43113,062.0512,099.1716,992.016448.08415,734.1716,328.646713.888916.75310,118.6512,597.79175.0917781.243
std1226.9942918.176452.4794735.80581422.756985.4345658.776386.34271291.61270.3168338.56112584.358392.6468
median4064.68211,893.411,546.3815,786.294779.04514,368.6615,664.86091.7057015.6749728.08612,133.018396.8637129.01
rank18913211123571064
C17-F27mean3363.1894366.5143897.6895125.1123448.7955237.0284903.7563470.7773717.3043693.7378470.6073854.5164433.001
best3320.6064286.1333848.5234720.7463366.6214761.124426.3063432.2613636.6373635.2818197.853698.9114182.635
worst3430.9444491.8723970.3485408.7183607.6095915.325499.8523571.6343830.1433756.6128860.9174102.2434668.505
std49.3645594.8850754.71087330.3539110.5841485.8052449.64267.5328487.2599255.50564322.7935173.0623198.6077
median3350.6044344.0253885.9445185.4913410.4755135.8354844.4333439.6063701.2183691.5288411.8313808.4564440.433
rank18711212103541369
C17-F28mean3304.0265927.0693619.74911,797.143373.5456674.6155136.2643310.3854426.4185582.8725201.7523869.9014966.713
best3279.1515470.6693530.72210,414.723341.6835799.4754614.2573301.5884091.0544696.4465136.0953456.6694889.397
worst3330.1526322.6163716.89515,468.783429.3778088.8875807.8943314.6524863.6956048.0665331.2944264.8885054.661
std21.30892374.697291.865492452.2738.517311032.791510.39596.127407361.6129603.628888.70492382.400275.04482
median3303.45957.4963615.68910,652.523361.5616405.055061.4533312.6494375.4625793.4895169.8093879.0234961.397
rank11141331282610957
C17-F29mean4311.5035508.0115680.66320,771.094536.48098.1928442.2545166.9414755.156354.8188566.635084.3676583.69
best4038.3424914.4245522.08210,858.344288.4116715.2616101.8244992.0534634.5816001.9887009.1274698.5296508.721
worst4509.9126458.3175834.62433,002.784775.3210,262.4910,228.645303.744937.4326842.11311,314.845645.8396630.906
std198.4451720.6455127.70439862.853200.541638.2441783.724129.1374137.3209361.71881938.806463.468652.64746
median4348.8785329.6515682.97419,611.634540.9347707.5078719.2745185.9844724.2936287.5857971.2744996.556597.567
rank16713210115381249
C17-F30mean2,025,56935,167,30423,274,8555.86 × 109 2,682,820 1.04 × 1092.59 × 10880,472,7571.24 × 1082.86 × 1081.97 × 1085990,56258,956,549
best1,181,08823,290,18814,200,5233.6 × 109 1,328,294 1.77 × 10898,570,18351,907,046736414001.95 × 1081.51 × 1081211,27845,838,053
worst2,875,32842,761,447 31,913,446 9.19 × 109 3,372,211 2.89 × 1094.62 × 1081.34 × 1081.7 × 1085.03 × 1082.58 × 10818,526,49966,979,535
std698,314.38,325,550 8701,063 2.41 × 109 918,657.5 1.26 × 1091.52 × 10837,734,53241,204,1601.47 × 108449,807,888,368,2969,118,859
median2,022,93037,308,790 23,492,726 5.32 × 109 3,015,388 5.51 × 1082.37 × 10868,043,7511.27 × 1082.22 × 1081.89 × 1082,112,23561,504,304
rank15413212107811936
Sum rank3221618236662325287115159256264157218
Mean rank1.1034487.4482766.27586212.620692.13793111.20699.8965523.9655175.4827598.8275869.1034485.4137937.517241
Total rank17613212113591048
Table 5. Optimization results of the CEC 2017 test suite (for the dimension d = 100 ).
Table 5. Optimization results of the CEC 2017 test suite (for the dimension d = 100 ).
SOAWSOAVOARSAMPATSAWOAMVOGWOTLBOGSAPSOGA
C17-F1mean160,265.11.19 × 10114.08 × 1092.48 × 10115 × 1081.31 × 10116.53 × 1010 67,475,871 5.52 × 10109.46 × 10101.4 × 10112.14 × 10106.99 × 1010
best138,023.51.11 × 10111.98 × 1092.44 × 10113.71 × 1081.18 × 10116.11 × 1010 57,348,883 3.89 × 10107.98 × 10101.18 × 10111.65 × 10106.09 × 1010
worst207,953.11.25 × 10115.86 × 1092.5 × 10117.33 × 1081.47 × 10116.77 × 1010 77,330,536 6.37 × 10101.12 × 10111.58 × 10112.72 × 10107.63 × 1010
std32,666.585.84 × 1091.59 × 1092.86 × 1091.6 × 1081.44 × 10102.97 × 109 10,406,022 1.16 × 10101.35 × 10101.71 × 10104.42 × 1096.58 × 109
median147,541.91.2 × 10114.23 × 1092.49 × 10114.47 × 1081.3 × 10116.61 × 1010 67,612,033 5.91 × 10109.31 × 10101.42 × 10112.1 × 10107.13 × 1010
rank11041331172691258
C17-F3mean155,460.5 284,175.9 347,477.9 343,395.2 163,200.2 375,331.1 977,486.3 500,947.0 389,167.5 329,791.4 365,707.5 607,827.5 654,898.7
best136,109.2 248,717.4 339,354.5 331,243.9 142,444.6 312,194.3 943,418.6 444,428.5 374,416.5 301,764.9 354,759.5 419,980.8 516,193.8
worst164,612.9 309,081.9 355,262.9 350,541.3 173,294.4 415,302.3 1,042,632.0 632,522.5 415,170.4 355,744.3 377,857.9 772,544 754,873.2
std13,110.98 28,313.85 6723.4319036.02914,518.0744,174.8546,210.5188,310.4517,980.2229,865.059617.27314,9726.7116,056.6
median160,559.9 289,452.2 347,647.2 345,897.8 168,530.8 386,913.8 961,947.5 463,418.4 383,541.7 330,828.1 365,106.3 619,392.6 674,264
rank13652813109471112
C17-F4mean707.842119,140.581615.54879,883.061078.9319,621.1213,056.76745.66675449.7699178.90836,332.133066.089469.057
best655.775914,643.471348.81172,405.031013.77714,895.8710,390.22702.47543735.4916841.93625,952.882030.3269196.622
worst793.672421,181.191785.13689,013.131187.74228,200.3314,,767.67773.18487492.23110,520.7245,851.154909.8199906.689
std60.214983087.898197.58326886.74475.376816081.2371906.10731.284241791.3531681.4838202.6821313.62306.1485
median690.9620,368.841664.12279,057.041057.117,694.1513,534.58753.50325285.6769676.48636,762.242662.0889386.458
rank11041331192671258
C17-F5mean1140.386 1416.959 1337.534 2014.712 1240.094 2115.385 1999.848 1237.633 1155.592 1860.505 1347.479 1399.646 1716.961
best1078.186 1389.332 1325.12 1977.422 1195.626 2036.245 1909.846 1113.65 1084.952 1792.429 1271.784 1348.014 1657.313
worst1174.568 1466.582 1347.384 2051.417 1308.247 2283.080 2096.681 1333.125 1222.116 1994.053 1469.717 1467.549 1744.248
std42.7894634.797469.31725137.3507549.9162911376.5765997.1832456.2831490.6875286.2464453.2503940.1947
median1154.3951405.9621338.8162015.0041228.2522071.1071996.4331251.8781157.6511827.7691324.2081391.511733.142
rank18512413113210679
C17-F6mean640.8564671.2647664.0195708.4028635.4878720.0932702.7487677.6025641.7199687.1312668.1679665.195667.7512
best638.4984665.4193659.8006703.2776632.7878708.063692.7168661.6266636.6993680.3454666.4344661.4675662.3638
worst645.8414687.6795668.4418711.4416640.1923730.9568716.0015693.8966645.3746696.9215669.6235670.454674.3769
