Front. Mech. Eng Frontiers in Mechanical Engineering Front. Mech. Eng 2297-3079 Frontiers Media S.A. 1126450 10.3389/fmech.2022.1126450 Mechanical Engineering Original Research Osprey optimization algorithm: A new bio-inspired metaheuristic algorithm for solving engineering optimization problems Dehghani and Trojovský 10.3389/fmech.2022.1126450 Dehghani Mohammad Trojovský Pavel * Department of Mathematics, Faculty of Science, University of Hradec Králové, Hradec Kralove, Czechia

Edited by: Debiao Meng, University of Electronic Science and Technology of China, China

Reviewed by: Hui Ma, Harbin Engineering University, China

Shiyuan Yang, University of Electronic Science and Technology of China, China

 *Correspondence: Pavel Trojovský, pavel.trojovsky@uhk.cz

This article was submitted to Engine and Automotive Engineering, a section of the journal Frontiers in Mechanical Engineering

 20 01 2023 2022 8 1126450 17 12 2022 30 12 2022 Copyright © 2023 Dehghani and Trojovský. 2023 Dehghani and Trojovský

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This paper introduces a new metaheuristic algorithm named the Osprey Optimization Algorithm (OOA), which imitates the behavior of osprey in nature. The fundamental inspiration of OOA is the strategy of ospreys when hunting fish from the seas. In this hunting strategy, the osprey hunts the prey after detecting its position, then carries it to a suitable position to eat it. The proposed approach of OOA in two phases of exploration and exploitation is mathematically modeled based on the simulation of the natural behavior of ospreys during the hunting process. The performance of OOA has been evaluated in the optimization of twenty-nine standard benchmark functions from the CEC 2017 test suite. Furthermore, the performance of OOA is compared with the performance of twelve well-known metaheuristic algorithms. The simulation results show that the proposed OOA has provided superior performance compared to competitor algorithms by maintaining the balance between exploration and exploitation. In addition, the implementation of OOA on twenty-two real-world constrained optimization problems from the CEC 2011 test suite shows the high capability of the proposed approach in optimizing real-world applications.

 exploitation exploration osprey metaheuristic bio-inspired optimization Univerzita Hradec Králové10.13039/100018512 1 Introduction

An optimization problem refers to a type of problem that has several feasible solutions. According to this definition, obtaining the best solution among these feasible solutions is called optimization (Xian et al., 2021). Every optimization problem has three main parts: decision variables, constraints, and objective function. Optimization aims to determine the values for the design variables respecting the constraints so that the value of the objective function is optimized. Numerous optimization problems in science, engineering, and real-world applications must be solved using optimization techniques (Assiri et al., 2020).

Techniques for solving optimization problems fall into two groups: deterministic and stochastic approaches. Deterministic approaches in two classes, gradient-based and non-gradient-based, have an appropriate performance in solving optimization problems of the following types: linear, convex, continuous, differentiable, and low-dimensional (Xue and Shen 2020). However, the deterministic approaches lose their capability against non-linear, non-convex, discontinuous, non-differentiable, and high-dimensional optimization problems. In this type of optimization problem, deterministic approaches provide unfavorable solutions by getting stuck in local optimal (Mirjalili et al., 2017).

The disadvantages and difficulties of deterministic approaches in solving optimization problems have led researchers to expand stochastic approaches. Metaheuristic algorithms are one of the most effective stochastic techniques based on random search in the problem-solving space using random operators and trial and error processes (Mirjalili 2015). Advantages such as efficiency in non-linear, non-convex, discontinuous, non-differentiable, NP-hard, complex, and high-dimensional problems, efficiency in non-linear and unknown search spaces, no need for differentiable information of the objective function and constraints, and no dependence on the type of problem, has led to the popularity of metaheuristic algorithms to deal with optimization problems (Cavazzuti 2013).

The nature of random search in metaheuristic algorithms means there is no guarantee to provide the global optimal using these techniques. However, the solutions obtained from metaheuristic algorithms are called quasi-optimal due to their proximity to the global optimal (Iba 1994).

Metaheuristic algorithms must be able to perform the search process well in the global and local problem-solving space to achieve a suitable solution. The search process at the global level with the concept of exploration leads to increasing the ability of the algorithm to identify the main optimal area and escape from local optima. The search process at the local level, with the concept of exploitation, leads to an increase in the ability of the algorithm to converge towards possible better solutions in promising areas (Mohar et al., 2022). The main key to the success of metaheuristic algorithms in solving optimization problems is balancing exploration and exploitation during the search process in the problem-solving space. Therefore, in comparing the performance of several metaheuristic algorithms on an optimization problem, an algorithm that provides a better quasi-optimal solution by better balancing exploration and exploitation is superior (Brunetti et al., 2022). The desire to obtain better solutions for optimization problems has led to the design of numerous metaheuristic algorithms by scientists.

The primary research question in the study of metaheuristic algorithms is, considering the numerous metaheuristic algorithms that have been introduced so far, is there still a need to introduce newer algorithms? In response to this question, the No Free Lunch (NFL) theorem (Wolpert and Macready 1997) explains that no unique metaheuristic algorithm is the best optimizer for all optimization problems. The proper performance of a metaheuristic algorithm in solving a set of optimization problems is not a guarantee for the similar performance of that algorithm in solving other optimization problems. According to the NFL theorem, an algorithm that successfully solves several optimization problems may even fail in solving another problem. Therefore, there is no assumption about the result of implementing a metaheuristic algorithm on optimization problems. Hence, the NFL theorem encourages scientists to search some more effective solutions for optimization problems by designing newer metaheuristic algorithms.

The innovation and novelty of this paper is in the design of a new metaheuristic algorithm called the Osprey Optimization Algorithm (OOA), which is used in solving optimization problems in various sciences. The main contributions of this paper are as follows:

• OOA is designed based on the simulation of osprey behavior in nature.

• The fundamental inspiration of OOA is the strategy of ospreys when hunting fish from the sea, during the steps of identifying the prey’s position, catching it, and transporting it to a suitable position for eating.

• The implementation steps of OOA in two phases of exploration and exploitation are mathematically modeled.

• OOA’s performance in optimization tasks is evaluated on twenty-nine benchmark functions from the CEC 2017 test suite.

• The performance of the proposed OOA approach is compared with the performance of twelve well-known algorithms.

• The ability of OOA to address real-world applications is tested on twenty-two constrained optimization problems from the CEC 2011 test suite.

The continuation of the paper is organized as follows: firstly, the literature review is presented in Section 2. Then, the proposed Osprey Optimization Algorithm (OOA) is introduced and mathematically modeled in Section 3. Next, simulation and evaluation studies on handling optimization tasks are presented in Section 4. The performance of the proposed OOA in solving real-world applications is evaluated in Section 5. Finally, conclusions and prospects for future studies are provided in Section 6.

 2 Literature review

Various natural phenomena, animal life in nature, biological sciences, physics, rules of games, human interactions, and other evolutionary phenomena have inspired metaheuristic algorithms. Based on the inspiration’s source used in the design, metaheuristic algorithms fall into five groups: swarm-based, evolutionary-based, physics-based, human-based, and game-based approaches.

Swarm-based metaheuristic algorithms have been introduced inspired by various natural swarming phenomena, such as the natural behaviors of animals, insects, aquatic animals, birds, plants, and other living organisms. Particle Swarm Optimization (PSO) (Kennedy and Eberhart 1995), Ant Colony Optimization (ACO) (Dorigo et al., 1996), and Artificial Bee Colony (ABC) (Karaboga and Basturk 2007), are among the most famous swarm-based algorithms. The main idea in PSO design is modeling the movement of flocks of birds and fish toward the food source. The design of ACO was inspired by the ability of ants to detect the shortest path between a nest and a food source. ABC is developed based on simulating the strategy of colony bees searching for food sources. Among the natural behaviors of animals, trying to obtain food through foraging and hunting has been a source of inspiration in the design of several swarm-based algorithms such as Golden Jackal Optimization (GJO) (Chopra and Ansari 2022), Coati Optimization Algorithm (COA) (Dehghani et al., 2023), Marine Predator Algorithm (MPA) (Faramarzi et al., 2020a), African Vultures Optimization Algorithm (AVOA) (Abdollahzadeh et al., 2021), Whale Optimization Algorithm (WOA) (Mirjalili and Lewis 2016), Pelican Optimization Algorithm (POA) (Trojovský and Dehghani 2022), Honey Badger Algorithm (HBA) (Hashim et al., 2022), Reptile Search Algorithm (RSA) (Abualigah et al., 2022), Grey Wolf Optimizer (GWO) (Mirjalili, Mirjalili, and Lewis 2014), White Shark Optimizer (WSO) (Braik et al., 2022a), and Tunicate Swarm Algorithm (TSA) (Kaur et al., 2020).

Evolutionary-based metaheuristic algorithms have been developed with inspiration from biological sciences, concepts of genetics, Darwin’s theory of evolution, survival of the fittest, and natural selection. Genetic Algorithm (GA) (Goldberg and Holland 1988) and Differential Evolution (DE) (Storn and Price 1997) are among the most famous evolutionary-based methods that are designed based on modeling the reproduction process and using random operators of selection, crossover, and mutation. Modeling the human immune system against disease and microbes is employed in the design of Artificial Immune Systems (AISs) (De Castro and Timmis 2003). Some other evolutionary-based metaheuristic algorithms are: Genetic programming (GP) (Koza and Koza 1992), Evolution Strategy (ES) (Beyer and Schwefel 2002), and Cultural Algorithm (CA) (Reynolds 1994).

Phenomena, forces, laws, and other physics concepts inspire physics-based metaheuristic algorithms. Simulated Annealing (SA) (Kirkpatrick et al., 1983) is one of the most famous physics-based techniques. SA is developed based on modeling the metal annealing process in which the metal is melted under heat and then slowly heated to achieve an ideal crystal.

Newton’s laws of motion and physical forces have been a source of inspiration in designing algorithms such as the Spring Search Algorithm (SSA) (Dehghani et al., 2017) based on spring tension force and Hooke’s law, Momentum Search Algorithm (MSA) (Dehghani and Samet 2020) based on momentum force, and Gravitational Search Algorithm (GSA) (Rashedi et al., 2009) based on gravitational force.

Various physical transformations in the natural water cycle have inspired the design of the Water Cycle Algorithm (WCA) (Eskandar et al., 2012). Other physics-based metaheuristic algorithms are, e.g., Multi-Verse Optimizer (MVO) (Mirjalili et al., 2016), Archimedes Optimization Algorithm (AOA) (Hashim et al., 2021), Equilibrium Optimizer (EO) (Faramarzi et al., 2020b), Electro-Magnetism Optimization (EMO) (Cuevas et al., 2012), Nuclear Reaction Optimization (NRO) (Wei et al., 2019), and Lichtenberg Algorithm (LA) (Pereira et al., 2021).

Human-based metaheuristic algorithms have been introduced with inspiration from human interactions, communication, thinking, and interaction in social and personal life. Teaching-Learning Based Optimization (TLBO) (Rao et al., 2011) is the most widely used human-based approach. Interactions between teachers and students in the classroom environment have been a source of inspiration in the design of TLBO. The strategy of the poor and the wealthy sections of society to improve their economic conditions has been the main idea used in the design of Poor and Rich Optimization (PRO) (Moosavi and Bardsiri 2019).

Some other human-based metaheuristic algorithms are, e.g., Gaining-Sharing Knowledge-based algorithm (GSK) (Mohamed et al., 2020), War Strategy Optimization (WSO) (Ayyarao et al., 2022), Teamwork Optimization Algorithm (TOA) (Dehghani and Trojovský 2021), Coronavirus Herd Immunity Optimizer (CHIO) (Al-Betar et al., 2021), Driving Training-Based Optimization (DTBO) (Dehghani et al., 2022), and Ali Baba and the Forty Thieves (AFT) (Braik et al., 2022b).

Game-based metaheuristic algorithms have been developed inspired by the strategies of players, coaches, referees, and the rules in different games. Mathematical modeling of competitions in different game leagues has been the main idea in designing algorithms such as Soccer league competition algorithm (SLC) (Dehghani et al., 2020) and Football Game Based Optimization (FGBO) (Dehghani et al., 2020) based on football league, and Volleyball Premier League (VPL) (Moghdani and Salimifard 2018) and based on volleyball league.

Analysis of existing optimization methods has shown that no metaheuristic algorithm is based on modeling the natural behavior of osprey. A study of the osprey’s fishing behavior shows that it is an intelligent process with great potential to design a new optimizer. In this regard and in order to address this research gap, in this paper, a new swarm-based metaheuristic algorithm based on the mathematical modeling of natural behaviors of osprey is designed, which is discussed in the next section.

 3 Osprey optimization algorithm

In this section, the proposed Osprey Optimization Algorithm (OOA) approach is introduced, then its mathematical modeling is presented.

3.1 Inspiration of OOA

The osprey, also known as the fish hawk, river hawk, and sea hawk, is a diurnal, fish-eating bird of prey with a cosmopolitan range. An osprey is 50–66 cm in length, 0.9–2.1 kg in weight, and 127–180 cm in wingspan. A picture of the osprey is shown in Figure 1. The appearance characteristics of the osprey are as follows (Strandberg 2013):

• The upperparts are a deep-glossy brown, while the breast is white and sometimes streaked with brown, and the underparts are pure white.

• The head is white with a black mask across the eyes, reaching to the sides of the neck.

• The irises of the eyes are golden to brown, and the transparent nictitating membrane is pale blue.

• The feet are white with black talons and bill is black with a blue cere.

• Osprey has narrow-long wings and a short tail.

Photo of a osprey; downloaded from free media Wikimedia Commons.

The osprey is a piscivorous bird, about 99% of its diet is fish (Grove et al., 2009). It usually catches alive fish weighing from 150 to 300 g and 25–35 cm in length. However, it can catch any fish from 50 g to 2 kg. Ospreys have the high vision to detect underwater objects. When the osprey is flying at a height of 10–40 m above the water’s surface, it detects the position of the fish underwater. Then it moves toward the fish, dips its feet into the water, and dives under the water to catch the fish (Poole et al., 2002). After the osprey catches its prey, it carries it to a nearby rock and begins to eat it (Szaro 1978).

The osprey’s strategy in hunting fish and carrying fish to a suitable position to eat it are intelligent natural behaviors that can be the basis of designing a new optimization algorithm. Therefore, the mathematical modeling of these intelligent osprey behaviors is employed in the design of the proposed OOA approach, which is discussed in the following part.

 3.2 Mathematical modeling

In this subsection, the initialization of OOA is described first, then the process of updating the position of ospreys in the two phases of exploration and exploitation based on the simulation of natural osprey behaviors is presented.

3.2.1 Initialization

The proposed OOA is a population-based approach that can provide a suitable solution based on the search power of its population members in the problem-solving space through a repetition-based process. Each osprey, as a member of the OOA population, determines values ​​for the problem variables based on its position in the search space. Therefore, each osprey is a candidate solution to the problem, mathematically modeled using a vector. Ospreys together form the OOA population, which can be modeled using a matrix according to (1). At the beginning of OOA implementation, the position of ospreys in the search space is randomly initialized using (2). X = X 1 ⋮ X i ⋮ X N N × m = x 1,1 ⋯ x 1 , j ⋯ x 1 , m ⋮ ⋱ ⋮ ⋰ ⋮ x i , 1 ⋯ x i , j ⋯ x i , m ⋮ ⋰ ⋮ ⋱ ⋮ x N , 1 ⋯ x N , j ⋯ x N , m N × m , x i , j = l b j + r i , j ∙ u b j − l b j ,   i = 1,2 ,   … ,   N ,   j = 1,2 ,   … , m , where X is the population matrix of ospreys’ locations, X i is the i th osprey (a candidate solution), x i , j is its j th dimension (problem variable), N is the number of ospreys, m is the number of problem variables, r i , j are random numbers in the interval 0,1 , l b j , and u b j are the lower bound and upper bound of the j th problem variable, respectively

Because each osprey is a candidate solution for the problem, corresponding to each osprey, the objective function can be evaluated. The evaluated values ​​for the objective function of the problem can be represented using a vector according to (3). F = F 1 ⋮ F i ⋮ F N N × 1 = F X 1 ⋮ F X i ⋮ F X N N × 1 , where F is the vector of the objective function values and F i is the obtained objective function value for the i th osprey.

The evaluated values ​​for the objective function are the main criteria for evaluating the quality of the candidate solutions. Therefore, the best value obtained for the objective function corresponds to the best candidate solution (i.e., the best member), and the worst value obtained for the objective function corresponds to the worst candidate solution (i.e., the worst member). Considering that the position of the ospreys in the search space is updated in each iteration, the best candidate solution must also be updated in each iteration.

 3.2.2 Phase 1: Position identification and hunting the fish (exploration)

Ospreys are mighty hunters able to detect the location of fish underwater due to their strong eyesight. After identifying the position of the fish, they attack it and hunt the fish by going underwater. The first phase of population update in OOA is modeled based on the simulation of this natural behavior of ospreys. Modeling the osprey attack on fish leads to significant changes in the position of the osprey in the search space, which increases the exploration power of OOA in identifying the optimal area and escaping from the local optima.

In OOA design, for each osprey, the positions of other ospreys in the search space that have a better objective function value are considered underwater fishes. The set of fish for each osprey is specified using (4). F P i = X k |   k ∈ 1,2 ,   … , N ∧ F k < F i   ∪ X b e s t   , where F P i is the set of fish positions for the i th osprey and X b e s t is the best candidate solution (the best osprey).

The osprey randomly detects the position of one of these fish and attacks it. Based on the simulation of the movement of the osprey towards the fish, a new position for the corresponding osprey is calculated using (5). This new position, if it improves the value of the objective function, replaces the previous position of the osprey according to (6). x i , j P 1 = x i , j + r i , j ∙ S F i , j − I i , j ∙ x i , j , x i , j P 1 = x i , j P 1 ,   l b j ≤ x i , j P 1 ≤ u b j ; l b j ,   x i , j P 1 < l b j ; u b j ,   x i , j P 1 > u b j . X i = X i P 1 ,   F i P 1 < F i ; X i ,   e l s e , where X i P 1 is the new position of the i th osprey based on the first phase of OOA, x i , j P 1 is its j th dimension, F i P 1 is its objective function value, S F i is the selected fish for i th osprey, S F i , j is the its j th dimension, r i , j are random numbers in the interval 0,1 , and I i , j are random numbers from the set 1,2 .

 3.2.3 Phase 2: Carrying the fish to the suitable position (exploitation)

After hunting a fish, the osprey carries it to a suitable (for him safe) position to eat it there. The second phase of updating the population in OOA is modeled based on the simulation of this natural behavior of osprey. The modeling of carrying the fish to the suitable position leads to the creation of small changes in the position of the osprey in the search space, which results in an increase in the exploitation power of the OOA in the local search and convergence towards better solutions near the discovered solutions.

In the design of OOA, to simulate this natural behavior of ospreys, first, for each member of the population, a new random position is calculated as a “suitable position for eating fish” using (7). Then, if the value of the objective function is improved in this new position, it replaces the previous position of the corresponding osprey according to (8). x i , j P 2 = x i , j + l b j + r ∙ u b j − l b j t ,   i = 1,2 ,   … ,   N ,   j = 1,2 ,   … , m ,   t = 1,2 ,   … ,   T , x i , j P 2 = x i , j P 2 ,   l b j ≤ x i , j P 2 ≤ u b j ; l b j ,   x i , j P 2 < l b j ; u b j ,   x i , j P 2 > u b j , X i = X i P 2 ,   F i P 2 < F i ; X i ,   e l s e , where X i P 2 is the new position of the i th osprey based on the second phase of OOA, x i , j P 2 is its j th dimension, F i P 2 is its objective function value, r i , j are random numbers in the interval 0,1 , t is the iteration counter of the algorithm, and T is the total number of iterations.

