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Frilled Lizard Optimization: A Novel Nature-Inspired Metaheuristic Algorithm for Solving Optimization Problems

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14 March 2024

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15 March 2024

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Abstract
This article introduces a novel nature-inspired metaheuristic algorithm called Frilled Lizard Optimization (FLO), which emulates the hunting behavior of frilled lizards in their natural habitat. FLO draws in-spiration from the sit-and-wait strategy observed in frilled lizards during hunting. The underlying theory of FLO is presented and mathematically formulated in two phases: (i) an exploration phase, simulating the frilled lizard's attack towards prey, and (ii) an exploitation phase, simulating the lizard's retreat to the top of the tree after feeding. To assess FLO's efficacy in solving optimization problems, the algorithm's performance is evaluated across fifty-two standard benchmark functions, encompassing unimodal, high-dimensional multimodal, fixed-dimensional multimodal, and the CEC 2017 test suite. Comparative analyses with twelve existing metaheuristic algorithms are conducted. The simulation results reveal that FLO, distinguished by its adeptness in exploration, exploitation, and balancing them during search process, outperforms competing algorithms. Additionally, FLO is implemented on twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems, demonstrating its effectiveness in addressing real-world optimization applications.
Keywords: 
Subject: Computer Science and Mathematics  -   Data Structures, Algorithms and Complexity

1. Introduction

In optimization, the aim is to determine a best solution from a set of options for a specific problem [1]. Mathematically, optimization problems consist of decision variables, constraints, and an objective function. The objective is to assign appropriate values to the decision variables in order to maximize or minimize the objective function while adhering to the problem's constraints [2]. Regarding such optimization problems, problem solving techniques can be partitioned into two main groups: deterministic and stochastic approaches [3]. Deterministic methods are particularly useful for solving linear, convex, low-dimensional, continuous, and differentiable problems [4]. However, as problems become more intricate and dimensions increase, deterministic approaches may struggle with being trapped in local optima and providing suboptimal solutions [5]. Conversely, within science, engineering, industry, technology, and practical applications, numerous intricate optimization problems exist that are characterized as non-convex, non-linear, discontinuous, non-differentiable, complex, and high-dimensional. Due to the inefficiencies and challenges associated with deterministic methods in addressing these optimization issues, scientists have turned to developing stochastic approaches [6].
Metaheuristic algorithms stand out as highly effective stochastic methods capable of offering viable solutions for optimization challenges, all without requiring derivative information. They rely on random exploration within the solution space, utilizing random operators and trial-and-error strategies. Their advantages include straightforward concepts, straightforward implementation, proficiency in tackling varied optimization problems, no matter how complex or high-dimensional they may be, as well as adaptability to nonlinear and unfamiliar search spaces. As a result, the popularity and extensive use of metaheuristic algorithms continue to grow [7]. In metaheuristic algorithms, the optimization process begins by randomly generating a set of candidate solutions at the start of the algorithm. These candidate solutions are then enhanced and modified by the algorithm during a certain number of iterations following its implementation steps. Upon completion of the algorithm, the best candidate solution found during its execution is put forward as the proposed solution to the problem [8]. This random search element in metaheuristic algorithms means that achieving a global optimum cannot be guaranteed using these methods. Nonetheless, the solutions derived from these algorithms, being near the global optimum, are deemed acceptable as quasi-optimal solutions [9].
For a metaheuristic algorithm to effectively carry out the optimization process, it needs to thoroughly explore the solution space on both a global and local scale. Global searching, through exploration, allows the algorithm to pinpoint the optimal area by extensively surveying all parts of the search space and avoiding narrow solutions. Local searching, through exploitation, helps the algorithm converge to solutions near a global optimum by carefully examining surrounding areas and promising solutions. Success in the optimization process hinges on striking a balance between exploration and exploitation during the search [10]. Researchers' desire to improve optimization outcomes has resulted in the development of many metaheuristic algorithms.
The key question at hand is whether, based on the available metaheuristic algorithms, there remains a need in scientific research to develop new metaheuristic algorithms. The concept of No Free Lunch (NFL) [11] addresses this by highlighting that while a metaheuristic algorithm may perform well in solving a particular set of optimization problems, it might not guarantee the same solution quality for different optimization problems. The NFL theorem suggests that there is no one-size-fits-all optimal metaheuristic algorithm for all types of optimization problems. It is conceivable that an algorithm may efficiently reach a global optimum for one problem but struggle to do so for another, possibly getting stuck at a local optimum. As a result, the success or failure of employing a metaheuristic algorithm for an optimization problem cannot be definitively assumed.
The novelty of this paper is the introduction of a new innovative bio-metaheuristic algorithm called Frilled Lizard Optimization (FLO) to solve optimization problems in different research fields and real-world applications. The main contributions of this investigation can be summarized as follows:
  • FLO is based on the imitation of the natural behavior of the frilled lizard in the wild.
  • The basic inspiration of FLO is derived from (i) the hunting strategy of the frilled lizard and (ii) the retreat of this animal to the top of the tree after feeding.
  • The theory of FLO is described and its implementation steps are mathematically modeled in two phases (i) exploration based on the simulation of the frilled lizard's attack towards the prey and (ii) exploitation based on the simulation of the retreat of the frilled lizard to the top of the tree after feeding.
  • The performance of FLO has been tested on fifty-two standard benchmark functions of various types of unimodal, high-dimensional multimodal, fixed-dimensional multimodal, and the CEC 2017 test suite.
  • FLO has been applied to real-world problems, and its performance is evaluated on twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems.
  • The results obtained from FLO are compared with the performance of other available metaheuristic algorithms.
This paper is organized as follows: Section 2 contains a review of the relevant literature. Section 3 describes the proposed Frilled Lizard Optimization (FLO) and gives a mathematical model. Then Section 4 presents the results of our simulation studies. Section 5 investigates the effectiveness of FLO in solving real-world applications, and Section 6 provides some conclusions and suggestions for future research.

2. Literature Review

Metaheuristic algorithms are created by drawing upon a range of influences from natural phenomena, behaviors of living organisms in nature, biological sciences, genetics, physical laws, human behavior, and other evolutionary processes. These algorithms are categorized into four groups, namely swarm-based, evolutionary-based, physics-based, and human-based approaches – depending on the inspiration behind their design.
Swarm-based metaheuristic algorithms leverage inspiration from the collective behavior and strategies observed in various natural systems, particularly those of animals, aquatic organisms, and insects in the wild. The most frequent swarm-based metaheuristics are Particle Swarm Optimization (PSO) [12], Ant Colony Optimization (ACO) [13], Artificial Bee Colony (ABC) [14], and Firefly Algorithm (FA) [15]. PSO has replicated the collective movements of birds and fish as they search for food. ACO has imitated the behavior of ants in identifying the best path of communication between their colony and a food source. ABC has mirrored the actions of honey bees within a colony when seeking out food sources. FA draws inspiration from how fireflies communicate through optical signals. The most prominent natural behaviors among animals are foraging, hunting, migration, digging, and chasing, which have been sources of inspiration in the design of several metaheuristic algorithms such as: Greylag Goose Optimization (GGO) [1], African Vultures Optimization Algorithm (AVOA) [16], Marine Predator Algorithm (MPA) [17], Gooseneck Barnacle Optimization Algorithm (GBOA) [18], Grey Wolf Optimizer (GWO) [19], electric eel foraging optimization (EEFO) [20], White Shark Optimizer (WSO) [21], Crested Porcupine Optimizer (CPO) [22], Tunicate Swarm Algorithm (TSA) [23], Orca Predation Algorithm (OPA) [24], Honey Badger Algorithm (HBA) [25], Reptile Search Algorithm (RSA) [26], Golden Jackal Optimization (GJO) [27], and Whale Optimization Algorithm (WOA) [28].
Evolutionary-based metaheuristics are introduced, drawing inspiration from the fundamental concepts of genetics, biology, natural selection, survival of the fittest, and Darwin's evolutionary theory. Notable examples within this category include Genetic Algorithm (GA) [29] and Differential Evolution (DE) [30] are among the most popular evolutionary-based metaheuristic algorithms that are developed by imitating the generation process, biological which stand out as widely adopted algorithms. These evolutionary-based metaheuristics emulate various biological processes such as the generation mechanism, principles of genetics, natural selection, and the incorporation of random operators like selection, mutation, and crossover. Additionally, the Artificial Immune Systems (AISs) algorithm is conceived, taking cues from the human body's immune system and its adept defense mechanisms against germs and diseases [31]. Other prominent members of evolutionary-based metaheuristics include Genetic programming (GP) [32], Cultural Algorithm (CA) [33], and Evolution Strategy (ES) [34].
Physics-based metaheuristic algorithms are introduced, drawing inspiration from the modeling of forces, laws, phenomena, and other fundamental concepts in physics. Simulated Annealing (SA) [35] , a widely employed physics-based metaheuristic algorithm, takes its design cues from the physical phenomenon of metal annealing. This process involves the melting of metals under heat, followed by a gradual cooling and freezing process to attain an ideal crystal structure. Gravitational Search Algorithm (GSA) [36] is crafted by modeling physical gravitational forces and applying Newton's laws of motion. Concepts derived from cosmology and astronomy serve as the foundation for algorithms like Multi-Verse Optimizer (MVO) [37] and Black Hole Algorithm (BHA) [38]. Some other physics-based metaheuristic algorithms are: Thermal Exchange Optimization (TEO) [39], Prism Refraction Search (PRS) [40], Equilibrium Optimizer (EO) [41], Archimedes Optimization Algorithm (AOA) [42], Lichtenberg Algorithm (LA) [43], Water Cycle Algorithm (WCA) [44], and Henry Gas Optimization (HGO) [45].
Human-based metaheuristic algorithms are introduced, seeking inspiration from the behaviors, decisions, thoughts, and various strategies exhibited by humans in both individual and social contexts. Teaching-Learning Based Optimization (TLBO) [46] is one of the most widely used human-based metaheuristics, which imitates the relationships and educational interactions in the classroom between the teacher and students and among the students. A Mother Optimization Algorithm (MOA) is designed based on modeling Eshrat's care of her children [9]. The strategy of the soldiers and their movements during a battle in ancient wars was the basic idea in the development of War Strategy Optimization (WSO) [47]. The efforts of both the rich and the poor in the society to improve their financial and economic status has been the main idea in the design of Poor and Rich Optimization (PRO) [48]. Some other human-based metaheuristic algorithms are: Coronavirus Herd Immunity Optimizer (CHIO) [49], Gaining Sharing Knowledge based Algorithm (GSK) [50], and Ali Baba and the Forty Thieves (AFT) [51].
Based on the above literature review, no metaheuristic algorithm does exist based on the simulation of the natural behavior of frilled lizard in the wild. Meanwhile, the strategy of the frilled lizard during hunting and retreating to the top of the tree after feeding are intelligent processes that can be the basis for the design of a new optimizer. In order to address this gap, based on the mathematical modeling of two natural behaviors of the frilled lizard: (i) attacking the prey and (ii) retreating to the top of the tree, a new metaheuristic algorithm has been developed, which is discussed in the subsequent section.

3. Frilled Lizard Optimization

In this section, the source of inspiration used in the development and theory of Frilled Lizard Optimization (FLO) is stated. Then the corresponding implementation steps are mathematically modeled in order to be used for the solution of optimization problems.

3.1. Inspiration of FLO

The frilled lizard (Chlamydosaurus kingii), is a species of lizard from the family Agamidae, which is native to southern New Guinea and northern Australia [52]. The frilled lizard is an arboreal species and diurnal that spends more than 90% of each day up in the trees [53]. During the short time that this animal is on the ground, it is busy with feeding, socializing or traveling to a new tree [52]. A frilled lizard can move bipedally and do this when hunting or escaping from predators. To keep balanced, it leans its head far back enough, so it lines up behind the tail base [52,54]. The total length of the frilled lizard is about 90 centimeters, a head-body length of 27 centimeters, and weighs up to 600 grams [55]. The frilled lizard has a special wide and big head with a long neck to accommodate the frill. It has long legs for running and a tail that makes most of the total length of this animal [56]. The male species is larger than the female species and has proportionally bigger jaw, head, and frill [57]. A picture of a frilled lizard is shown in Figure 1.
The main diet of the frilled lizard are insects and other invertebrates, although it also rarely feeds on vertebrates. Prominent prey includes centipedes, ants, termites, and moth larvae [58]. The frilled lizard is a sit-and-wait predator that looks for potential prey. After seeing the prey, the frilled lizard runs fast on two legs and attacks the prey to catch it and feed on it. After feeding, the frilled lizard retreats back up a tree [52].
Among the frilled lizard's natural behaviors, its sit-and-wait hunting strategy to catch prey and retreat to the top of the tree after feeding is much more prominent. These natural behaviors of frilled lizard are intelligent processes that are the fundamental inspiration in designing the proposed FLO approach.

3.2. Initialization of the Algorithm

The proposed FLO method is a metaheuristic algorithm that considers frilled lizards as its members. FLO efficiently discovers optimal solutions for optimization challenges by leveraging the search capabilities of its members within the problem-solving space. Each frilled lizard establishes value assignments for the decision variables according to its particular location in the problem-solving space. Consequently, every frilled lizard represents a potential solution that can be interpreted mathematically through a vector. Collectively, the frilled lizards constitute the FLO population, which can be mathematically characterized as a matrix using Equation (1). The initial placements of the frilled lizards within the problem-solving space are established through random initialization using Equation (2):
X = X 1 ⋮ X i ⋮ X N N × m = x 1,1 ⋯ x 1 , d ⋯ x 1 , m ⋮ ⋱ ⋮ ⋰ ⋮ x i , 1 ⋯ x i , d ⋯ x i , m ⋮ ⋰ ⋮ ⋱ ⋮ x N , 1 ⋯ x N , d ⋯ x N , m N × m
x i , d = l b d + r · ( u b d − l b d )
Here X denotes the FLO population matrix, X i represents the i th frilled lizard (candidate solution), x i , d denotes its d th dimension in the search space (decision variable), N gives the number of frilled lizards, m is the number of decision variables, r represents a random number from the interval 0,1 , l b d , and u b d are a lower bound and an upper bound on the d th. decision variable, respectively.
Considering that each frilled lizard represents a candidate solution for the problem, corresponding to each candidate solution, the corresponding objective function value can be calculated for the problem. The set of determined objective function values can be represented mathematically using the vector given in Equation (3):
F = F 1 ⋮ F i ⋮ F N N × 1 = F ( X 1 ) ⋮ F ( X i ) ⋮ F ( X N ) N × 1
Here F denotes the vector of calculated objective function values and F i gives the evaluated objective function value corresponding to the i th frilled lizard.
The determined objective function values are appropriate criteria for measuring the quality of the population individuals (i.e., candidate solutions). In particular, the best evaluated value for the objective function corresponds to the best individual of the population (i.e., the best candidate solution) and similarly, the worst evaluated value for the objective function corresponds to the worst individual of the population (i.e., the worst candidate solution). Since in each iteration of FLO, the position of the frilled lizards is updated in the solution space, new values are also evaluated for the objective function of the problem. Consequently, in each iteration the position of the best individual (i.e., the best candidate solution) must also be updated. At the end of the FLO implementation, the best candidate solution obtained during the iterations of the algorithm is presented as the solution to the problem.