std3.41240710.946713.5618523.6084023.40055710.0727310.1694513.633773.9249447.3048141.695354.1264965.003959
median639.5429665.98663.9178709.446634.4854720.6764701.1382677.4434642.4029685.629668.3068664.4292667.132
rank28412113119310756
C17-F7mean1893.5523374.3513157.7093828.4872006.6733664.6883488.4852248.0662145.7312970.4652998.8312589.662744.412
best1757.9323209.7412991.0563733.3971831.2813130.5383237.0592030.1482005.5742855.1612786.072157.7522606.048
worst2019.1253753.7463297.5333909.3722119.1334341.9173644.5852476.4992342.9763102.6493161.2472829.4852825.746
std114.7911257.1896153.15975.52237124.2497501.5494182.1772189.4768147.1537121.8524178.6652315.2845100.1396
median1898.5763266.9593171.1243835.5892038.1393593.1483536.1482242.8072117.1872962.0263024.0042685.7032772.926
rank11091321211437856
C17-F8mean1431.6691802.6041766.2542528.821483.6712517.5012210.1271482.5021540.9772240.1391883.4471919.5212069.916
best1288.8251707.7651707.5422502.5571317.8062416.132128.7621356.2161467.9442124.1341857.3221747.0852014.745
worst1505.1451846.7351795.7212544.9611637.4922615.0762313.5521578.7111603.1752340.5971932.5382074.1042101.56
std97.5696763.8661640.4981818.33389131.538983.696679.9196597.9354856.0792695.5303835.38708164.730639.18582
median1466.3531827.9591780.8772533.8811489.6932519.3992199.0961497.5411546.3952247.9121871.9651928.4482081.68
rank16513312102411789
C17-F9mean19,364.46 82,669.58 27,172.11 79,373.17 22,818.43 137,756.3 59,269.65 68,461.86 45,236.99 69,329.59 26,731.7 30,740.75 48,051.43
best16,807.53 75,191.45 22,605.98 76,723.33 17,418.05 108,475.1 45,027.85 49,383.76 33,072.81 65,899.36 24,236.3 25,972.61 45,151.63
worst21,107.66 91,644.35 30,594.59 81,543.18 25,883.72 152,399.4 83,531.69 94,754.6 54,921.68 73,999.44 30,955.57 36,306.87 53,358.57
std2103.88 7174.092 3328.994 2100.287 3868.815 19,968.74 17,085.29 19,622.14 9095.293 3435.528 2919.456 5374.775 3799.396
median19,771.33 81,921.27 27,743.93 79,613.09 23,985.98 145,075.3 54,259.52 64,854.53 46,476.73 68,709.78 25,867.47 30,341.75 46,847.75
rank11241121389610357
C17-F10mean12,718.41 25,000.53 16,130.00 32,371.07 14,957.25 29,288.9 27,809.40 16,429.39 20,918.75 32,258.28 16,555.1 17,941.46 26,975.78
best12,215.97 17,070.00 13,311.48 31,423.28 13,758.01 28,082.1 25,632.71 12,929.29 16,553.94 31,909.45 15,209.4 16,127.12 26,382.45
worst13,595.73 31,663.57 18,484.04 32,801.76 15,544.55 30,431.1 29,064.27 20,010.88 32,186.21 32,592.30 17,295.4 20,010.14 27,903.87
std603.51 7406.58 2244.45 647.84 825.34 969.3 1538.77 2909.97 7526.20 360.10 923.7 1735.06 702.90
median12,530.97 25,634.28 16,362.23 32,629.62 15,263.23 29,321.1 28,270.31 16,388.69 17,467.43 32,265.69 16,857.76 17,814.29 26,808.41
rank18313211104712569
C17-F11mean2576.63966,539.9866,864.48215,441.74466.55474,278.42233,670.55121.71373,277.2754,827.31182,410.780,069.76163,604.5
best2351.24548,807.4660,071.57164,806.43731.43557,060.97147,832.14503.9759,898.2849,308.24150,790.350,850.22122,067.2
worst2838.8592,451.6779,917.87307,014.45502.6383,711.11404,045.36212.61987,020.9663,160.31215,076.397,950.76207,411.1
std224.038519,508.049184.96664,071.68763.263511,773.96115,695.8775.77611,363.385914.5327,214.4822,176.8241,642.89
median2558.23262450.463,734.24194,972.94316.07678,170.81191,402.34885.13173,094.9153,420.35181,888.285,739.03162,469.8
rank15612281337411910
C17-F12mean21,615,544 5.33 × 10106.87 × 1081.79 × 10112.66 × 1086.03 × 10101.25 × 10105.51 × 1081.78 × 10102.51 × 10106.17 × 10107.41 × 1091.34 × 1010
best10,630,715 4.58 × 10103.65 ×1081.34 × 10111.03 × 1084.46 × 10109.28 × 1092.54 × 1081.19 × 10101.69 × 10104.86 × 10102.54 × 1091.17 × 1010
worst32,253,244 6.1 × 10101.1 × 1092.08 × 10113.55 × 1088.39 × 10101.46 × 10109.23 × 1083.18 × 10103.8 × 10106.66 × 10101.75 × 10101.58 × 1010
std9525,992 6.22 × 1093.16 × 1083.4 × 10101.13 × 1081.72 × 10102.28 × 1092.95 × 1089.38 × 1099.05 × 1098.78 × 1096.81 × 1092.01 × 109
median21,789,109 5.33 × 10106.44 × 1081.87 × 10113.04 × 1085.64 × 10101.31 × 10105.13 × 1081.38 × 10102.28 × 10106.59 × 10104.8 × 1091.3 × 1010
rank11041321163891257
C17-F13 mean44,203.57.73 × 109102,552.14.46 × 101070,549.512.06 × 10108.69 × 108440,253.31.66 × 1094.43 × 1099.98 × 1091.63 × 1081.74 × 108
best39,417.162.91 × 10972,417.293.45 × 101039,047.231.37 × 10104.27 × 108385,351.83490,8141.73 × 1098.96 × 10963,314.6861,879,640
worst47,640.751.2 × 1010139,843.75.05 × 101089,174.523.1 × 10101.71 × 109549,081.55.13 × 1097.98 × 1091.11 × 10105.76 × 1082.79 × 108
std3447.1094.32 × 10928,590.467.42 × 10923,479.337.5 × 1095.8 × 10873,870.142.35 × 1092.61 × 1099.87 × 1082.76 × 10892177490
median44,878.038.02 × 10998,973.744.67 × 101076,988.151.88 × 10106.72 × 108413,289.97.45 × 1084.01 × 1099.92 × 109383545721.78 × 108
rank11031321274891156
C17-F14 mean38,247.02 8,398,313 6,983,904 83,370,942 37,992.77 7574,523 23,051,859 2,406,385 7,632,300 12,005,340 8,966,839 3,232,258 16,151,979
best19,691.01 5,689,698 4,235,073 76,038,636 19,526.17 3959,310 16,617,918 1,206,786 4,435,632 8,814,373 6,670,450 720,653 10,615,559
worst70,208.67 11,219,705 11,593,673 91,264,970 69,787.65 15,774,376 35,953,797 4,757,866 13,293,148 15,067,242 14,374,476 7,262,728 21,599,852
std22,041.78 2,526,169 3,241,606 7,314,111 21,928.96 5,505,061 8787,473 1,618,096 3,949,408 2,628,175 3,652,301 2,811,928 4561,649
median31,544.2 8,341,924 6,053,435 83,090,081 31,328.63 5,282,204 19,817,861 1,830,443 6,400,211 12,069,874 7,411,215 2,472,826 16,196,252
rank28513161237109411
C17-F15 mean33,097.944.19 × 10888,181.732.46 × 101034,635.71.03 × 10101.95 × 108145,183.42.72 × 1087.94 × 1081.72 × 1091.18 × 1099,645,827
best16,382.2123,910,67972,089.621.76 × 101017,471.386.25 × 10969,888,98980,674.011.24 × 1084.89 × 1085.36 × 10858117.716,699,332
worst44,575.11.22 × 109110,726.83.07 × 101048,568.521.53 × 10104.94 × 108208,724.25.69 × 1081.14 × 1094.14 × 1092.91 × 10912,373,368
std13,525.925.44 × 10818,472.616.5 × 10914,309.743.92 × 1092.01 × 10853,444.252.01 × 1082.9 × 1081.64 × 1091.44 × 1092,319,862