 3.3 Repetitions process, flowchart, and pseudocode of OOA

The proposed OOA is an iteration-based approach the first iteration of OOA is completed by updating all ospreys’ positions based on the first and second phases. Then, the best candidate solution is updated based on comparing the objective function values. After that, the algorithm enters the next iteration with the updated positions for the ospreys, and the algorithm update process continues until the last iteration based on (4) to (8). Finally, after the full implementation of the algorithm, OOA presents the best candidate solution stored during the iterations as a solution to the problem. The implementation steps of OOA are presented as the flowchart in Figure 2 and its pseudocode in Algorithm 1.

Flowchart of the proposed OOA.

Pseudocode of OOA.

Start OOA.

Input: The problem information (variables, objective function, and constraints).

Set OOA population size (N) and the total number of iterations (T).

Generate the initial population matrix at random using (1) and (2).

Evaluate the objective function by (3).

For t = 1 to T

For i = 1 to N

Phase 1: Position identification and hunting the fish

Update fish positions set for the i th OOA member using (4). F P i = X k |   k ∈ 1,2 ,   … , N ∧ F k < F i   ∪ X b e s t   .

Determine the selected fish by the i th osprey at random.

Calculate new position of the i th OOA member based on the first phase of OOA using (5-a). x i , j P 1 ← x i , j + r i , j ∙ S F i , j − I i , j ∙ x i , j .

Check the boundary conditions for the new position of OOA members using (5-b). x i , j P 1 ← x i , j P 1 ,   l b j ≤ x i , j P 1 ≤ u b j ; l b j ,   x i , j P 1 < l b j ; u b j ,   x i , j P 1 > u b j .

Update the i th OOA member using (6). X i ← X i P 1 ,   F i P 1 < F i ; X i ,   e l s e .

Phase 2: Carrying the fish to the suitable position

Calculate new position of the i th OOA member based on the second phase of OOA using (7-a). x i , j P 2 ← x i , j + l b j + r i , j ∙ u b j − l b j / t .

Check the boundary conditions for the new position of OOA members using (7-b). x i , j P 2 ← x i , j P 2 ,   l b j ≤ x i , j P 2 ≤ u b j ; l b j ,   x i , j P 2 < l b j ; u b j ,   x i , j P 2 > u b j .

Update the i th OOA member using (8). X i ← X i P 2 ,   F i P 2 < F i ; X i ,   e l s e .

end

Save the best candidate solution found so far.

End OOA

3.4 Computational complexity

In this subsection, the computational complexity of the proposed OOA approach is evaluated. OOA initialization for a problem with dimension m equals O N m , where m is the number of problem variables, and N is the population size (i.e., the number of ospreys). In each iteration of the algorithm, each osprey is updated in two phases exploration and exploitation. This population update process has a complexity equal to O 2 N m T , where T is the total number of algorithm iterations. Therefore, the total computational complexity of OOA equals O N m 1 + 2 T .

 4 Simulation studies and discussion

This section presents simulation studies on the performance of the proposed OOA in solving optimization problems. For this purpose, the efficiency of OOA is evaluated on the CEC 2017 test suite. Also, in order to analyze the quality of OOA in providing appropriate solutions, the results obtained from the proposed approach are compared with the performance of twelve well-known metaheuristic algorithms, including GA, PSA, GSA, TLBO, GWO, MVO, WOA, TSA, MPA, RSA, WSO, and AVOA. The values ​​of the control parameters for competitor algorithms are specified in Table 1. Simulation results are reported using six statistical indicators: mean, best, worst, standard deviation (std), and rank. The ranking criterion for metaheuristic algorithms in solving each benchmark function is to provide a better value for the mean index.

Parameter values for the competitor algorithms.

Algorithm Parameter Value
GA
Type Real coded
Selection Roulette wheel (Proportionate)
Crossover Whole arithmetic ( P r o b a b i l i t y = 0.8 ; α ∈ − 0.5 ,   1.5 )
Mutation Gaussian ( P r o b a b i l i t y = 0.05 )
PSO
Topology Fully connected
Cognitive and social constant C 1 ,   C 2 = 2 ,   2 .
Inertia weight Linear reduction from 0.9 to 0.1
Velocity limit 10% of the dimension range
GSA
Alpha, G 0 , R norm , R power 20, 100, 2, 1
TLBO
T F : the teaching factor T F = r o u n d   1 + r a n d
random number rand rand is a random number from the interval 0,1 .
GWO
Convergence parameter (a) a: Linear reduction from 2 to 0
MVO
wormhole existence probability (WEP) Min W E P = 0.2 and Max W E P = 1
Exploitation accuracy over the iterations (p) p = 6
WOA
Convergence parameter a a: Linear reduction from 2 to 0
Parameters r and l r is a random vector in 0,1 ,
l is a random number in − 1,1 .
TSA
P min and P max 1, 4
c 1 , c 2 , c 3 random numbers lie in the range 0,1 .
MPA
Constant number P = 0.5
Random vector R is a vector of uniform random numbers from 0,1 .
Fish Aggregating Devices (FADs) F A D s = 0.2
Binary vector U = 0 or 1
RSA
Sensitive parameter β = 0.01
Sensitive parameter α = 0.1
Evolutionary Sense (ES) ES are randomly decreasing values between 2 and − 2
AVOA
L 1 L 2 L 1 , L 2 = 0.8,0.2 .
w w = 2.5
P 1 , P 2 P 3 P 1 , P 2 , P 3 = 0.6 ,   0.4 ,   0.6
WSO
F min and F max F min , F max = 0.07 ,   0.75 .
τ , a 0 , a 1 , a 2 τ , a 0 , a 1 , a 2 = 4.125 ,   6.25 ,   100 ,   0.0005 .
4.1 Evaluation of the CEC 2017 test suite

In this subsection, the OOA’s performance in solving optimization problems is evaluated on the CEC 2017 test suite. This test suite has thirty 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 composition functions of C17-F21 to C17-F30. The C17-F2 function has been excluded from the simulation studies due to its unstable behavior. The full description of the CEC 2017 test suite is provided in (Awad et al., 2016). The proposed OOA approach and competitor algorithms are applied in handling the CEC 2017 test suite for dimensions equal to 10, 30, 50, and 100. For all of the functions in the CEC 2017 test suite, OOA and competitor algorithms are employed for the maximal number of function evaluations (FEs) of 10000 ∙ D , where D is the number of variables (dimensions of the problem). The number of runs is 51 for each problem in each dimension, and the stop criterion is set to the maximal number of FEs.

The implementation results of the proposed OOA and competitor algorithms on the CEC 2017 test suite for D = 10 ( D is the dimension of the problem, i.e., the number of decision variables) are presented in Table 2. The convergence curves under this experiment for dimensions equal to 10 are plotted in Figure 3. Based on the obtained results, the proposed OOA approach is the first best optimizer for C17-F1, C17-F3 to C17-F21, C17-F23, C17-F24, and C17-F27 to C17-F30.

Performance of optimization algorithms on the CEC 2017 test suite ( D = 10 ).