3.3. Mathematical Modelling of FLO

In the FLO design, in each iteration, the position of the frilled lizard in the problem solving space is updated in two phases (i) exploration based on the simulation of the frilled lizard’s movement towards the prey during hunting and (ii) exploitation based on the simulation of the frilled lizard’s movement towards the top of the tree after feeding.

3.3.1. Phase 1: Hunting Strategy (Exploration)

One of the most characteristic natural behaviors of the frilled lizard is the hunting strategy of this animal. The frilled lizard is a sit-and-wait predator that attacks its prey after seeing it. The simulation of frilled lizard's movement towards the prey leads to extensive changes in the position of the population members in the problem-solving space and as a result increases the exploration power of the algorithm for global search. In the first phase of FLO, the position of the population individuals in the solution space of the problem is updated based on the frilled lizard's hunting strategy. In the design of FLO, for each frilled lizard, the position of other population members who have a better objective function value is considered as the prey position. According to this, the set of candidate preys' positions for each frilled lizard is determined using Equation (4):
C P i = X k : F k < F i   a n d   k ≠ i ,   i = 1,2 ,   … ,   N   a n d   k ∈ 1,2 ,   … ,   N
Here, C P is the candidate preys set for the i th frilled lizard, X k is the population member with a better objective function value than the i th frilled lizard, and F k is its objective function value.
In the FLO design, it is assumed that the frilled lizard randomly chooses one of these candidate preys and attacks it. Based on the modeling of the frilled lizard's movement towards the chosen prey, a new position for each individual of the population has been calculated using Equation (5). Then, if the objective function value is better, this new position replaces the previous position of the corresponding individual using Equation (6):
x i , d P 1 = x i , d + r · S P i , d − I · x i , d ,     i = 1,2 ,   … ,   N ,   a n d   d = 1,2 ,   … , m
X i = X i P 1 ,     F i P 1 < F i X i ,     e l s e
Here, X i P 1 denotes the new suggested position of ith frilled lizard based on the first phase of FLO, x i , d P 1 represents its d th dimension, F i P 1 denotes its objective function value, r is a random number with a normal distribution from the interval 0,1 , S P i , d denotes the d th dimension of the selected prey for the i th frilled lizard, I is a random number from the set 1,2 , N is the number of frilled lizards, and m gives the number of decision variables.

3.3.2. Phase 2: Moving up the Tree (Exploitation)

After feeding, the frilled lizard retreats to the top of a tree near its position. Simulating the movement of the frilled lizard to the top of the tree leads to small changes in the position of the population individuals in the solution space of the problem and as a result, increasing the exploitation power of the algorithm for local search. In the second phase of FLO, the position of the population individuals in the solution space is updated based on the frilled lizard's strategy when retreating to the top of the tree after feeding.
Based on modeling the movement of the frilled lizard to the top of the nearby tree, a new position for each population individual is calculated using Equation (7). Then this new position, if it improves the objective function value, replaces the previous position of the corresponding individual using Equation (8):
x i , d P 2 = x i , d + 1 − 2 r · u b d − l b d t ,   i = 1,2 ,   … ,   N ,     d = 1,2 ,   … , m ,     a n d   t = 1,2 ,   … ,   T
X i = X i P 2 ,     F i P 2 < F i X i ,     e l s e
Here X i P 2 denotes the new suggested position of the i th frilled lizard based on the second phase of FLO, x i , d P 2 represents its d th dimension, F i P 2 gives its objective function value, t represents the iteration counter of the algorithm, and T describes the maximum number of iterations of the algorithm.

3.4. Repetition Process, Pseudocode, and Flowchart of FLO

The first iteration of FLO is completed after updating the position of all frilled lizards in the problem solving space based on the first and second phases. After that, with the new updated values, the algorithm starts with the next iteration and the process of updating the position of the frilled lizards continues until the algorithm finishes using Equations (4) to (8). In each iteration, the best candidate solution is also updated and stored based on the comparison of the obtained objective function values. After the complete implementation of the algorithm, the best candidate solution obtained during the iterations of the algorithm is presented as the FLO solution for the given problem. The implementation steps of FLO are shown as a flowchart in Figure 2, and its pseudocode is presented in Algorithm 1.
Algorithm 1. Pseudocode of FLO.
Start FLO.
1. Input the problem information: variables, constraints, and the objective function.
2. Set the FLO population size (N) and number of iterations (T).
3. Generate the initial population matrix using Equation (2). x i , d ← l b d + r · ( u b d − l b d )
4. Evaluate the objective function.
5. For t = 1 to T
6. For   i = 1 to N
7. Phase 1: Hunting strategy (exploration)
8. Determine the candidate preys set using Equation (4). C P i ← X k i : F k i < F i   a n d   k i ≠ i
9. Choose the prey for the ith frilled lizard at random.
10. Calculate the new position of ith frilled lizard using Equation (5). x i , d P 1 ← x i , d + r · S P i , d − I · x i , d
11. Update ith FLO member using Equation (6). X i ← X i P 1 ,     F i P 1 < F i X i ,     e l s e
12. Phase 2: Moving up the tree (exploitation)
13. Calculate the new position of ith frilled lizard using Equation (7). x i , d P 2 ← x i , d + ( 1 − 2 r ) · u b d − l b d t
14. Update ith frilled lizard using Equation (8). X i ← X i P 2 ,     F i P 2 < F i X i ,     e l s e
15. end
16. Save the best candidate solution so far.
17. end
18. Output the best quasi-optimal solution obtained with the FLO.
End FLO.

3.5. Computational Complexity of FLO

In this subsection, the computational complexity of FLO is evaluated. The preparation and initialization steps of FLO have a computational complexity of O(Nm), where N denotes the number of frilled lizards and m gives the number of decision variables of the problem. In the FLO design, the position of each frilled lizard is updated in each iteration in two phases of exploration and exploitation. Therefore, the update process in FLO has a computational complexity of O(2TNm), where T is the maximum number of iterations of the algorithm. According to this, the overall computational complexity of the proposed FLO approach is O ( N m ( 1 + 2 T ) ) .

4. Simulation Studies and Results

In this section, the performance of the developed FLO algorithm in dealing with optimization tasks is evaluated. For this purpose, a set of fifty-two standard benchmark functions of unimodal, high-dimensional multimodal, fixed-dimensional multimodal types [59], and the CEC 2017 test suite [60] have been employed. The results obtained by using FLO have been compared with the performance of twelve well-known metaheuristic algorithms: GA [29], PSO [12], GSA [36], TLBO [46], MVO [37], GWO [19], WOA [28], MPA [17], TSA [23], RSA [26], AVOA [16], and WSO [21]. The values of the control parameters of the metaheuristic algorithms are determined in Table 1. In order to optimize the objective functions F1 to F23, FLO and each of the competitive algorithms have been employed in 30 independent runs, where each run includes 30,000 function evaluations (FEs) and a population size of 30 is chosen. In handling the CEC 2017 test suite, the FLO approach and each of the competitive algorithms have been implemented in 51 independent runs, where each independent run includes 10,000 · m ( m is the number of variables) of Fs and a population size of 30. The simulation outcomes are presented through six statistical indicators, namely mean, best, worst, standard deviation (std), median, and rank. To establish the ranking of the metaheuristic algorithms for optimizing each benchmark function, a comparison of the mean index values has been employed.

4.1. Evaluation of Unimodal Functions

The unimodal variables F1 to F7, due to not having a local optimum, are suitable criteria for measuring the ability of the metaheuristic algorithms in local exploitation and search. The implementation results of FLO and the competitive algorithms for the functions F1 to F7 are reported in Table 2. Based on the results, FLO with high capability in exploitation and local search has been able to converge to the global optimum for the functions F1 to F6. Also, in solving the F7 function, FLO is the first best optimizer for this function. The comparison of the simulation results indicates that FLO, with its high exploitation ability, has delivered a superior performance in handling unimodal functions F1 to F7 against the competitive algorithms.

4.2. Evaluation of High-Dimensional Multimodal Functions

The high-dimensional multimodal functions F8 to F13, due to having multiple local optima, are suitable criteria for challenging the ability of the metaheuristic algorithms in global exploration and search. The optimization results for the functions F8 to F13 using FLO and the competitive algorithms are reported in Table 3. Based on the obtained results, FLO with its high ability in exploration has been able to provide a global optimum for these functions by discovering the main optimal region in dealing with the F9 and F11 functions. Also, in order to optimize the functions F8, F10, F12, and F13, FLO is the first best optimizer for these functions. The analysis of the simulation results shows that FLO with high capability in exploration and global search in order to cross local optima and discover the main optimal area turned out to be superior in competition with the compared algorithms.

4.3. Evaluation of Fixed-Dimensional Multimodal Functions

The fixed-dimension multimodal functions F14 to F23, having a smaller number of local optima compared to functions F8 to F13, are suitable criteria for measuring the ability of the metaheuristic algorithms in balancing exploration and exploitation. The results of employing FLO and the competitive algorithms for the functions F14 to F23 are reported in Table 4. It turned out that FLO is the best optimizer for functions F14 to F23. In cases where FLO has the same value for the mean index with some competitive algorithms, it has provided a more effective performance by providing a better value for the std index. The simulation results show that FLO, with an appropriate ability to balance exploration and exploitation, has a superior performance by providing better results for the benchmark functions in comparison with the competitive algorithms.
The convergence curves resulting from the execution of FLO and the competitive algorithms for the functions F1 to F23 are drawn in Figure 3.

4.4. Evaluation of the CEC 2017 Test Suite

In this subsection, the efficiency of FLO in handling the CEC 2017 test suite is evaluated. The CEC 2017 test suite includes thirty standard benchmark functions consisting of three unimodal functions of C17-F1 to C17-F3, seven multimodal functions of C17-F4 to C17-F10, ten hybrid functions of C17-F11 to C17-F20, and ten composite functions of C17-F21 to C17-F30. From this test suite, the C17-F2 function was not included in the simulation studies due to the instability of the behavior. A complete description and the details of the CEC 2017 test suite can be found in [60]. The optimization results of the CEC 2017 test suite using FLO and the competitive algorithms for the mentioned dimensions are reported in Table 5. Also, the boxplot diagrams resulting from the performance of the metaheuristic algorithms on this test suite are shown in Figure 4. Based on the obtained optimization results, FLO is the best approach for the functions: C17-F1, C17-F3 to C17-F21, C17-F23, C17-F24, and C17-F27 to C17-F30.
The optimization outcomes indicate that FLO has achieved favorable results for the benchmark functions due to its strong capabilities in both exploration and exploitation, as well as effectively balancing them throughout the search process. Through the comparison of the simulation results, it is clear that FLO surpasses the competitive algorithms for most benchmark functions, establishing itself as the top optimizer overall and demonstrating a superiority in handling the CEC 2017 test suite.

4.5. Statistical Analysis

A statistical analysis has been made to check whether the superiority of FLO against the competitive algorithms is significant from a statistical point of view. For this purpose, the non-parametric Wilcoxon rank sum test [61] has been used, which is useful in determining a significant difference between the averages of two data samples. In the Wilcoxon rank sum test, it is investigated whether there is a significant difference between the performance of two algorithms by using an index called p-value. The results of implementing the Wilcoxon rank sum test on the results of FLO compared to each of the competitive algorithms are given in Table 6. Based on the results, in cases where the p-value is less than 0.05, FLO has a statistically significant advantage in competition with the alternative metaheuristic algorithms. Basically, based on the statistical analysis, it is obvious that FLO has a significant statistical superiority compared to the competitive algorithms in handling all the studied benchmark functions.

5. FLO for Real-World Applications

In this section, the efficiency of FLO for the solution of optimization problems in real-world applications is investigated on twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems.

5.1. Evaluation of the CEC 2011 Test Suite

In this subsection, the performance of FLO and the competitive algorithms in handling optimization tasks in real-world applications is evaluated on the CEC 2011 test suite. This test suite has twenty-two constrained optimization problems under the headings: A full description and the details of the CEC 2011 test suite are provided in [62]. FLO and each of the competitive algorithms with a population size of 30 are employed to handle the CEC 2011 test suite in 25 independent runs, where each independent run contains 150,000 FEs. The results of implementing FLO and the competitive algorithms on the CEC 2011 test suite are reported in Table 7. The boxplot diagrams obtained from the performance of the metaheuristic algorithms are plotted in Figure 5. Based on the optimization results, FLO has been the best algorithm for the problems C11-F1 to C11-F22. The comparison of the simulation results shows that FLO has provided better results for the majority of the problems and has delivered a superior performance in handling the CEC 2011 test suite compared to the competitive algorithms. In addition, the values obtained for the p-value of the Wilcoxon rank sum test show that FLO leads to a significant statistical superiority over the competitive algorithms.

5.2. Pressure Vessel Design Problem

The pressure vessel design is an optimization problem with the schematic representation displayed in Figure 6, whose main design goal is to minimize construction cost. The mathematical model of this design can be given as follows [63]:
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 a n d 10 ≤ x 3 , x 4 ≤ 200 .
The pressure vessel design optimization results using FLO and the competitive algorithms are presented in Table 8 and Table 9. The convergence curve of FLO while reaching the solution during the iterations of the algorithm is plotted in Figure 7. The obtained results show that FLO has obtained an optimal design with the values of the design variables equal to (0.7780271, 0.3845792, 40.312284, 200). The corresponding objective function value is 5882.9013. The simulation results show that FLO has a better performance for the pressure vessel design by achieving superior results in comparison with the competitive algorithms.

5.3. Speed Reducer Design Problem

The speed reducer design is an optimization problem with the schematic shown in Figure 8, whose main design goal is the minimization of the weight of the speed reducer. The mathematical model of this design can be given as follows [64,65]:
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 , a n d 5 ≤ x 7 ≤ 5.5 .
The results of handling the speed reducer design using FLO and the competitive algorithms are presented in Table 10 and Table 11. The convergence curve of FLO while achieving the optimal design is drawn in Figure 9. The obtained results show that FLO has provided an optimal design, where the values of the design variables are equal to (3.5, 0.7, 17, 7.3, 7.8, 3.3502147, 5.2866832). The corresponding objective function value is equal to 2996.3482. The simulation results indicate that FLO has presented a better performance for the speed reducer design in comparison with the competitive algorithms.