median35,717.232.16 × 10884,955.272.51 × 101036,251.469.75 × 1091.07 × 108145,667.71.98 × 1087.73 × 1081.11 × 1099.01 × 1089,755,304
rank18313212647911105
C17-F16 mean5289.6678075.6197346.9423,585.125556.75412,659.1316,572.816401.6566685.50212,217.0810,417.546132.96311,108.18
best4897.67396.5926037.36418,423.114859.18711,295.5513,966.615961.5996185.13410,721.769217.0535449.2759424.787
worst5710.2768970.7298134.84226,458.226867.36114,021.1418,721.126668.8977453.34113,860.2211,540.536832.96711,940.21
std332.3395704.2064923.01353635.585892.97251121.2771963.62327.4517604.65481289.506962.7863582.53511138.671
median5275.3957967.5787607.77824,729.575250.23512,659.9216,801.766488.0636551.76612,143.1710,456.296124.80511,533.86
rank17613211124510839
C17-F17 mean4400.55810,409.145990.3388,696,8524366.873151,808.615,333.775168.3065408.6239341.009228,490.87064.666756.812
best4055.4415815.15796.7422,357,2544026.54332,785.6611,709.254917.8664332.3098097.86842,794.526170.7996191.594
worst4671.85416,795.956476.64220,011,8524628.367375,45318,769.635547.3756488.27810,527.61741,374.47828.8367423.221
std260.72334852.693326.23538,306,042255.0694158,483.73908.91267.6024896.4995999.1039342,134.4692.6501514.2877
median4437.4689512.7665843.9856,209,1524406.29299,497.8915,428.15103.9925406.9539369.27864,897.147129.5026706.217
rank29513111103481276
C17-F18 mean188,979.97,212,6062,951,0491.08 × 108235,943.218,747,09620,370,3063,891,6439,340,65720,568,8688,970,7562,463,6377,316,713
best168,960.22,281,7791,467,40741,911,753178,664.29,857,55012,477,2912,117,3312,729,39412,444,2284,277,344834,378.15,157,171
worst217,287.514,425,9824,665,8421.97 × 108379,81328,848,85831,924,1357,854,02615,928,69538,379,79911,074,7203,768,39510,061,294
std22,970 5,140,198 1,449,149 65,562,281 96,626.92 7,801,882 8256,895 2,663,425 5,635,043 12,131,498 3,152,116 1,271,998 2,343,646
median184,836 6071,331 2,835,474 96,283,852 192,647.90 18,140,988 18,539,900 2,797,608 9,352,270 15,725,723 10,265,479 2,625,888 7,024,194
rank16413210115912837
C17-F19 mean240,362.6 8.76 × 1083,017,804 2.34 × 1010253,780.5 6.88 × 1091.99 × 10819,860,187 5.05 × 1089.51 × 1081.38 × 1095.71 × 10815,651,499
best106,471.1 33,323,5261,155,276 1.71 × 1010112,995.9 6.17 × 109734693476,784,792 13,099,4584.24 × 1085.24 × 10814207435,304,591
worst317,561.9 2.12 × 1095,555,441 2.92 × 1010335,639.9 8.01 × 1093.14 × 10831,723,596 1.91 × 1091.98 × 1092.57 × 1091.49 × 10935,863,459
std92,219.0 9.6 × 1081,859,799 4.98 × 10997,113.3 8.03 × 1081.01 × 10810,221,012 9.38 × 1086.96 × 1089.4 × 1086.8 × 10814,375,122
median268,708.6 6.73 × 1082,680,249 2.37 × 1010283,243.1 6.67 × 1092.05 × 10820,466,180 48,052,4657.03 × 1081.22 × 1093.96 × 10810,718,973
rank19313212657101184
C17-F20 mean4329.037 5209.792 6336.396 7771.453 4716.199 7113.265 6925.121 5395.872 4996.863 7387.463 5974.363 5170.578 6642.64
best3983.977 4821.265 5904.681 7612.301 4388.633 6767.715 6587.829 5094.733 4850.608 7234.322 5242.187 4710.582 6076.979
worst4539.248 5659.555 6676.751 7841.025 4978.188 7576.378 7332.808 5543.659 5163.653 7497.733 6314.946 5782.027 7037.92
std240.729 355.8995343.9901106.793252.3931386.9691368.0945203.5619148.5221125.1266496.6382474.2739420.3355
median4396.461 5179.175 6382.075 7816.242 4748.988 7054.484 6889.924 5472.548 4986.596 7408.899 6170.159 5094.852 6727.831
rank15813211106312749
C17-F21 mean2736.167 3745.407 3729.792 4508.594 2838.871 4334.393 4315.533 3017.185 3055.55 3655.302 4547.086 3650.019 3629.856
best2679.86 3690.774 3506.549 4427.007 2768.495 4120.796 3872.558 2972.154 2900.463 3567.526 4387.706 3423.076 3557.698
worst2813.182 3796.665 3875.17 4568.519 2890.906 4510.86 4750.77 3054.39 3136.03 3719.88 4719.22 3851.77 3712.51
std58.078743.77476159.051161.5853558.50085166.9908385.413635.2477105.795470.19626135.5839178.164171.99652
median2725.81 3747.09 3768.73 4519.43 2848.04 4352.96 4319.40 3021.10 3092.85 3666.90 4540.71 3662.61 3624.61
rank19812211103471365
C17-F22 mean16,754.43 22,635.78 20,677.34 34,479.41 19,077.25 32,106.98 30,302.25 19,243.96 18,528.36 34,487.37 20,703.73 20,184.32 29,011.38
best15,651.14 21,193.53 19,202.46 34,027.03 18,107.02 29,368.93 29,139.08 17,859.86 17,156.79 33,489 19,451.88 17,961.85 28,178.06
worst17,268.55 23,665.76 22,710.46 35,121.42 20,362.55 33,216.89 31,342.78 20,447.75 19,437 35,320.04 21,879.51 21,096.56 30,184.34
std754.64181037.438 1555.121 491.9479953.84981830.945 1029.101 1331.092 1001.715 894.53231039.709 1497.787 894.7833
median17,049.01 22,841.92 20,398.22 34,384.6 18,919.72 32,921.05 30,363.58 19,334.13 18,759.82 34,570.22 20,741.76 20,839.44 28,841.56
rank18612311104213759
C17-F23 mean3330.207 5009.335 4180.505 5475.558 3302.335 5508.514 5381.020 3553.843 3630.727 4364.402 7806.769 5035.024 4491.184
best3220.897 4915.964094.7595190.5843192.09 5331.809 5167.214 3462.843 3529.972 4221.49 7299.394 4779.893 4285.936
worst3407.188 5150.114 4271.186 5700.211 3376.761 5801.627 5828.837 3670.808 3713.651 4491.014 8053.976 5209.153 4806.089
std84.12463101.509983.55801210.950383.8719203.2764302.635690.0293975.82554126.8273353.6352191.2664233.4911
median3346.372 4985.632 4178.037 5505.718 3320.244 5450.31 5264.014 3540.861 3639.643 4372.552 7936.852 5075.525 4436.355
rank28511112103461397
C17-F24 mean3838.394 6619.833 5501.153 10,952.73 4019.698 7130.901 6669.652 4156.191 4463.04 4946.4 10,577.72 6083.567 5573.038
best3804.994 6393.899 5264.016 7257.544 3965.063 6181.58 5802.949 3992.141 4329.01 4779.776 10,344.84 6038.202 5498.956
worst3880.947 6956.558 5697.902 13,415.8 4076.189 7864.329 7222.435 4387.281 4672.438 5127.243 10,849.330 6123.814 5654.398
std33.5964240.8529193.12162980.52255.04538703.9283626.4185170.6274146.9678153.8419268.481540.9405569.16896
median3833.817 6564.438 5521.346 11,568.79 4018.771 7238.848 6826.611 4122.67 4425.356 4939.289 10,558.35 6086.125 5569.398
rank19613211103451287
C17-F25 mean3419.878 12,914.68 4200.03 22,463.79 3727.179 10,109.35 7602.511 3458.415 6612.48 9840.617 13,003.17 6315.64 8226.567
best3270.776 10,959.75 3793.055 20,802.69 3639.544 7389.433 7500.673 3391.467 5802.505 7855.257 11,232.45 5406.982 7683.57