OOA WSO AVOA RSA MPA TSA WOA MVO GWO TLBO GSA PSO GA
C17-F1 mean 100 5.88E + 09 4013.006 1.07E+10 64541720 1.82E+09 6741740 7,856.682 92,201,959 1.54E+08 775.8252 3,282.289 12,388,229
best 100 4.87E + 09 116.3249 9.22E + 09 20,390.56 3.90E + 08 4908968 4,995.766 29,049.83 6,85,32,154 100.0201 356.7806 64,15,095
worst 100 7.55E + 09 12,447.47 1.27E + 10 2.34E + 08 3.96E + 09 88,76,332 11,579 3.35E + 08 3.71E + 08 1,866.593 9,727.858 17784373
std 0.00E + 00 1.14E + 09 5690.299 1.56E + 09 1.12E + 08 1.57E + 09 1658695 3046.673 1.61E + 08 1.44E + 08 755.0732 4286.7 4694555
median 100 5.56E + 09 1744.113 1.04E + 10 11829649 1.46E + 09 6590830 7425.982 16898644 87873367 568.3437 1522.258 12676723
rank 1 12 4 13 8 11 6 5 9 10 2 3 7
C17-F3 mean 300 8901.041 301.9788 10068.59 2325.391 11693.03 1794.181 300.0571 3193.416 745.447 10706.01 300 15424.57
best 300 4498.536 300 5422.718 1198.474 4444.41 633.6523 300.0133 1583.535 478.9383 6732.014 300 4531.794
worst 300 11914.76 304.2326 13475.69 4387.107 16537.31 3466.904 300.1299 6138.724 919.541 14556.37 300 24388.24
std 0.00E + 00 3205.813 2.268088 3637.573 1.45E + 03 5073.04 1318.659 0.050646 2076.194 190.8099 3188.063 4.59E-14 10247.01
median 300 9595.433 301.8413 10687.98 1857.991 12895.19 1538.083 300.0425 2525.702 791.6544 10767.84 300 16389.12
rank 1 9 4 10 7 12 6 3 8 5 11 2 13
C17-F4 mean 400 957.888 404.9693 1394.55 410.7178 584.5092 426.3024 403.4875 412.2763 409.5913 404.7619 421.2443 415.3942
best 400 708.735 401.2981 865.3083 404.4693 481.4117 406.7374 401.6671 406.3689 408.7705 403.7249 400.1105 412.2142
worst 400 1182.602 406.826 1912.946 420.7999 704.8832 476.9316 405.1198 429.6616 410.1096 406.3548 473.6029 419.2849
std 0 213.4578 2.572911 441.7709 7.396089 108.2126 33.44582 1.773533 11.4502 0.567212 1.191513 34.83844 3.057003
median 400 970.1075 405.8765 1399.972 408.801 575.871 410.7703 403.5815 406.5373 409.7426 404.484 405.632 415.0388
rank 1 12 4 13 6 11 10 2 7 5 3 9 8
C17-F5 mean 501.2464 567.4359 546.4592 576.8394 514.5862 567.9134 543.2107 524.9737 513.7034 535.9086 556.8219 529.4119 529.53
best 500.9951 552.2557 528.2967 561.3407 509.1961 545.6081 524.7203 510.7503 508.9499 530.1252 551.6979 511.7206 524.571
worst 501.9917 577.3946 566.3238 592.6999 521.2896 601.7087 581.0582 540.0197 521.413 539.6513 569.2487 554.6231 535.6298
std 0.490881 11.3307 19.73016 17.18769 5.554223 24.58643 26.06952 12.07391 5.31579 4.136386 8.296054 19.5701 4.94214
median 500.9993 570.0465 545.6081 576.6584 513.9295 562.1684 533.5322 524.5623 512.2253 536.9289 553.1704 525.652 528.9596
rank 1 11 9 13 3 12 8 4 2 7 10 5 6
C17-F6 mean 600 634.3963 618.3666 643.1669 601.6245 626.3311 624.5677 602.2798 601.1951 607.2775 618.2456 607.879 610.8805
best 600 630.2308 617.3015 639.7602 601.076 615.9859 607.9813 600.5006 600.6321 605.0464 603.0926 601.4365 607.3226
worst 600 637.8967 621.0724 647.6775 602.9185 642.8641 647.9323 604.574 601.8229 610.7564 638.3277 620.4237 615.3818
std 0 3.554897 1.787183 3.514845 0.866795 11.45464 16.61899 1.808355 0.48699 2.571713 16.10331 8.50812 3.529157
median 600 634.729 617.5462 642.615 601.2518 623.2373 621.1786 602.0223 601.1628 606.6536 615.781 604.8279 610.4088
rank 1 12 9 13 3 11 10 4 2 5 8 6 7
C17-F7 mean 711.1267 808.7141 768.8554 810.0082 726.7612 835.5864 765.1706 732.0761 726.9165 754.5335 717.4829 734.0534 738.434
best 710.6726 787.3283 745.9056 795.9949 720.1534 793.084 753.5139 717.5222 717.841 749.7496 715.0157 726.4152 727.4159
worst 711.7995 826.3722 798.2962 823.4679 738.3295 879.6027 796.3526 752.5328 745.5196 763.1776 721.473 746.3446 743.3223
std 0.505957 16.26902 23.82059 12.74505 8.085852 37.13584 20.5967 14.54833 12.56493 5.945114 2.758161 8.969209 7.372599
median 711.0174 810.578 765.6099 810.2851 724.2809 834.8294 755.4079 729.1246 722.1528 752.6035 716.7214 731.7268 741.499
rank 1 11 10 12 3 13 9 5 4 8 2 6 7
C17-F8 mean 801.4928 851.4892 832.938 856.8894 815.8214 851.1497 838.4982 812.467 816.731 839.9276 820.9936 824.0754 817.7315
best 800.995 842.9416 821.4712 844.9551 810.9035 833.9359 819.6608 807.8228 811.1105 832.6263 812.6957 816.5959 813.5324
worst 801.9912 860.4205 849.7478 862.4836 819.0522 871.6601 851.4164 817.5787 822.055 848.4403 829.2716 830.8931 826.035
std 0.567911 7.858445 11.80419 7.985664 3.557128 16.56874 13.4719 3.96306 4.522294 7.992517 6.968316 6.994147 5.564979
median 801.4926 851.2974 830.2666 860.0594 816.665 849.5013 841.4579 812.2331 816.8792 839.3219 821.0036 824.4063 815.6793
rank 1 12 8 13 3 11 9 2 4 10 6 7 5
C17-F9 mean 900 1454.819 1205.691 1501.902 909.3144 1410.034 1404.287 900.8504 912.6632 912.5493 900 904.5013 905.4237
best 900 1302.583 957.0056 1400.258 900.53 1184.376 1084.748 900.0011 900.6083 907.6741 900 900.9542 902.9691
worst 900 1604.736 1707.615 1647.775 924.7041 1715.875 1702.526 903.3049 935.1553 921.227 900 913.0719 909.6329
std 0 133.9345 343.6003 104.1144 1.09E + 01 227.2 256.9198 1.617041 16.01129 5.884131 0 5.716437 2.977433
median 900 1455.978 1079.072 1479.788 906.0117 1369.942 1414.936 900.0477 907.4447 910.648 900 901.9895 904.5464
rank 1 11 8 12 5 10 9 2 7 6 1 3 4
C17-F10 mean 1006.179 2369.887 1816.758 2658.303 1687.833 2084.667 2076.678 1819.878 1761.675 2231.198 2342.712 1993.442 1751.437
best 1000.284 2094.46 1507.356 2478.328 1504.795 1795.636 1471.344 1478.853 1565.484 1820.23 2048.463 1588.159 1434.859
worst 1012.668 2563.274 2484.999 3036.783 1896.677 2349.493 2628.11 2347.506 2042.318 2534.398 2454.347 2420.523 2167.228
std 6.575908 209.9802 453.5094 256.7411 158.7545 288.9921 552.2516 415.8416 200.1676 299.9799 194.1695 337.7744 310.1715
median 1005.882 2410.907 1637.338 2559.051 1674.929 2096.769 2103.629 1726.576 1719.45 2285.082 2434.018 1982.542 1701.831
rank 1 12 5 13 2 9 8 6 4 10 11 7 3
C17-F11 mean 1100 3997.365 1150.914 4127.568 1145.428 5676.276 1153.49 1128.874 1158.019 1153.442 1141.141 1145.692 2446.535
best 1100 2691.465 1117.896 1476.434 1120.373 5520.68 1113.604 1105.822 1122.697 1139.712 1120.621 1133.854 1115.793
worst 1100 5259.248 1206.845 6746.097 1201.544 5761.647 1176.737 1151.345 1234.755 1175.901 1172.026 1168.255 6221.771
std 0 1140.192 38.67705 2339.679 37.65748 105.8026 28.8003 22.46952 51.60553 15.43246 21.67382 15.3044 2486.946
median 1100 4019.373 1139.458 4143.87 1129.898 5711.388 1161.81 1129.165 1137.313 1149.078 1135.959 1140.329 1224.288
rank 1 11 6 12 4 13 8 2 9 7 3 5 10
C17-F12 mean 1352.959 3.72E + 08 1158306 7.42E + 08 1043793 1094197 2477570 1083075 1489573 5317863 1073890 8443.507 636709.5
best 1318.646 83388866 374631.4 1.65E + 08 34796.03 567385.7 180695.2 9224.785 47751.04 1423075 499376.5 2581.009 184374.6
worst 1438.176 6.50E + 08 2100635 1.30E + 09 1633880 1343368 4110247 3402232 2331654 9414272 1816170 14572.97 1124017
std 56.60449 2.83E + 08 797531.4 5.66E + 08 696087.2 361487 1804563 1548367 994452.6 4181611 550650.1 5397.789 381157.9
median 1327.506 3.77E + 08 1078979 7.53E + 08 1253248 1233017 2809669 460422 1789444 5217052 990006.9 8310.027 619223.1
rank 1 12 8 13 4 7 10 6 9 11 5 2 3
C17-F13 mean 1305.324 18096300 19224.71 36181435 8490.319 13336.78 7907.686 7012.701 10770.4 17534.3 10530.73 6899.771 57236.95
best 1303.114 1509879 2797.376 3003814 5490.652 7917.031 3384.574 1390.317 6780.111 16553.42 5243.304 2435.238 8923.086
worst 1308.508 60065247 32989.77 1.20E + 08 11056.06 21170.17 15879.1 12959.9 15073.2 19930.73 14860.49 17522.46 189419.1
std 2.245246 27700233 15419.68 55398529 2381.677 5650.759 5626.796 5920.152 3357.506 1593.289 4015.727 7073.631 87105.22
median 1304.837 5405037 20555.85 10802981 8707.281 12129.97 6183.535 6850.295 10614.14 16826.52 11009.57 3820.696 15302.78
rank 1 12 10 13 5 8 4 3 7 9 6 2 11
C17-F14 mean 1400.746 4132.154 2055.135 5558.351 2266.925 3493.002 1525.39 1581.095 2397.109 1601.045 5788.637 3081.174 13581.11
best 1400 3248.102 1693.97 4856.237 1458.079 1492.599 1486.174 1424.301 1465.616 1522.321 4773.477 1434.271 3851.459
worst 1400.995 5651.76 2904.613 7191.996 4109.713 5807.657 1567.324 2025.362 5154.382 1632.942 7883.393 7135.644 27138.06
std 0.491454 1096.309 563.6311 1083.921 1240.519 2267.558 40.95033 292.6607 1815.952 52.09687 1439.473 2691.813 9745.123
median 1400.995 3814.378 1810.979 5092.585 1749.954 3335.877 1524.031 1437.358 1484.218 1624.458 5248.839 1877.39 11667.46
rank 1 10 5 11 6 9 2 3 7 4 12 8 13
C17-F15 mean 1500.331 10751.75 5501.39 14536.8 5471.727 7297.181 6470.957 1544.072 6045.284 1720.393 25078.78 9398.54 4713.312
best 1500.001 3084.974 2103.124 2800.156 3969.523 2363.442 2042.013 1527.29 3680.991 1588.624 11746.99 2945.015 1911.483
worst 1500.5 18908.97 13222.65 31904.23 6702.584 13137.14 14086.81 1556.748 7188.984 1814.857 37679.46 15506.07 8362.85
std 0.232574 6523.699 5124.631 12553.82 1113.178 4573.809 5186.651 12.71076 1592.372 109.6817 12238.7 5186.186 3168.595
median 1500.413 10506.52 3339.894 11721.41 5607.4 6844.073 4877.502 1546.124 6655.58 1739.045 25444.33 9571.536 4289.458
rank 1 11 6 12 5 9 8 2 7 3 13 10 4
C17-F16 mean 1600.76 2034.461 1820.711 2038.594 1728.993 2071.12 1969.139 1827.739 1735.311 1680.961 2098.37 1940.866 1813.107
best 1600.356 1958.053 1644.471 1830.998 1648.699 1876.344 1773.893 1733.216 1616.611 1653.606 1965.56 1834.305 1725.011
worst 1601.12 2197.971 1943.576 2327.058 1791.629 2265.545 2104.18 1892.713 1837.448 1738.137 2303.605 2109.139 1845.824
std 0.312085 108.9644 124.4788 206.9655 58.70989 174.4498 155.0947 66.64265 90.01747 38.9192 151.8365 125.7682 58.06975
median 1600.781 1990.909 1847.399 1998.161 1737.822 2071.294 1999.242 1842.514 1743.592 1666.05 2062.157 1910.01 1840.796
rank 1 10 6 11 3 12 9 7 4 2 13 8 5
C17-F17 mean 1700.099 1824.573 1753.718 1824.731 1757.367 1807.68 1849.517 1850.445 1772.211 1761.522 1854.652 1755.178 1758.992
best 1700.02 1814.876 1736.238 1806.914 1726.832 1791.663 1777.567 1782.711 1725.759 1750.842 1750.517 1748.159 1755.699
worst 1700.332 1830.094 1800.113 1834.454 1832.663 1819.163 1899.523 1963.932 1880.822 1772.01 1987.778 1762.23 1761.564
std 0.153284 6.661755 30.63523 12.0928 49.7366 11.6701 52.33144 84.76388 71.89705 10.35641 119.5278 5.944272 2.620464
median 1700.022 1826.661 1739.262 1828.778 1734.987 1809.948 1860.49 1827.569 1741.131 1761.617 1840.156 1755.161 1759.353
rank 1 9 2 10 4 8 11 12 7 6 13 3 5
C17-F18 mean 1805.36 3002166 12367.56 5987025 17226.94 12580.35 24399.03 21918.55 20824.69 30910.66 10114.73 22895.02 13373.88
best 1800.003 153812 4999.356 296295.4 5704.958 7753.988 6684.609 9051.44 6554.048 25117.67 6625.996 2935.762 3519.428
worst 1820.451 8700828 16297.36 1.74E + 07 26124.1 17023.98 38376 35331.55 35203.57 38683.41 12367.24 42715.76 19330
std 9.941474 3910776 5004.059 7.82E + 06 9988.101 3808.451 15085.81 12222.39 14352.66 6164.374 2420.751 20287.36 6822.131
median 1800.492 1577013 14086.77 3135817 18539.35 12771.71 26267.75 21645.61 20770.57 29920.77 10732.85 22964.27 15323.05
rank 1 12 3 13 6 4 10 8 7 11 2 9 5
C17-F19 mean 1900.445 416604.2 6952.688 739541 6884.348 131794.7 36476.33 1915.482 5560.851 4838.776 42379.53 26116.33 6400.24
best 1900.039 26213.49 2190.883 48044.1 2387.655 1951.727 7952.233 1909.898 1946.858 2050.465 11570.77 2661.409 2229.189
worst 1901.559 880029.5 13820.02 1588787 12016.72 263352.4 66856.63 1925.538 14413.53 13027.13 61513.11 80690.36 10287.94
std 0.735595 363914.2 5586.445 686643.6 4697.35 148091.1 23888.81 7.294323 5891.559 5392.982 22095.98 36346.34 3284.642
median 1900.09 380086.8 5899.924 660666.3 6566.509 130937.3 35548.24 1913.247 2941.508 2138.756 48217.12 10556.78 6541.917
rank 1 12 7 13 6 11 9 2 4 3 10 8 5
C17-F20 mean 2000.312 2225.94 2179.197 2234.325 2145.849 2217.885 2217.062 2146.623 2178.519 2075.639 2266.556 2177.535 2052.745
best 2000.312 2166.285 2032.905 2172.857 2117.502 2112.101 2103.303 2049.306 2137.541 2064.053 2197.289 2152.186 2037.613
worst 2000.312 2299.729 2309.427 2292.56 2200.427 2337.172 2302.475 2259.975 2258.47 2086.588 2364.382 2210.998 2060.904
std 0 54.45673 122.8932 58.19579 36.72394 94.18969 94.05653 85.4547 53.85427 9.333807 80.30884 28.87632 10.60866
median 2000.312 2218.874 2187.229 2235.941 2132.735 2211.134 2231.234 2138.606 2159.033 2075.958 2252.277 2173.479 2056.232
rank 1 11 8 12 4 10 9 5 7 3 13 6 2
C17-F21 mean 2200 2297.878 2214.518 2270.58 2295.882 2331.616 2315.515 2255.881 2319.129 2304.823 2376.981 2324.917 2303.224
best 2200 2248.092 2204.341 2225.19 2291.937 2222.324 2219.34 2200.008 2314.704 2203.911 2358.604 2316.441 2227.926
worst 2200 2325.371 2241.022 2296.388 2300.117 2380.993 2361.984 2313.154 2324.354 2345.503 2395.201 2332.858 2339.654
std 0 35.57146 17.50932 31.10755 3.360515 73.22109 64.14039 63.74518 3.920714 66.93679 15.1092 7.977182 50.22458
median 2200 2309.025 2206.355 2280.371 2295.737 2361.574 2340.369 2255.181 2318.728 2334.938 2377.061 2325.184 2322.659
rank 1 6 2 4 5 12 9 3 10 8 13 11 7
C17-F22 mean 2300.073 2759.652 2309.448 2948.005 2307.827 2735.473 2325.039 2285.03 2309.046 2320.591 2300.001 2313.959 2318.861
best 2300 2627.622 2304.591 2728.184 2301.157 2456.93 2320.134 2225.754 2301.333 2313.992 2300 2300.671 2315.814
worst 2300.29 2905.985 2311.729 3109.327 2316.925 2954.198 2333.087 2305.553 2323.58 2332.946 2300.006 2347.843 2323.552
std 0.143325 126.8546 3.24115 158.5521 6.508385 219.2463 5.719468 39.05074 10.10884 8.565301 0.002867 22.36311 3.265565
median 2300 2752.501 2310.737 2977.255 2306.613 2765.381 2323.468 2304.406 2305.635 2317.714 2300 2303.661 2318.04
rank 3 12 6 13 4 11 10 1 5 9 2 7 8
C17-F23 mean 2600.919 2702.3 2644.345 2706.013 2616.616 2730.078 2651.35 2621.31 2614.436 2644.846 2802.158 2646.682 2659.182
best 2600.003 2658.113 2632.078 2675.359 2611.65 2636.222 2632.537 2607.533 2608.072 2633.416 2733.615 2638.993 2638.185
worst 2602.87 2727.808 2663.101 2748.835 2621.263 2776.96 2672.529 2633.542 2621.547 2654.553 2948.093 2659.303 2668.032
std 1.304343 32.16667 14.4216 33.92966 4.584374 62.80396 21.35488 11.20767 6.825764 9.271253 99.601 9.057713 14.04045
median 2600.403 2711.639 2641.1 2699.929 2616.776 2753.566 2650.167 2622.082 2614.062 2645.707 2763.462 2644.216 2665.256
rank 1 10 5 11 3 12 8 4 2 6 13 7 9
C17-F24 mean 2630.488 2785.506 2775.021 2861.754 2706.431 2670.333 2767.648 2686.173 2755.175 2762.612 2753.792 2772.867 2728.127
best 2516.677 2721.56 2745.428 2829.747 2692.666 2514.52 2746.082 2500.511 2735.735 2755.975 2500.868 2756.913 2527.453
worst 2732.32 2864.718 2804.956 2921.527 2720.353 2816.479 2795.053 2761.171 2776.077 2768.672 2906.529 2789.452 2815.512
std 115.0901 67.0623 26.43805 40.27187 11.48417 159.3105 20.18173 122.6274 17.25984 5.293828 173.2975 13.39486 132.9355
median 2636.477 2777.873 2774.851 2847.87 2706.353 2675.166 2764.728 2741.505 2754.444 2762.9 2803.886 2772.55 2784.771
rank 1 12 11 13 4 2 9 3 7 8 6 10 5
C17-F25 mean 2932.639 3172.085 2912.47 3295.1 2929.286 3144.314 2906.204 2921.541 2939.019 2933.577 2921.72 2922.841 2953.287
best 2898.047 3073.752 2899.151 3222.244 2918.393 2903.855 2755.097 2898.695 2919.99 2914.597 2900.46 2898.701 2940.339
worst 2945.793 3390.408 2949.345 3377.004 2936.509 3698.097 2962.384 2944.924 2946.088 2952.782 2943.394 2946.594 2963.808
std 22.81025 145.1648 24.29665 63.27114 7.613289 368.726 99.64557 26.0023 12.57023 20.30113 24.22675 26.53174 9.871435
median 2943.359 3112.09 2900.692 3290.576 2931.12 2987.651 2953.668 2921.272 2944.999 2933.464 2921.514 2923.034 2954.5
rank 7 12 2 13 6 11 1 3 9 8 4 5 10
C17-F26 mean 2900 3638.702 2984.147 3802.372 3155.204 3659.559 3198.238 2900.156 3284.966 3223.122 3913.384 2904.278 2897.067
best 2900 3276.142 2802 3461.088 2942.976 3157.35 2928.678 2900.119 2972.95 2912.703 2802 2802 2697.055
worst 2900 3897.259 3170.625 4157.559 3633.147 4343.293 3631.459 2900.204 3961.353 3928.241 4427.01 3015.112 3120.882
std 3.67E-13 294.9394 207.8071 296.6646 316.9464 572.8857 303.5414 0.037246 449.5678 467.4405 743.8456 86.0872 212.0623
median 2900 3690.703 2981.981 3795.421 3022.346 3568.797 3116.409 2900.15 3102.78 3025.771 4212.262 2900 2885.165
rank 2 10 5 12 6 11 7 3 9 8 13 4 1
C17-F27 mean 3089.518 3214.802 3121.644 3238.743 3113.99 3184.372 3200.595 3091.742 3117.541 3116.468 3233.373 3138.58 3163.798
best 3089.518 3163.375 3095.618 3129.242 3093.945 3103.124 3183.846 3089.721 3094.702 3095.701 3220.557 3097.503 3120.949
worst 3089.518 3292.134 3185.842 3441.154 3163.781 3228.902 3212.991 3095.257 3181.476 3175.664 3256.086 3188.445 3225.9
std 2.59E-13 54.02256 42.4041 136.5134 32.9318 56.29399 12.01633 2.572715 42.14892 38.9956 15.61935 37.78113 43.83476
median 3089.518 3201.85 3102.557 3192.288 3099.117 3202.73 3202.772 3090.994 3096.994 3097.254 3228.424 3134.186 3154.172
rank 1 11 6 13 3 9 10 2 5 4 12 7 8
C17-F28 mean 3100 3650.317 3243.258 3814.894 3302.112 3611.82 3296.617 3246.015 3357.808 3336.911 3469.172 3316.468 3254.039
best 3100 3598.481 3100 3728.262 3200.348 3428.915 3155.46 3100.131 3199.666 3219.959 3455.215 3181.164 3147.237
worst 3100 3694.959 3405.585 3877.429 3349.496 3832.012 3406.123 3405.586 3428.423 3405.84 3488.569 3405.812 3535.105
std 0 39.91454 133.5314 68.39759 6.79E + 01 206.5341 127.2636 166.702 104.9689 87.64807 15.26821 100.6178 185.8137
median 3100 3653.913 3233.724 3826.943 3329.303 3593.177 3312.443 3239.172 3401.571 3360.922 3466.453 3339.448 3166.908
rank 1 12 2 13 6 11 5 3 9 8 10 7 4
C17-F29 mean 3132.241 3338.849 3292.773 3388.331 3246.266 3241.876 3360.601 3206.511 3272.384 3217.004 3357.428 3273.303 3242.928
best 3130.076 3320.568 3214.749 3313.223 3185.239 3167.863 3241.421 3143.177 3192.939 3167.591 3239.055 3169.945 3191.43
worst 3134.841 3358.76 3377.836 3459.255 3326.402 3315.826 3515.39 3294.941 3392.631 3240.799 3662.207 3360.534 3294.792
std 2.452283 19.14621 82.95056 74.34003 64.69228 59.8292 113.533 63.51536 93.81494 33.93212 201.5651 85.54342 42.93781
median 3132.023 3338.033 3289.253 3390.423 3236.712 3241.908 3342.796 3193.963 3251.983 3229.814 3264.226 3281.367 3242.744
rank 1 10 9 13 6 4 12 2 7 3 11 8 5
C17-F30 mean 3418.734 2329979 309087.1 3856834 728531.3 644628.9 1040867 317639.2 981763.2 63452.35 821101.4 406173.7 1602388
best 3394.682 1420187 109668.3 868214.9 26053.54 117676.4 4516.352 7639.062 35077.78 30568.26 631231.4 6540.001 551547.9
worst 3442.907 3497681 805467.6 6091908 1061109 1363214 3929844 1211523 1420932 106604.5 1048619 805506.1 3650707
std 27.43434 853522.5 327803.1 2160680 472437.7 522852.9 1905032 588898.3 643256.9 36686.7 171347.8 454921.7 1443151
median 3418.673 2201025 160606.2 4233608 913481.5 548812.7 114553.4 25697.17 1235521 58318.33 802277.8 406324.5 1103649
rank 1 12 3 13 7 6 10 4 9 2 8 5 11
Sum rank 38 319 173 350 137 280 236 111 187 189 236 180 191
Mean rank 1.310345 11 5.965517 12.06897 4.724138 9.655172 8.137931 3.827586 6.448276 6.517241 8.137931 6.206897 6.586207
Total rank 1 11 4 12 3 10 9 2 6 7 9 5 8

Boxplot of OOA and competitor algorithms in optimization of the CEC-2017 test suite (D = 10).

The results of employing OOA and competitor algorithms in handling the CEC 2017 test suite for D = 30 are reported in Table 3. The convergence curves while reaching the solution for the dimension D equal to 30 are shown in Figure 4. Based on the simulation results, OOA is the first best optimizer for C17-F1, C17-F3 to C17-F22, C17-F24, C17-F5, and C17-F27 to C17-F30.

Performance of optimization algorithms on the CEC 2017 test suite ( D = 30 ).