5.4. Welded Beam Design

Welded beam design is an optimization problem of real-world applications with the schematic shown in Figure 10, whose main design goal is the minimization of the fabrication cost of the welded beam. The mathematical model of this design can be formulated as follows [28]:
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 ,
where
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 · 1 0 6 x 4 x 3 3 , p c x = 4.013 30 · 1 0 6 x 3 2 x 4 6 36 196 1 − x 3 28 30 · 1 0 6 4 ( 12 · 1 0 6 ) ,
with
0.1 ≤ x 1 , x 4 ≤ 2 a n d 0.1 ≤ x 2 , x 3 ≤ 10 .
The results of FLO and the competitive algorithms for the welded beam design are given in Table 12 and Table 13. The convergence curve of FLO while reaching the solution is plotted in Figure 11. The obtained results show that FLO has provided an optimal design, where the values of the design variables are equal to (0.2057296, 3.4704887, 9.0366239, 0.2057296). The corresponding objective function value is 1.7246798. From the analysis of the simulation results, it is obvious that FLO has a superior performance in dealing with welded beam design in comparison with the competitive algorithms.

5.5. Tension/Compression Spring Design

Tension/compression spring design is an optimization problem from real-world applications with the schematic representation displayed in Figure 12, where the main design objective is to minimize the weight of tension/compression spring. The mathematical model of this design can be formulated in the following way [28]:
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 a n d 2 ≤ x 3 ≤ 15
The results of employing FLO and the competitive algorithms to optimize the tension/compression spring design are presented in Table 14 and Table 15. The convergence curve of FLO while reaching the solution is drawn in Figure 13. The results show that FLO has obtained an optimal design, where the values of the design variables are equal to (0.0516891, 0.3567177, 11.288966) and the corresponding objective function value is equal to (0.0126019). It can be seen from the simulation results that FLO gives better results than the competitive algorithms for the tension/compression spring design problem.

5. Conclusions and Future Works

In this paper, a new bio-metaheuristic algorithm called Frilled Lizard Optimization (FLO) was introduced based on imitating the natural behavior of frilled lizard in the wild. The fundamental inspiration for FLO is derived from (i) the frilled lizard's sit-and-wait strategy during hunting and (ii) the frilled lizard's retreat to the top of the tree after feeding. The FLO theory was described and its implementation steps were mathematically modeled in two phases of exploration and exploitation based on simulating the natural behavior of frilled lizard in the wild. FLO was tested on fifty-two standard benchmark functions covering unimodal, high-dimensional multimodal, fixed-dimensional multimodal, and CEC 2017 test suite. The findings demonstrated that FLO effectively produced suitable results for the given objective functions, showcasing strong capabilities in exploration, exploitation, and maintaining a balance between them in problem-solving environments. These results were then contrasted with the performance of twelve well-known algorithms. The simulation results showed that by providing better results for the majority of the benchmark functions, FLO has been able to reach a superior performance in comparison to the competitive algorithms by being ranked as the best optimizer. Testing FLO on twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design challenges revealed its efficacy in handling real-world scenarios. A statistical analysis on the performance of FLO compared to the competitive algorithms showed that the proposed approach has a significant statistical superiority over the competitive algorithms.
The introduction of FLO enables several research tasks for further research in the future. The most special research potentials of this study are the design of binary and multi-objective versions of the proposed FLO approach. Employing FLO for the solution of optimization problems arising in science and real-world applications are other research proposals of this paper.