worst3570.643 15,484.74 4584.833 26,151.87 3817.394 12,792.82 7828.874 3519.4 7077.109 13,190.3 14,733.21 7810.14 8659.75
std127.21282106.496 327.33142514.302 84.076392611.297 152.069367.05617559.2212384.22 1600.964 1069.279 439.551
median3419.046 12,607.124211.116 21,450.29 3725.889 10,127.58 7540.248 3461.397 6785.152 9158.458 13,023.5 6022.719 8281.474
rank11141331072691258
C17-F26 mean12,161.53 33,589.15 25,812.52 47,915.4 12,417.5 36,882.86 39,697.02 13,011.43 16,796.16 26,337.9 36,037.68 26,179.06 23,813.48
best11,570.58 32,040.89 22,771.24 45,204.13 12,010.75 34,702.39 36,284.73 12,173.32 16,104.17 23,268.42 34,617.49 23,436.07 18,794.69
worst13,140.16 35,062.47 28,918.45 49,592 13,166.37 39,236.38 46,080.27 14,038 18,257.86 31,568.31 38,296.26 29,012.3 30,130.64
std714.78951291.344 2627.91 2110.57 517.05922305.593 4571.018 774.4961986.14793755.775 1584.211 2288.35 5129.716
median11,967.7 33,626.62 25,780.19 48,432.75 12,246.44 36,796.33 38,211.53 12,917.19 16,411.3 25,257.43 35,618.49 26,133.95 23,164.3
rank19613211123481075
C17-F27 mean3573.983 7203.719 4220.107 12,874.15 3543.702 6590.686 5549.819 3636.355 4351.534 4365.099 14,390.47 4068.458 5427.951
best3550.019 6374.999 4035.689 9574.128 3515.753 5450.785 5255.732 3538.111 4047.274 4231.086 12,460.66 3698.791 5163.581
worst3617.97 8029.224 4525.976 16,299.41 3586.981 7337.146 6286.109 3743.107 4728.147 4538.464 16,773.18 4534.344 5728.043
std30.36471874.2169212.08543625.498 30.41211825.7145492.403587.09758298.1583144.91641847.757 369.1118238.5253
median3563.971 7205.326 4159.381 12,811.52 3536.036 6787.406 5328.717 3632.1 4315.358 4345.422 14,164.01 4020.349 5410.089
rank21151211093671348
C17-F28 mean3582.808 16,810.98 4799.097 29,879.23 3850.162 16,533.6 11,426.62 3532.959 10765.5311,980.69 17,423.16 6348.414 13,091.11
best3420.284 15,333.26 4474.553 26,738.46 3777.292 14,884.29 9122.899 3454.509 8221.553 9544.848 15,852.57 4956.941 12,054.66
worst3659.831 18,085.58 5038.336 33,810.2 3972.019 18,597.99 13,140.11 3636.308 14,812.54 14,940.48 19,363.45 7865.142 14,211.12
std110.80761313.643 238.64272965.76 86.657581651.567 1959.457 75.762932827.535 2230.037 1666.674 1328.374 1159.838
median3625.557 16,912.53 4841.749 29,484.12 3825.668 16,326.07 11,721.74 3520.51 10,014.01 11,718.71 17,238.3 6285.787 13,049.32
rank21141331071681259
C17-F29 mean6785.358 11,200.52 9945.999 371,294.3 7011.439 29,447.41 16,930.69 8701.23 8650.05 14,358.73 21,205.48 8372.447 12,438.74
best6237.576 10,204.38591.844 199,089.7 6512.523 15,174.79 13,802.42 7941.767 7491.625 13,504.12 16,011.96 7665.661 12,035.03
worst7296.052 11,863.42 10,788.87 515,537.6 7399.818 66,248.24 21,278.25 9252.116 10,249.87 15,398.75 24,203.99 8683.496 13,143.18
std489.398 754.6761947.033 134,866.2 422.667824,574.85 3132.304 552.05851345.612 871.2183595.977 482.8703485.136
median6803.901 11,367.19 10,201.64 385,275.00 7066.708 18,183.31 16,321.06 8805.519 8429.351 14,266.02 22,302.99 8570.316 12,288.36
rank17613212105491138
C17-F30mean5,554,314 2.47 × 10929,428,850 4 × 10106,188,795 1.46 × 10101.17 × 1091.19 × 1082.32 × 1093.05 × 1091.18 × 10103.93 × 1086.24 × 108
best2,780,134 1.38 × 10816,769,683 3.74 × 10102,886,968 1.32 × 10108.85 × 10885,223,9275.3 × 1081.18 × 1098.16 × 10973585894.39 × 108
worst7,910,687 6.19 × 10951,751,386 4.33 × 10108,330,799 1.64 × 10101.6 × 1091.65 × 1085.4 × 1094.23 × 1091.56 × 10101.52 × 1091.04 × 109
std2,148,328 2.6 × 10915,661,651 2.53 × 1092,560,052 1.43 × 1093.08 × 10834,020,1422.12 × 1091.31 × 1093.05 × 1097.51 × 1082.83 × 108
median5,763,217 1.77 × 10924,597,166 3.97 × 10106,768,707 1.45 × 10101.1 × 1091.13 × 1081.68 × 1093.4 × 1091.17 × 101022,288,3285.08 × 108
rank19313212748101156
Sum rank3524414435961318275116159255281172220
Mean rank1.2068978.4137934.96551712.379312.10344810.965529.48275945.4827598.7931039.6896555.9310347.586207
Total rank18413212103591167
Table 6. Optimization results of the CEC 2019 test suite.
Table 6. Optimization results of the CEC 2019 test suite.
SOAWSOAVOARSAMPATSAWOAMVOGWOTLBOGSAPSOGA
C19-F1mean171,754.4311120,197.7411,700,9001,132,77127,189.4268.360276.2 × 108149,550.87,917,670
best11160.8911112.75372,455,600376,289.61.0038111.0000031.49 × 10814,365.342,787,375
worst1205,465.811180,61916,679,0002,285,09394,712.58270.4151.07 × 109475,601.416,682,079
std092,622.5500040,280.896,367,086841,143.645,495.01134.70324.66 × 108218,241.36,534,856
median140,195.511184.613,834,500934,851.27022.041.0130286.33 × 10854,118.36,100,613
rank15111397421068
C19-F2mean3.1203172.93674.69410953.569175800.596139.5563.8434454.4571667.105427,745.11363.9978857.5801
best2.60781398.733514.30386852.798006629.092908286.6219210.57415.5640199851.265226.8182577.9538
worst3.596576266.8291554.344219937.159275.1838.1335658.09281190.36242,469.39544.86261135.415
std0.40911170.349080.3598645.13 × 10−160.631625136.55412632.46227.506184.6327537.80414,988.08133.4324290.0151
median3.138405163.09224.73628553.567238818.066187.45565.309474.5807736.247829,329.9342.1552858.4759
rank15342101287913611
C19-F3mean1.2501391.8921552.1269988.1383151.4239226.452556.13948.9607751.8899484.7166044.0031683.7775455.779726
best1.0459651.4097891.4091966.2945171.1963441.74083.06427.7117741.1411244.1495362.7634221.4091353.599784
worst1.4681142.3770113.3427919.211891.5630649.71439.647310.708553.4075555.8016785.9113277.6870417.700128
std0.235580.5553520.9203081.2995980.1653623.3979942.7405171.4984991.0262280.7510391.3828053.0103911.849209
median1.2432391.8909091.8780028.5234261.468147.177555.923058.711391.5055564.4576013.6689623.0070025.909496
rank14512211101338769
C19-F4mean8.99204817.4255835.3258973.9634811.4468553.103559.7737525.55723.203835.4313751.9914218.9092222.35515
best3.01918413.935348.95966755.89967.964741.45932.98422.8900410.1058132.6892139.8032810.9495919.10672
worst12.9686821.8946357.712394.3719414.92965.75792.98529.5735238.8084441.0653160.6971936.8183827.48026
std4.531744.09273720.2361116.552283.49392711.0697126.374132.95131513.705013.9313499.09621312.049493.586093
median9.99016316.9361737.3157972.7911811.4468552.59956.56324.8822221.9504833.9854753.7326113.9344621.41682