OOA WSO AVOA RSA MPA TSA WOA MVO GWO TLBO GSA PSO GA
C17-F1 mean 100 2.74E + 10 3246.782 4.29E + 10 27955.3 1.87E + 10 1.77E + 09 561459.3 1.74E + 09 6.44E + 09 10969548 1.47E + 09 1.86E + 08
best 100 2.36E + 10 287.7475 3.83E + 10 12862.65 1.17E + 10 1.40E + 09 436189.9 2.87E + 08 4.07E + 09 2638.178 3907.173 1.39E + 08
worst 100 3.43E + 10 7988.686 5.28E + 10 4.25E + 04 2.55E + 10 2.20E + 09 714120.5 5.25E + 09 9.60E + 09 38295792 5.86E + 09 2.57E + 08
std 8.20E-15 4.99E + 09 3609.784 6.69E + 09 1.43E+04 6.43E+09 4.12E+08 137372.1 2.35E+09 2.31E+09 18406047 2.93E+09 50984062
median 100 2.59E+10 2355.347 4.03E+10 28226.58 1.88E+10 1.75E+09 547763.4 7.19E+08 6.05E+09 2789882 3337182 1.74E+08
rank 1 12 2 13 3 11 9 4 8 10 5 7 6
C17-F3 mean 300 101633.6 46663.59 76843.79 1137.963 49271.14 242151.8 1840.629 43494.28 36200.9 100078.6 33287.44 174605
best 300 92811.92 25329.62 59505.44 875.4228 46681.09 200336.3 1443.303 37995.06 30818.09 86159.6 23746.43 132120.5
worst 300 111590.3 60348.45 83475.86 1404.115 51916.92 278190.7 2533.228 48576.67 39208.31 110212 42756.91 242606.2
std 0.00E + 00 9268.696 15015.25 11596.23 2.37E + 02 2623.438 32383.18 483.401 4341.018 3777.964 10848.73 8.66E + 03 52440.39
median 300 101066.1 50488.14 82196.94 1136.157 49243.27 245040.1 1692.993 43702.69 37388.59 101971.5 33323.2 161846.7
rank 1 11 7 9 2 8 13 3 6 5 10 4 12
C17-F4 mean 458.5616 6713.376 517.17 10240.74 494.6831 4726.869 874.719 498.4769 576.9153 927.8779 600.8366 631.5183 827.8094
best 458.5616 3761.644 493.297 6556.459 483.7374 1073.544 806.7569 490.2673 518.7869 711.5748 579.5763 518.2477 773.3242
worst 458.5616 9095.208 536.1458 14323.08 517.4113 7871.644 959.5679 512.5597 609.669 1345.835 625.2714 828.5023 852.6342
std 0 2210.584 17.77946 3223.857 15.4642 2870.887 69.72336 9.858853 39.93277 284.0201 19.92651 142.4119 37.16562
median 458.5616 6998.326 519.6185 10041.72 488.7918 4981.143 866.2756 495.5404 589.6027 827.051 599.2493 589.6616 842.6397
rank 1 12 4 13 2 11 9 3 5 10 6 7 8
C17-F5 mean 502.4874 860.8119 735.8015 901.8293 586.9968 807.5434 838.1713 624.8871 627.4941 782.8762 733.0872 638.6644 711.6994
best 500.995 839.7705 697.0525 874.4106 563.4297 777.4863 807.8111 609.4272 585.2307 759.2274 712.6215 613.1615 660.4988
worst 503.9798 883.1903 797.4113 937.5737 611.1767 842.804 852.7472 661.673 657.7094 810.5974 760.4182 690.2534 777.5973
std 1.284487 18.09831 45.37773 30.17291 20.01431 30.70331 20.48856 24.65663 35.82491 24.80316 21.30231 35.05297 48.57437
median 502.4874 860.1434 724.3711 897.6664 586.6904 804.9415 846.0635 614.2241 633.5182 780.84 729.6545 625.6214 704.3507
rank 1 12 8 13 2 10 11 3 4 9 7 5 6
C17-F6 mean 600 683.2593 648.6906 686.5586 603.4092 680.2796 679.4806 625.433 612.3978 645.163 658.9108 648.9292 631.4651
best 600 681.8765 646.6203 681.0214 602.0831 664.2888 668.1048 613.024 604.8794 637.741 658.1333 636.2289 624.1297
worst 600 684.6334 651.9283 693.5103 604.8734 689.6805 685.134 638.6693 619.7951 657.2588 659.9437 660.1326 636.2644
std 6.56E-14 1.136731 2.29414 5.768166 1.21244 11.93951 7.770707 12.07229 6.138977 8.580254 0.798015 10.58212 5.286354
median 600 683.2637 648.1069 685.8514 603.3402 683.5746 682.3417 625.0193 612.4583 642.8261 658.7831 649.6776 632.7332
rank 1 12 7 13 2 11 10 4 3 6 9 8 5
C17-F7 mean 733.478 1323.18 1165.029 1365.755 852.1581 1245.969 1332.559 859.8309 892.8667 1091 982.064 885.2218 978.1606
best 732.8186 1273.344 1043.834 1351.453 823.8626 1094.001 1287.148 803.9986 819.4373 998.7227 932.6094 863.0263 936.6552
worst 734.5199 1361.67 1334.31 1390.085 908.4055 1402.942 1417.143 937.2911 934.9791 1171.318 1055.847 913.8816 1035.328
std 0.754023 38.21647 128.3635 17.16868 38.22291 134.0122 60.35374 57.06996 50.55656 90.10826 54.04342 22.03435 41.29017
median 733.2867 1328.854 1140.987 1360.742 838.1821 1243.467 1312.972 849.017 908.5253 1096.979 969.8995 881.9897 970.3298
rank 1 11 9 13 2 10 12 3 5 8 7 4 6
C17-F8 mean 803.3298 1097.714 956.6115 1136.745 895.0252 1070.617 1040.781 897.8914 896.4964 1032.66 968.9946 929.4506 994.7715
best 801.2023 1081.688 924.8903 1115.143 887.944 1024.064 981.5952 865.7409 889.1471 1012.772 943.4161 917.0864 977.8659
worst 804.1574 1118.894 979.4359 1165.336 903.5246 1179.456 1084.49 929.1027 905.1587 1067.236 997.0403 946.3199 1016.214
std 1.420826 17.07768 24.73911 25.43645 6.423142 73.2693 44.03211 27.70034 6.962854 23.80721 23.62993 13.01299 19.43129
median 803.9798 1095.137 961.06 1133.251 894.3161 1039.475 1048.52 898.3611 895.8399 1025.317 967.761 927.198 992.5034
rank 1 12 6 13 2 11 10 4 3 9 7 5 8
C17-F9 mean 900 11412.3 4998.426 11057.92 1093.089 11959.73 11477.1 5657.162 2130.215 5989.381 4220.894 3665.519 1310.801
best 900 9745.192 3683.123 10784.39 931.1253 7270.656 8763.171 4497.485 1567.431 4315.456 3659.405 2171.299 1088.838
worst 900 12982.02 5701.215 11196.4 1252.781 16169.69 13694.57 8671.674 2952.773 9063.199 5085.903 5609.363 1531.093
std 6.56E-14 1346.646 903.1946 185.5277 1.49E + 02 3676.321 2480.859 2014.662 672.2215 2148.751 628.3843 1458.571 207.6059
median 900 11460.99 5304.683 11125.45 1094.224 12199.29 11725.33 4729.744 2000.329 5289.434 4069.135 3440.707 1311.636
rank 1 11 7 10 2 13 12 8 4 9 6 5 3
C17-F10 mean 2293.267 7451.423 5601.947 8167.99 4071.219 6761.467 6694.942 4761.605 4907.207 8188.548 4969.302 5170.261 6325.003
best 1851.756 6864.804 4816.341 7290.565 3691.225 5323.381 5758.367 4461.471 4363.013 7800.169 4706.526 4907.06 5812.799
worst 2525.027 7777.8 6126.721 8822.181 4563.196 7383.453 8112.533 5152.101 5274.913 8368.287 5396.6 5639.503 6940.578
std 300.3743 404.5651 630.3589 641.3718 402.1601 965.5237 1046.016 332.1405 389.0474 261.5497 319.9676 322.3387 531.2767
median 2398.142 7581.545 5732.362 8279.606 4015.227 7169.517 6454.435 4716.423 4995.45 8292.867 4887.042 5067.24 6273.318
rank 1 11 7 12 2 10 9 3 4 13 5 6 8
C17-F11 mean 1102.987 7803.855 1265.381 9163.709 1173.015 5319.484 8130.774 1324.41 2246.467 2029.444 2982.756 1256.511 9547.915
best 1100.995 6412.414 1194.89 7450.369 1123.147 3759.725 5828.095 1278.937 1403.435 1611.866 2296.384 1225.683 3469.333
worst 1105.977 8946.259 1332.604 10320.02 1208.056 8057.429 12061.62 1368.514 4488.456 2799.519 3685.573 1286.111 17980.45
std 2.152498 1114.02 57.41383 1314.787 36.73152 1930.596 2717.032 50.27274 1496.726 525.8601 654.8211 29.32658 6219.375
median 1102.487 7928.373 1267.016 9442.224 1180.429 4730.39 7316.687 1325.094 1546.988 1853.196 2974.533 1257.126 8370.939
rank 1 10 4 12 2 9 11 5 7 6 8 3 13
C17-F12 mean 1744.553 7.36E + 09 21848906 1.14E + 10 22582.89 5.31E + 09 2.59E + 08 11763262 55054538 3.17E + 08 2.09E + 08 2685476 8052480
best 1721.81 6.08E + 09 3074431 1.02E + 10 16103.81 2.74E + 09 66358101 5462275 5343709 2.02E + 08 40320435 290167.8 5575599
worst 1764.937 9.35E + 09 53361483 1.44E + 10 28838.95 6.95E + 09 5.18E+08 28461227 1.15E+08 5.50E+08 6.66E+08 5339385 10540140
std 20.15277 1.40E+09 22135561 1.99E+09 5426.481 1.82E+09 2.08E+08 11148928 48004259 1.58E+08 3.05E+08 2177968 2251710
median 1745.733 7.00E+09 15479855 1.06E+10 22694.39 5.78E+09 2.27E+08 6564773 49711051 2.57E+08 63924215 2556176 8047090
rank 1 12 6 13 2 11 9 5 7 10 8 3 4
C17-F13 mean 1315.791 5.98E + 09 156642.2 1.10E + 10 1916.786 1.53E + 09 947267.4 95212.36 790648.9 92358820 38150.3 33845.58 12479237
best 1314.587 2.92E + 09 86693.23 5.80E+09 1629.136 20663451 446934 38085.85 95403.66 64138702 30933.95 13962.84 3385725
worst 1318.646 8.38E + 09 247788.6 1.36E + 10 2480.548 5.32E + 09 1400607 191329.3 2453604 1.36E + 08 55854.54 76556.25 26842916
std 1.936281 2.26E + 09 66898.98 3.55E + 09 384.8803 2.55E + 09 497232.3 71965.13 1123018 31175388 11934.24 28806.38 10052501
median 1314.967 6.32E + 09 146043.5 1.24E + 10 1778.731 3.95E + 08 970764.4 75717.15 306794 84553056 32906.35 22431.62 9844153
rank 1 12 6 13 2 11 8 5 7 10 4 3 9
C17-F14 mean 1423.017 1982497 283653.6 2297446 1441.218 1228615 2325947 21206.89 557593.6 146256.1 1196423 19560.92 2099859
best 1422.014 1222507 39610.11 1154687 1437.754 879123 37494.25 5154.89 35881.92 84969.46 776319.1 3255.266 347509.5
worst 1423.993 2509614 656794.9 3421163 1446.222 1735736 7105793 36153.83 1195148 168290.8 1806474 35774.83 3540250
std 0.808176 602440.2 272316.4 1090285 3.906552 393127.7 3245872 13352.63 588524.3 40873.69 484796.1 14205.74 1472523
median 1423.03 2098934 219104.7 2306968 1440.449 1149801 1080250 21759.42 499672.1 165882.1 1101450 19606.79 2255838
rank 1 10 6 12 2 9 13 4 7 5 8 3 11
C17-F15 mean 1503.129 3.18E + 08 39085.05 6.25E + 08 1624.216 15028021 5273750 44659.54 16546043 5367162 16731.91 4928.184 999014.1
best 1502.462 2.75E + 08 11358.6 5.39E + 08 1585.012 5920046 242861.2 25821.73 102650.6 1218626 11864.76 1932.732 183311.8
worst 1504.265 3.52E+08 63518.56 6.90E + 08 1641.841 34958331 17123462 73931.02 61951220 10103264 22716.54 9221.236 2238398
std 0.855557 38176645 22028.08 73866219 26.34809 13404148 8007632 20851.46 30285039 3642993 4536.727 3228.17 940410
median 1502.893 3.22E + 08 40731.52 6.35E + 08 1635.005 9616854 1864339 39442.71 2065152 5073378 16173.18 4279.384 787173.5
rank 1 12 5 13 2 10 8 6 11 9 4 3 7
C17-F16 mean 1663.469 4439.696 3062.344 5127.072 2044.419 3345.697 4362.256 2630.907 2584.694 3546.992 3758.016 2989.162 3008.96
best 1614.72 4082.878 2598.582 4302.954 1738.317 2906.393 3566.611 2375.662 2431.252 3348.493 3566.236 2724.525 2651.41
worst 1744.118 4726.006 3607.169 5858.072 2313.973 3612.159 5256.341 2912.394 2716.235 3796.306 3932.464 3280.186 3379.142
std 61.972 294.6568 414.7725 831.0934 259.3752 310.7514 695.6954 231.5981 146.6984 199.2798 163.2425 279.3156 355.369
median 1647.519 4474.951 3021.813 5173.631 2062.693 3432.118 4313.035 2617.786 2595.645 3521.584 3766.682 2975.969 3002.644
rank 1 12 7 13 2 8 11 4 3 9 10 5 6
C17-F17 mean 1728.099 3489.234 2511.869 3807.821 1871.468 3348.346 2904.1 2100.516 1945.661 2219.538 2564.256 2367.184 2179.495
best 1718.761 2857.879 2358.989 3407.936 1755.856 2245.888 2400.724 2045.651 1810.568 1980.588 2459.396 2118.963 2129.738
worst 1733.659 4270.993 2634.028 4514.491 1935.97 6235.042 3246.258 2258.637 2101.661 2531.9 2722.967 2780.395 2253.289
std 6.708055 601.71 120.3358 500.8459 79.456 1927.199 361.504 105.4266 138.9375 233.8376 128.7943 296.9954 56.38169
median 1729.987 3414.033 2527.23 3654.429 1897.023 2456.226 2984.71 2048.889 1935.208 2182.832 2537.33 2284.689 2167.477
rank 1 12 8 13 2 11 10 4 3 6 9 7 5
C17-F18 mean 1825.696 29710394 2769111 34160751 1900.212 37987428 6168952 668885.9 438453.4 1741398 538191 143342.3 3810887
best 1822.524 8558506 294804.7 11044148 1876.324 1392881 2078826 168246.2 81901.12 808377.6 301687.1 101966.8 2975129
worst 1828.42 57699276 5525101 6.71E + 07 1913.847 71988170 12732773 1810901 1126704 2189284 1047952 170095.9 5586097
std 2.701927 21723931 2451257 2.38E + 07 16.91978 39202488 4578261 766186.5 491704 634910.2 344271.9 29786.51 1197059
median 1825.92 26291898 2628270 29243157 1905.34 39284331 4932105 348198.3 272604.2 1983966 401562.3 150653.2 3341160
rank 1 11 8 12 2 13 10 6 4 7 5 3 9
C17-F19 mean 1910.989 6.07E + 08 70677.22 1.02E + 09 1924.439 3.08E + 08 14977184 981927.3 4216074 6012116 85386.09 46425.81 1694555
best 1908.84 4.54E + 08 14998.1 7.39E + 08 1921.777 3822398 1948825 24686.78 73930.76 3120615 46271.28 9070.513 669506.3
worst 1913.095 7.90E + 08 157546.8 1.55E + 09 1929.726 8.52E + 08 25861520 2207830 13595581 8546193 114938 139224 3010453
std 1.932009 1.69E + 08 62094.74 3.60E + 08 3.595445 3.92E + 08 10904711 1062424 6295643 2668748 28586.25 62079.66 987306.8
median 1911.01 5.92E + 08 55082.01 9.02E + 08 1923.126 1.88E + 08 16049195 847596.4 1597391 6190829 90167.52 18704.33 1549130
rank 1 12 4 13 2 11 10 6 8 9 5 3 7
C17-F20 mean 2065.787 2943.157 2666.047 2999.881 2182.582 2891.635 2878.627 2633.973 2391.736 2837.133 3059.535 2573.027 2497.368
best 2029.521 2848.33 2500.843 2813.717 2062.415 2742.377 2658.25 2392.295 2210.293 2737.083 2667.859 2520.355 2448.1
worst 2161.126 3065.811 2915.867 3118.188 2270.667 3053.535 3072.02 3056.258 2573.153 2968.631 3558.832 2714.428 2525.393
std 63.64645 90.99524 182.3155 131.4166 87.69513 129.9414 176.8356 291.1121 148.1372 106.4891 373.4816 94.55795 34.96127
median 2036.25 2929.243 2623.74 3033.81 2198.623 2885.314 2892.119 2543.671 2391.748 2821.409 3005.723 2528.663 2507.99
rank 1 11 7 12 2 10 9 6 3 8 13 5 4
C17-F21 mean 2308.456 2639.625 2448.327 2700.07 2369.638 2547.413 2626.832 2410.58 2394.555 2505.718 2584.537 2441.547 2502.684
best 2304.034 2540.722 2212.582 2617.632 2359.68 2308.477 2545.196 2373.04 2357.178 2493.17 2565.333 2421.377 2467.965
worst 2312.987 2704.875 2613.57 2797.969 2386.406 2688.337 2696.816 2442.675 2411.112 2516.37 2622.078 2456.436 2555.916
std 4.459041 77.49394 168.9789 78.46193 11.76068 167.6737 74.41901 28.88315 25.34246 11.41997 25.4703 17.17562 37.53347
median 2308.402 2656.452 2483.579 2692.339 2366.232 2596.42 2632.658 2413.303 2404.965 2506.666 2575.369 2444.187 2493.427
rank 1 12 6 13 2 9 11 4 3 8 10 5 7
C17-F22 mean 2300 8290.945 5959.627 8035.034 2303.085 9123.626 7681.703 4043.833 2725.83 5867.839 6544.994 5024.178 2723.901
best 2300 7933.19 2303.22 6949.061 2302.013 8880.025 6673.195 2306.853 2589.53 2746.59 4093.179 2466.247 2644.929
worst 2300 8849.181 7372.316 9126.495 2304.897 9238.634 8581.424 6200.904 3006.029 9328.484 7622.516 7499.707 2784.931
std 0 391.4643 2441.527 935.8416 1.293614 168.516 792.9999 2033.845 190.489 3583.826 1645.262 2314.809 69.38582
median 2300 8190.705 7081.486 8032.289 2302.715 9187.922 7736.097 3833.787 2653.881 5698.141 7232.14 5065.378 2732.873
rank 1 12 8 11 2 13 10 5 4 7 9 6 3
C17-F23 mean 2655.081 3224.03 2943.409 3282.201 2645.53 3229.193 3070.858 2742.56 2757.12 2918.998 3834.557 2915.417 2994.767
best 2653.745 3133.016 2827.683 3224.44 2460.819 3103.231 2883.035 2695.565 2736.247 2895.554 3719.839 2880.184 2963.876
worst 2657.377 3310.768 3126.654 3365.761 2715.998 3438.488 3176.805 2773.514 2778.949 2972.291 3948.084 2969.723 3062.066
std 1.653199 83.61115 130.984 61.45503 123.4634 147.9262 130.4108 33.28019 18.40235 36.19027 120.9484 41.42868 45.26803
median 2654.6 3226.168 2909.648 3269.301 2702.651 3187.527 3111.795 2750.58 2756.641 2904.074 3835.152 2905.88 2976.564
rank 2 10 7 12 1 11 9 3 4 6 13 5 8
C17-F24 mean 2831.409 3344.884 3191.747 3451.053 2886.787 3308.553 3133.959 2910.382 2926.476 3055.298 3396.878 3149.531 3250.619
best 2829.992 3304.551 3044.455 3356.13 2870.121 3192.344 3065.747 2859.258 2913.188 3030.073 3357.472 3068.781 3150.106
worst 2832.366 3428.397 3356.572 3615.985 2894.104 3363.195 3161.648 2934.559 2934.13 3093.988 3437.006 3270.637 3334.659
std 1.145563 56.38206 137.1339 120.0691 11.26823 79.67988 45.70028 34.52858 9.367664 27.22707 35.16428 86.46064 85.67692
median 2831.64 3323.294 3182.981 3416.047 2891.461 3339.338 3154.221 2923.854 2929.292 3048.567 3396.518 3129.352 3258.855
rank 1 11 8 13 2 10 6 3 4 5 12 7 9
C17-F25 mean 2886.698 4003.369 2910.017 4666.579 2891.558 3504.551 3093.569 2910.887 2999.714 3085.991 3001.979 2895.453 3120.923
best 2886.691 3603.934 2894.928 4025.646 2884.402 3103.52 3054.856 2884.399 2959.348 2957.822 2989.708 2887.66 3103.11
worst 2886.707 4302.456 2951.527 5520.858 2898.129 3920.941 3113.921 2978.946 3074.876 3231.746 3015.29 2913.74 3133.671
std 0.007606 291.6233 27.69321 622.7309 6.203473 399.5395 27.77419 45.49544 53.57291 130.9921 10.56349 12.25018 13.50433
median 2886.698 4053.542 2896.806 4559.906 2891.851 3496.872 3102.749 2890.102 2982.317 3077.198 3001.458 2890.207 3123.456
rank 1 12 4 13 2 11 9 5 6 8 7 3 10
C17-F26 mean 3578.65 9505.856 7547.774 10128.37 2913.927 9034.652 8656.39 4857.652 4622.618 6060.802 7701.708 4922.634 4441.597
best 3559.841 9052.672 6182.311 9237.341 2913.384 8332.571 7878.942 4487.659 4205.25 4597.446 6588.236 3542.85 4035.18
worst 3607.686 10319.54 8348.706 11702.51 2914.623 9478.824 9561.172 5526.7 5270.944 7453.751 8281.429 6560.541 4942.653
std 2.28E + 01 590.9343 953.4032 1157.036 0.612786 491.2266 691.009 481.7705 455.0595 1309.68 789.9883 1409.09 380.781
median 3573.536 9325.606 7830.04 9786.803 2913.851 9163.607 8592.723 4708.124 4507.14 6096.006 7968.583 4793.572 4394.277
rank 2 12 8 13 1 11 10 5 4 7 9 6 3
C17-F27 mean 3207.018 3635.083 3364.435 3799.851 3215.073 3489.858 3441 3232.89 3252.443 3324.708 5079.067 3283.025 3475.121
best 3200.749 3571.91 3272.328 3501.824 3199.978 3348.212 3260.611 3212.628 3243.403 3243.069 4600.363 3241.679 3392.776
worst 3210.656 3742.409 3444.906 4105.559 3237.047 3754.78 3574.937 3260.918 3268.115 3401.347 5428.053 3327.891 3523.117
std 4.65E + 00 76.02481 90.87128 258.816 16.89199 181.5433 134.7922 20.23104 10.80413 65.49988 405.1245 37.16087 56.90489
median 3208.335 3613.006 3370.252 3796.01 3211.634 3428.22 3464.226 3229.007 3249.127 3327.207 5143.927 3281.264 3492.296
rank 1 11 7 12 2 10 8 3 4 6 13 5 9
C17-F28 mean 3100 4882.76 3275.922 5848.123 3220.859 4222.822 3459.263 3266.155 3627.651 3704.529 3547.219 3343.147 3612.527
best 3100 4631.999 3243.079 5519.028 3203.244 3629.749 3394.391 3227.732 3415.823 3544.755 3471.443 3200.091 3557.078
worst 3100 5154.567 3310.077 6192.818 3252.993 4827.838 3517.889 3301.239 4145.174 4070.207 3706.247 3562.298 3672.727
std 2.63E-13 223.9419 27.39059 321.8779 2.22E + 01 554.7615 53.75108 30.21632 347.0962 246.5379 107.3151 167.6314 54.95501
median 3100 4872.238 3275.267 5840.323 3213.6 4216.85 3462.387 3267.825 3474.803 3601.576 3505.594 3305.1 3610.152
rank 1 12 4 13 2 11 6 3 9 10 7 5 8
C17-F29 mean 3353.75 5524.92 4391.314 5758.417 3676.206 5356.856 5193.275 3873.064 3817.422 4581.658 5167.765 4218.386 4344.553
best 3325.385 5044.708 4017.813 5086.959 3512.962 4781.459 4905.636 3739.462 3729 4233.958 4877.069 4005.304 3939.263
worst 3370.797 6031.908 4618.791 6682.215 3822.95 6297.168 5375.072 4002.304 3947.313 5095.035 5441.424 4478.436 4723.806
std 19.68889 476.764 266.0991 783.7085 136.5606 710.3936 201.2981 111.148 97.68969 367.9411 303.3514 195.0757 351.0088
median 3359.41 5511.532 4464.326 5632.247 3684.455 5174.398 5246.195 3875.245 3796.688 4498.82 5176.284 4194.901 4357.572
rank 1 12 7 13 2 11 10 4 3 8 9 5 6
C17-F30 mean 5007.854 1.50E + 09 1498936 2.97E + 09 7823.141 40397211 41223012 3251418 6705220 39796839 2378375 286240.9 737698
best 4955.449 1.11E + 09 528393.4 2.13E + 09 6445.621 13811024 8220703 583519.5 1495507 21302710 2076120 7723.578 203915.2
worst 5086.396 1.65E + 09 2654566 3.28E + 09 10507.56 94391198 66056116 4655317 18107182 83477825 2861687 1084352 1411438
std 58.9744 2.65E + 08 888936.3 5.59E + 08 1902.204 36577105 24108427 1816057 7672186 29284758 338189.3 532335.2 587913.9
median 4994.785 1.63E + 09 1406393 3.23E + 09 7169.694 26693311 45307615 3883418 3609095 27203411 2287846 26444.16 667719.3
rank 1 12 5 13 2 10 11 7 8 9 6 3 4
Sum rank 31 334 182 361 57 305 284 128 151 232 231 139 204
Mean rank 1.068966 11.51724 6.275862 12.44828 1.965517 10.51724 9.793103 4.413793 5.206897 8 7.965517 4.793103 7.034483
Total rank 1 12 6 13 2 11 10 3 5 9 8 4 7

Boxplot of OOA and competitor algorithms in optimization of the CEC-2017 test suite ( D = 30 ).