Author Contributions

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

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable

Data Availability Statement

The datasets are available from the authors on request.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Frilled lizard taken from: free media Wikimedia Commons.
Figure 1. Frilled lizard taken from: free media Wikimedia Commons.
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Figure 2. Flowchart of FLO.
Figure 2. Flowchart of FLO.
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Figure 3. Convergence curves of FLO and competitive algorithms performances for F1 to F23.
Figure 3. Convergence curves of FLO and competitive algorithms performances for F1 to F23.
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Figure 4. Boxplot diagrams of FLO and the performance of the competitive algorithms for the CEC 2017 test suite.
Figure 4. Boxplot diagrams of FLO and the performance of the competitive algorithms for the CEC 2017 test suite.
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Figure 5. Boxplot diagrams of FLO and the performances of the competitor algorithms on the CEC 2011 test suite.
Figure 5. Boxplot diagrams of FLO and the performances of the competitor algorithms on the CEC 2011 test suite.
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Figure 6. Schematic of the pressure vessel design.
Figure 6. Schematic of the pressure vessel design.
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Figure 7. FLO’s performance convergence curve for the pressure vessel design.
Figure 7. FLO’s performance convergence curve for the pressure vessel design.
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Figure 8. Schematic of the speed reducer design.
Figure 8. Schematic of the speed reducer design.
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Figure 9. FLO’s performance convergence curve for the speed reducer design.
Figure 9. FLO’s performance convergence curve for the speed reducer design.
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Figure 10. Schematic of the welded beam design.
Figure 10. Schematic of the welded beam design.
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Figure 11. FLO’s performance convergence curve for the welded beam design.
Figure 11. FLO’s performance convergence curve for the welded beam design.
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Figure 12. Schematic of the tension/compression spring design.
Figure 12. Schematic of the tension/compression spring design.
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Figure 13. FLO’s performance convergence curve for the tension/compression spring.
Figure 13. FLO’s performance convergence curve for the tension/compression spring.
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Table 1. Control parameter values.
Table 1. Control parameter values.
Algorithm Parameter Value
GA
Selection Roulette wheel (Proportionate)
Crossover Whole arithmetic (Probability = 0.8, α ∈ − 0.5 ,   1.5 )
Mutation Gaussian (Probability = 0.05)
PSO
Cognitive and social constant (C1, C2) = ( 2 ,   2 )
Velocity limit 10% of dimension range
Inertia weight Linear reduction from 0.9 to 0.1
GSA
Alpha, G0, Rnorm, Rpower 20, 100, 2, 1
TLBO
Teaching Factor (TF) TF = round ( 1 + r a n d )
MVO
wormhole existence probability (WEP) Min(WEP) = 0.2 and Max(WEP)=1.
Exploitation accuracy over the iterations (p) p = 6 .
GWO
Convergence parameter (a) a: Linear reduction from 2 to 0.
WOA
Convergence parameter (a) a: Linear reduction from 2 to 0.
l is a random number in the range of − 1,1 .
TSA
Pmin and Pmax 1, 4
c1, c2, c3 random numbers lie in the range of 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) 𝐹𝐴𝐷𝑠=0.2
Binary vector U= 0 or 1
RSA
Sensitive parameter α = 0.1
Sensitive parameter β = 0.01
Evolutionary Sense (ES) ES: randomly decreasing values between 2 and −2
WSO
Fmin and Fmax 0.07, 0.75
τ, ao, a1, a2 4.125, 6.25, 100, 0.0005
AVOA
w 2.5
P1, P2, P3 0.6, 0.4, 0.6
L1, L2 0.8, 0.2
Table 2. Optimization results for the unimodal functions.
Table 2. Optimization results for the unimodal functions.
FLO AVOA WSO RSA MPA TSA WOA GWO MVO TLBO GSA PSO GA
F1 best 0 4.822882 0 0 3.47E-52 1.32E-50 8.50E-171 0.096099 1.36E-61 5.35E-77 4.88E-17 0.000443 16.32806
mean 0 60.02965 0 0 1.75E-49 4.24E-47 1.30E-151 0.13629 1.61E-59 2.30E-74 1.21E-16 0.091953 27.78154
median 0 41.36897 0 0 3.79E-50 3.89E-48 2.00E-159 0.137102 9.80E-60 1.54E-75 1.03E-16 0.008853 25.68391
worst 0 217.602 0 0 1.51E-48 3.01E-46 2.40E-150 0.183344 7.03E-59 2.36E-73 3.40E-16 1.27308 51.8506
std 0 49.03061 0 0 3.65E-49 9.30E-47 5.60E-151 0.02579 1.99E-59 5.72E-74 6.65E-17 0.288752 9.721273
rank 1 11 1 1 5 6 2 9 4 3 7 8 10
F2 best 0 0.603391 1.20E-306 0 1.68E-29 1.84E-30 7.20E-118 0.145798 4.44E-36 8.04E-40 3.18E-08 0.041243 1.589689
mean 0 1.948988 9.90E-277 0 6.34E-28 1.92E-28 2.30E-105 0.236058 1.23E-34 6.16E-39 5.00E-08 0.815635 2.539698
median 0 1.39396 6.00E-290 0 3.20E-28 1.80E-29 3.10E-108 0.244414 5.92E-35 4.53E-39 4.67E-08 0.532063 2.497037
worst 0 6.781435 2E-275 0 4.29E-27 1.66E-27 2.50E-104 0.332 7.21E-34 2.22E-38 1.12E-07 2.270937 3.46705
std 0 1.648521 0.00E+00 0 1.02E-27 4.92E-28 6.40E-105 0.058527 1.82E-34 5.19E-39 1.74E-08 0.671498 0.506175
rank 1 11 2 1 7 6 3 9 5 4 8 10 12
F3 best 0 947.6498 0 0 5.64E-19 1.25E-21 1880.714 5.44143 2.15E-19 2.00E-29 224.0264 19.82675 1297.164
mean 0 1626.99 0 0 2.29E-12 1.08E-10 18179.06 14.54867 1.98E-14 3.50E-24 433.0901 353.5142 1975.532
median 0 1419.306 0 0 1.66E-13 9.79E-14 18511.54 10.81976 4.25E-16 3.68E-26 364.629 266.9079 1913.339
worst 0 3227.104 0 0 1.31E-11 1.78E-09 31594.58 44.57484 3.69E-13 3.28E-23 1080.509 933.9386 3150.434
std 0 583.2962 0 0 4.07E-12 4.05E-10 7950.636 10.00187 8.38E-14 1.01E-23 204.6704 267.9873 594.3514
rank 1 9 1 1 4 5 11 6 3 2 8 7 10
F4 best 0 10.85214 0.00E+00 0 2.75E-20 0.0000879 0.823893 0.242208 5.97E-16 5.29E-32 9.01E-09 2.085999 2.018782
mean 0 15.75337 3E-265 0 2.71E-19 4.03E-03 47.1994 0.49832 1.12E-14 1.67E-30 1.13E+00 5.719781 2.577042
median 0 16.18755 1.80E-282 0 2.36E-19 0.001339 50.48116 0.483681 5.78E-15 5.94E-31 0.826058 5.357815 2.53522
worst 0 21.70983 4.1E-264 0 8.75E-19 0.032632 83.53016 0.877153 5.23E-14 7.40E-30 4.488194 12.16864 3.636626
std 0 2.682766 0.00E+00 0 2.13E-19 0.007381 27.51566 0.178584 1.35E-14 2.23E-30 1.288828 2.325016 0.433841
rank 1 11 2 1 4 6 12 7 5 3 8 10 9
F5 best 0 1227.144 1.27E-06 7.93E-29 20.77433 23.38145 24.33873 25.16727 23.28628 23.30644 23.57596 23.93699 208.4007
mean 0 9836.204 1.30E-05 1.18E+01 21.24372 25.93746 24.87395 87.63958 24.21079 24.39873 40.1211 4200.597 542.2831
median 0 5109.367 8.55E-06 1.12E-28 21.21726 26.2519 24.67096 27.34075 23.89208 23.97967 23.99658 78.41897 433.1568
worst 0 84446.57 5.38E-05 26.40458 21.90432 26.31483 26.17246 344.199 24.734 26.18823 152.3277 82043.31 2055.752
std 0 18645.98 1.35E-05 1.37E+01 0.361087 0.732269 0.53677 94.27249 0.489025 0.869977 41.18185 18690.79 394.8645
rank 1 13 2 3 4 8 7 10 5 6 9 12 11
F6 best 0 15.44096 6.47E-09 3.336533 7.36E-10 2.325128 0.009582 0.072166 0.224723 0.212329 5.03E-17 0.00000173 14.21997
mean 0 91.90698 4.53E-08 5.881906 1.64E-09 3.353519 0.074298 0.137535 0.601908 1.1489 9.53E-17 5.78E-02 31.10185
median 0 63.37101 4.20E-08 6.270887 1.46E-09 3.457431 0.028788 0.145872 0.662447 1.108843 8.63E-17 0.001874 28.85646
worst 0 348.3797 1.24E-07 6.603377 4.37E-09 4.360664 0.297605 0.227803 1.140588 1.971716 1.65E-16 0.493414 57.16884
std 0 88.71011 3.05E-08 0.955084 8.69E-10 0.644215 0.094418 0.044022 0.284877 0.461983 3.45E-17 0.138033 12.58959
rank 1 13 4 11 3 10 6 7 8 9 2 5 12
F7 best 2.35E-06 1.22E-05 2.30E-06 5.58E-06 0.000102 0.001361 0.0000218 0.003619 0.000166 0.0000829 0.012869 0.062864 0.002764
mean 2.54E-05 8.43E-05 5.93E-05 2.97E-05 0.0005 0.003958 1.17E-03 0.010581 0.000759 1.40E-03 0.048101 0.16772 0.009646
median 1.83E-05 0.0000619 0.0000382 0.0000148 0.000488 0.003393 0.000748 0.010309 0.000772 0.001376 0.047213 0.161882 0.009274
worst 6.89E-05 3.10E-04 2.44E-04 1.23E-04 0.00082 0.009088 0.00492 0.020558 0.001783 0.002684 0.087051 0.374669 0.019985
std 2.02E-05 8.29E-05 6.81E-05 3.21E-05 0.0002 0.002175 0.001343 0.004677 0.000433 0.000817 0.023188 0.073415 0.004477
rank 1 4 3 2 5 9 7 11 6 8 12 13 10
Sum rank 7 15 72 20 32 50 48 36 59 35 54 65 74
Mean rank 1 2.142857 10.28571 2.857143 4.571429 7.142857 6.857143 5.142857 8.428571 5 7.714286 9.285714 10.57143
Total rank 1 2 12 3 4 8 7 6 10 5 9 11 13
Table 3. Optimization results for the high-dimensional multimodal functions.
Table 3. Optimization results for the high-dimensional multimodal functions.
FLO AVOA WSO RSA MPA TSA WOA GWO MVO TLBO GSA PSO GA
F8 best -12622.8 -9319.44 -12574.2 -6277.23 -10663.1 -7789.11 -12572.1 -9494.05 -7348.84 -7525.59 -4717.57 -8584.04 -9939.5
mean -12498.6 -7535.73 -12471.8 -6064.7 -9936.78 -6704.97 -11191.6 -8247.67 -6650.74 -6212.41 -3646.54 -7076.8 -8783.73
median -12577.8 -7475.79 -12561.2 -6088.96 -9959.64 -6677.51 -12087.2 -8140.19 -6654.03 -6229.64 -3578.49 -7214.2 -8749.63
worst -11936.3 -6637.72 -11957.2 -5569.4 -9389.03 -5089.32 -8167.12 -7387.54 -5723.04 -5269.79 -3021.3 -5668.31 -7523.94
std 194.2272 686.0549 179.0354 209.0513 341.0971 682.0853 1611.369 683.5917 439.8347 568.8056 464.0183 688.4831 597.0402
rank 1 7 2 12 4 9 3 6 10 11 13 8 5
F9 best 0 13.31572 0 0 0 81.74052 0 48.07879 0.00E+00 0 12.68706 36.24875 21.1603
mean 0 22.43339 0 0 0 157.6833 0 89.10431 1.55E-14 0 25.96315 61.67496 49.80422
median 0 20.66512 0 0 0 151.8098 0 88.42416 0.00E+00 0 24.01479 59.26511 47.92176
worst 0 41.85228 0 0 0 262.4813 0 135.9663 1.04E-13 0 44.40468 104.3443 70.04209
std 0 8.007302 0 0 0 47.39195 0 23.41107 3.02E-14 0 8.516422 17.5057 12.82893
rank 1 3 1 1 1 8 1 7 2 1 4 6 5
F10 best 8.88E-16 3.081216 8.88E-16 8.88E-16 8.88E-16 7.36E-15 8.88E-16 0.091628 7.36E-15 4.12E-15 4.24E-09 1.542411 2.62492
mean 8.88E-16 4.819446 8.88E-16 8.88E-16 3.96E-15 1.13E+00 3.80E-15 0.526357 1.53E-14 4.12E-15 7.48E-09 2.483992 3.256237
median 8.88E-16 4.717522 8.88E-16 8.88E-16 4.12E-15 2.03E-14 4.12E-15 0.176984 1.38E-14 4.12E-15 7.03E-09 2.490083 3.305859
worst 8.88E-16 7.467465 8.88E-16 8.88E-16 4.12E-15 3.072576 7.36E-15 2.290859 2.03E-14 4.12E-15 1.32E-08 4.606032 4.227951
std 0.00E+00 1.134893 0.00E+00 0.00E+00 7.38E-16 1.46E+00 2.11E-15 0.629188 3.30E-15 8.26E-31 2.17E-09 0.796999 0.36853
rank 1 11 1 1 3 8 2 7 5 4 6 9 10
F11 best 0 1.005423 0 0 0 0 0 0.231481 0 0 2.728462 0.002156 1.173211
mean 0 1.563093 0 0 0 0.008054 0 0.364028 0.00122 0 6.565133 0.168742 1.342053
median 0 1.458192 0 0 0 0.008191 0 0.379369 0 0 6.659052 0.111443 1.318588
worst 0 2.991765 0 0 0 0.018714 0 0.488181 0.017145 0 11.51061 0.797732 1.57193
std 0 0.504151 0 0 0 0.005847 0 0.076055 0.004166 0 2.528054 0.212292 0.115089
rank 1 7 1 1 1 3 1 5 2 1 8 4 6
F12 best 1.57E-32 0.868125 3.67E-10 0.700576 4.73E-11 0.944381 0.001117 0.00091 0.011442 0.021959 4.33E-19 0.0000973 0.055414
mean 1.57E-32 2.978079 2.35E-09 1.200098 1.85E-10 5.276134 0.018304 0.833065 0.036322 0.064967 1.91E-01 1.37E+00 0.250377
median 1.57E-32 2.63405 2.18E-09 1.265478 1.87E-10 3.920951 0.005268 0.382795 0.034529 0.062564 0.073046 1.170635 0.24084
worst 1.57E-32 6.729691 7.13E-09 1.499107 3.47E-10 12.87521 0.124691 3.504838 0.079043 0.123083 0.848666 4.753719 0.592794
std 2.86E-48 1.699748 1.54E-09 0.28233 8.93E-11 3.605398 0.037167 1.111915 0.01982 0.019465 0.285629 1.194504 0.128821
rank 1 12 3 10 2 13 4 9 5 6 7 11 8
F13 best 1.35E-32 12.567 1.04E-09 6.06E-32 9.07E-10 1.832961 0.033885 0.005868 0.0000427 0.536004 4.24E-18 0.008719 1.176729
mean 1.35E-32 3278.627 9.13E-09 2.86E-31 2.28E-03 2.474572 0.195464 0.029851 4.68E-01 1.00371 5.16E-02 3.285858 2.466324
median 1.35E-32 40.28554 5.94E-09 3.66E-31 2.57E-09 2.309059 0.15101 0.021526 0.471026 1.015205 1.62E-17 3.010954 2.611495
worst 1.35E-32 56617.17 3.47E-08 4.96E-31 0.023056 3.382691 0.637881 0.083455 0.865379 1.403745 0.872898 11.46312 3.588802