rank13813211127691045
C19-F5mean1.0508361.6883481.20134288.276021.03447529.612252.03991.3924421.5954062.9342971.1369181.2135111.725773
best1.0474711.2104391.1085266.198281.007413.5551.69411.2392891.3473282.638661.0662251.1795851.550766
worst1.0571752.2650971.285169108.99161.061662.0672.35911.7561751.8938823.1435061.2051431.2536341.841835
std0.0045770.4566410.0758617.492180.02857822.854610.2798080.2466180.2584450.2255160.0705910.0382280.125116
median1.0493491.6389291.20584188.957081.0344521.41352.05321.2871531.5702072.9775111.1381521.2104121.755246
rank28413112106711359
C19-F6mean1.0744982.9013747.0911549.9425981.1926987.85717510.68213.0235983.4942854.5765214.1111913.6435963.227866
best1.0364861.8012156.2141339.4561871.0744.39649.37541.2266041.3359623.5871611.1393571.5314742.490765
worst1.1791653.6035719.32256510.75121.32162410.54311.9694.3637594.9953925.2300965.6527046.2407274.391409
std0.069880.7942521.4964190.6064010.1044122.5984361.0624261.3709341.5486550.7228282.1099741.9809840.835364
median1.041173.1003556.413969.7815041.1875848.2446510.6923.2520143.8228944.7444124.8263523.4010923.014646
rank13101221113469875
C19-F7mean246.8228487.53511082.7941730.98302.09541406.91141.9051056.6151177.531175.0921660.5431151.383707.8292
best150.4273282.8179803.75541577.123165.39731022702.62723.02971021.721708.64851502.922658.1489490.5917
worst328.0506758.7951546.2821815.608451.111685.71794.31850.8271376.2271521.9111797.541606.7771093.745
std73.04863211.9359352.957108.8781117.496279.3649463.0885533.2837167.1757370.338133.3842444.184282.0167
median254.4067454.2637990.56861765.594295.93721459.951035.35826.30161156.0861234.9051670.8561170.303623.4902
rank13613211751091284
C19-F8mean2.9817743.7308564.4686534.8183383.35394.15044.849454.0590963.6092614.3015785.2169254.6081374.529276
best2.8252943.6764263.9272884.5381893.0105523.56114.6073.5616313.3662844.0558445.1197084.3696114.30402
worst3.1044123.842464.894834.9853933.68114.90295.0824.4842654.0360744.8700755.3732094.9783184.669163
std0.1315050.075410.4560510.2005630.3389510.5629380.2105160.4363970.3026510.3817540.122120.2648860.164295
median2.9986943.7022694.5262474.8748853.3619744.06884.85444.0952453.5173424.1401985.1873914.5423094.571962
rank14811261253713109
C19-F9mean1.0818971.1758381.3932243.1056561.0923471.3507751.38031.1764421.2204591.3240731.1962671.1938621.138063
best1.0631021.1221861.0783362.4012671.05121.19811.19171.1501861.10621.2536091.0788011.0896271.11366
worst1.1155021.2424741.6041683.6841311.1231.56071.65081.2036331.3558021.3849951.3159111.2631661.184152
std0.0240240.0542440.224060.5460420.0339780.1615330.216880.0247860.1132890.069440.115060.0750430.031644
median1.0744911.1693461.4451973.1686131.0975951.322151.339351.1759741.2099171.3288441.1951781.2113271.12722
rank14121321011589763
C19-F10mean18.0338317.8951721.0362821.4678421.0007521.4137521.21221.0274321.4756421.4306923.2498921.0322721.26732
best6.8115927.3742220.9716521.375142121.35521.04721.0073921.4604121.3986120.9979920.9996921.10764
worst22.0154721.5415821.1720721.5473321.00321.53821.59421.0406121.4966921.5020424.5471221.1294621.39193
std7.4839527.0146660.0915580.0707980.00150.0842150.2566570.0141640.0176140.0488141.6661640.0647910.126423
median21.6541421.3324521.000721.474452121.38121.103521.0308721.4727321.4110723.7272220.9999721.28486
rank21611397412101358
Sum rank1240631031994103646683966371
Mean rank1.246.310.31.99.410.36.46.68.39.66.37.1
Total rank1341129115681047
Table 7. Wilcoxon rank sum test results.
Table 7. Wilcoxon rank sum test results.
Compared AlgorithmTypes of Objective Functions
CEC 2017CEC 2019
d = 10 d = 30 d = 50 d = 100
SOA vs. WSO1.93 × 10−152.02 × 10−212.02 × 10−211.97 × 10−212.63 × 10−7
SOA vs. AVOA6.1 × 10−201.16 × 10−201.97 × 10−211.97 × 10−212.07 × 10−5
SOA vs. RSA2.02 × 10−211.97 × 10−211.97 × 10−211.97 × 10−212.79 × 10−7
SOA vs. MPA0.0253181.36 × 10−51.86 × 10−121.74 × 10−150.000237
SOA vs. TSA2.24 × 10−211.97 × 10−211.97 × 10−211.97 × 10−211.18 × 10−7
SOA vs. WOA1.97 × 10−211.97 × 10−211.97 × 10−211.97 × 10−211.48 × 10−7
SOA vs. MVO2.77 × 10−195.13 × 10−191.56 × 10−202.8 × 10−204.98 × 10−7
SOA vs. GWO7.23 × 10−204.62 × 10−213.69 × 10−214.74 × 10−211.71 × 10−6
SOA vs. TLBO3.78 × 10−211.97 × 10−211.97 × 10−211.97 × 10−211.97 × 10−7
SOA vs. GSA7.23 × 10−201.97 × 10−212.73 × 10−211.97 × 10−213.57 × 10−8
SOA vs. PSO2.42 × 10−207.79 × 10−212.07 × 10−211.97 × 10−217.06 × 10−7
SOA vs. GA4.18 × 10−212.02 × 10−211.97 × 10−211.97 × 10−211.71 × 10−7
Table 8. Optimization results of the CEC 2011 test suite.
Table 8. Optimization results of the CEC 2011 test suite.
SOAWSOAVOARSAMPATSAWOAMVOGWOTLBOGSAPSOGA
C11-F1mean3.70326715.0255725.7456226.004124.06995217.5227521.8983113.494215.56820.1373925.8697521.4981125.49193
best2.68 × 10−1011.7567324.8390125.156832.85 × 10−108.51038517.241326.54896110.7223617.3914824.191417.4910424.75272
worst14.8130719.2917726.9422527.7876916.2798121.6313824.8261116.513122.0710622.5615827.7654725.5857926.07053
std7.4065343.1295991.0511121.2094798.1399056.0773773.2575214.678195.3899972.1790181.5241993.7538190.558841
median1.37 × 10−914.5268825.6006125.535971.48 × 10−919.9746222.762915.4573614.739320.2982525.7610721.4578125.57223
rank14111326935712810
C11-F2 mean−25.5074−22.3944−11.6301−8.06028−24.8695−6.49474−15.4353−9.34108−21.6587−8.61104−9.40185−22.0875−13.1955
best−26.343−22.8813−14.0599−8.64276−25.7669−9.42193−20.4428−12.3034−24.5033−9.39253−13.4981−24.4758−14.0636
worst−24.2869−21.4054−9.06418−7.42364−23.6464−4.02889−11.7159−7.01703−15.7941−7.55467−6.03461−19.675−12.6002
std1.0040790.6749672.0434590.5505650.9964612.5426354.0650622.3333233.9603150.8588253.0994452.3346020.625598
median−25.6998−22.6454−11.6981−8.08736−25.0324−6.26406−14.7912−9.02192−23.1687−8.74848−9.03735−22.0996−13.0591
rank13812213610511947
C11-F4 mean−34.1366−29.4024−17.5227−16.7272−33.0629−27.8822−23.8592−27.1652−31.4271−14.1604−26.2837−7.58−8.27727
best−34.4821−33.8574−19.4025−19.341−34.2193−31.2664−27.0822−31.5712−34.2156−20.773−34.1647−11.3148−12.0614