The optimization results of the CEC 2017 test suite for D = 50 , using OOA and competitor algorithms are presented in Table 4. The performance convergence curves of the algorithms on the CEC 2017 test suite for the dimension D equal to 50 are drawn in Figure 5. The simulation results show that OOA is the first best optimizer for C17-F1, C17-F3 to C17-F25, and C17-F27 to C17-F30.

Performance of optimization algorithms on the CEC 2017 test suite ( D = 50 ).

OOA WSO AVOA RSA MPA TSA WOA MVO GWO TLBO GSA PSO GA
C17-F1 mean 100 6.23E + 10 9633539 9.76E + 10 5869184 3.97E + 10 8.02E + 09 4236811 9.75E + 09 2.16E + 10 1.79E + 10 2.64E + 09 1.08E + 10
best 100 5.56E + 10 1146908 8.54E + 10 2265189 3.65E + 10 4.74E + 09 3032349 7.03E + 09 1.47E + 10 1.42E + 10 1.08E + 09 1.03E + 10
worst 100 6.67E + 10 25492683 1.07E + 11 1.49E + 07 4.27E + 10 1.20E + 10 5273674 1.33E + 10 2.92E + 10 2.14E + 10 3.52E + 09 1.17E + 10
std 0.00E + 00 4.88E + 09 10817285 9.29E + 09 6.05E + 06 2.55E + 09 3.44E + 09 922454.4 2.63E + 09 7.02E + 09 2.91E + 09 1.07E + 09 6.35E + 08
median 100 6.35E + 10 5947282 9.92E + 10 3164863 3.98E + 10 7.68E + 09 4320610 9.31E + 09 2.13E + 10 1.79E + 10 2.98E + 09 1.07E + 10
rank 1 12 4 13 3 11 6 2 7 10 9 5 8
C17-F3 mean 300 164990.9 152600.7 164380.1 18661.51 113601.9 243601.6 48146.73 135309.7 102338.5 185297.2 150706.5 274277.7
best 300 141509 117250.3 149115.9 16118.97 99806.91 183717.9 38171.73 118871.5 77391.5 167333.8 113263.4 228606.7
worst 300 189752.4 185670.2 179187.8 22027.1 121120.2 371611.5 59884.37 151887 116770.7 209357.4 196385.7 315163.7
std 0.00E + 00 20312.69 30901.56 13351.7 2.65E + 03 9850.809 88493.05 9054.928 13491.26 17981.65 20314.17 3.61E + 04 35447.28
median 300 164351.1 153741.1 164608.3 18249.99 116740.2 209538.4 47265.41 135240.1 107595.8 182248.9 146588.4 276670.2
rank 1 10 8 9 2 5 12 3 6 4 11 7 13
C17-F4 mean 470.3679 15348.71 706.9316 24716.34 533.5922 8634.859 2000.401 566.9454 1475.71 2890.43 3170.708 1038.473 1568.726
best 428.5127 11914.91 688.9078 16303.31 498.7435 6917.086 1255.35 530.6005 1094.805 1627.809 2647.03 694.6241 1354.896
worst 525.7252 17475.53 737.8169 29521.96 585.5478 11157.34 2403.438 650.5981 1814.452 4961.15 3379.846 1873.597 1697.292
std 49.57162 2488.603 21.84865 6035.32 40.44038 1792.421 514.0172 57.03692 323.3367 1464.804 350.9684 559.0229 151.6567
median 463.6168 16002.2 700.5009 26520.05 525.0388 8232.503 2171.407 543.2916 1496.792 2486.381 3327.979 792.8352 1611.358
rank 1 12 4 13 2 11 8 3 6 9 10 5 7
C17-F5 mean 504.7261 1123.301 871.7653 1153.706 745.1988 1172.288 973.809 747.8744 734.0444 1018.402 817.9617 800.2022 907.1433
best 503.9798 1089.318 840.1738 1133.945 660.411 1024.094 931.673 671.4986 705.6612 974.8783 763.3441 743.5681 875.9634
worst 505.9698 1165.376 914.6958 1167.384 811.6015 1290.821 1000.001 865.2826 763.424 1047.139 855.9935 867.1823 929.2374
std 0.952601 36.58288 32.18295 15.34658 63.43604 129.7059 30.40081 86.76501 30.9783 32.26537 43.62829 50.89626 25.25519
median 504.4773 1119.255 866.0957 1156.747 754.3913 1187.118 981.781 727.3583 733.5462 1025.795 826.2546 795.0293 911.6862
rank 1 11 7 12 3 13 9 4 2 10 6 5 8
C17-F6 mean 600 698.6303 662.3682 700.7781 611.7616 693.173 701.4466 639.0237 623.6346 666.5519 660.0374 655.5293 650.3066
best 600 695.5222 657.3779 698.4178 608.8762 672.0678 695.8802 628.4342 617.7485 653.3697 654.9504 653.1633 636.9216
worst 600 703.7772 668.0267 703.8157 615.5752 710.4695 709.9702 663.9289 633.8645 675.531 663.0595 659.2186 663.4004
std 0.00E + 00 3.811353 4.919607 2.526735 2.869191 17.06203 6.120002 16.9075 7.23156 9.483047 3.568297 2.736949 11.08088
median 600 697.6108 662.0342 700.4394 611.2974 695.0775 699.9679 631.8659 621.4626 668.6534 661.0698 654.8676 650.4523
rank 1 11 8 12 2 10 13 4 3 9 7 6 5
C17-F7 mean 756.7298 1831.732 1700.36 1935.27 1039.036 1717.665 1743.988 1065.09 1077.354 1505.879 1435.645 1216.161 1327.914
best 754.7543 1805.652 1626.499 1851.094 980.1498 1559.621 1678.718 1025.576 1053.374 1373.672 1260.791 1050.449 1246.374
worst 758.3522 1864.937 1769.289 2042.793 1088.944 1873.178 1833.88 1096.7 1097.02 1570.261 1569.795 1457.924 1381.316
std 1.553328 24.68384 60.69443 82.21055 52.75884 146.2221 72.43668 30.1216 20.46653 89.30069 139.2221 175.9512 59.43138
median 756.9065 1828.17 1702.826 1923.597 1043.525 1718.93 1731.678 1069.041 1079.511 1539.792 1455.997 1178.136 1341.982
rank 1 12 9 13 2 10 11 3 4 8 7 5 6
C17-F8 mean 805.721 1443.403 1136.498 1472.085 1018.376 1461.466 1344.601 1030.224 1042.684 1341.947 1153.084 1066.128 1275.052
best 802.9849 1384.167 1088.523 1439.291 986.5417 1358.405 1205.917 988.3925 1006.955 1283.386 1144.293 1022.045 1231.089
worst 810.9445 1489.032 1185.558 1495.149 1051.378 1601.119 1458.423 1104.113 1082.781 1399.705 1167.814 1134.884 1299.788
std 3.575859 47.16846 55.12104 23.48974 33.62207 104.6383 104.2461 50.74067 33.8052 48.1601 10.47402 53.66591 30.16378
median 804.4773 1450.206 1135.956 1476.95 1017.793 1443.171 1357.032 1014.196 1040.499 1342.348 1150.114 1053.791 1284.666
rank 1 11 6 13 2 12 10 3 4 9 7 5 8
C17-F9 mean 900 37435.21 13732.69 37634.97 3412.42 39270.78 34182.47 20347.96 7089.402 24884.17 11023.85 10629.64 13227.11
best 900 35953.22 13086.26 35356.66 2116.199 36194.69 31811.4 10870.21 6158.35 19155.41 10040.24 9844.681 10875.33
worst 900 40887.05 14631.28 39493.29 4963.594 43807.91 39990.16 26924.48 8079.887 29287.22 11907.98 12086.39 15240.81
std 9.28E-14 2338.477 668.233 1958.867 1.18E + 03 3279.618 3884.13 7550.383 997.8825 4207.754 776.3777 1008.051 2313.445
median 900 36450.28 13606.61 37844.96 3284.943 38540.25 32464.16 21798.57 7059.685 25547.04 11073.59 10293.75 13396.16
rank 1 11 7 12 2 13 10 8 3 9 5 4 6
C17-F10 mean 4347.157 13343.2 8494.952 14609.52 6635.739 12073.65 12081.7 7804.173 8847.779 14389.25 8778.309 7938.476 12000.08
best 3555.132 12849.01 7942.324 14351.97 5691.888 11063.73 10814.98 6474.801 6693.825 13685.33 7898.014 7689.561 11398.43
worst 5099.795 14077.96 9092.32 14975.91 7289.844 13144.07 13233.91 8924.646 14254.59 15002.14 9882.736 8499.551 12758.01
std 644.7565 600.4906 486.1522 292.3419 769.5238 888.9848 1065.232 1060.221 3626.029 692.8408 827.7477 376.9283 602.0036
median 4366.851 13222.91 8472.582 14555.11 6780.611 12043.4 12138.96 7908.622 7221.351 14434.77 8666.244 7782.397 11921.93
rank 1 11 5 13 2 9 10 3 7 12 6 4 8
C17-F11 mean 1128.435 15970.37 1625.263 21792.43 1260.434 13409.08 5257.146 1586.94 6332.42 5273.682 14727.83 1694.101 24939.55
best 1121.25 14709.64 1500.05 19379.48 1210.457 11518.2 4626.927 1429.111 3782.438 4941.775 13810.56 1410.45 14555.84
worst 1133.132 16771.11 1785.034 23622.66 1292.829 16108.7 6591.487 1746.66 11030.99 5875.754 16699.11 2034.056 33463.02
std 5.442974 911.4777 131.4877 1773.027 36.5879 1978.961 903.272 138.1628 3347.852 429.6829 1328.437 266.9856 7819.297
median 1129.678 16200.36 1607.984 22083.79 1269.225 13004.71 4905.085 1585.995 5258.125 5138.601 14200.82 1665.948 25869.67
rank 1 11 4 12 2 9 6 3 8 7 10 5 13
C17-F12 mean 2905.102 4.55E + 10 76488687 7.42E + 10 15008871 2.70E + 10 1.38E + 09 82569413 9.98E + 08 5.27E + 09 2.26E + 09 1.67E + 09 2.13E + 08
best 2527.376 3.82E + 10 32398991 5.41E + 10 14138065 1.14E + 10 1.14E + 09 44472138 1.57E + 08 2.97E + 09 7.45E + 08 13236089 67230979
worst 3168.37 5.45E + 10 1.18E + 08 1.02E + 11 15712646 4.54E + 10 1.87E + 09 1.31E + 08 1.85E + 09 1.04E + 10 4.07E + 09 4.83E + 09 2.95E + 08
std 273.7079 7.37E + 09 45956701 2.19E + 10 735042.8 1.40E + 10 3.39E + 08 36537981 8.47E + 08 3.46E + 09 1.37E + 09 2.24E + 09 1.00E + 08
median 2962.331 4.46E + 10 77682587 7.05E + 10 15092386 2.56E + 10 1.25E + 09 77208577 9.91E + 08 3.87E + 09 2.12E + 09 9.22E + 08 2.45E + 08
rank 1 12 3 13 2 11 7 4 6 10 9 8 5
C17-F13 mean 1340.1 2.56E + 10 155301 4.49E + 10 16965.85 1.05E + 10 98948884 251427.3 3.72E + 08 6.10E + 08 19317832 4.98E + 08 43282065
best 1333.781 1.48E + 10 35663.11 2.27E + 10 8949.719 5.59E + 09 74390411 156850.3 1.69E + 08 4.97E + 08 32490.3 53050.75 28213323
worst 1343.015 3.50E + 10 342287.3 6.46E + 10 19982.83 1.64E + 10 1.12E + 08 392376.6 9.36E + 08 8.34E + 08 65117269 1.26E + 09 57848352
std 4.28228 8.86E + 09 131117.5 1.76E + 10 5349.655 4.56E + 09 16790760 100212 3.76E + 08 1.52E + 08 31056900 6.13E + 08 13234381
median 1341.801 2.64E + 10 121626.7 4.62E + 10 19465.43 1.01E + 10 1.05E + 08 228241.2 1.92E + 08 5.55E + 08 6060785 3.67E + 08 43533292
rank 1 12 3 13 2 11 7 4 8 10 5 9 6
C17-F14 mean 1429.458 27081175 1275709 50490844 1568.61 2803162 4974020 199054.8 1201322 902849.7 15803387 598687 11695661
best 1425.995 8845673 395100.5 15485581 1555.038 740420.8 4403782 126067.8 93478.08 744436.2 3582984 215014.8 5755160
worst 1431.939 53015295 3038603 1.02E + 08 1594.054 4446096 5910871 386437.7 2318258 1041691 25947767 958913.5 20129344
std 2.621295 18640032 1201625 36897657 17.90616 1536508 650814.5 125348.6 908290.7 155051.3 10139162 304451.2 6068894
median 1429.95 23231865 834565.9 42124750 1562.675 3013065 4790713 141856.9 1196777 912635.7 16841399 610409.8 10449070
rank 1 12 7 13 2 8 9 3 6 5 11 4 10
C17-F15 mean 1530.66 2.72E + 09 39576.52 4.37E + 09 2292.413 1.78E + 09 10351874 126572.6 6207179 73625584 2.06E + 08 11258.85 8947884
best 1526.359 1.92E + 09 24404.91 3.41E + 09 2153.345 6.11E + 08 954087.8 52361.59 44097.2 43172084 19916.79 2829.437 3040558
worst 1532.953 3.56E + 09 72936.75 5.17E + 09 2446.375 3.87E + 09 19328956 188825.4 16348954 95837478 7.99E + 08 22190.16 19419750
std 2.934025 7.70E + 08 22496.07 7.81E + 08 155.0294 1.51E + 09 8076778 60658.75 7114578 22024463 3.96E + 08 8605.455 7243715
median 1531.664 2.69E + 09 30482.21 4.44E + 09 2284.967 1.32E + 09 10562226 132551.8 4217832 77746388 12459587 10007.9 6665614
rank 1 12 4 13 2 11 8 5 6 9 10 3 7
C17-F16 mean 2062.891 6467.786 4454.613 7825.815 2783.346 4754.701 5642.343 3383.174 3380.144 4652.575 4038.784 3397.153 3995.017
best 1728.6 5611.632 4131.023 5825.29 2604.917 4186.789 4602.031 3129.154 2952.051 4211.721 3671.68 2951.009 3374.03
worst 2242.663 8293.805 4880.918 11751.18 3064.367 5061.045 6327.528 3626.933 3982.258 4990.093 4449.79 3869.221 4536.026
std 232.9126 1263.466 359.7965 2700.737 207.3689 397.3114 766.949 208.9199 502.8265 323.8354 369.5609 445.7504 506.6747
median 2140.15 5982.854 4403.256 6863.395 2732.05 4885.484 5819.906 3388.304 3293.133 4704.244 4016.834 3384.191 4035.007
rank 1 12 8 13 2 10 11 4 3 9 7 5 6
C17-F17 mean 2021.151 7864.94 3625.516 11447.28 2582.195 4042.008 4643.48 3116.308 3009.652 4241.893 3898.861 3409.074 3652.934
best 1900.43 5968.502 3167.427 8335.531 2515.43 3229.366 4159.634 2547.325 2839.56 3582.174 3430.184 3163.152 3415.859
worst 2138.267 9672.379 4171.06 14909.34 2632.889 4514.153 4869.736 3641.457 3288.941 4634.222 4212.706 3750.094 3920.045
std 134.2279 1525.7 483.7474 2708.446 49.19338 561.9249 333.0093 450.8387 194.6068 464.657 342.8432 274.2753 236.3956
median 2022.954 7909.439 3581.789 11272.13 2590.23 4212.255 4772.276 3138.225 2955.053 4375.588 3976.277 3361.525 3637.916
rank 1 12 6 13 2 9 11 4 3 10 8 5 7
C17-F18 mean 1830.62 79038917 2517385 1.17E + 08 27318.18 36598429 47170477 2756941 5976370 8562679 8780429 860476.2 9889243
best 1822.239 63247715 326038.7 52717001 3827.566 3289654 12774753 1624221 1139398 5888888 4151457 366707 3543460
worst 1841.673 93201878 4611003 1.63E + 08 40893.25 1.05E + 08 85386017 4291725 11921610 11902506 16408722 1411087 23773873
std 8.144613 12939086 2171210 5.41E + 07 16192.76 46542242 35911954 1276269 5626104 2544077 5587586 478991.4 9344049
median 1829.285 79853038 2566250 1.27E + 08 32275.95 19271300 45260570 2555908 5422237 8229661 7280767 832055.4 6119819
rank 1 12 4 13 2 10 11 5 6 7 8 3 9
C17-F19 mean 1925.185 2.84E + 09 271089.4 4.01E + 09 2088.842 2.79E + 09 7143483 5350787 1214177 52932263 471848.4 410897.5 1035399
best 1924.437 1.36E + 09 95090.16 2.70E + 09 2024.769 10209677 1074349 4072948 594255.2 44937557 271296 2941.646 809903.2
worst 1926.121 4.75E + 09 559193.5 4.96E + 09 2120.518 8.15E + 09 16836853 6635876 1866908 67217170 1034110 1026432 1402612
std 0.791342 1.43E + 09 201412.2 1.00E + 09 43.73606 3.64E+09 6773460 1046384 532090.3 9917798 374976.7 488021.6 279710.9
median 1925.091 2.64E+09 215036.9 4.19E+09 2105.04 1.50E+09 5331366 5347162 1197773 49787163 290993.9 307108 964540
rank 1 12 3 13 2 11 9 8 7 10 5 4 6
C17-F20 mean 2160.172 3902.726 3314.196 4182.46 2681.379 3491.135 3820.777 3328.773 2644.017 3847.343 4125.237 3338.356 3211.827
best 2104.423 3553.251 2680.948 3891.026 2388.332 3018.461 3511.001 3086.079 2435.374 3718.643 3815.715 2914.562 3144.381
worst 2323.891 4094.792 3876.296 4349.867 2980.722 3703.867 4437.649 3821.882 2852.034 4030.649 4421.364 3526.903 3349.373
std 109.1545 243.3125 511.9094 201.3674 248.5529 318.6519 423.8782 338.0152 220.2416 133.9882 248.3652 285.41 93.24512
median 2106.186 3981.43 3349.771 4244.474 2678.232 3621.106 3667.229 3203.565 2644.33 3820.04 4131.934 3455.98 3176.777
rank 1 11 5 13 3 8 9 6 2 10 12 7 4
C17-F21 mean 2314.895 3024.982 2776.534 3065.379 2455.399 2988.402 2978.859 2585.629 2530.781 2846.15 2867.33 2674.985 2770.557
best 2309.045 2987.311 2649.011 2955.469 2433.168 2876.912 2858.967 2549.236 2473.707 2819.125 2793.088 2599.744 2746.032
worst 2329.683 3060.959 2972.716 3156.551 2481.493 3166.5 3079.581 2625.475 2576.118 2893.219 2908.841 2786.272 2791.041
std 9.88675 36.98458 139.5481 95.42996 24.40496 124.7929 94.3526 40.11014 43.56859 33.99151 52.11661 82.32307 21.65294
median 2310.426 3025.828 2742.205 3074.748 2453.467 2955.097 2988.443 2583.903 2536.649 2836.128 2883.695 2656.963 2772.577
rank 1 12 7 13 2 11 10 4 3 8 9 5 6
C17-F22 mean 3095.169 15546.07 11523.72 16875.49 5460.247 14247.34 14170.59 9274.339 9143.685 16297.58 11828.71 10070 9101.677
best 2300 15291.34 9115.39 16683.2 2321.173 13837.27 13569.66 7048.728 8080.949 15702.47 11547.9 9258.166 4110.231
worst 5480.678 15761.44 13374.28 17116.93 8836.816 14936.95 14590.05 10684.71 9619.678 16958.76 12072.52 10655.4 14120.16
std 1590.339 193.3974 1950.5 206.8345 3593.354 477.1638 432.7474 1562.212 714.1168 624.4732 225.7436 593.7372 5460.19
median 2300 15565.74 11802.61 16850.92 5341.499 14107.58 14261.32 9681.96 9437.056 16264.53 11847.21 10183.21 9088.16
rank 1 11 7 13 2 10 9 5 4 12 8 6 3
C17-F23 mean 2743.354 3879.447 3321.484 3959.549 2897.778 3798.275 3800.951 3002.351 3035.017 3310.82 4865.912 3411.604 3396.276
best 2729.988 3796.923 3232.017 3908.274 2883.059 3576.962 3608.486 2957.277 2950.092 3216.594 4659.793 3341.618 3259.398
worst 2752.657 3982.27 3407.106 4004.274 2918.496 4155.531 3907.366 3081.474 3182.177 3383.798 5046.783 3473.215 3542.759
std 10.01651 81.40284 82.73573 40.06302 15.11385 276.1892 133.7199 58.27082 101.5194 69.42439 158.742 70.02127 116.1106
median 2745.387 3869.299 3323.406 3962.825 2894.779 3730.303 3843.976 2985.327 3003.899 3321.445 4878.536 3415.792 3391.474
rank 1 11 6 12 2 9 10 3 4 5 13 8 7
C17-F24 mean 2919.043 4286.238 3547.924 4577.944 3074.006 4068.707 3883.269 3147.951 3215.296 3479.24 4467.891 3494.903 3707.905
best 2909.046 4016.606 3429.98 4062.096 3042.344 3966.333 3765.025 3107.082 3110.284 3396.664 4430.594 3324.53 3664.824
worst 2924.412 4891.139 3744.643 5843.684 3114.143 4216.513 3940.888 3185.211 3353.722 3541.622 4525.056 3660.481 3812.488
std 6.824073 406.7445 136.3239 852.3545 32.47015 114.5252 81.01454 34.09211 101.6004 67.38126 43.92506 149.8783 69.9721
median 2921.358 4118.604 3508.536 4202.999 3069.768 4045.991 3913.582 3149.755 3198.588 3489.338 4457.956 3497.3 3677.154
rank 1 11 7 13 2 10 9 3 4 5 12 6 8
C17-F25 mean 2983.145 8913.238 3189.114 12434.36 3072.796 6174.478 4218.573 3058.97 4091.536 4449.958 4348.249 3129.198 4107.028
best 2980.235 7309.81 3159.605 9950.985 3051.24 4987.73 3787.054 3022.776 3883.116 3936.757 3980.719 3082.062 3993.713
worst 2991.831 9928.409 3238.276 13959.87 3092.602 7303.543 4544.754 3079.673 4307.26 5076.341 5045.736 3181.651 4236.998
std 5.790467 1160.733 34.07083 1883.712 17.13166 995.1887 322.1558 25.8861 220.723 575.0947 500.8538 50.91036 100.3565
median 2980.257 9207.366 3179.288 12913.29 3073.671 6203.32 4271.242 3066.715 4087.884 4393.366 4183.27 3126.539 4098.701
rank 1 12 5 13 3 11 8 2 6 10 9 4 7
C17-F26 mean 3776.432 14680.4 11390.88 15721.37 3301.982 13135.48 14395.37 5906.71 6673.057 10066.4 11994.83 8374.443 9289.183
best 3748.807 14428.29 10848.9 15065.61 3087.756 10893.91 13411.27 5387.351 6258.372 9209.102 11625.73 7762.897 7310.614
worst 3793.643 14881.59 11932.88 16727.86 3607.293 14475.55 16221.49 6187.468 7065.22 10862.89 12421.32 8969.642 11852.52
std 1.95E+01 208.8096 443.4593 721.243 236.001 1554.866 1246.032 362.9424 420.5216 693.4693 331.9833 540.4058 2160.858
median 3781.639 14705.85 11390.87 15545.99 3256.439 13586.22 13974.35 6026.011 6684.319 10096.81 11966.13 8382.617 8996.796
rank 2 12 8 13 1 10 11 3 4 7 9 5 6
C17-F27 mean 3251.26 4887.468 3884.761 5087.635 3391.321 4792.654 4529.274 3368.675 3662.495 3864.077 8366.22 3668.289 4512.883
best 3227.701 4551.506 3836.254 4691.033 3278.398 4039.14 3921.791 3326.967 3614.569 3661.421 8098.447 3390.082 4396.484
worst 3313.631 5112.054 3955.495 5366.816 3499.636 5312.282 5140.685 3445.227 3717.117 4038.56 8748.4 3922.207 4668.892
std 4.17E+01 248.0112 53.68705 324.1841 90.4861 554.8304 574.8573 53.68336 55.23699 168.6275 315.9437 242.0103 115.3615
median 3231.854 4943.156 3873.647 5146.345 3393.626 4909.597 4527.31 3351.252 3659.146 3878.164 8309.017 3680.434 4493.078
rank 1 11 7 12 3 10 9 2 4 6 13 5 8
C17-F28 mean 3258.849 9039.799 3612.531 11626.37 3357.711 7481.465 4910.869 3287.562 4468.446 5362.586 5162.894 3907.406 5142.044
best 3258.849 8136.512 3525.284 10271.61 3318.801 6022.699 4266.446 3264.397 4181.808 4704.857 5098.55 3570.553 4873.83
worst 3258.849 11322.77 3707.734 15224.58 3405.491 8982.664 5157.508 3307.624 4828.807 5941.318 5289.845 4453.539 5340.596
std 0.00E+00 1534.376 90.02818 2403.224 4.25E+01 1500.662 430.9645 21.19846 301.7903 508.1125 86.93082 382.1522 226.2349
median 3258.849 8349.955 3608.552 10504.65 3353.275 7460.25 5109.76 3289.114 4431.585 5402.085 5131.59 3802.765 5176.874
rank 1 12 4 13 3 11 7 2 6 10 9 5 8
C17-F29 mean 3263.038 14218.62 5632.311 20420.93 4143.015 7110.393 9375.189 4929.302 4968.983 6724.479 8460.558 4932.794 6316.421
best 3247.132 9312.287 5477.19 10706.72 3766.573 6635.55 6256.789 4454.759 4752.572 5768.286 6934.52 4689.737 5986.254
worst 3278.787 19596.96 5782.906 32407.69 4403.306 7667.373 12342.09 5565.908 5285.425 7773.026 11153.49 5025.851 6959.8
std 17.45672 4732.831 124.9251 9665.478 287.62 427.7218 2506.204 465.3867 244.3511 950.7145 1899.848 162.2628 455.0945
median 3263.116 13982.62 5634.574 19284.66 4201.091 7069.324 9450.938 4848.271 4918.967 6678.302 7877.111 5007.795 6159.814
rank 1 12 6 13 2 9 11 3 5 8 10 4 7
C17-F30 mean 623575.2 3.42E+09 22821830 5.74E+09 1705474 1.73E+09 1.66E+08 73642941 1.46E+08 3.14E+08 1.93E+08 5002045 61053462
best 582411.6 2.64E+09 13929371 3.52E+09 1280832 2.12E+08 1.12E+08 66548195 70487353 2.18E+08 1.48E+08 3485746 49252148
worst 655637.4 4.65E+09 31288289 9.01E+09 2798738 3.52E+09 2.29E+08 84714113 2.16E+08 3.97E+08 2.53E+08 6965674 85718115
std 32665.89 8.74E+08 8527252 2.36E+09 732080.7 1.70E+09 58549245 7884159 73601328 74915265 44081620 1719705 16882161
median 628125.9 3.20E+09 23034829 5.21E+09 1371162 1.60E+09 1.61E+08 71654727 1.48E+08 3.20E+08 1.85E+08 4778379 54621793
rank 1 12 4 13 2 11 8 6 7 10 9 3 5
Sum rank 30 335 166 367 63 294 269 112 144 248 254 150 207
Mean rank 1.034483 11.55172 5.724138 12.65517 2.172414 10.13793 9.275862 3.862069 4.965517 8.551724 8.758621 5.172414 7.137931
Total rank 1 12 6 13 2 11 10 3 4 8 9 5 7