std 2.86E-48 12872.05 8.16E-09 2.09E-31 5.89E-03 0.518014 0.170514 0.02303 0.239555 0.214971 1.99E-01 2.816182 0.701
rank 1 13 3 2 4 11 7 5 8 9 6 12 10
Sum rank 6 11 53 27 15 52 18 32 39 32 44 50 44
Mean rank 1 1.83334 8.83334 4.5 2.5 8.66667 3 5.33334 6.5 5.33334 7.33334 8.33334 7.33334
Total rank 1 2 11 5 3 10 4 6 7 6 8 9 8
Table 4. Optimization results for the fixed-dimensional multimodal functions.
Table 4. Optimization results for the fixed-dimensional multimodal functions.
FLO AVOA WSO RSA MPA TSA WOA GWO MVO TLBO GSA PSO GA
F14 best 0.998004 0.998004 0.998004 0.998033 0.998004 1.903374 0.998004 0.998004 0.998004 0.998004 0.998004 0.998004 0.998004
mean 0.998004 1.089592 1.089412 2.920195 1.009791 7.965725 3.455667 2.430619 0.999055 0.999056 3.333745 3.365148 1.045199
median 0.998004 0.998004 0.998004 2.115668 0.998004 10.76083 2.805144 0.998014 0.998004 0.998004 2.722809 1.903377 0.998008
worst 9.98E-01 1.90E+00 2.81E+00 1.16E+01 1.23E+00 1.42E+01 9.89E+00 9.89E+00 1.02E+00 1.02E+00 1.09E+01 1.16E+01 1.90E+00
std 0 0.284 0.412 2.84 0.0537 4.69 3.47 2.74 0.00479 0.00479 2.56 3.52 0.206
rank 1 7 6 9 4 13 12 8 2 3 10 11 5
F15 best 0.000307 0.000308 0.000316 0.000772 0.000309 0.000317 0.000317 0.000324 0.000317 0.000327 0.000844 0.000308 0.000861
mean 0.000307 0.001349 0.000436 0.001135 0.001212 0.015074 0.003178 0.000849 0.002523 0.000654 0.002255 0.002388 0.014128
median 0.000307 0.000429 0.000429 0.001026 0.0016 0.000874 0.000429 0.000704 0.000736 0.000438 0.002087 0.000429 0.013087
worst 3.07E-04 1.87E-02 6.94E-04 2.77E-03 1.67E-03 1.01E-01 1.87E-02 2.20E-03 1.86E-02 1.29E-03 6.49E-03 1.87E-02 6.11E-02
std 2.59E-19 0.00417 0.0000907 0.000435 0.000552 0.0279 0.00679 0.000471 0.00562 0.000382 0.00128 0.0057 0.0151
rank 1 7 2 5 6 13 11 4 10 3 8 9 12
F16 best -1.03163 -1.03163 -1.03163 -1.03161 -1.03163 -1.03163 -1.03163 -1.03163 -1.03163 -1.03163 -1.03163 -1.03163 -1.03163
mean -1.03163 -1.0313 -1.0313 -1.02928 -1.02916 -1.02986 -1.0313 -1.0313 -1.0313 -1.03129 -1.0313 -1.0313 -1.03129
median -1.03163 -1.03163 -1.03163 -1.03119 -1.0316 -1.03163 -1.03163 -1.03163 -1.03163 -1.03162 -1.03163 -1.03163 -1.03162
worst -1.03E+00 -1.03E+00 -1.03E+00 -1.00E+00 -1.00E+00 -1.00E+00 -1.03E+00 -1.03E+00 -1.03E+00 -1.03E+00 -1.03E+00 -1.03E+00 -1.03E+00
std 1.87E-16 0.000853 0.000853 0.00652 0.00708 0.00657 0.000853 0.000853 0.000853 0.000853 0.000853 0.000853 0.000853
rank 1 6 2 10 11 9 4 3 5 8 2 2 7
F17 best 0.397887 0.397887 0.397887 0.398697 0.397887 0.397893 0.397888 0.397887 0.397887 0.397897 0.397887 0.397887 0.397887
mean 0.397887 0.397919 0.397919 0.409491 0.398387 0.397952 0.397919 0.397919 0.397919 0.397985 0.397919 0.713742 0.459977
median 0.397887 0.397894 0.397894 0.403269 0.397974 0.397917 0.397894 0.397894 0.397894 0.397969 0.397894 0.397913 0.397943
worst 3.98E-01 3.98E-01 3.98E-01 4.77E-01 4.01E-01 3.98E-01 3.98E-01 3.98E-01 3.98E-01 3.98E-01 3.98E-01 2.58E+00 1.63E+00
std 0 0.0000644 0.0000644 0.0181 0.000932 0.0000842 0.0000644 0.0000643 0.0000644 0.0000895 0.0000644 0.659 0.281
rank 1 4 2 10 9 7 6 5 3 8 2 12 11
F18 best 3 3.001243 3.001243 3.002335 3.013933 3.001249 3.001246 3.001243 3.001243 3.001244 3.001243 3.001243 3.00321
mean 3 3.265013 3.265014 5.79232 6.144686 11.00853 3.265025 3.265037 3.265013 3.265014 3.265013 3.265013 7.18414
median 3 3.035691 3.035691 3.08846 3.563655 3.099528 3.0357 3.035714 3.035692 3.035692 3.035691 3.035691 3.161867
worst 3.00E+00 5.41E+00 5.41E+00 2.88E+01 3.00E+01 8.41E+01 5.41E+00 5.41E+00 5.41E+00 5.41E+00 5.41E+00 5.41E+00 3.22E+01
std 1.19E-15 0.582 0.582 7.91 6.49 24.3 0.582 0.582 0.582 0.582 0.582 0.582 9.71
rank 1 2 6 10 11 13 8 9 5 7 4 3 12
F19 best -3.86278 -3.86278 -3.86278 -3.85352 -3.86278 -3.86268 -3.86278 -3.86277 -3.86278 -3.86251 -3.86278 -3.86278 -3.86276
mean -3.86278 -3.85019 -3.85019 -3.82664 -3.72454 -3.84982 -3.8488 -3.84804 -3.85019 -3.84918 -3.85019 -3.85019 -3.85004
median -3.86278 -3.85056 -3.85056 -3.83066 -3.72574 -3.85052 -3.84988 -3.84899 -3.85056 -3.85015 -3.85056 -3.85056 -3.85049
worst -3.86E+00 -3.81E+00 -3.81E+00 -3.77E+00 -3.29E+00 -3.81E+00 -3.81E+00 -3.81E+00 -3.81E+00 -3.81E+00 -3.81E+00 -3.81E+00 -3.81E+00
std 2.32E-15 0.0123 0.0123 0.0237 0.14 0.0121 0.0124 0.012 0.0123 0.0117 0.0123 0.0123 0.0124
rank 1 2 3 10 11 6 8 9 4 7 2 2 5
F20 best -3.322 -3.31333 -3.2804 -3.0278 -3.22483 -3.31126 -3.31333 -3.30816 -3.31333 -3.29698 -3.31333 -3.31333 -3.23904
mean -3.322 -3.23202 -3.19953 -2.74117 -2.52925 -3.18729 -3.19091 -3.18259 -3.20485 -3.17608 -3.24826 -3.196 -3.16292
median -3.322 -3.24933 -3.19492 -2.82466 -2.58954 -3.17741 -3.19778 -3.18393 -3.22077 -3.17676 -3.25667 -3.2116 -3.17485
worst -3.32E+00 -3.14E+00 -3.09E+00 -1.70E+00 -1.78E+00 -3.06E+00 -3.00E+00 -3.04E+00 -3.08E+00 -2.92E+00 -3.18E+00 -3.03E+00 -2.97E+00
std 4.53E-16 0.0502 0.0636 0.297 0.344 0.0699 0.0882 0.0802 0.0703 0.0927 0.0342 0.0841 0.0658
rank 1 3 5 12 13 8 7 9 4 10 2 6 11
F21 best -10.1532 -10.1437 -10.1531 -5.50974 -10.1515 -10.1221 -10.1529 -10.1524 -10.153 -9.43287 -10.1531 -10.1362 -9.56481
mean -10.1532 -8.33089 -9.92179 -5.27848 -7.55875 -6.07089 -9.22698 -9.2225 -8.76716 -6.91569 -7.22664 -5.79638 -6.37604
median -10.1532 -9.88105 -9.95235 -5.30903 -7.90122 -5.07671 -9.88043 -9.8783 -9.77271 -7.16601 -9.69851 -5.15726 -7.0501
worst -1.02E+01 -2.89E+00 -9.70E+00 -5.06E+00 -5.06E+00 -2.83E+00 -5.10E+00 -5.08E+00 -5.06E+00 -3.65E+00 -2.89E+00 -2.87E+00 -2.62E+00
std 2.12E-15 2.97 0.187 0.187 2.09 3.04 1.73 1.75 2.08 1.93 3.26 2.66 2.63
rank 1 6 2 13 7 11 3 4 5 9 8 12 10
F22 best -10.4029 -10.4027 -10.4027 -5.56152 -10.4005 -10.3106 -10.4025 -10.3741 -10.376 -9.77163 -10.4027 -10.3804 -9.99024
mean -10.4029 -9.84679 -10.1952 -5.35402 -8.0883 -6.9905 -10.1947 -8.10544 -8.40252 -7.96089 -9.94596 -6.53375 -7.43449
median -10.4029 -10.1786 -10.2819 -5.44069 -9.04577 -7.67583 -10.2816 -9.92632 -9.98379 -8.27344 -10.211 -5.21856 -7.85052
worst -1.04E+01 -3.41E+00 -9.93E+00 -5.09E+00 -5.09E+00 -2.12E+00 -9.93E+00 -2.17E+00 -3.29E+00 -4.32E+00 -5.26E+00 -2.97E+00 -2.89E+00
std 3.58E-15 1.56 0.189 0.189 2.13 3.38 0.189 2.82 2.54 1.58 1.14 3.28 1.86
rank 1 5 2 13 8 11 3 7 6 9 4 12 10
F23 best -10.5364 -10.5286 -10.5286 -5.60303 -10.4492 -10.4288 -10.5284 -10.5277 -10.5286 -9.73892 -10.5286 -10.5196 -9.73355
mean -10.5364 -10.4131 -10.4131 -5.48753 -9.15348 -7.57014 -10.4127 -8.63436 -9.43441 -8.1814 -10.1863 -6.66461 -6.60937
median -10.5364 -10.4482 -10.4482 -5.52257 -9.54713 -9.95964 -10.4479 -10.3968 -10.4202 -8.70553 -10.4482 -4.32836 -7.12733
worst -1.05E+01 -1.01E+01 -1.01E+01 -5.13E+00 -5.13E+00 -3.11E+00 -1.01E+01 -2.33E+00 -5.17E+00 -4.67E+00 -5.88E+00 -2.97E+00 -3.04E+00
std 2.82E-15 0.134 0.134 0.134 1.5 3.18 0.134 3.07 2.08 1.54 1.04 3.57 2.39
rank 1 2 3 13 7 10 4 8 6 9 5 11 12
Sum rank 10 33 44 105 87 101 66 66 50 73 47 80 95
Mean rank 1.00E+00 3.30E+00 4.40E+00 1.05E+01 8.70E+00 1.01E+01 6.60E+00 6.60E+00 5.00E+00 7.30E+00 4.70E+00 8.00E+00 9.50E+00
Total rank 1 2 3 12 9 11 6 6 5 7 4 8 10
Table 5. Optimization results for the CEC 2017 test suite.
Table 5. Optimization results for the CEC 2017 test suite.
FLO WSO AVOA RSA MPA TSA WOA MVO GWO TLBO GSA PSO GA
C17-F1 mean 100 5.47E+09 3736.741 9.92E+09 34277291 1.69E+09 6265768 7309.046 85692339 1.43E+08 728.1107 3057.613 11513604
best 100 4.53E+09 115.1723 8.57E+09 10886.23 3.62E+08 4562393 4650.116 27005.92 63693665 100.0187 338.6514 5962184
worst 100 7.01E+09 11575.72 1.18E+10 1.25E+08 3.68E+09 8249654 10768.56 3.11E+08 3.45E+08 1741.869 9048.114 16528771
std 0 1.13E+09 5637.763 1.54E+09 63646439 1.56E+09 1643381 3018.544 1.59E+08 1.43E+08 748.1019 4247.123 4651212
median 100 5.16E+09 1628.036 9.64E+09 6282818 1.36E+09 6125512 6908.755 15705576 81669353 535.2778 1421.844 11781730
rank 1 12 4 13 8 11 6 5 9 10 2 3 7
C17-F3 mean 300 8293.792 301.8391 9378.914 1375.654 10888.66 1688.689 300.053 2989.135 713.9977 9971.33 300 14356.74
best 300 4202.111 300 5061.044 777.166 4151.807 610.0958 300.0123 1492.915 466.305 6277.902 300 4233.022
worst 300 11094.74 303.9338 12545.46 2470.6 15390.93 3243.315 300.1207 5726.5 875.8003 13549.84 300 22687.57
std 0 3176.215 2.247148 3603.989 822.8102 5026.203 1306.484 0.050178 2057.025 189.0482 3158.628 4.89E-14 10152.4
median 300 8939.158 301.7113 9954.573 1127.425 12005.95 1450.672 300.0395 2368.563 756.9427 10028.79 300 15253.2
rank 1 9 4 10 6 12 7 3 8 5 11 2 13
C17-F4 mean 400 918.5001 404.6184 1324.333 406.5383 571.4825 424.4454 403.2412 411.4095 408.9142 404.4257 419.7445 414.3073
best 400 686.9377 401.2064 832.4566 402.378 475.6638 406.2617 401.5494 405.9193 408.1513 403.4619 400.1027 411.3519
worst 400 1127.349 406.3441 1806.129 411.0611 683.3579 471.5001 404.7584 427.5674 409.3958 405.9062 468.4064 417.9233
std 0 211.487 2.549157 437.6922 4.510676 107.2135 33.13703 1.757159 11.34448 0.561975 1.180512 34.5168 3.028779
median 400 929.8569 405.4616 1329.372 406.357 563.4541 410.0099 403.3286 406.0757 409.0548 404.1674 405.2343 413.977
rank 1 12 4 13 5 11 10 2 7 6 3 9 8
C17-F5 mean 501.2464 562.7628 543.267 571.5024 512.6851 563.2066 540.248 523.2985 512.8239 533.4614 552.8981 527.4234 527.5331
best 500.9951 548.6366 526.3694 557.1506 508.2448 542.4586 523.0456 510.0618 508.3883 528.0685 548.1185 510.9634 522.9067
worst 501.9917 572.071 561.7117 586.2257 517.6984 594.6685 575.476 537.3349 519.9718 536.9224 564.4298 550.8372 533.1848
std 0.523294 11.24763 19.528 17.00721 5.239914 24.39291 25.86418 11.99271 5.257986 4.097238 8.204426 19.37243 4.881442
median 500.9993 565.1717 542.4935 571.3166 512.3987 557.8496 531.2352 522.8986 511.4678 534.4273 549.5221 523.9465 527.0204
rank 1 11 9 13 2 12 8 4 3 7 10 5 6
C17-F6 mean 600 631.9679 617.0699 640.1193 601.1766 624.4721 622.8332 602.1188 601.1108 606.7637 616.9574 607.3227 610.1123
best 600 628.0964 616.08 636.953 600.7006 614.8572 607.4178 600.4653 600.5875 604.6901 602.8743 601.3351 606.8056
worst 600 635.2211 619.5846 644.3114 602.3635 639.8378 644.5482 604.2511 601.6942 609.997 635.6217 618.9817 614.2958
std 0 3.522076 1.770682 3.482394 0.835501 11.34888 16.46555 1.791659 0.482493 2.54797 15.95463 8.429568 3.496574
median 600 632.277 616.3074 639.6063 600.8212 621.5967 619.6833 601.8795 601.0807 606.1838 614.6668 604.4871 609.6739
rank 1 12 9 13 3 11 10 4 2 5 8 6 7
C17-F7 mean 711.1267 801.8243 764.7796 803.027 724.4458 826.7993 761.355 730.597 725.8017 751.4689 717.0341 732.4347 736.5061
best 710.6726 781.9245 743.4263 789.9792 720.2997 787.2738 750.5297 717.1182 717.3753 746.9989 714.7886 725.3833 726.3134
worst 711.7995 818.2036 792.1502 815.5044 728.797 867.6759 790.383 749.5856 743.0676 759.4706 720.7105 743.8261 741.0172
std 0.539366 16.11626 23.59666 12.6169 3.767224 36.78314 20.43924 14.38877 12.42733 5.871453 2.702113 8.863158 7.268468
median 711.0174 803.5845 761.771 803.3123 724.3433 826.1237 752.2536 727.8421 721.382 749.7031 716.3187 730.2647 739.3469
rank 1 11 10 12 3 13 9 5 4 8 2 6 7
C17-F8 mean 801.4928 847.9594 830.7179 852.9783 812.5179 847.6438 835.8856 811.6922 815.6551 837.214 819.6168 822.481 816.585
best 800.995 840.0504 820.0255 841.9218 808.7429 831.6806 818.3429 807.3408 810.3963 830.393 811.8696 815.4944 812.6473
worst 801.9912 856.225 846.3057 858.1424 814.642 866.671 847.9269 816.4079 820.5681 845.0906 827.2753 828.8525 824.2671
std 0.605411 7.772196 11.68425 7.882006 2.862269 16.39951 13.38181 3.92383 4.482335 7.915127 6.904175 6.97152 5.495759
median 801.4926 847.7811 828.2702 855.9245 813.3434 846.1118 838.6363 811.51 815.8281 836.6862 819.6612 822.7886 814.7128
rank 1 12 8 13 3 11 9 2 4 10 6 7 5
C17-F9 mean 900 1415.647 1184.109 1459.407 905.126 1374.025 1368.683 900.7903 911.7692 911.6633 900 904.1835 905.0408
best 900 1274.159 952.9809 1364.939 900.323 1164.299 1071.705 900.001 900.5653 907.1323 900 900.8869 902.7594
worst 900 1554.98 1650.596 1594.98 913.158 1658.273 1645.866 903.0715 932.6733 919.7283 900 912.149 908.9528
std 0 132.6979 340.4279 103.1532 6.084099 225.1024 254.5478 1.602111 15.86347 5.829805 0 5.66366 2.949944