worst−33.4793−27.2841−16.1919−13.8236−31.812−25.793−21.0792−24.2788−26.2364−10.1154−21.2736−3.0134−4.22734
std0.4523453.0684821.377312.8489360.9988882.513442.4668163.1067493.576374.8883315.7490633.4285483.376156
median−34.2925−28.234−17.2482−16.8722−33.1102−27.2347−23.6377−26.4055−32.6282−12.8767−24.8482−7.99588−8.41014
rank14910258631171312
C11-F5 mean−27.1938−19.3848−10.0921−10.5136−25.6126−12.3726−14.19420−22.2141−5.21334−14.282800
best−27.4298−23.0059−14.7254−11.5854−26.5001−19.429−15.08780−26.8272−12.1923−16.912400
worst−26.4862−16.8457−7.09818−9.51564−23.00590−12.42190−19.51160−10.677500
std0.4717522.6471733.3827541.1533671.737998.5050391.2567103.4887446.190063.10917400
median−27.4297−18.8438−9.27249−10.4768−26.4723−15.0308−14.63360−21.2588−4.33054−14.770700
rank14982761131051111
C11-F6 mean0.890731.147542.0247552.1550710.902341.6437252.0520310.9657171.0481961.8745351.0820471.1808482.052041
best0.8664060.9980871.66141.830690.8419221.5604051.7937150.8335450.9387641.6143060.9055941.0088461.855107
worst0.9346291.2585332.1879472.3455920.9665921.8312182.3097431.2471271.1468282.0076361.3876681.4239342.215405
std0.0309070.1298350.2470210.2270850.0537490.1258780.2340160.1903550.0851770.176570.2240940.1768040.15728
median0.8809431.166772.1248362.2220010.9004231.5916382.0523330.8910971.0535951.9380991.0174641.1453062.068825
rank16101328113495712
C11-F7 mean231.7975223.6592276.5349.25220234291.25220224.5227231.5329.9552229.8177
best220220230299220220238220220220220220220
worst239.4414234.6368323404220258333220238238248535.8208259.2707
std8.3878187.31837640.0208343.06874018.1107739.55903098.71779813.89244141.268319.63537
median233.8743220276.5347220229297220220225229282220
rank7291218101346115
C11-F8 mean7306.049 735,450.4 1,029,600 1301,689 7878.01896,263.48 401,545.2 127,251.3 29,143.13 549,851 998,962.5 1,637,763 2,186,754
best4355.355 547,467.2 914,559 848,186.4 4665.333 69,896.59 323,827.5 88,847.38 17,099.15 377,823.8 955,606.7 616,975.9 2,097,586
worst11,321.75 873,535.9 1,112,084 1,528,229 12,237.1 134,708.9 547,008.1 164,393.9 42,638.2 747,382 1,052,476 3,466,500 2,273,623
std3118.179 147,088.4 86,808.81 308,299.6 3430.68 27,889.49 103,098.1 30,939.54 12,355.31 152,022.1 42,600.87 1256,071 72,053.94
median6773.544 760,399.2 1,045,878 1,415,170 7304.821 90,224.2 367,672.5 127,881.9 28,417.59 537,099 993,883.9 1,233,788 2,187,904
rank18101124653791213
C11-F9 mean−21.1519−16.4886−8.89083−10.1972−19.7224−12.2849−9.69672−17.4831−11.0572−10.2549−11.3843−10.2803−10.0705
best−21.5356−18.7915−9.07622−10.5958−20.7818−15.2414−10.2871−21.2049−12.9979−10.2865−11.8241−10.364−10.1474
worst−20.6824−14.5174−8.70287−9.81793−16.9694−10.5739−8.59883−12.3942−10.3885−10.2262−10.935−10.2224−9.99439
std0.4184532.0996640.194970.3178581.8435032.1106510.7883874.0540171.2940260.0272170.4468270.060440.066287
median−21.1947−16.3228−8.89211−10.1875−20.5693−11.6621−9.95046−18.1667−10.4213−10.2535−11.389−10.2674−10.0701
rank14131025123796811
C11-F10 mean275,953.6 293,843.2 1,728,003 10,733,179 507,841.1 6814,247 1668,923 1969,755 4265,332 5,689,705 1,570,136 5710,499 6,730,921
best111,906.1 119,680.6 1,671,593 10,440,242 443,295.4 5637,001 1596,987 1584,500 3731,297 5,689,705 1,439,379 5689,705 6,711,548
worst435,202.3 457,974.2 1,774,439 10,907,934 577,384.2 8,101,877 1718,805 2559,982 4585,233 5,689,705 1,703,594 5748,435 6,752,137
std133,300.2 140,150.7 43,954.16 203,662.6 57,922.72 1,169,664 51,360.73 465,329.2 378,888.6 0 128,768.8 27,792.26 17,639.59
median278,353 298,858.9 1,732,991 10,792,271 505,342.4 6,759,054 1,679,950 1,867,268 4,372,400 5,689,705 1,568,786 5,701,929 6,729,999
rank12613312578941011
C11-F11 mean1244,166 3917,084 7200,862 15,954,845 1,205,197 6,003,707 6,372,956 1,356,302 1,588,951 15,505,519 6777,124 2,450,810 15,381,122
best1115,988 3701,366 6,994,873 14,807,307 1,087,931 4,992,934 5,869,317 1,237,960 1,378,205 14,999,220 6594,614 2,214,801 14,242,937
worst1331,317 4098,419 7,301,082 16,959,124 1,287,550 7,191,303 6,835,128 1,627,660 2,025,660 15,937,381 7026,405 2,558,421 16,446,911
std91,442.38 176,574.8 139,296.1 882,680.7 84,212.59 903,553.7 417,842.9 184,571.9 300,145.8 397,440.9 207,663.9 158,822.7 935,503.1
median1,264,680 3,934,275 7,253,747 16,026,475 1,222,653 5,915,295 6,393,689 1,279,795 1,475,969 15,542,737 6,743,739 2,515,010 15,417,320
rank26101317834129511
C11-F12 mean 15,444.2 15,775.42 15,454.67 16,506.66 16,390.34 15,566.71 15,506.98 15,488.7 15,519.23 15,935.78 129,995.2 15,515.88 16,052.38
best 15,444.19 15,444.23 15,451.9 15,996.18 15,444.21 15,529.77 15,484.73 15,469.18 15,500.74 15,514.42 113,048.8 15,477.7 15,489.97
worst 15,444.21 16,732.14 15,457.72 17,774.94 16,966.72 15,613.13 15,533.89 15,505.51 15,547.46 16,670.71 151,977.1 15,555.92 17,025.63
std0.010596637.97262.451103852.4752667.719535.9562724.01314.913320.19086541.7847 16,644.88 33.017694.0509
median 15,444.19 15,462.66 15,454.53 16,127.77 16,575.22 15,561.97 15,504.65 15,490.05 15,514.36 15,778.99 127477.5 15,514.94 15,846.96
rank18212117436913510
C11-F13 mean18,808.95 18,401.87 18,913.78 276,941.8 18,308.93 19,373.02 19,262.54 19,316.49 19,545.76 108,292.5 19,295.41 19,116.86 19,139.07
best18,583.4 18,212.02 18,679.04 202,857.5 18,281.42 19,118.13 19,044.16 19,136.48 19,340.25 42,350.89 19,218.27 18,963.55 18,965.28
worst18,941.42 18,524.11 19,086.98 400,961 18,342.71 19,749.48 19,529.18 19,493.59 19,714.25 254,303.9 19,414.24 19,255.93 19,268.32
std156.5688134.1286181.804 88,734.65 31.09058267.474210.8087146.6933196.8206 98,161.29 84.17887157.4519144.5224
median18,855.48 18,435.69 18,944.54 251,974.4 18,305.8 19,312.23 19,238.41 19,317.94 19,564.27 68,257.65 19,274.56 19,123.97 19,161.34
rank32413110791112856
C11-F14mean 33,436.9 33,006.71 239,954.4 2,343,498 32,840.39 58,551.43 55,559.51 32,996.16 33,227.12 9,328,370 406,885.7 33,333.04 6,772,673
best 33,126.18 32,844.61 61,174.85 975,713.9 32,786.83 33,205.38 33,070.23 32,880.13 33,076.3 5,245,604 353,167.1 33,320.61 4,619,131
worst 33,831.7 33,126.57 439,827.3 6,129,448 32,874.34 133,374.7 122,653.1 33,127.33 33,382.38 13,754,921 429,323.5 33,357.17 10,429,108