Boxplot of OOA and competitor algorithms in optimization of the CEC-2017 test suite ( D = 50 ).

The results of using OOA and competitor algorithms in handling the CEC 2017 test suite for D = 100 are reported in Table 5. The convergence curves under the tests of the algorithms for the dimension D equal to 100 are drawn in Figure 6. Based on the obtained results, OOA is the first best optimizer for C17-F1, and C17-F3 to C17-F30.

Performance of optimization algorithms on the CEC 2017 test suite ( D = 100 ).

OOA WSO AVOA RSA MPA TSA WOA MVO GWO TLBO GSA PSO GA
C17-F1 mean 1.00E+02 1.74E+11 3.99E+09 2.43E+11 5.42E+08 1.32E+11 6.55E+10 68678335 5.96E+10 9.52E+10 1.42E+11 2.09E+10 5.85E+10
best 1.00E+02 1.70E+11 1.94E+09 2.39E+11 4.11E+08 1.16E+11 6.18E+10 57227179 5.16E+10 9.06E+10 1.31E+11 1.41E+10 5.54E+10
worst 1.00E+02 1.79E+11 5.75E+09 2.45E+11 6.85E+08 1.47E+11 7.33E+10 80426511 6.75E+10 1.05E+11 1.52E+11 2.85E+10 6.62E+10
std 1.16E-14 3.57E+09 1.56E+09 2.80E+09 1.32E+08 1.29E+10 5.27E+09 11329664 7.50E+09 6.60E+09 9.07E+09 7.89E+09 5.12E+09
median 1.00E+02 1.74E+11 4.14E+09 2.44E+11 5.37E+08 1.32E+11 6.34E+10 68529824 5.97E+10 9.27E+10 1.43E+11 2.06E+10 5.63E+10
rank 1 12 4 13 3 10 8 2 7 9 11 5 6
C17-F3 mean 300 446688 340534.4 336533.3 164664.5 379261.4 823214.7 486188 384092.3 308928.1 358205.8 564202.5 602214.5
best 300 407043.7 332573.4 324625 126043 303917 720501.8 403851.3 351464.8 289773.8 331548.2 427413.5 577504.8
worst 300 467184.3 348163.6 343536.5 199250.2 433127.5 953343.5 582037.9 420620.5 326886.3 392059.9 792030 621795.5
std 0.00E+00 28122.02 6588.962 8855.309 3.18E+04 54720.06 100372.6 90722.49 36997.81 15166.91 25109.58 1.69E+05 19561.07
median 300 456262 340700.2 338985.8 166682.5 390000.6 809506.8 479431.3 382142 309526.3 354607.5 518683.2 604778.9
rank 1 9 5 4 2 7 13 10 8 3 6 11 12
C17-F4 mean 602.1722 46425.06 1595.28 78297.44 1036.547 16665.98 11382.73 767.1665 4632.319 11174.66 35541.63 2552.569 9573.019
best 592.0676 42721.73 1334.08 70969.18 919.7862 10896.85 9691.342 710.4 3550.563 10645.77 28255.08 1532.072 9045.943
worst 612.2769 50903.66 1761.275 87244.71 1157.91 22167.69 12501.8 830.7531 6975.431 12085.21 40226.75 3228.695 10172.28
std 11.66782 3517.165 193.5368 6748.829 114.9462 4657.53 1195.509 50.31302 1576.528 684.2829 5781.367 731.3458 524.4966
median 602.1722 46037.43 1642.883 77487.94 1034.245 16799.69 11668.88 763.7565 4001.641 10983.82 36842.34 2724.755 9536.924
rank 1 12 4 13 3 10 9 2 6 8 11 5 7
C17-F5 mean 512.9345 2016.061 1321.042 1984.676 1229.716 2173.477 1858.398 1240.48 1186.243 1894.235 1343.618 1423.306 1596.092
best 510.9445 1996.281 1308.836 1948.152 1098.912 2147.495 1758.588 1128.356 1128.005 1866.353 1308.664 1320.509 1445.49
worst 514.9244 2028.261 1330.675 2020.607 1317.823 2203.685 2014.507 1312.65 1236.095 1924.526 1375.832 1601.033 1686.437
std 1.816538 13.81528 9.143136 36.5698 105.6913 25.82916 111.1346 82.91296 47.25802 23.79415 34.97224 131.6567 108.363
median 512.9345 2019.851 1322.328 1984.973 1251.065 2171.363 1830.25 1260.457 1190.437 1893.03 1344.989 1385.84 1626.221
rank 1 12 5 11 3 13 9 4 2 10 6 7 8
C17-F6 mean 600 707.9979 662.7391 706.2348 637.8589 712.636 705.4486 675.6992 640.7707 682.4373 664.8946 662.2912 663.9558
best 600 705.2448 658.6046 701.2121 633.9668 700.0829 695.4872 668.8272 635.6752 673.4656 662.2483 654.9442 656.4549
worst 600 710.5746 667.073 709.2128 644.4488 721.2849 723.0693 682.0943 647.2093 687.8391 669.1866 668.281 669.6147
std 0.00E+00 2.405659 3.490615 3.536234 4.968795 10.23301 12.38019 5.698777 5.002692 6.873829 3.060808 6.335007 6.625553
median 600 708.0861 662.6395 707.2571 636.5099 714.588 701.6191 675.9376 640.0992 684.2223 664.0718 662.9698 664.8767
rank 1 12 5 11 2 13 10 8 3 9 7 4 6
C17-F7 mean 811.392 3650.012 3110.783 3768.145 1852.472 3472.709 3620.631 2019.959 2035.7 3127.589 3153.412 2496.513 2596.172
best 810.0205 3560.732 2947.498 3674.969 1792.523 3289.848 3497.126 1854.037 1842.155 2979.588 3019.382 2223.826 2493.591
worst 813.1726 3754.355 3247.79 3847.385 1935.614 3645.438 3800.358 2145.656 2178.357 3249.226 3375.012 2617.996 2820.7
std 1.461146 79.52358 150.0855 74.00421 61.81474 160.5773 139.2818 121.3792 140.701 111.3592 155.1109 186.8218 151.6415
median 811.1874 3642.48 3123.922 3775.113 1840.876 3477.776 3592.52 2040.072 2061.143 3140.771 3109.627 2572.115 2535.198
rank 1 12 7 13 2 10 11 3 4 8 9 5 6
C17-F8 mean 812.437 2438.962 1747.178 2494.492 1437.028 2415.901 2334.726 1461.053 1523.371 2268.597 1836.65 1715.25 2045.045
best 808.9546 2386.801 1689.729 2468.844 1262.642 2346.597 2130.213 1305.878 1415.268 2202.808 1752.914 1672.251 1992.605
worst 816.9143 2500.607 1775.985 2510.32 1545.619 2505.19 2492.139 1647.002 1666.346 2321.834 1969.831 1812.534 2097.835
std 3.398432 48.76287 39.62258 17.91963 124.0899 76.55454 185.4982 140.8902 113.0837 51.68063 97.13552 65.28887 45.02509
median 811.9395 2434.22 1761.499 2499.402 1469.926 2405.908 2358.276 1445.666 1505.934 2274.873 1811.928 1688.108 2044.87
rank 1 12 6 13 2 11 10 3 4 9 7 5 8
C17-F9 mean 900 90764.2 26646.67 77803.71 22561.33 121332.2 77313.46 59631.12 36181.74 74955.71 23704.95 33074.4 46287.42
best 900 81019.54 22171.86 75206.87 20994.98 99454.95 60105.71 50272.15 22365.37 71785.25 22059.83 27958.73 41915.23
worst 900 104864.2 30000.7 79930.32 23270.35 151359 97456.55 67831.59 49224.87 76641.35 24978.89 36845.85 52148.43
std 9.28E-14 10288.47 3262.414 2058.281 1.05E+03 21792.63 18649.98 7230.081 13112.33 2216.189 1220.703 3951.421 4297.465
median 900 88586.55 27207.05 78038.83 22990 117257.4 75845.79 60210.36 36568.37 75698.13 23890.54 33746.52 45543.02
rank 1 12 4 11 2 13 10 8 6 9 3 5 7
C17-F10 mean 11023.04 30595.17 16027.86 31944.11 13904.48 29669.86 28610.77 17066.4 15236.31 31953.93 17304.19 17149.28 26342.05
best 9625.608 30282.5 13282.43 31019.71 13168.74 28880.58 27704.45 16396.22 14174.4 30762.38 15424.2 15246.95 25898.25
worst 11858.81 30854.79 18339.25 32382.9 14716.98 30614.68 29995.84 17655.74 15824.2 33029.07 18334.84 18554.1 26856.79
std 968.8655 239.4268 2197.965 636.4917 640.6625 747.9646 1036.742 520.6368 743.9012 950.3788 1318.387 1384.967 393.9588
median 11303.87 30621.69 16244.87 32186.92 13866.09 29592.1 28371.39 17106.82 15473.32 32012.14 17728.85 17398.04 26306.58
rank 1 11 4 12 2 10 9 5 3 13 7 6 8
C17-F11 mean 1162.329 168249.9 65550.44 211156.1 4873.88 66824.43 213122.4 4666.962 89141.96 73371.33 176673 53204.82 142435.3
best 1139.568 130568.3 58892.97 161533.1 3832.317 30431.1 123997.3 4058.33 74036.37 61877.78 147212.3 24172.28 108681.1
worst 1220.662 195828.5 78342.3 300896.9 5829.576 95622.42 343667.6 4943.856 100456.7 93559.24 206123.3 108812.4 196401.5
std 39.03582 28058.47 9000.9 62789.92 860.1431 27023.48 102188.9 410.1504 11303.93 13892.48 24331.69 37853.07 38428.36
median 1144.542 173301.3 62483.24 191097.1 4916.814 70622.09 192412.4 4832.832 91037.38 69024.15 176678.3 39917.28 132329.3
rank 1 10 5 12 3 6 13 2 8 7 11 4 9
C17-F12 mean 5.97E+03 1.08E+11 6.74E+08 1.76E+11 2.67E+08 5.82E+10 1.35E+10 3.40E+08 1.17E+10 2.24E+10 6.84E+10 1.03E+10 1.26E+10
best 5.38E+03 7.67E+10 3.57E+08 1.31E+11 1.49E+08 2.98E+10 1.10E+10 2.16E+08 8.11E+09 1.76E+10 5.93E+10 1.34E+09 1.15E+10
worst 6.57E+03 1.20E+11 1.08E+09 2.04E+11 3.20E+08 9.65E+10 1.54E+10 5.34E+08 1.39E+10 3.09E+10 8.04E+10 1.96E+10 1.49E+10
std 494.4731 2.09E+10 3.10E+08 3.33E+10 79248544 2.78E+10 1.88E+09 1.40E+08 2.51E+09 6.07E+09 8.79E+09 8.33E+09 1.55E+09
median 5.97E+03 1.17E+11 6.31E+08 1.84E+11 2.99E+08 5.32E+10 1.38E+10 3.05E+08 1.24E+10 2.06E+10 6.69E+10 1.02E+10 1.20E+10
rank 1 12 4 13 2 10 8 3 6 9 11 5 7
C17-F13 mean 1407.28 2.85E+10 100529.2 4.37E+10 99147.86 2.19E+10 5.35E+08 362483.5 9.70E+08 2.88E+09 8.94E+09 1.81E+09 1.79E+08
best 1371.145 2.48E+10 70996.36 3.38E+10 42435.99 1.55E+10 3.81E+08 319413.3 8.36E+07 1.99E+09 5.50E+09 1.99E+08 1.40E+08
worst 1439.935 3.16E+10 137075.5 4.95E+10 246335.9 2.62E+10 7.24E+08 422619.2 2.56E+09 3.49E+09 1.15E+10 3.27E+09 2.15E+08
std 34.75018 3.54E+09 28019.23 7.27E+09 98706.36 4.51E+09 1.77E+08 45176.93 1.15E+09 6.82E+08 2.51E+09 1.51E+09 38881835
median 1409.02 2.88E+10 97022.51 4.57E+10 53909.78 2.29E+10 5.18E+08 353950.7 6.16E+08 3.02E+09 9.39E+09 1.88E+09 1.80E+08
rank 1 12 3 13 2 11 6 4 7 9 10 8 5
C17-F14 mean 1467.509 46573127 6844256 81703552 93142.92 9123209 14921942 3111444 9864026 14264672 11792012 835861.8 10771886
best 1458.803 40220080 4150401 74517892 26553.57 4143920 8589788 938970 6239142 10628070 9089036 397194.6 6027073
worst 1472.733 53202617 11361828 8.94E+07 197961.1 17800434 20397661 4282899 14787691 18228855 17684938 1735572 15868325
std 6.045415 5702604 3176774 7167829 76757.13 6017123 4854894 1490724 3747137 3974355 3973287 609228.1 4096540
median 1469.25 46434906 5932396 81428308 74028.5 7274241 15350159 3611953 9214635 14100880 10197036 605340.4 10596073
rank 1 12 5 13 2 6 11 4 7 10 9 3 8
C17-F15 mean 1609.893 1.58E+10 86450.29 2.41E+10 57374.46 1.24E+10 71895592 129687.1 5.14E+08 1.22E+09 1.27E+09 3.42E+08 13014150
best 1551.154 1.46E+10 70680.49 1.72E+10 16508.42 2.57E+08 40042584 88710.64 33727236 4.08E+08 5.10E+08 63001.02 8391451
worst 1652.294 1.78E+10 108545.3 3.01E+10 87155.7 2.32E+10 1.38E+08 190873.5 1.54E+09 2.61E+09 1.63E+09 1.35E+09 22174867
std 44.16216 1.37E+09 18103.24 6.37E+09 29825.57 9.95E+09 44792419 44980.64 6.97E+08 9.65E+08 5.18E+08 6.72E+08 6261238
median 1618.063 1.54E+10 83287.7 2.46E+10 62916.86 1.30E+10 54697797 119582.1 2.41E+08 9.35E+08 1.48E+09 8853798 10745140
rank 1 12 3 13 2 11 6 4 8 9 10 7 5
C17-F16 mean 2711.795 19365.41 7254.238 23167.65 5602.853 14877.15 16592.22 6703.045 6195.429 11728.21 11278.52 6588.173 10751.32
best 2171.69 17960.39 5984.563 18122.6 5501.583 12288 13497.6 5927.57 5590.138 11183.5 9761.669 6244.448 9629.085
worst 3397.326 20009.8 8023.667 25972.49 5757.693 17806.25 18399.38 7229.866 6944.472 12894.25 13113.34 6863.71 11594.5