median 900 1416.725 1066.429 1438.854 903.5116 1336.763 1378.581 900.0444 906.9191 909.8963 900 901.849 904.2254
rank 1 11 8 12 5 10 9 2 7 6 1 3 4
C17-F10 mean 1006.179 2273.607 1759.53 2541.66 1504.452 2008.524 2001.099 1762.429 1708.336 2144.709 2248.35 1923.739 1698.821
best 1000.284 2018.084 1472.325 2374.849 1382.178 1739.505 1438.96 1445.087 1526.348 1763.109 1975.334 1547.422 1405.051
worst 1012.668 2452.924 2381.05 2893.002 1577.649 2255.004 2513.204 2252.39 1968.77 2426.108 2351.688 2320.273 2084.862
std 7.010122 207.8178 449.5863 254.1133 96.96531 286.3359 547.0173 411.8901 198.0574 296.8361 192.0592 334.2969 307.0471
median 1005.882 2311.71 1592.372 2449.394 1528.991 2019.793 2026.115 1676.12 1669.112 2194.81 2333.189 1913.63 1652.685
rank 1 12 5 13 2 9 8 6 4 10 11 7 3
C17-F11 mean 1100 3792.805 1147.32 3913.816 1126.386 5353.182 1149.714 1126.836 1153.923 1149.669 1138.236 1142.466 2351.467
best 1100 2579.105 1116.633 1449.857 1112.878 5208.571 1112.643 1105.411 1121.094 1136.909 1119.165 1131.464 1114.678
worst 1100 4965.598 1199.302 6347.472 1157.346 5432.526 1171.319 1147.72 1225.241 1170.543 1166.94 1163.436 5860.164
std 0 1129.665 38.31996 2318.078 22.11378 104.8258 28.5344 22.26206 51.12907 15.28998 21.47371 15.1631 2463.985
median 1100 3813.26 1136.672 3928.967 1117.659 5385.816 1157.446 1127.106 1134.679 1145.613 1133.42 1137.481 1215.513
rank 1 11 6 12 2 13 8 3 9 7 4 5 10
C17-F12 mean 1352.959 3.46E+08 1076623 6.9E+08 555218.2 1017040 2302745 1006704 1384502 4942507 998167 7942.901 591852.2
best 1318.646 77501553 348274.8 1.53E+08 19458.1 527421.5 168030.9 8666.689 44473.99 1322697 464212.7 2491.975 171450.5
worst 1438.176 6.04E+08 1952421 1.21E+09 868884.4 1248617 3820158 3162122 2167137 8749709 1688039 13645.63 1044753
std 60.34215 2.8E+08 790170 5.61E+08 394089.6 358149.2 1787906 1534071 985273.1 4143007 545566.9 5351.454 377639
median 1327.506 3.51E+08 1002899 7E+08 666265.1 1146061 2611395 428012.7 1663199 4848812 920208 7817.002 575602.7
rank 1 12 8 13 3 7 10 6 9 11 5 2 4
C17-F13 mean 1305.324 16818760 17959.57 33627051 5343.993 12487.34 7441.547 6609.75 10102.15 16388.5 9879.403 6504.793 53288.07
best 1303.114 1403371 2692 2791832 3667.876 7450.198 3237.74 1384.282 6393.425 15476.72 4965.5 2355.43 8385.223
worst 1308.508 55824623 30752.64 1.12E+08 6529.18 19767.52 14850.39 12137.03 14101.13 18615.71 13903.43 16377.46 176137.9
std 2.393502 27444488 15277.22 54887056 1437.183 5598.534 5574.972 5865.593 3326.546 1578.577 3978.516 7008.283 86300.99
median 1304.837 5023524 19196.82 10040363 5589.459 11365.82 5839.029 6458.844 9957.02 15730.79 10324.34 3643.14 14314.57
rank 1 12 10 13 2 8 5 4 7 9 6 3 11
C17-F14 mean 1400.746 3939.312 2008.934 5264.816 1928.918 3345.285 1516.59 1568.362 2326.763 1586.904 5478.844 2962.532 12721.15
best 1400 3117.693 1673.286 4612.291 1434.307 1486.131 1480.161 1422.656 1461.054 1513.755 4535.374 1431.851 3678.382
worst 1400.995 5351.578 2798.455 6783.07 2872.227 5496.468 1555.51 1981.281 4889.386 1616.566 7425.724 6730.767 25320.97
std 0.523906 1086.154 558.4401 1073.877 710.2403 2246.597 40.54732 289.9705 1799.198 51.60308 1426.2 2666.975 9655.175
median 1400.995 3643.988 1781.998 4831.952 1704.569 3199.271 1515.345 1434.756 1478.307 1608.646 4977.139 1843.756 10942.63
rank 1 10 6 11 5 9 2 3 7 4 12 8 13
C17-F15 mean 1500.331 10098.58 5218.908 13616.4 3924.269 6887.913 6120.022 1540.983 5724.402 1704.856 23414.1 8840.912 4486.47
best 1500.001 2973.107 2060.565 2708.398 3187.379 2302.517 2003.782 1525.386 3527.032 1582.367 11023.57 2843.03 1882.431
worst 1500.5 17679.9 12395.04 29757.67 4821.345 12315.56 13198.16 1552.777 6787.366 1792.663 35125.15 14517.22 7878.356
std 0.247931 6463.467 5077.324 12437.92 713.8488 4531.581 5138.748 12.60171 1577.67 108.6858 12125.71 5138.289 3139.349
median 1500.413 9870.658 3210.011 10999.77 3844.176 6466.789 4639.074 1542.885 6291.604 1722.197 23753.83 9001.7 4092.546
rank 1 11 6 12 4 9 8 2 7 3 13 10 5
C17-F16 mean 1600.76 2003.841 1805.182 2007.682 1682.475 2037.911 1943.131 1811.714 1725.811 1675.298 2063.238 1916.854 1798.115
best 1600.356 1932.833 1641.383 1814.741 1640.895 1856.913 1761.641 1723.889 1615.517 1649.88 1939.83 1817.788 1716.236
worst 1601.12 2155.779 1919.398 2275.751 1712.433 2218.582 2068.663 1872.098 1820.735 1728.436 2253.981 2073.252 1828.527
std 0.332693 107.9397 123.3381 205.0407 32.40876 172.8169 153.6816 66.01326 89.17165 38.56257 150.4328 124.6204 57.53511
median 1600.781 1963.376 1829.974 1970.119 1688.287 2038.075 1971.11 1825.435 1733.496 1661.439 2029.57 1888.188 1823.848
rank 1 10 6 11 3 12 9 7 4 2 13 8 5
C17-F17 mean 1700.099 1815.785 1749.933 1815.932 1735.005 1800.085 1838.968 1839.831 1767.119 1757.185 1843.74 1751.289 1754.835
best 1700.02 1806.767 1733.703 1799.367 1721.462 1785.215 1772.092 1776.895 1723.964 1747.255 1746.952 1744.783 1751.768
worst 1700.332 1820.911 1793.046 1824.963 1773.372 1810.751 1885.438 1945.3 1868.057 1766.927 1967.462 1757.838 1757.219
std 0.163405 6.604172 30.34816 11.98474 26.95023 11.55193 51.8522 83.97523 71.2283 10.25543 118.4177 5.880496 2.593649
median 1700.022 1817.731 1736.491 1819.698 1722.593 1802.187 1849.171 1818.564 1738.228 1757.279 1830.274 1751.268 1755.175
rank 1 9 3 10 2 8 11 12 7 6 13 4 5
C17-F18 mean 1805.36 2790335 11621.85 5564458 10833.91 11819.61 22803.87 20498.52 19481.89 28855.77 9528.075 21406.05 12557.12
best 1800.003 143079.7 4773.511 275503.5 4103.869 7333.66 6341.191 8540.919 6218.404 23471.4 6286.716 2855.611 3398.068
worst 1820.451 8086661 15273.85 16153226 16174.54 15949.14 35793.67 32964.2 32845.25 36080.81 11621.21 39828.48 18092.38
std 10.59792 3874670 4958.103 7747506 5781.245 3773.409 14945.94 12109.03 14219.8 6108.088 2397.694 20100.54 6759.369
median 1800.492 1465801 13220.02 2914551 11528.62 11997.83 24540.31 20244.49 19431.95 27935.43 10102.19 21470.05 14369.02
rank 1 12 4 13 3 5 10 8 7 11 2 9 6
C17-F19 mean 1900.445 387325.3 6595.99 687462.2 5511.926 122623.9 34035.21 1914.421 5302.419 4631.324 39521.63 24406.65 6082.546
best 1900.039 24496.92 2170.349 44786.25 2308.105 1948.078 7524.944 1909.202 1943.66 2039.952 10888 2607.662 2205.951
worst 1901.559 818032 12978.45 1476750 9240.37 244893.5 62270.58 1923.745 13530.05 12241.53 57304.33 75127.73 9695.844
std 0.784167 360554.3 5534.839 680304 3721.021 146723.9 23668.24 7.235854 5837.141 5343.17 21892 36010.83 3254.361
median 1900.09 353386.2 5617.582 614156.2 5249.615 121827 33172.67 1912.368 2867.982 2121.906 44947.1 9945.598 6214.195
rank 1 12 7 13 5 11 9 2 4 3 10 8 6
C17-F20 mean 2000.312 2210.011 2166.568 2217.803 2090.167 2202.524 2201.759 2136.293 2165.937 2070.321 2247.759 2165.023 2049.043
best 2000.312 2154.567 2030.604 2160.675 2071.051 2104.208 2096.032 2045.847 2127.852 2059.553 2183.382 2141.464 2034.979
worst 2000.312 2278.59 2287.603 2271.927 2119.96 2313.389 2281.142 2241.642 2240.244 2080.497 2338.678 2196.123 2056.626
std 0 53.95395 121.7585 57.65849 22.07053 93.32007 93.18815 84.66573 53.35706 9.247632 79.56738 28.60972 10.51071
median 2000.312 2203.443 2174.032 2219.306 2084.828 2196.25 2214.931 2128.842 2147.827 2070.617 2234.488 2161.253 2052.284
rank 1 11 8 12 4 10 9 5 7 3 13 6 2
C17-F21 mean 2200 2290.968 2213.493 2265.597 2255.897 2322.324 2307.36 2251.936 2310.718 2297.422 2364.486 2316.097 2295.936
best 2200 2244.697 2204.034 2223.411 2253.464 2220.748 2217.975 2200.007 2306.605 2203.635 2347.406 2308.22 2225.954
worst 2200 2316.52 2238.126 2289.583 2258.376 2368.215 2350.548 2305.165 2315.574 2335.231 2381.419 2323.478 2329.794
std 0 35.24304 17.34766 30.82035 2.18896 72.54506 63.5482 63.15665 3.884515 66.31879 14.9697 7.903532 49.76087
median 2200 2301.327 2205.907 2274.697 2255.874 2350.166 2330.458 2251.285 2310.346 2325.411 2364.56 2316.346 2313.999
rank 1 6 2 5 4 12 9 3 10 8 13 11 7
C17-F22 mean 2300.073 2727.205 2308.786 2902.26 2304.896 2704.733 2323.277 2286.092 2308.412 2319.143 2300.006 2312.979 2317.535
best 2300 2604.512 2304.267 2697.974 2300.923 2445.851 2318.712 2230.996 2301.239 2313.024 2300 2300.624 2314.697
worst 2300.29 2863.201 2310.901 3052.187 2309.155 2908.01 2330.751 2305.182 2321.915 2330.62 2300.026 2344.466 2321.909
std 0.152789 125.676 3.213404 157.0784 3.653123 217.2198 5.66243 38.69394 10.01534 8.480752 0.013627 22.1532 3.245621
median 2300 2720.553 2309.989 2929.44 2304.752 2732.534 2321.822 2304.095 2305.248 2316.463 2300 2303.413 2316.766
rank 3 12 6 13 4 11 10 1 5 9 2 7 8
C17-F23 mean 2600.919 2695.142 2641.279 2698.593 2614.053 2720.959 2647.789 2619.87 2613.481 2641.744 2787.95 2643.451 2655.069
best 2600.003 2654.05 2630.016 2670.241 2611.708 2633.704 2630.257 2607.041 2607.705 2631.074 2724.222 2636.443 2635.506
worst 2602.87 2718.801 2658.663 2738.344 2616.681 2764.466 2667.61 2631.191 2620.042 2650.904 2923.517 2655.116 2663.229
std 1.39047 31.84495 14.22616 33.5504 2.494325 62.22928 21.21048 11.06898 6.713412 9.259933 98.62473 8.912385 13.94899
median 2600.403 2703.859 2638.218 2692.893 2613.911 2742.834 2646.645 2620.625 2613.089 2642.5 2752.031 2641.123 2660.77
rank 1 10 5 11 3 12 8 4 2 6 13 7 9
C17-F24 mean 2630.488 2774.562 2764.817 2845.426 2630.649 2667.52 2757.964 2682.241 2746.372 2753.283 2745.087 2762.814 2721.233
best 2516.677 2721.424 2731.165 2822.869 2614.606 2529.897 2729.886 2501.653 2720.269 2739.08 2503.872 2753.308 2541.917
worst 2732.32 2855.174 2784.603 2907.973 2639.658 2810.341 2790.428 2758.938 2759.651 2765.909 2894.23 2785.223 2809.443
std 122.6896 68.4245 26.78397 43.96585 11.87973 158.2364 26.36986 127.588 19.50185 13.53941 177.0559 15.83832 127.6645
median 2636.477 2760.824 2771.75 2825.431 2634.167 2664.92 2755.771 2734.188 2752.784 2754.072 2791.123 2756.364 2766.787
rank 1 12 11 13 2 3 9 4 7 8 6 10 5
C17-F25 mean 2932.639 3155.18 2913.894 3269.51 2918.209 3129.369 2908.071 2922.324 2938.568 2933.51 2922.491 2923.532 2951.829
best 2898.047 3064.544 2899.073 3202.552 2914.394 2906.642 2768.56 2901.85 2921.811 2915.723 2903.487 2898.655 2937.353
worst 2945.793 3355.646 2948.92 3343.189 2923.782 3641.612 2957.842 2943.71 2945.838 2952.119 2943.394 2946.537 2962.362
std 24.31643 142.1273 24.70364 61.2439 4.370192 363.6379 98.02209 24.7485 11.84326 21.01351 23.0419 27.38159 11.26241
median 2943.359 3100.264 2903.792 3266.149 2917.329 2984.611 2952.94 2921.869 2943.312 2933.1 2921.543 2924.469 2953.8
rank 7 12 2 13 3 11 1 4 9 8 5 6 10
C17-F26 mean 2900 3586.548 2978.206 3738.663 3009.357 3605.933 3177.182 2900.145 3257.787 3200.309 3841.837 2903.976 2897.274
best 2900 3249.585 2808.919 3421.475 2892.278 3139.18 2926.653 2900.111 2967.8 2911.806 2808.919 2808.919 2711.383
worst 2900 3826.851 3151.519 4068.773 3285.46 4241.393 3579.816 2900.189 3886.42 3855.646 4319.2 3006.985 3105.287
std 3.91E-13 292.2164 205.8885 293.9256 194.7487 567.5965 300.739 0.036902 445.4171 463.1248 736.9779 85.29239 210.1045
median 2900 3634.878 2976.193 3732.203 2929.845 3521.579 3101.13 2900.14 3088.464 3016.891 4119.614 2900 2886.213
rank 2 10 5 12 6 11 7 3 9 8 13 4 1
C17-F27 mean 3089.518 3205.957 3119.375 3228.207 3104.379 3177.675 3192.753 3091.585 3115.563 3114.565 3223.217 3135.116 3158.554
best 3089.518 3158.16 3095.187 3126.438 3092.187 3102.163 3177.187 3089.706 3094.336 3095.264 3211.305 3096.94 3118.73
worst 3089.518 3277.829 3179.042 3416.328 3132.899 3219.061 3204.273 3094.852 3174.984 3169.582 3244.326 3181.461 3216.271
std 2.76E-13 53.52379 42.0126 135.253 20.16865 55.77425 11.90539 2.548962 41.75977 38.63556 15.47514 37.43231 43.43005
median 3089.518 3193.919 3101.636 3185.032 3096.215 3194.737 3194.776 3090.89 3096.466 3096.708 3218.617 3131.032 3149.607
rank 1 11 6 13 3 9 10 2 5 4 12 7 8
C17-F28 mean 3100 3611.463 3233.144 3764.422 3215.961 3575.685 3282.736 3235.706 3339.606 3320.184 3443.108 3301.185 3243.164
best 3100 3563.288 3100 3683.905 3165.474 3405.693 3151.545 3100.121 3192.63 3211.49 3430.136 3175.433 3143.902
worst 3100 3652.954 3384.01 3822.542 3240.311 3780.33 3384.51 3384.011 3405.236 3384.247 3461.135 3384.221 3504.385