std309.7042134.8848 203,421.2 2,527,893 37.99649 49,883.49 44,729.12 107.7502132.081 3,483,336 36,212.51 17.05479 2,694,081
median 33,394.87 33,027.82 229,407.7 1,134,415 32,850.19 33,812.82 33,257.37 32,988.58 33,224.9 9,156,476 422,526.2 33,327.18 6,021,226
rank63911187241310512
C11-F15mean 131,896.2 140,692.8 138,044.1 2362,861 141,030 139,185.7 149,714.5 142,078 137,989.1 98,216,083 16,661,057 82,739,645 90,489,805
best 129,599.2 134,258.9 134,145.7 551,007.8 136,822.8 134,983.1 143,326.3 137,708.7 134,926 81,712,714 145,480.7 67,023,611 77,222,102
worst 133,406.6 151,586.1 142,366.4 5916,825 143,594.4 144,166.6 156,177.1 149,494.5 143,091.5 1.11 × 10842,521,5881 × 1081.04 × 108
std 1692.118 7711.282 3367.715 2,413,071 3052.919 3843.03 5629.417 5157.713 3897.759 12,641,427 18,783,856 16,142,420 11,527,058
median 132,289.6 138,463.1 137,832.2 1491,805 141,851.4 138,796.5 149,677.3 140,554.5 136,969.5 99,861,336 11,988,581 81,736,076 90,566,521
rank15396487213101112
C11-F16 mean 1,992,168 4,080,675 6.39 × 1091.91 × 1010 1,931,384 1.7 × 1091.12 × 1010 3,335,018 2,977,526 2.45 × 10101.37 × 10102.05 × 10102.24 × 1010
best 1,966,581 1,935,090 5.81 × 1091.37 × 1010 1,899,652 9.95 × 1088.91 × 109 2,994,846 2,450,995 1.85 × 10101.1 × 10101.67 × 10102.1 × 1010
worst 2,004,837 9,745,854 7 × 1092.33 × 1010 1,956,433 2.5 × 1091.29 × 1010 3,937,955 3,345,029 2.98 × 10101.55 × 10102.44 × 10102.39 × 1010
std 17,340.9 3,793,542 5.24 × 1084.12 × 109 24,490.05 6.31 × 1081.81 × 109 435,811.6 379,800.1 5.09 × 1091.91 × 1093.47 × 1091.26 × 109
median 1,998,628 2,320,879 6.38 × 1091.96 × 1010 1,934,726 1.66 × 1091.15 × 1010 3,203,636 3,057,041 2.48 × 10101.41 × 10102.04 × 10102.24 × 1010
rank25710168431391112
C11-F17 mean 966,310.6 1,616,414 16,449,493 1.45 × 108944,296.71909,74612,427,380986,840.31100,25629,913,77114,001,5211.87 × 1081.28 × 108
best 952,773.8 1,094,971 9,228,474 1 × 108941,062.21271,7744919,875958,253.11018,90726,861,27611,017,3301.75 × 1081.26 × 108
worst 979,957.6 2,285,712 29,396,167 1.66 × 108947,649.92280,11725,716,4141,030,1021,340,21637,140,31817,250,1122.03 × 1081.32 × 108
std 11,741.73 494,288.8 9,390,234 30,701,4192718.591440,403.59,750,59130,602.88159,976.74,871,6722,620,74414,016,7963,160,298
median 966,255.5 1,542,486 13,586,665 1.57 × 108944,237.32,043,5469,536,616979,5031,020,95127,826,74513,869,3211.85 × 1081.27 × 108
rank25912167341081311
C11-F18 mean 1,000,924 2,590,116 16,582,334 1.42 × 108 1,069,181 2,937,110 14,575,361 1,710,525 1,589,011 35,041,852 15,310,972 1.81 × 1081.27 × 108
best 943,538.7 1,954,981 14,730,898 1.23 × 108 997,145.1 2,414,231 5831,462 1,393,338 1,167,771 26,831,644 9,362,833 1.66 × 1081.26 × 108
worst 1,061,517 3,891,235 20,548,870 1.79 × 108 1130,546 3,393,560 22,239,505 2,349,957 1,889,412 48,044,844 18,324,799 1.98 × 1081.28 × 108
std 52,438.1 879,404 2678,190 26,103,571 55,567.61 467,088.6 7,003,567 433,187.8 318,048.5 9,787,189 4,155,256 13,395,124674,274.6
median 999,320.4 2,257,125 15,524,785 1.34 × 108 1,074,517 2,970,325 15,115,238 1,549,403 1,649,430 32,645,460 16,778,129 1.81 × 1081.26 × 108
rank15912267431081311
C11-F19 mean 955,907 3,058,450 14,619,890 1.54 × 108 943,673 2,198,561 9,996,685 1,012,821 1,152,320 32,955,731 15,021,597 1.71 × 1081.27 × 108
best 946,161.6 1,019,342 12,702,812 1.34 × 108 939,631 1,982,053 7,065,299 967,576.5 1,006,006 28,464,019 8,991,833 1.42 × 1081.25 × 108
worst 965,274.1 8,406,301 16,683,672 1.83 × 108 945,155.6 2,462,708 16,377,179 1,042,643 1,410,250 38,222,795 20,803,805 1.98 × 1081.29 × 108
std 7877.253 3,580,771 1,652,290 20,568,980 2698.357 216,671.7 4,296,432 32,544.64 184,022.2 4361,221 5,235,721 26,365,90721,75,679
median 956,096.1 1,404,078 14,546,539 1.49 × 108 944,952.7 2,174,742 8,272,131 1,020,533 1,096,512 32,568,056 15,145,374 1.73 × 1081.27 × 108
rank26812157341091311
C11-F20 mean9.89113518.4348437.3899591.0135510.647833.2694850.3114330.1409731.4193106.503442.4290699.89802120.8118
best8.4870817.1045635.0819266.458989.07532827.2610637.9449624.4449220.00596.6801236.6161194.66104117.9591
worst12.1750120.7730738.49468115.65112.9616740.0670454.8316534.7737739.31062119.989548.12816104.208122.6439
std1.5979041.6369281.60245921.73811.6592655.3798668.2613724.5698238.42928710.427335.6296444.1401732.131154
median9.45122517.9308737.991690.9721110.2771132.8749254.2345630.6725933.18078104.67242.48599100.3615121.3222
rank13710269451281113
C11-F21 mean15.5639227.4808947.4150774.3044516.7397934.4850350.0550327.4699227.2507497.5284877.1208196.6437790.22154
best11.9623425.2695341.0009952.0345712.776527.4996444.7897920.4201124.6076188.3353563.2660383.1447955.52377
worst19.3718932.6484556.3206185.3879120.5770646.0143359.9626934.3075430.17099110.6057103.444115.6542116.5376
std3.0988763.4754447.08541615.105453.2400048.0546317.083436.2575892.84348310.4716517.9198713.6990726.29488
median15.4607326.0027946.1693379.8976616.802832.2130747.7338127.5760127.1121995.5864570.886693.8880494.41237
rank15792684313101211
C11-F22 mean3.70326715.0255725.7456226.004124.06995217.5227521.8983113.494215.56820.1373925.8697521.4981125.49193
best2.68 × 10−1011.7567324.8390125.156832.85 × 10−108.51038517.241326.54896110.7223617.3914824.191417.4910424.75272
worst14.8130719.2917726.9422527.7876916.2798121.6313824.8261116.513122.0710622.5615827.7654725.5857926.07053
std7.4065343.1295991.0511121.2094798.1399056.0773773.2575214.678195.3899972.1790181.5241993.7538190.558841
median1.37 × 10−914.5268825.6006125.535971.48 × 10−919.9746222.762915.4573614.739320.2982525.7610721.4578125.57223
rank14111326935712810
Sum rank3891161226481401549691205166189213
Mean rank1.8095244.3333337.66666710.76192.2857146.6666677.3333334.5714294.3333339.7619057.904762910.14286
Total rank1371225643108911
Wilcoxon: p-value1.81 × 10−71.09 × 10−141.71 × 10−150.5220698.49 × 10−152.42 × 10−146.09 × 10−102.71 × 10−125.56 × 10−157.92 × 10−158.57 × 10−147.92 × 10−15
Table 9. Performance of optimization algorithms on the pressure vessel design problem.
Table 9. Performance of optimization algorithms on the pressure vessel design problem.