std 509.9574 946.9546 895.4879 3553.475 110.0055 2267.179 2181.479 564.2757 681.3093 805.1832 1491.139 255.8628 899.6235
median 2639.081 19745.73 7504.36 24287.76 5576.068 14707.17 17235.94 6827.372 6123.553 11417.55 11119.54 6622.266 10890.84
rank 1 12 6 13 2 10 11 5 3 9 8 4 7
C17-F17 mean 2716.564 4332312 5924.863 8522970 4696.486 224479.3 17417.27 5025.573 5576.485 8902.419 47578.57 6184.81 7271.438
best 2275.021 1269631 5729.125 2310154 4496.558 10394.46 10682.9 4585.989 4468.008 8775.635 31096.29 5902.201 7022.396
worst 3429.127 9856790 6402.033 19611684 4947.147 596595.7 29492.83 5375.651 7206.416 9023.651 77425.76 6474.548 7479.366
std 514.4523 4048021 321.3756 8139930 214.9931 256147.1 8461.496 375.9154 1205.47 137.3921 20490.79 251.9735 193.888
median 2581.054 3101414 5784.147 6085020 4671.12 145463.6 14746.67 5070.326 5315.758 8905.195 40896.11 6181.246 7291.996
rank 1 12 5 13 2 11 9 3 4 8 10 6 7
C17-F18 mean 1903.746 59966445 2892066 1.06E+08 238237 15302010 12324557 5040123 11253997 16639374 12071761 6609639 6199867
best 1881.15 27167009 1438096 41073557 166073.9 5731498 9167098 3732608 3543887 12261147 5565364 4082002 4971094
worst 1919.921 1.08E+08 4572563 1.93E+08 429403.1 3.13E+07 14600247 8466334 18186245 23519874 26837047 9522450 8975554
std 19.37923 34719523 1420166 6.43E+07 127838.5 11506761 2476086 2293111 6024617 4836245 10027545 2527999 1884653
median 1906.955 52127545 2778803 9.44E+07 178735.6 12103511 12765441 3980776 11642929 15388238 7942317 6417053 5426409
rank 1 12 3 13 2 10 9 4 7 11 8 6 5
C17-F19 mean 1972.839 1.31E+10 2957487 2.30E+10 287612.8 5.19E+09 1.38E+08 17096238 3.70E+08 6.87E+08 1.62E+09 2.77E+08 13150284
best 1967.139 1.15E+10 1132210 1.68E+10 60458.19 2.30E+09 54627419 9971400 2940146 2.98E+08 2.92E+08 46042259 6713112
worst 1977.869 1.54E+10 5444372 2.86E+10 487240.3 1.03E+10 2.32E+08 27174288 1.11E+09 1.58E+09 3.06E+09 5.99E+08 23782834
std 4.536847 1.74E+09 1822603 4.88E+09 177142.4 3.54E+09 82196892 8492386 5.19E+08 6.03E+08 1.38E+09 2.68E+08 7574634
median 1973.174 1.27E+10 2626684 2.33E+10 301376.3 4.07E+09 1.32E+08 15619633 1.82E+08 4.35E+08 1.57E+09 2.31E+08 11052595
rank 1 12 3 13 2 11 6 5 8 9 10 7 4
C17-F20 mean 3192.04 7418.691 6273.509 7679.865 4538.934 7152.329 7165.301 5897.533 6168.53 7376.317 6422.898 5444.145 6372.713
best 2806.762 7242.682 5859.83 7529.468 4494.4 6528.975 6668.73 5543.679 4856.908 6562.859 5918.058 4708.61 5762.988
worst 3662.121 7574.972 6599.351 7740.777 4633.288 7922.074 7539.882 6485.392 7098.94 7768.853 6722.411 6310.776 6901.902
std 439.3601 137.7297 332.5845 100.6237 63.98214 596.0402 384.9015 408.4228 1110.196 549.9021 361.9903 680.7944 542.3505
median 3149.639 7428.555 6317.427 7724.606 4514.024 7079.133 7226.296 5780.529 6359.135 7586.777 6525.562 5378.596 6412.982
rank 1 12 6 13 2 9 10 4 5 11 8 3 7
C17-F21 mean 2342.155 4338.067 3702.04 4465.266 2846.188 4168.578 4274.302 3261.155 2992.89 3744.253 4776.928 3615.156 3446.495
best 2338.689 4289.402 3483.22 4385.387 2799.799 4019.499 3963.356 3189.708 2908.12 3576.85 4203.103 3420.594 3409.766
worst 2346.015 4411.347 3844.587 4524.025 2882.485 4269.559 4514.665 3396.715 3048.549 3933.761 5238.547 3982.56 3498.016
std 3.368829 58.11544 155.9035 60.32082 35.09706 123.0249 246.6719 92.99767 59.76649 151.9949 431.767 254.683 38.12764
median 2341.959 4325.759 3740.176 4475.825 2851.235 4192.628 4309.593 3229.098 3007.446 3733.201 4833.03 3528.734 3439.098
rank 1 11 7 12 2 9 10 4 3 8 13 6 5
C17-F22 mean 11739 32314.86 20498.57 34024.6 18895.97 31278.92 29594.43 17444.52 23663.76 33898.24 21428.82 22214.76 29245.76
best 11119.08 31498.22 19055.67 33598.52 17433.73 30100.68 28065.17 16309.54 18654.34 32906.97 20650.39 20725 28192.66
worst 12601.6 32796.71 22478.63 34656.25 20660.7 32370.82 30796.12 18189.42 35247.78 34406.11 21731.2 23892.02 30069.05
std 652.7204 584.6414 1520.604 480.0723 1360.105 929.0826 1209.045 909.3989 7855.797 672.6938 521.5104 1336.676 877.7121
median 11617.67 32482.26 20229.99 33921.82 18744.73 31322.1 29758.21 17639.55 20376.47 34139.94 21666.84 22121.02 29360.67
rank 1 11 4 13 3 10 9 2 7 12 5 6 8
C17-F23 mean 2877.697 5421.356 4154.449 5423.6 3311.825 5546.504 5228.023 3506.002 3644.759 4259.125 8070.116 4938.627 4313.384
best 2872.107 5159.061 4070.365 5144.214 3294.854 4753.305 5078.216 3410.529 3611.339 4203.784 7447.068 4394.694 4243.985
worst 2884.013 5729.211 4243.355 5643.709 3344.345 6621.872 5379.187 3629.972 3692.991 4341.396 8510.217 5226.769 4381.782
std 5.215893 255.0332 81.86927 206.7708 22.11191 836.0171 143.7173 92.91948 37.4283 58.53447 480.6159 374.7417 74.74386
median 2877.334 5398.576 4152.038 5453.239 3304.05 5405.42 5227.344 3491.752 3637.352 4245.66 8161.589 5066.523 4313.884
rank 1 10 5 11 2 12 9 3 4 6 13 8 7
C17-F24 mean 3327.407 8744.654 5457.678 10800.23 3731.939 6810.582 6506.829 3995.082 4334.743 4811.761 11127.75 6070.298 5460.761
best 3295.518 6782.951 5225.896 7178.303 3679.174 6298.858 6066.236 3924.907 4082.576 4565.078 10446.32 5675.875 5363.251
worst 3357.991 10077.26 5650.136 13213.68 3799.514 7152.369 7177.36 4112.718 4562.486 5056.26 12926 6572.385 5641.715
std 29.61997 1577.532 188.7213 2921.1 56.65989 363.0739 484.2087 89.17051 245.9371 201.538 1200.317 400.8331 126.2704
median 3328.059 9059.201 5477.339 11404.46 3724.534 6895.551 6391.861 3971.351 4346.955 4812.854 10569.34 6016.467 5419.04
rank 1 11 6 12 2 10 9 3 4 5 13 8 7
C17-F25 mean 3185.232 15767.5 4179.734 22078.21 3707.002 10777.75 7475.354 3417.941 6567.276 9146.095 11358.81 4180.841 8084.274
best 3137.371 14986.58 3781.284 20451.87 3525.711 10096.58 6828.902 3353.337 6394.801 7860.821 10468.28 3896.766 7342.328
worst 3261.571 17606.64 4555.884 25692.93 3831.904 11211.33 7881.516 3488.168 6993.903 10861.85 12958.88 4633.007 8850.794
std 59.9065 1239.237 320.1035 2464.125 129.1747 508.9357 474.6206 58.65314 286.1755 1372.124 1105.777 349.7519 784.6
median 3170.992 15238.39 4190.885 21084.03 3735.196 10901.55 7595.5 3415.129 6440.2 8930.857 11004.04 4096.796 8071.987
rank 1 12 4 13 3 10 7 2 6 9 11 5 8
C17-F26 mean 5757.621 40886.19 25411.42 47072.25 11876.21 34438.92 35091.97 12093.63 17324.98 24611.8 34990 21323.72 23711.84
best 5645.905 40318.38 22430.83 44412.96 11114.22 33121 31425.59 10679.1 15351.15 20080 33529.85 19077.91 22005.03
worst 5844.642 41410.54 28456.97 48717.05 12662.69 35273.64 38186.61 14592.8 18986.83 30373.32 36897.07 23411.95 24861.79
std 8.39E+01 455.5407 2576.448 2069.674 761.8059 925.5767 3341.621 1719.955 1532.501 4270.467 1414.82 1826.615 1229.822
median 5769.969 40907.92 25378.94 47579.49 11863.97 34680.51 35377.83 11551.31 17480.97 23996.93 34766.54 21402.51 23990.26
rank 1 12 8 13 2 9 11 3 4 7 10 5 6
C17-F27 mean 3309.493 9592.663 4201.894 12682.85 3544.987 6751.874 6128.369 3638.737 4117.472 4375.483 14528.59 4109.501 5584.987
best 3278.01 8043.474 4020.935 9449.536 3501.09 6432.688 5394.53 3596.249 3939.051 4082.75 14171.06 3894.824 5309.717
worst 3344.5 11151.23 4501.806 16039.78 3581.318 7147.152 6944.5 3742.724 4255.656 4857.651 14812.72 4326.92 6001.893
std 2.84E+01 1689.157 207.9817 3552.725 33.22976 309.1239 838.4094 70.09844 155.9076 344.9133 290.8187 235.9997 296.3851
median 3307.732 9587.972 4142.418 12621.05 3548.77 6713.829 6087.224 3607.987 4137.59 4280.765 14565.29 4108.13 5514.17
rank 1 11 6 12 2 10 9 3 5 7 13 4 8
C17-F28 mean 3322.242 21688.67 4769.56 29348.09 3791.774 16254.9 10690.8 3465.288 9537.435 11524.89 19493.14 7839.022 11852.83
best 3318.742 20180.54 4451.619 26270.06 3661.77 12715.2 9106.46 3376.52 8048.721 8959.263 16791.95 5261.171 10776.61
worst 3327.816 24479.73 5003.944 33200.56 3884.657 18937 11729.1 3551.826 11660.97 13776.33 21530.88 12212.41 13045.63
std 4.38E+00 1955.599 233.8001 2906.514 9.37E+01 2982.302 1116.37 72.17256 1523.241 2245.475 1982.038 3168.049 1215.14
median 3321.205 21047.21 4811.338 28960.86 3810.334 16683.7 10963.83 3466.402 9220.023 11681.98 19824.86 6941.254 11794.54
rank 1 12 4 13 3 10 7 2 6 8 11 5 9
C17-F29 mean 4450.696 191101.4 9836.093 363957.5 6980.052 19038.39 17055.8 8855.915 8461.958 12758.76 25807.94 8818.204 12140.68
best 4169.151 108773.9 8506.522 195194.4 6139.017 14518.99 14205.32 7850.134 8293.249 11882.08 21254.62 8136.039 11936.68
worst 4829.521 260817 10656.48 505323.4 7789.152 24193.52 19593.29 9541.22 8795.287 13394.65 33955.81 9778.475 12630.65
std 282.3422 64748.72 928.1423 132171.8 674.6827 4031.095 2645.123 737.4808 235.1403 644.4315 5908.187 785.245 328.1646
median 4402.056 197407.3 10090.68 377656 6996.021 18720.52 17212.29 9016.154 8379.647 12879.16 24010.65 8679.15 11997.7
rank 1 12 6 13 2 10 9 5 3 8 11 4 7
C17-F30 mean 5407.166 2.41E+10 28840381 3.92E+10 4883520 1.39E+10 1.56E+09 1.07E+08 1.91E+09 3.94E+09 7.64E+09 6.29E+08 6.92E+08
best 5337.48 2.11E+10 16434396 3.66E+10 2175972 8.48E+09 1.28E+09 65884676 7.85E+08 1.48E+09 5.45E+09 1.53E+08 5.78E+08
worst 5557.155 2.62E+10 50716469 4.24E+10 7974458 1.73E+10 2.11E+09 1.32E+08 2.50E+09 7.30E+09 9.26E+09 1.95E+09 7.42E+08
std 101.1571 2.13E+09 15348420 2.48E+09 2679742 3.84E+09 3.77E+08 29325356 7.71E+08 2.92E+09 1.61E+09 8.82E+08 77135828
median 5367.014 2.45E+10 24105330 3.89E+10 4691824 1.50E+10 1.43E+09 1.15E+08 2.18E+09 3.48E+09 7.93E+09 2.07E+08 7.25E+08
rank 1 12 3 13 2 11 7 4 8 9 10 5 6
Sum rank 29 336 140 355 65 293 265 114 156 249 272 162 203
Mean rank 1 11.58621 4.827586 12.24138 2.241379 10.10345 9.137931 3.931034 5.37931 8.586207 9.37931 5.586207 7
Total rank 1 12 4 13 2 11 9 3 5 8 10 6 7

Boxplot of OOA and competitor algorithms in optimization of the CEC-2017 test suite ( D = 100 ).

The analysis of the simulation results shows that the proposed OOA approach has provided better results in most of the benchmark functions of the CEC 2017 test suite and for dimensions 10, 30, 50, and 100. Compared to competitor algorithms, the proposed OOA has been ranked first as the best optimizer in handling the CEC 2017 test suite.