std 0 39.54602 132.2986 67.7661 36.47716 204.6273 126.0886 165.1629 103.9998 86.83885 15.12725 99.68882 184.0982
median 3100 3614.806 3224.283 3775.62 3229.03 3558.358 3297.444 3229.346 3380.28 3342.5 3440.581 3322.542 3162.184
rank 1 12 3 13 2 11 6 4 9 8 10 7 5
C17-F29 mean 3132.241 3324.262 3281.439 3370.25 3201.677 3234.136 3344.478 3201.267 3262.49 3211.02 3341.53 3263.344 3235.113
best 3130.076 3307.388 3208.777 3300.298 3165.245 3165.464 3233.56 3142.258 3188.769 3164.942 3231.698 3167.136 3187.368
worst 3134.841 3342.951 3360.613 3436.015 3242.35 3302.718 3488.455 3283.301 3374.431 3233.318 3624.637 3344.532 3283.168
std 2.61421 19.00098 82.36137 73.67541 35.72216 59.15374 112.573 62.88155 93.01789 33.76393 199.5837 84.79596 42.46644
median 3132.023 3323.354 3278.183 3372.344 3199.556 3234.181 3327.949 3189.755 3243.379 3222.909 3254.892 3270.854 3234.958
rank 1 10 9 13 3 5 12 2 7 4 11 8 6
C17-F30 mean 3418.734 2165720 287506.3 3584776 404550 599358.3 967621 295454.7 912690.3 59213.86 763371.5 377738.5 1489498
best 3394.682 1320162 102168.6 807160.4 15619.96 109607.9 4440.547 7339.401 32840.9 28649.75 586908.4 6321.34 512847.3
worst 3442.907 3250978 748843.2 5662047 597037.7 1267208 3652633 1126231 1320855 99321.06 974824 748878.9 3393200
std 29.24586 845640.7 324777.7 2140730 278078.7 518025.2 1887445 583462.4 637318.9 36349.97 169764.2 450721.6 1429827
median 3418.673 2045870 149506.7 3934949 502771.1 510308.6 106705.4 24124.27 1148533 54442.32 745876.8 377876.9 1025972
rank 1 12 3 13 6 7 10 4 9 2 8 5 11
Sum rank 38 319 177 351 106 284 239 116 188 191 238 183 197
Mean rank 1.310345 11 6.103448 12.10345 3.655172 9.793103 8.241379 4 6.482759 6.586207 8.206897 6.310345 6.793103
Total rank 1 12 4 13 2 11 10 3 6 7 9 5 8
Table 6. Wilcoxon rank sum test results.
Table 6. Wilcoxon rank sum test results.
Compared Algorithm Test Functions
F1 to F7 F8 to F13 F14 to F23 CEC 2017
FLO vs. AVOA 2.93E-14 4.68E-08 1.39E-37 3.65E-22
FLO vs. WSO 1.79E-27 1.91E-24 2.02E-37 1.96E-24
FLO vs. RSA 4.12E-10 1.58E-14 1.39E-37 1.91E-24
FLO vs. MPA 9.78E-28 1.01E-17 2.02E-37 1.94E-21
FLO vs. TSA 9.78E-28 1.27E-23 1.39E-37 9.20E-24
FLO vs. WOA 2.36E-27 5.94E-14 1.39E-37 9.20E-24
FLO vs. GWO 9.78E-28 5.17E-19 1.39E-37 5.07E-24
FLO vs. MVO 9.78E-28 1.91E-24 1.39E-37 8.75E-22
FLO vs. TLBO 9.78E-28 6.76E-18 1.39E-37 3.57E-24
FLO vs. GSA 9.78E-28 1.91E-24 1.39E-37 1.55E-21
FLO vs. PSO 9.78E-28 1.91E-24 1.39E-37 1.49E-22
FLO vs. GA 9.78E-28 1.91E-24 1.39E-37 2.63E-22
Table 7. Optimization results for the CEC 2011 test suite.
Table 7. Optimization results for the CEC 2011 test suite.
FLO AVOA WSO RSA MPA TSA WOA GWO MVO TLBO GSA PSO GA
C11-F1 best 2E-10 15.71076 9.066921 20.62179 0.380394 17.94272 8.419541 1.141181 11.67985 17.18251 20.08481 10.70815 22.81227
mean 5.92E+00 17.99435 13.14302 22.38356 7.610637 18.74465 13.4423 10.99171 14.21507 18.77743 22.09735 18.27145 23.83473
median 5.687176 17.74842 13.22952 22.04179 8.683689 18.4701 13.92203 12.50987 14.38094 18.7476 22.36096 18.89069 23.29871
worst 12.30606 20.76982 17.04611 24.82885 12.69478 20.09569 17.50561 17.80591 16.41857 20.43201 23.58269 24.59626 25.92925
std 7.196379 2.60606 4.637594 2.136403 5.931504 1.059314 4.389729 7.430618 2.408351 1.40313 1.543546 6.81674 1.494991
rank 1 7 4 12 2 9 5 3 6 10 11 8 13
C11-F2 best -27.0676 -15.5758 -21.5126 -11.7957 -25.7104 -14.8607 -21.9772 -24.7158 -10.5976 -11.8759 -20.5101 -23.9972 -15.0976
mean -26.3179 -14.211 -20.9668 -11.3516 -25.073 -11.0667 -18.5091 -22.5833 -8.54387 -10.6677 -15.3823 -22.633 -12.7312
median -26.3856 -14.1395 -21.0593 -11.3616 -25.4223 -10.2821 -18.8021 -23.3354 -8.29251 -10.5943 -14.8825 -23.1576 -12.4044
worst -25.4328 -12.9891 -20.2358 -10.8875 -23.7372 -8.84169 -14.4552 -18.9464 -6.99281 -9.60622 -11.254 -20.2195 -11.0182
std 0.738935 1.387208 0.593111 0.509167 0.966972 2.990523 4.075175 2.675921 1.64456 0.990412 4.427235 1.741838 2.015308
rank 1 8 5 10 2 11 6 4 13 12 7 3 9
C11-F4 best 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05
mean 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05
median 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05
worst 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05
std 2.00E-19 2.22E-11 2.55E-09 5.00E-11 1.25E-15 2.39E-14 6.15E-19 3.74E-15 9.98E-13 7.86E-14 2.00E-19 6.08E-20 2.77E-18
rank 1 11 13 12 6 8 4 7 10 9 3 2 5
C11-F4 best 0 0 0 0 0 0 0 0 0 0 0 0 0
mean 0 0 0 0 0 0 0 0 0 0 0 0 0
median 0 0 0 0 0 0 0 0 0 0 0 0 0
worst 0 0 0 0 0 0 0 0 0 0 0 0 0
std 0 0 0 0 0 0 0 0 0 0 0 0 0
rank 1 1 1 1 1 1 1 1 1 1 1 1 1
C11-F5 best -34.7494 -25.9018 -29.1581 -22.0228 -33.8571 -31.5428 -27.7524 -34.1779 -31.7223 -12.7456 -31.5218 -11.9996 -10.7279
mean -34.1274 -24.7516 -28.0779 -19.864 -33.2723 -27.0943 -27.5969 -31.5621 -26.9534 -10.5939 -27.3127 -8.41031 -9.27774
median -34.1871 -24.6441 -27.771 -19.9721 -33.646 -27.5565 -27.7204 -32.2815 -25.8032 -10.3414 -26.7975 -7.48141 -9.39582
worst -33.3862 -23.8166 -27.6116 -17.489 -31.9401 -21.7214 -27.1943 -27.5074 -24.4848 -8.94709 -24.1342 -6.67878 -7.59142
std 0.589989 0.95518 0.770371 2.52328 0.939346 4.252721 0.282677 2.997963 3.555513 1.70007 3.400906 2.637435 1.455951
rank 1 9 4 10 2 7 5 3 8 11 6 13 12
C11-F6 best -27.4298 -14.5571 -20.4029 -13.6442 -25.7465 -16.4981 -22.9899 -22.38 -17.3952 -2.44646 -26.6323 -5.94001 -9.20345
mean -24.1119 -13.9676 -19.0042 -12.965 -22.6108 -7.43437 -19.9336 -19.6085 -9.4219 -2.15053 -21.8798 -3.02392 -3.93842
median -23.0059 -13.7835 -19.2017 -13.1341 -21.6869 -4.54604 -21.9278 -19.0489 -9.12028 -2.05189 -21.5738 -2.05189 -2.24917
worst -23.0059 -13.7463 -17.2104 -11.9475 -21.3227 -4.1473 -12.8889 -17.9562 -2.05189 -2.05189 -17.7395 -2.05189 -2.05189
std 2.324951 0.414304 1.546268 0.826562 2.226773 6.363543 5.036499 2.223563 8.734288 0.207362 4.026194 2.0434 3.694555
rank 1 7 6 8 2 10 4 5 9 13 3 12 11
C11-F7 best 0.582266 1.546644 1.142773 1.679132 0.757596 1.129698 1.628575 0.814227 0.817307 1.528266 0.88491 0.831913 1.350761
mean 0.860699 1.607366 1.284792 1.921733 0.929758 1.302666 1.745163 1.067876 0.881104 1.720185 1.079947 1.123948 1.741991
median 0.91775 1.582628 1.284988 1.950134 0.974943 1.206637 1.716675 1.081482 0.875765 1.744987 1.076935 1.148661 1.835193
worst 1.025027 1.717562 1.426421 2.107532 1.011549 1.667694 1.918728 1.294312 0.955577 1.862501 1.28101 1.366556 1.946817
std 0.211503 0.081267 0.161509 0.187417 0.123219 0.258951 0.129822 0.207943 0.071645 0.153083 0.188418 0.290556 0.283994
rank 1 9 7 13 3 8 12 4 2 10 5 6 11
C11-F8 best 220 258.6912 223.6432 284.7586 220 220 245.4116 220 220 220 220 248.2351 220
mean 220 285.3233 240.5843 325.95 222.4592 257.5026 266.3147 227.3776 224.0986 224.0986 246.4917 470.865 222.5047
median 220 281.17 240.5843 324.1056 222.4592 227.3776 253.6089 227.3776 220 220 236.0957 531.6483 220
worst 220 320.262 257.5254 370.8302 224.9184 355.2553 312.6294 234.7551 236.3946 236.3946 293.7756 571.9284 230.0189
std 0 28.33492 15.32562 37.10878 2.984731 68.88765 32.70744 8.954192 8.616175 8.616175 36.77287 161.0301 5.26544
rank 1 10 6 11 2 8 9 5 4 4 7 12 3
C11-F9 best 5457.674 374781.4 336665.2 697814.5 11041.59 47806.58 208627.9 18499.88 75972.25 340166.3 708937.6 874293.6 1873402
mean 8789.286 560697.8 380695.1 1068617 20271.76 66589.3 377005.8 43236.71 134161.7 411177.8 828454.6 1089209 1954852
median 7828.591 611878.7 388154.1 1161468 20599.99 66996.02 330330.8 39412.95 128749.4 388492.1 856508.3 1074211 1938336
worst 14042.29 644252.5 409807.3 1253718 28845.46 84558.58 638733.6 75621.04 203175.8 527560.9 891864.2 1334122 2069334
std 3889.181 133563.1 33735.78 264941.7 8281.714 16458.05 206133.7 25372.04 55157.42 86679.05 85588.09 258343.4 101384.9
rank 1 9 7 11 2 4 6 3 5 8 10 12 13
C11-F10 best -21.8299 -15.1474 -17.0907 -12.6209 -19.405 -18.8457 -13.5067 -14.5551 -21.1801 -11.3313 -13.6317 -11.3867 -11.0857
mean -21.4889 -13.9334 -16.8991 -12.2327 -19.0152 -14.3551 -12.8304 -14.0677 -14.6685 -11.236 -13.1122 -11.3363 -11.0392
median -21.669 -13.6326 -16.9924 -12.1747 -19.0175 -13.2966 -12.7347 -14.4077 -13.0427 -11.2352 -13.2437 -11.332 -11.0534
worst -20.7878 -13.321 -16.521 -11.9607 -18.6207 -11.9813 -12.3453 -12.9002 -11.4084 -11.1423 -12.3297 -11.2945 -10.9644
std 0.498616 0.873428 0.27595 0.300903 0.421114 3.248101 0.512428 0.827904 4.63444 0.085067 0.667544 0.04007 0.055176
rank 1 7 3 10 2 5 9 6 4 12 8 11 13
C11-F11 best 260837.9 5557994 774810 8605193 1549827 4971008 1107924 3655950 612213.8 5205505 1268560 5226548 6108796
mean 571712.3 5828813 992977 8901363 1664311 5971920 1218465 3849509 1311819 5233232 1415206 5244366 6150774
median 598725.2 5779629 1011965 8954672 1654404 5848479 1192994 3765694 945640.2 5235642 1400232 5241956 6135506
worst 828560.9 6198000 1173167 9090917 1798611 7219712 1379948 4210699 2743783 5256141 1591802 5267002 6223290
std 260922.1 311888.7 182652.4 218241.7 126103.5 976503 121814.2 259907.3 1016990 23271.59 139894.1 21402.87 53468.09
rank 1 10 2 13 6 11 3 7 4 8 5 9 12
C11-F12 best 1155937 8077880 3283202 12348593 1198994 4777223 5416874 1260376 1175696 13547393 5521409 2148081 14426897
mean 1199805 8426155 3386135 13294942 1274918 5048294 5837448 1425143 1328064 14393299 5812159 2319287 14555006
median 1196965 8445689 3403596 13352032 1273202 5111608 5942811 1438043 1331765 14488324 5853035 2299989 14553136
worst 1249353 8735363 3454144 14127111 1354273 5192738 6047295 1564112 1473028 15049154 6021158 2529089 14686857
std 47157.58 286877.7 78506.39 766755.2 71443.68 202702 305338.9 132403.6 127721.8 662190.6 226324.5 165169.6 111676.6
rank 1 10 6 11 2 7 9 4 3 12 8 5 13
C11-F13 best 15444.19 15673.41 15447.01 15896.67 15460.95 15480.47 15492.04 15494.46 15487.9 15628.53 93207.96 15473.77 15460.54
mean 15444.2 15859.93 15448.01 16315.13 15463.27 15490.45 15535.98 15501.42 15508.25 15935.9 128965.6 15491.09 30083.63
median 15444.2 15727.24 15447.96 16004.54 15462.43 15489.09 15528.33 15498.88 15499.07 15806.47 122594.5 15481.38 15637.37
worst 15444.21 16311.82 15449.13 17354.77 15467.29 15503.13 15595.2 15513.47 15546.97 16502.13 177465.4 15527.81 73599.23
std 0.009091 319.7317 0.936378 734.5073 2.951184 11.77287 50.4487 8.855953 28.7804 415.6116 39873.02 26.01087 30492.98
rank 1 9 2 11 3 4 8 6 7 10 13 5 12
C11-F14 best 18241.58 85957.61 18404.38 169588 18524.65 19280.26 19076.34 19090.78 19324.11 30320.43 18806.64 18965.34 18831.17
mean 18295.35 113024.3 18520.66 230315.4 18609.59 19538.42 19231.08 19238.06 19425.84 312498.8 19097.29 19130.04 19117.21
median 18275.87 104003.8 18529.52 209852.9 18613.95 19391.67 19248.13 19218.59 19435.9 308198.1 19136.96 19138.94 19109.76
worst 18388.08 158132.1 18619.22 331967.8 18685.8 20090.09 19351.71 19424.28 19507.46 603278.4 19308.62 19276.94 19418.16
std 71.59938 33932.57 106.2459 76452.06 72.70982 390.523 133.2113 154.6878 81.27908 289123 228.3532 134.2236 252.1723
rank 1 11 2 12 3 10 7 8 9 13 4 6 5
C11-F15 best 32782.17 376496.4 43237.1 805730.9 32870.41 33048.29 33010.83 33043.73 33016.97 3251765 266808.2 33282.14 3635414
mean 32883.58 913742.5 108324.5 1926961 32948.76 54692.41 219807.3 33079.11 33101.38 15519775 301493.4 33290.62 7987262
median 32897.86 490250.6 104456.3 935843.3 32952.78 33191.73 265754.4 33064.22 33113.92 17841852 306962.8 33289.27 7312427
worst 32956.46 2297973 181148.2 5030428 33019.06 119337.9 314709.5 33144.26 33160.73 23143632 325240 33301.81 13688781
std 76.94696 973487 77908.34 2178087 64.00355 45299.42 133672.1 49.07491 66.50997 9507027 28571.3 8.604748 4845152
rank 1 10 7 11 2 6 8 3 4 13 9 5 12
C11-F16 best 131374.2 289911.4 133666.6 475920.9 135600.3 142488.4 136342.4 143409.4 133204.9 87188553 9569677 66243039 62144961
mean 133550 950389 135187.3 1960981 137683.4 145199.6 142217.9 145950.3 141860.1 89472496 18842170 80082023 76892038
median 133257.5 632638.2 135643.3 1246618 136879.5 145567.6 142484.3 144445.2 141757.3 89326420 15854250 79194589 73536101
worst 136310.8 2246368 135796 4874767 141374.4 147174.8 147560.5 151501.4 150721 92048591 34090502 95695874 98350989