AlgorithmOptimum VariablesOptimum Cost
TsThRL
SOA0.7780270.38457940.312282005882.901
WSO0.7780270.38457940.312282005882.901
AVOA0.7780270.38457940.312282005882.901
RSA0.8738040.74365441.801672007974.077
MPA0.7780270.38457940.312282005882.901
TSA0.7805890.39078740.347952005925.31
WOA0.9565710.4832749.13929105.25536351.3
MVO0.8604960.42636144.58435148.04646044.069
GWO0.7798110.3872640.39606198.84515892.513
TLBO1.9973460.97659375.7257377.8660224,264.99
GSA1.1297840.55845258.53792108.22029777.026
PSO1.3818560.69689260.63946169.947916,744.76
GA0.9622252.05086647.54579183.933514,893.63
Table 10. Statistical results of optimization algorithms on pressure vessel design problem.
Table 10. Statistical results of optimization algorithms on pressure vessel design problem.
AlgorithmMeanBestWorstStdMedianRank
SOA5882.9015882.9015882.9011.87 × 10−125882.9011
WSO5906.9615882.9016118.93658.510935882.9013
AVOA6318.025883.4537029.777356.80456324.1896
RSA12,261.317974.07720,310.053514.1911,063.859
MPA5889.2375883.8565893.8362.7081825888.8982
TSA6257.6255925.317013.933368.4926102.7275
WOA7683.1516351.311,250.021293.8397295.0168
MVO6668.446044.0697322.993415.19496676.8027
GWO6112.1935892.5137112.804416.03915909.9544
TLBO42,144.2724,264.9983,025.0916,815.7238,698.0712
GSA21,921.339777.02635,302.667701.70721,288.7210
PSO42,957.1316,744.7675,536.8815,181.1746,304.6513
GA40,300.8214,893.6398,448.0218,294.2035,672.0711
Table 11. Performance of optimization algorithms on the speed reducer design problem.
Table 11. Performance of optimization algorithms on the speed reducer design problem.
AlgorithmOptimum VariablesOptimum Cost
bMpb1b2d1d2
SOA3.50.7177.37.83.3502155.2866832996.348
WSO3.50.7177.3000037.8000063.3502155.2866832996.349
AVOA3.50.7177.37.83.3502155.2866832996.348
RSA3.60.7177.38.33.3513455.53188.612
MPA3.50.7177.37.83.3502155.2866832996.348
TSA3.5020040.7177.5313617.9188443.3539675.2928293006.665
WOA3.5242150.700895177.8594277.9572613.3512785.3173213038.216
MVO3.5031850.7177.4258177.8394153.3741155.2869343005.882
GWO3.5007890.7177.4682657.8185223.3516425.2867642998.965
TLBO3.5750750.71037626.877477.4125547.9338673.4010065.3080595360.781
GSA3.5557410.70128617.262898.1531218.1590383.4433035.3482423150.666
PSO3.5778730.70314517.694957.756698.0489953.7363365.28893289.236
GA3.570420.70061520.378758.0290338.0555973.4339435.3625343758.934
Table 12. Statistical results of optimization algorithms on the speed reducer design problem.
Table 12. Statistical results of optimization algorithms on the speed reducer design problem.
AlgorithmMeanBestWorstStdMedianRank
SOA2996.3482996.3482996.3489.33 × 10−132996.3481
WSO2996.8452996.3493004.8061.8766912996.3672
AVOA3000.5072996.3483010.1094.6654882999.1254
RSA3273.6283188.6123363.87360.876043258.2388
MPA2999.1182996.443002.0781.8411972999.3383
TSA3038.4843006.6653059.81714.52613039.6877
WOA3354.0823038.2165582.461653.64653113.0289
MVO3032.4623005.8823058.96416.874543032.8466
GWO3004.4462998.9653009.5523.5423393004.0595
TLBO6.67 × 10135360.7812.32 × 10147.04 × 10134.58 × 101312
GSA3498.0233150.6664508.259317.10423414.03710
PSO1.32 × 10143289.2361.32 × 10152.95 × 10144.5 × 101313
GA6.44 × 10133758.9345.09 × 10141.34 × 10141.05 × 101311
Table 13. Performance of optimization algorithms on the welded beam design problem.
Table 13. Performance of optimization algorithms on the welded beam design problem.
AlgorithmOptimum VariablesOptimum Cost
hltb
SOA0.205733.4704899.0366240.205731.724852
WSO0.205733.4704899.0366240.205731.724852
AVOA0.205733.4704899.0366240.205731.724852
RSA0.1501344.75063100.2063131.979431
MPA0.205733.4704899.0366240.205731.724852
TSA0.2013533.5849299.0450020.2057821.735242
WOA0.204883.5461888.9216860.2110651.754013
MVO0.2031463.5392769.036610.2058191.730769
GWO0.2054623.4799.0367290.2057591.72583
TLBO0.2365317.251667.680740.298462.791973
GSA0.1923984.1857349.3458580.2193641.964869
PSO0.2406864.0958097.0001390.5980263.90663
GA0.1339496.6133119.4701630.3961973.852019
Table 14. Statistical results of optimization algorithms on the welded beam design problem.
Table 14. Statistical results of optimization algorithms on the welded beam design problem.
AlgorithmMeanBestWorststdMedianRank
SOA1.7248521.7248521.7248526.83 × 10−161.7248521
WSO1.7248521.7248521.7248544.32 × 10−71.7248522
AVOA1.7592091.7251171.8924830.0418771.746567
RSA2.2791391.9794312.7511240.2118742.2587558
MPA1.7265151.7254411.7280920.0009721.7263463
TSA1.7468631.7352421.7615330.0068751.7469785
WOA2.5094991.7540134.6296880.8753012.1104710
MVO1.7490391.7307691.8072430.0174141.7450696
GWO1.7272891.725831.7300010.0010781.7270854
TLBO4.32 × 10132.7919734.69 × 10141.24 × 10145.33106612
GSA2.4552321.9648692.8761070.2585652.3916029
PSO7.27 × 10133.906633.02 × 10141.16 × 10145.09 × 101213
GA3.15 × 10133.8520192.37 × 10147.65 × 10135.3604511
Table 15. Performance of optimization algorithms on the tension/compression spring design problem.
Table 15. Performance of optimization algorithms on the tension/compression spring design problem.
AlgorithmOptimum VariablesOptimum Cost
dDP
SOA0.0516890.35671811.288970.012665
WSO0.0516890.35671611.289060.012665
AVOA0.0516890.35671811.288970.012665
RSA0.050.310493150.013196
MPA0.0516880.35670311.289820.012665
TSA0.0515040.35215211.577710.012683
WOA0.0521210.3672110.699360.012669
MVO0.0613210.6352213.9763560.014275
GWO0.051710.35711811.272380.012674
TLBO0.0692750.93972420.018039
GSA0.055140.4399268.9214180.014608
PSO0.0689940.93343220.017773
GA0.0693080.94096120.01808
Table 16. Statistical results of optimization algorithms on the tension/compression spring design problem.
Table 16. Statistical results of optimization algorithms on the tension/compression spring design problem.
AlgorithmMeanBestWorstStdMedianRank
SOA0.0126650.0126650.0126651.19 × 10−180.0126651
WSO0.0126830.0126650.0128514.4 × 10−50.0126663
AVOA0.0133160.0126840.0148680.0006550.0130497
RSA0.0267370.0131960.1526460.0330290.01333311
MPA0.0126760.0126660.012697.04 × 10−60.0126742
TSA0.0129670.0126830.0136630.000280.0128835
WOA0.0132660.0126690.015580.0007390.0130416
MVO0.0177160.0142750.0184340.0009850.0179888
GWO0.0127690.0126740.0131950.000120.0127274
TLBO0.0186330.0180390.0192210.0003320.0187119
GSA0.0191850.0146080.0360670.0047350.01765810
PSO1.99 × 10130.0177733.97 × 10148.88 × 10130.01777313
GA3.38 × 10120.018085.31 × 10131.19 × 10130.02621912
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Dehghani, M.; Trojovský, P. Serval Optimization Algorithm: A New Bio-Inspired Approach for Solving Optimization Problems. Biomimetics 2022, 7, 204. https://doi.org/10.3390/biomimetics7040204

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Dehghani M, Trojovský P. Serval Optimization Algorithm: A New Bio-Inspired Approach for Solving Optimization Problems. Biomimetics. 2022; 7(4):204. https://doi.org/10.3390/biomimetics7040204

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Dehghani, Mohammad, and Pavel Trojovský. 2022. "Serval Optimization Algorithm: A New Bio-Inspired Approach for Solving Optimization Problems" Biomimetics 7, no. 4: 204. https://doi.org/10.3390/biomimetics7040204

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