std 2392.2 924912.3 1073.35 2079444 2709.018 2414.887 4923.103 3938 7730.783 2140935 11144663 13343828 16165669
rank 1 8 2 9 3 6 5 7 4 13 10 12 11
C11-F17 best 1916953 7.69E+09 2.12E+09 1.12E+10 1957612 1.06E+09 6.96E+09 2038930 2299063 2.16E+10 9.93E+09 1.85E+10 2.06E+10
mean 1926615 9.02E+09 2.33E+09 1.56E+10 2293290 1.29E+09 9.76E+09 3026639 3119727 2.25E+10 1.13E+10 2.10E+10 2.20E+10
median 1923412 9.20E+09 2.33E+09 1.61E+10 2151018 1.31E+09 9.55E+09 2583866 3212688 2.24E+10 1.16E+10 2.06E+10 2.13E+10
worst 1942685 1.00E+10 2.55E+09 1.91E+10 2913511 1.47E+09 1.30E+10 4899892 3754468 2.34E+10 1.20E+10 2.42E+10 2.49E+10
std 12003.53 1.08E+09 2.00E+08 3.55E+09 450632.8 2.22E+08 2.66E+09 1354721 706077.7 7.96E+08 9.67E+08 2.72E+09 2.04E+09
rank 1 7 6 10 2 5 8 3 4 13 9 11 12
C11-F18 best 938416.2 38056233 3961154 8.23E+07 949848.4 1798555 4141903 967157.9 964035.1 24725187 8352117 1.14E+08 1.11E+08
mean 942057.5 55335975 6591790 119000000 971938.3 2057336 9635664 1031700 988411.5 31196699 11194144 1.36E+08 1.15E+08
median 942553.5 60171362 5547873 1.29E+08 953682.2 2014229 8740133 978717.1 994949.5 33157306 11150777 1.39E+08 1.15E+08
worst 944706.9 62944940 11310259 136000000 1030540 2402332 16920486 1202207 999712.1 33746997 14122905 151000000 120000000
std 2774.139 12250909 3597267 2.65E+07 41194.04 305963.2 5671597 119728.3 17307.31 4553484 2710018 1.73E+07 3.64E+06
rank 1 10 6 12 2 5 7 4 3 9 8 13 11
C11-F19 best 967927.7 46475624 6108745 1.01E+08 1068411 2231819 2066743 1233805 1129215 25080259 2395060 1.58E+08 1.13E+08
mean 1025341 54468087 6693128 1.17E+08 1138554 2472748 10278052 1364982 1479593 35816486 6301409 1.74E+08 1.16E+08
median 983146.6 51071115 6276995 1.10E+08 1096048 2369288 10210584 1339636 1411616 36753610 7266724 1.68E+08 1.15E+08
worst 1167142 69254493 8109779 147000000 1293711 2920598 18624296 1546849 1965925 44678464 8277126 201000000 119000000
std 99675.04 10803670 999513.5 2.25E+07 109726.8 322083.5 8192328 137939.2 368407.8 8923056 2806570 1.97E+07 2.73E+06
rank 1 10 7 12 2 5 8 3 4 9 6 13 11
C11-F20 best 936143.2 50954706 5223024 1.10E+08 957152.3 1647208 6898584 977723.6 962978.5 34027686 9534816 1.46E+08 1.10E+08
mean 941250.4 57915854 5924708 1.26E+08 960470.6 1832300 7322355 998911.1 973018.6 34790803 14357706 1.60E+08 1.16E+08
median 940995.9 56061928 5900417 1.22E+08 961081.7 1770969 7251433 1001253 972340.5 34759730 12833683 1.60E+08 1.17E+08
worst 946866.6 68584856 6674976 150000000 962566.8 2140056 7887969 1015414 984414.7 35616065 22228640 174000000 120000000
std 5013.552 7896396 633442.1 1.77E+07 2451.631 245964.6 444598.3 17075.25 9907.006 694445.5 5830855 1.61E+07 4.38E+06
rank 1 10 6 12 2 5 7 4 3 9 8 13 11
C11-F21 best 9.974206 41.53049 20.3649 57.17792 13.78391 26.56174 35.6313 20.64111 24.52369 48.57262 35.96567 91.90592 59.07491
mean 12.71443 50.50227 21.70431 76.8985 15.95743 29.94194 38.96962 22.44344 27.63915 101.2915 40.89389 106.3608 103.21
median 12.95425 50.18166 21.46124 76.90776 15.89929 30.83809 38.56438 22.17047 27.67676 103.6709 41.89081 107.5927 113.8538
worst 14.97499 60.11528 23.52986 96.60055 18.24724 31.52984 43.11841 24.79172 30.67941 149.2516 43.82828 118.3517 126.0577
std 2.412667 8.418648 1.420534 18.29913 2.179408 2.41647 3.47749 1.927552 3.642786 43.35039 3.696548 13.67187 32.75307
rank 1 9 3 10 2 6 7 4 5 11 8 13 12
C11-F22 best 11.50133 40.80883 22.24877 46.17906 16.22029 28.20055 40.22051 23.95676 24.842 66.75931 39.13532 89.94588 92.17511
mean 16.12513 47.02927 27.52481 63.81642 19.10191 32.24992 46.54553 25.05548 32.40714 103.2154 46.90524 107.2767 93.11538
median 16.72317 47.34708 27.50942 67.79995 19.45731 33.00267 47.31893 25.18537 33.63111 111.9199 46.28905 110.3475 92.79237
worst 19.55286 52.6141 32.83162 73.4867 21.27275 34.79379 51.32375 25.89443 37.52432 122.2625 55.90755 118.4661 94.70167
std 4.197797 5.316853 5.254729 12.72251 2.533066 2.999636 5.265485 0.938547 5.947232 26.22423 7.253922 13.5143 1.170916
rank 1 9 4 10 2 5 7 3 6 12 8 13 11
Sum rank 22 109 191 231 55 146 145 97 118 222 157 198 224
Mean rank 1 4.954545 8.681818 10.5 2.5 6.636364 6.590909 4.409091 5.363636 10.09091 7.136364 9 10.18182
Total rank 1 12 2 4 13 3 11 6 9 7 10 5 8
Wilcoxon: p-value 9.77E-15 1.71E-15 1.71E-15 7.09E-15 3.66E-15 1.71E-15 7.10E-15 3.99E-12 5.36E-15 8.52E-15 2.54E-15 5.36E-15
Table 8. Performance of the optimization algorithms for the pressure vessel design problem.
Table 8. Performance of the optimization algorithms for the pressure vessel design problem.
Algorithm Optimum variables Optimum cost
Ts Th R L
FLO 0.7780271 0.3845792 40.312284 200 5882.8955
WSO 0.7780271 0.3845790 40.312281 200 5882.9011
AVOA 0.7780312 0.3845812 40.3125 199.99699 5882.9085
RSA 1.2457527 0.6715044 63.011657 29.541317 7988.1974
MPA 0.7780271 0.3845792 40.312284 200 5882.9013
TSA 0.7796786 0.3859699 40.39555 200 5912.596
WOA 0.9277593 0.4591628 46.950913 125.78933 6318.0671
MVO 0.8412891 0.4202776 43.571299 159.51516 6018.566
GWO 0.7785126 0.3859628 40.321643 199.96009 5891.0999
TLBO 1.6576806 0.4930712 48.594397 115.47982 11406.549
GSA 1.1728405 1.2516962 44.572154 189.66668 12726.817
PSO 1.6439969 0.6521499 65.916976 31.507574 10499.421
GA 1.4828762 0.8317876 60.436345 58.62914 11533.208
Table 9. Statistical results of the optimization algorithms for the pressure vessel design problem.
Table 9. Statistical results of the optimization algorithms for the pressure vessel design problem.
Algorithm mean best worst std median rank
FLO 5882.8955 5882.8955 5882.8955 1.92E-12 5882.8955 1
WSO 5892.2389 5882.9011 5975.0303 25.597613 5882.9017 3
AVOA 6260.4989 5882.9085 7187.8793 405.93922 6067.7468 5
RSA 13203.718 7988.1974 21708.462 3602.5764 12075.035 9
MPA 5882.9013 5882.9013 5882.9013 4.24E-06 5882.9013 2
TSA 6318.3698 5912.596 7078.0209 383.81952 6175.3376 6
WOA 8256.0687 6318.0671 13647.68 1937.652 7786.0827 8
MVO 6595.3898 6018.566 7192.4617 369.02769 6656.059 7
GWO 6028.1203 5891.0999 6766.8855 275.78721 5900.4533 4
TLBO 30997.684 11406.549 66934.242 15893.46 27298.577 12
GSA 22439.592 12726.817 35296.066 7732.2649 21527.451 10
PSO 32584.002 10499.421 56166.913 14879.262 35973.439 13
GA 27805.883 11533.208 50355.085 12475.479 24579.373 11
Table 10. Performance of the optimization algorithms for the speed reducer design problem.
Table 10. Performance of the optimization algorithms for the speed reducer design problem.
Algorithm Optimum variables Optimum cost
b M p l1 l2 d1 d2
FLO 3.5 0.7 17 7.3 7.8 3.3502147 5.2866832 2996.3482
WSO 3.5000005 0.7 17 7.3000098 7.8000004 3.3502148 5.2866833 2996.3483
AVOA 3.5 0.7 17 7.3000007 7.8 3.3502147 5.2866832 2996.3482
RSA 3.591081 0.7 17 8.2108102 8.2554051 3.3555991 5.4809743 3180.6287
MPA 3.5 0.7 17 7.3 7.8 3.3502147 5.2866832 2996.3482
TSA 3.512746 0.7 17 7.3 8.2554051 3.3505367 5.2901744 3013.6687
WOA 3.5864383 0.7 17 7.3 8.0068571 3.3614768 5.2867549 3037.755
MVO 3.5022252 0.7 17 7.3 8.0658639 3.3693655 5.2868795 3008.0939
GWO 3.5006336 0.7 17 7.3050825 7.8 3.3637852 5.2887849 3001.4528
TLBO 3.5554343 0.7039501 26.213531 8.0919072 8.1411238 3.6597328 5.3387355 5243.4107
GSA 3.5226393 0.7027207 17.364783 7.8143829 7.8885518 3.4080837 5.3847634 3167.6764
PSO 3.5080871 0.7000711 18.082718 7.3978687 7.867227 3.5925551 5.3433467 3298.9223
GA 3.5770929 0.7054996 17.804212 7.7373556 7.8551847 3.6974126 5.3456293 3.34E+03
Table 11. Statistical results of the optimization algorithms for the speed reducer design problem.
Table 11. Statistical results of the optimization algorithms for the speed reducer design problem.
Algorithm mean best worst std median rank
FLO 2996.3482 2996.3482 2996.3482 9.58E-13 2996.3482 1
WSO 2996.6283 2996.3483 2998.7703 0.593604 2996.3642 3
AVOA 3000.8027 2996.3482 3010.9013 4.0273974 3000.7044 4
RSA 3273.4732 3180.6287 3331.0939 58.376714 3288.1754 9
MPA 2996.3482 2996.3482 2996.3482 3.23E-06 2996.3482 2
TSA 3031.7102 3013.6687 3045.2791 10.291508 3033.477 7
WOA 3148.2393 3037.755 3439.8167 107.88934 3115.2947 8
MVO 3029.4277 3008.0939 3069.3067 13.455894 3029.8624 6
GWO 3004.5239 3001.4528 3010.4183 2.5448163 3004.0121 5
TLBO 6.873E+13 5243.4107 4.975E+14 1.175E+14 2.692E+13 12
GSA 3449.2391 3167.6764 4062.9379 266.13006 3321.1203 10
PSO 1.014E+14 3298.9223 5.139E+14 1.258E+14 7.256E+13 13
GA 4.884E+13 3342.7177 3.152E+14 7.902E+13 1.957E+13 11
Table 12. Performance of the optimization algorithms for the welded beam design problem.
Table 12. Performance of the optimization algorithms for the welded beam design problem.
Algorithm Optimum variables Optimum cost
h l t b
FLO 0.2057296 3.4704887 9.0366239 0.2057296 1.7248523
WSO 0.2057296 3.4704887 9.0366239 0.2057296 1.7248523
AVOA 0.204974 3.4868751 9.0365185 0.2057344 1.7259067
RSA 0.1968043 3.5338976 9.9140767 0.2176511 1.972399
MPA 0.2057296 3.4704887 9.0366239 0.2057296 1.7248523
TSA 0.2042143 3.4950752 9.0638538 0.2061512 1.7337349
WOA 0.2136308 3.3314508 8.9745837 0.2208114 1.8201429
MVO 0.2059899 3.4648795 9.0445874 0.2060516 1.7283218
GWO 0.2055937 3.4736068 9.0362448 0.2057979 1.7255154
TLBO 0.313913 4.4099301 6.8250934 0.4224048 3.0076755
GSA 0.2927572 2.7308917 7.4410075 0.3066903 2.0800575
PSO 0.3704896 3.4252427 7.3653863 0.5694254 3.9945673
GA 0.2240807 6.8722146 7.7790615 0.3031552 2.7482044
Table 13. Statistical results for the optimization algorithms for the welded beam design problem.
Table 13. Statistical results for the optimization algorithms for the welded beam design problem.
Algorithm mean best worst std median rank
FLO 1.7246798 1.7246798 1.7246798 2.34E-16 1.7246798 1
WSO 1.7248526 1.7248523 1.7248578 1.269E-06 1.7248523 3
AVOA 1.7607647 1.7259067 1.8412254 0.0369935 1.7470447 7
RSA 2.1759749 1.972399 2.519308 0.1462171 2.1512469 8
MPA 1.7248523 1.7248523 1.7248523 3.40E-09 1.7248523 2
TSA 1.7429229 1.7337349 1.7520018 0.0056864 1.743018 6
WOA 2.3034718 1.8201429 4.0174423 0.6509517 2.08131 9
MVO 1.7410203 1.7283218 1.7744279 0.013955 1.7370004 5
GWO 1.7272229 1.7255154 1.7312168 0.0013824 1.7269807 4
TLBO 3.285E+13 3.0076755 3.17E+14 8.229E+13 5.6424231 12
GSA 2.4348133 2.0800575 2.7395728 0.1942722 2.4639914 10
PSO 4.53E+13 3.9945673 2.742E+14 8.886E+13 6.6680784 13
GA 1.112E+13 2.7482044 1.203E+14 3.506E+13 5.609433 11
Table 14. Performance of the optimization algorithms for the tension/compression spring design problem.
Table 14. Performance of the optimization algorithms for the tension/compression spring design problem.
Algorithm Optimum variables Optimum cost
d D P
FLO 0.0516891 0.3567177 11.288966 0.0126652
WSO 0.0516871 0.3566707 11.291725 0.0126652
AVOA 0.0511979 0.3450265 12.012341 0.0126701
RSA 0.0501506 0.3146926 14.669014 0.0131519
MPA 0.0516907 0.3567578 11.286619 0.0126652
TSA 0.0509975 0.3403048 12.334688 0.0126818
WOA 0.0511727 0.3444345 12.0511 0.0126706
MVO 0.0501506 0.3204164 13.852933 0.0127487
GWO 0.0519529 0.3630808 10.930013 0.0126706
TLBO 0.0675319 0.8850682 2.828481 0.0174182
GSA 0.055068 0.4400717 7.8639345 0.0130684
PSO 0.0674505 0.8819948 2.828481 0.0173176
GA 0.0679933 0.8927686 2.828481 0.0178071
Table 15. Statistical results for the optimization algorithms for the tension/compression spring design problem.
Table 15. Statistical results for the optimization algorithms for the tension/compression spring design problem.
Algorithm mean best worst std median rank
FLO 0.0126019 0.0126019 0.0126019 7.07E-18 0.0126019 1
WSO 0.0126761 0.0126652 0.012822 3.588E-05 0.0126656 3
AVOA 0.0133257 0.0126701 0.014115 0.000558 0.0132591 8
RSA 0.0132315 0.0131519 0.0133718 6.945E-05 0.013211 6
MPA 0.0126652 0.0126652 0.0126652 2.85E-09 0.0126652 2
TSA 0.0129549 0.0126818 0.0135043 0.0002418 0.0128831 5
WOA 0.013257 0.0126706 0.0144524 0.0006048 0.0130638 7
MVO 0.0163776 0.0127487 0.0177786 0.0016487 0.0172702 9
GWO 0.0127215 0.0126706 0.0129391 5.535E-05 0.0127191 4
TLBO 0.0179362 0.0174182 0.0185278 0.0003583 0.0178931 10
GSA 0.0192519 0.0130684 0.0315728 0.0042637 0.0188366 11
PSO 2.039E+13 0.0173176 3.618E+14 8.314E+13 0.0173176 13
GA 1.593E+12 0.0178071 1.647E+13 4.885E+12 0.0252274 12
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