 4.2 Statistical analysis

In this subsection, statistical analysis is presented on the performance of OOA and competitor algorithms to determine whether OOA has a significant statistical superiority or not. For this purpose, Wilcoxon rank sum test (Wilcoxon 1992), a non-parametric statistical test to determine the significant difference between the average of two data samples, is employed. In the Wilcoxon rank sum test, using an index called a p -value, it is determined whether the superiority of OOA against any of the competitor algorithms is significant from a statistical point of view.

The results of the statistical analysis on the performance of OOA and competitor algorithms are reported in Table 6. Based on these results, in cases where the p -value is less than 0.05, OOA has a significant statistical superiority compared to the corresponding algorithm.

p -values obtained from Wilcoxon rank sum test.

Compared algorithm Objective function type
D = 10 D = 30 D = 50 D = 100
OOA vs WSO 2.02E-21 1.97E-21 1.97E-21 1.97E-21
OOA vs AVOA 3.77E-19 3.02E-21 1.97E-21 1.97E-21
OOA vs RSA 1.97E-21 1.97E-21 1.97E-21 1.97E-21
OOA vs MPA 9.43E-20 1.56E-16 6.62E-18 1.97E-21
OOA vs TSA 9.5E-21 1.97E-21 1.97E-21 1.97E-21
OOA vs WOA 9.5E-21 1.97E-21 1.97E-21 1.97E-21
OOA vs MVO 9.03E-19 2.13E-21 1.97E-21 1.97E-21
OOA vs GWO 5.23E-21 1.97E-21 1.97E-21 1.97E-21
OOA vs TLBO 3.69E-21 1.97E-21 1.97E-21 1.97E-21
OOA vs GSA 1.6E-18 2.02E-21 1.97E-21 1.97E-21
OOA vs PSO 1.54E-19 2.35E-21 1.97E-21 1.97E-21
OOA vs GA 2.71E-19 1.97E-21 1.97E-21 1.97E-21
5 OOA for real-world applications

In this section, the effectiveness of the proposed OOA approach in solving optimization problems in real-world applications is tested. In this regard, OOA is implemented on the CEC 2011 test suite. This test suite has twenty-two up-to-date test functions for real-world constrained optimization problems. The IEEE CEC-2011 test suite details and complete information are stated in (Das and Suganthan 2010). For OOA and each competitor algorithm, the maximum number of FEs for all 22 test functions is set to 150,000, with 25 independent runs in all experiments. Furthermore, the stop criterion for the proposed OOA is set to the maximum number of function evaluations (MFEs).

The employment results of the proposed OOA approach and competitor algorithms in solving the CEC 2011 test suite are reported in Table 7. The convergence curves of the performance of the algorithms while achieving the solution for different problems of the CEC 2011 test suite are plotted in Figure 7. Based on the simulation results, OOA is the first best optimizer for C11-F1, C11-F4 to C11-F8, C11-F10, C11-F13, C11-F14, and C11-F16 to C11-F22. What is evident from the comparison of the simulation results is that OOA has provided better results in most of the CEC 2011 test suite benchmark functions and has provided superior performance in handling this test suite compared to competitor algorithms. Also, the results obtained from the Wilcoxon sum rank test show that OOA has a significant statistical superiority in handling the CEC 2011 test suite compared to competitor algorithms.

Performance of optimization algorithms on the CEC 2011 test suite.

OOA WSO AVOA RSA MPA TSA WOA MVO GWO TLBO GSA PSO GA
C11-F1 mean 5.92E+00 1.89E+01 1.37E+01 2.36E+01 1.10E+01 1.97E+01 1.40E+01 14.8452 1.14E+01 1.98E+01 2.33E+01 1.92E+01 2.52E+01
best 2.00E-10 1.69E+01 9.76E+00 2.22E+01 9.96E+00 1.84E+01 9.06E+00 11.70306 1.23E+00 1.76E+01 2.16E+01 1.15E+01 2.40E+01
worst 1.23E+01 2.15E+01 1.75E+01 2.59E+01 1.17E+01 2.08E+01 1.80E+01 17.66581 1.92E+01 2.20E+01 2.45E+01 2.55E+01 2.70E+01
std 6.85E+00 2.17E+00 4.23E+00 1.71E+00 7.50E-01 1.05E+00 4.01E+00 2.610069 7.49E+00 1.88E+00 1.34E+00 6.83E+00 1.28E+00
median 5.69E+00 1.86E+01 1.38E+01 2.32E+01 1.11E+01 1.98E+01 1.45E+01 15.00597 1.26E+01 1.97E+01 2.36E+01 1.99E+01 2.49E+01
rank 1 7 4 12 2 9 5 6 3 10 11 8 13
C11-F2 mean −26.3179 −13.2913 −20.5603 −10.2147 −26.7129 −9.90809 −17.9159 −7.19366 −22.2996 −9.47882 −14.5515 −22.353 −11.699
best −27.0676 −14.827 −21.0907 −10.7066 −27.4388 −14.0575 −21.7147 −9.37908 −24.6614 −10.7219 −20.0119 −23.7965 −14.3125
worst −25.4328 −11.9522 −19.7495 −9.69101 −26.2216 −7.48974 −13.4971 −5.53885 −18.4005 −8.40395 −10.1238 −19.6993 −9.79905
std 7.03E-01 1.468953 0.613286 0.503179 5.16E-01 3.11281 4.22075 1.653768 2.74601 0.962131 4.518559 1.81E+00 2.101387
median −26.3856 −13.193 −20.7005 −10.2306 −26.5956 −9.04253 −18.226 −6.92836 −23.0682 −9.39471 −14.0352 −22.9582 −11.3423
rank 2 8 5 10 1 11 6 13 4 12 7 3 9
C11-F3 mean 1.15E-05 1.15E-05 -0.00354 1.15E-05 2.30E-07 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05
best 1.15E-05 1.15E-05 −0.00354 1.15E-05 2.30E-07 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 −0.00354 1.15E-05 2.30E-07 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 1.91E-19 2.28E-11 2.50E-19 5.12E-11 3.80E-21 2.45E-14 6.43E-19 1.02E-12 3.82E-15 8.04E-14 2.06E-19 6.01E-20 2.84E-18
median 1.15E-05 1.15E-05 -0.00354 1.15E-05 2.30E-07 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05 1.15E-05
rank 3 12 1 13 2 9 6 11 8 10 5 4 7
C11-F4 mean 0 0 0 0 0 0 0 0 0 0 0 0 0
best 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
median 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 mean −34.1274 −24.0394 −27.6184 −18.7805 −29.4055 −26.56 −27.1008 −26.4084 −31.3672 −8.80614 −26.7951 −6.45671 −7.39003
best −34.7494 −25.2701 −28.7334 −21.0964 −31.7317 −31.4028 −27.2613 −31.5374 −34.2381 −11.1777 −31.3217 −10.3165 −8.94813
worst −33.3862 −23.0312 −27.1144 −16.2229 −23.8644 −20.7317 −26.7239 −23.8086 −27.0023 −6.98701 −23.4313 −4.5488 −5.52836
std 5.61E-01 0.989761 0.755405 2.555151 3.734357 4.3945 0.253933 3.626498 3.08384 1.782197 3.459885 2.710558 1.501269
median −34.1871 −23.9282 −27.3129 −18.9013 −31.013 −27.0528 −27.2089 −25.1438 −32.1142 −8.52994 −26.2136 −5.48077 −7.54181
rank 1 9 4 10 3 7 5 8 2 11 6 13 12
C11-F6 mean −24.1119 −13.197 −18.6161 −12.1182 −20.917 −6.16746 −19.6162 −8.30597 −19.2664 −0.48224 −21.7103 −1.42197 −2.40594
best −27.4298 −13.9153 −19.8691 −12.933 −23.0059 −16.0037 −22.9887 −16.9689 −21.9964 −0.5486 −26.9078 −4.30754 −8.15494
worst −23.0059 −12.7147 −16.7701 −11.1075 −17.9859 −2.7147 −12.1204 −0.46012 −17.5726 −0.46012 −17.0034 −0.46012 −0.46012
std 2.211922 0.509588 1.482046 0.892726 2.508209 6.562134 5.134849 9.036013 2.130979 0.044238 4.244118 1.92371 3.832891
median −23.0059 −13.0789 −18.9127 −12.2162 −21.338 −2.97571 −21.6779 −7.89742 −18.7483 −0.46012 −21.465 −0.46012 −0.50436
rank 1 7 6 8 3 10 4 9 5 13 2 12 11
C11-F7 mean 0.860699 1.664086 1.317009 2.002335 0.962583 1.336241 1.812352 0.882654 1.083614 1.785476 1.096603 1.143946 1.808939
best 0.582266 1.616063 1.165353 1.731485 0.850449 1.171283 1.688059 0.804191 0.798213 1.569158 0.907901 0.819907 1.409139
worst 1.025027 1.770171 1.459576 2.189765 1.042937 1.719177 1.986618 0.963936 1.328403 1.92612 1.314091 1.412034 2.019504
std 0.201221 0.072353 0.153197 0.194267 0.080731 0.256775 0.127811 0.084878 0.220699 0.156603 0.186737 0.30992 0.275769
median 0.91775 1.635056 1.321553 2.044045 0.978472 1.227251 1.787365 0.881244 1.10392 1.823314 1.082209 1.171921 1.903555
rank 1 9 7 13 3 8 12 2 4 10 5 6 11
C11-F8 mean 220 290.2856 242.148 333.9985 220 260.3515 269.833 224.41 227.938 224.41 248.5042 489.922 222.695
best 220 261.6304 223.92 289.678 220 220 247.342 220 220 220 220 250.38 220
worst 220 327.8784 260.376 382.288 220 365.53 319.666 237.64 235.876 237.64 299.38 598.6627 230.78
std 0.00E+00 29.00521 15.68816 37.98663 0.00E+00 70.51727 33.48117 8.82 9.166013 8.82 37.64277 164.8394 5.39
median 220 285.8168 242.148 332.014 220 227.938 256.162 220 227.938 220 237.3184 555.3226 220
rank 1 9 5 10 1 7 8 3 4 3 6 11 2
C11-F9 mean 8789.286 602623.6 408947 1149127 3386.759 70980.08 404977.4 143685.7 45853.51 441745.3 890720.6 1171284 2102685
best 5457.674 402185 361825.4 749757.4 2202.958 51023.62 223920.9 81109.58 19349.73 365373.1 761725.4 940153.9 2014648
worst 14042.29 692778.6 440382.8 1348542 4140.039 90348.17 686621.2 217543.4 80951.01 566570.4 959059.3 1434402 2225897
std 3700.105 136986.5 34468.96 271487.7 909.0091 16698.95 211097.9 56267.25 26035.62 88487.46 87867.94 264341.2 103858.4
median 7828.591 657765.4 416789.9 1249104 3602.02 71274.27 354683.8 138044.8 41556.65 417518.9 921048.8 1155289 2085097
rank 2 9 7 11 1 4 6 5 3 8 10 12 13
C11-F10 mean −21.4889 −13.3594 −16.5505 −11.5296 −12.8771 −13.8131 −12.1726 −14.1504 −13.5039 −10.4572 −12.4759 −10.5651 −10.2454
best −21.8299 −14.6547 −16.7402 −11.9214 −12.9073 −18.6339 −12.9536 −21.1403 −14.0025 −10.5434 −13.0185 −10.6029 −10.2695
worst −20.7878 −12.6842 −16.1969 −11.2205 −12.8225 −11.2332 −11.6343 −10.6317 −12.2368 −10.4096 −11.6229 −10.5297 −10.2182
std 0.474376 0.894754 0.247359 0.294081 0.038835 3.326357 0.557534 4.735187 0.850081 0.062339 0.670202 0.030904 0.022699
median −21.669 −13.0495 −16.6324 −11.4882 −12.8894 −12.6928 −12.0513 −12.4148 −13.8882 −10.4378 −12.6311 −10.5639 −10.247
rank 1 6 2 10 7 4 9 3 5 12 8 11 13
C11-F11 mean 5.72E+05 6.23E+06 1.02E+06 9.53E+06 6.29E+04 6.38E+06 1.27E+06 1.37E+06 4.10E+06 5.59E+06 1.48E+06 5.60E+06 6.57E+06
best 2.61E+05 5.94E+06 8.14E+05 9.24E+06 5.67E+04 5.31E+06 1.15E+06 6.05E+05 3.90E+06 5.58E+06 1.33E+06 5.59E+06 6.54E+06
worst 8.29E+05 6.62E+06 1.20E+06 9.72E+06 6.82E+04 7.71E+06 1.42E+06 2.91E+06 4.51E+06 5.59E+06 1.66E+06 5.61E+06 6.66E+06
std 248237.1 3.06E+05 1.68E+05 2.06E+05 5104.022 9.93E+05 1.12E+05 1.05E+06 2.80E+05 4.96E+03 1.38E+05 1.26E+04 5.64E+04
median 5.99E+05 6.18E+06 1.04E+06 9.59E+06 6.33E+04 6.25E+06 1.25E+06 9.76E+05 3.99E+06 5.59E+06 1.47E+06 5.60E+06 6.55E+06
rank 2 10 3 13 1 11 4 5 7 8 6 9 12
C11-F12 mean 1199805 8.98E+06 3552219 1.42E+07 1075197 5.34E+06 6.19E+06 1337807 1.44E+06 1.54E+07 6.16E+06 2.40E+06 1.56E+07
best 1155937 8.60E+06 3444800 1.32E+07 1073495 5.05E+06 5.74E+06 1171904 1.27E+06 1.45E+07 5.85E+06 2.22E+06 1.54E+07
worst 1249353 9.30E+06 3621631 1.51E+07 1076709 5.50E+06 6.42E+06 1497116 1.59E+06 1.61E+07 6.39E+06 2.63E+06 1.57E+07
std 44864.97 2.92E+05 77325.21 7.82E+05 1317.465 2.11E+05 3.15E+05 132828.4 1.33E+05 6.81E+05 2.34E+05 1.72E+05 112097.4
median 1196965 9.00E+06 3571223 1.43E+07 1075291 5.41E+06 6.30E+06 1341104 1.46E+06 1.55E+07 6.21E+06 2.38E+06 1.56E+07
rank 2 10 6 11 1 7 9 3 4 12 8 5 13
C11-F13 mean 15444.2 15891.51 15448.3 16381.29 15446.7 15493.96 15542.95 15513.12 15505.77 15973.25 137589.2 15494.65 31195.71
best 15444.19 15690.82 15447.22 15931.05 15444.21 15483.23 15495.68 15491.22 15498.27 15642.53 99115.29 15476.01 15461.79
worst 15444.21 16377.73 15449.5 1.75E+04 15449.25 15507.6 15606.67 15554.77 15518.73 16582.5 189773.3 15534.16 78016.98
std 0.008649 327.2954 0.958783 751.8829 2.595291 12.05117 51.64227 29.46168 9.065538 425.4429 40816.26 26.62637 31214.32
median 15444.2 15748.75 15448.25 16047.11 15446.68 15492.5 15534.72 15503.23 15503.03 15833.99 130734.2 15484.21 15652.04
rank 1 9 3 11 2 4 8 7 6 10 13 5 12
C11-F14 mean 18295.35 1.20E+05 18537.77 2.46E+05 18309.64 1.96E+04 19302.16 19511.72 1.93E+04 3.35E+05 1.92E+04 1.92E+04 19179.64
best 18241.58 9.11E+04 18416.33 1.81E+05 18228.42 1.94E+04 19139.75 19406.35 19155.29 3.12E+04 1.88E+04 19015.52 18871.16
worst 18388.08 1.69E+05 18636.78 3.56E+05 18454.01 2.02E+04 1.94E+04 19603.22 1.95E+04 6.48E+05 1.94E+04 1.93E+04 19507.13
std 68.11851 3.47E+04 104.2335 7.83E+04 103.3697 4.02E+02 134.4807 82.00062 1.58E+02 2.96E+05 2.34E+02 1.35E+02 259.8922
median 18275.87 1.11E+05 18548.99 2.24E+05 18278.07 1.95E+04 19316.63 19518.66 1.93E+04 3.30E+05 1.92E+04 19206.91 19170.15
rank 1 11 3 12 2 10 7 9 8 13 4 6 5
C11-F15 mean 32883.58 980657.1 114055.3 2070845 32826.89 56349.12 234007 33117.93 33093.96 16696239 321898.4 33321.54 8591518
best 32782.17 402606.7 44018.07 864448 32749.87 33055.26 33028.2 33034.8 33054.35 3496287 284586 33310.88 3909088
worst 32956.46 2470035 192410.5 5410062 32896.01 125903.9 336116.9 33181.16 33158.53 24899241 347448.3 33328.04 14726152
std 73.20611 996511.9 79750.21 2229608 69.17515 46370.02 136829.2 64.61464 46.43207 9731927 29242.56 7.993635 4959765
median 32897.86 524993.4 109896.4 1004436 32830.83 33218.68 283441.4 33127.88 33081.48 19194713 327779.7 33323.63 7865415
rank 2 10 7 11 1 6 8 4 3 13 9 5 12
C11-F16 mean 133550 1012440 135311.7 2099802 133558.3 146084.5 142876.3 142491.4 146892.3 96259139 20263372 86155318 82723005
best 131374.2 301718.4 133840.8 501858.1 131941.8 143332.7 136483.5 133107.7 144087.4 93801811 10286608 71265169 66855780
worst 136310.8 2406659 136082.1 5234724 135855.5 148325.2 148415.1 152190.7 152655.3 99030856 36670211 1.03E+08 1.06E+08
std 2275.9 946672.1 1013.646 2128515 1943.812 2460.669 4993.537 8040.181 3906.059 2191541 11408377 13659390 16548000
median 133257.5 670691.9 135662 1331312 133218 146340.1 143303.4 142333.6 145413.2 96101944 17048334 85200517 79112111
rank 1 8 3 9 2 6 5 4 7 13 10 12 11
C11-F17 mean 1926615 9.70E+09 2.51E+09 1.68E+10 2018066 1.39E+09 1.05E+10 3210362 3110202 2.42E+10 1.21E+10 2.26E+10 2.37E+10
best 1916953 8.27E+09 2.28E+09 1.21E+10 1921673 1.14E+09 7.49E+09 2326135 2048196 2.32E+10 1.07E+10 1.99E+10 2.21E+10
worst 1942685 1.08E+10 2.74E+09 2.05E+10 2280655 1.59E+09 1.40E+10 3893330 5126235 2.52E+10 1.29E+10 2.61E+10 2.67E+10
std 11419.96 1.10E+09 2.05E+08 3.64E+09 175323.5 2.27E+08 2.72E+09 723382.4 1386878 8.15E+08 9.90E+08 2.78E+09 2.09E+09
median 1923412 9.89E+09 2.50E+09 1.73E+10 1934968 1.41E+09 1.03E+10 3310991 2633188 2.41E+10 1.25E+10 2.21E+10 2.29E+10
rank 1 7 6 10 2 5 8 4 3 13 9 11 12
C11-F18 mean 942057.5 5.95E+07 7020972 1.28E+08 945809.1 2.14E+06 1.03E+07 991932.8 1.04E+06 3.35E+07 1.20E+07 1.46E+08 1.24E+08
best 938416.2 4.09E+07 4190482 8.85E+07 943930.4 1.86E+06 4385257 965503.4 969046.1 2.65E+07 8.91E+06 1.23E+08 1.19E+08
worst 944706.9 6.77E+07 12097679 1.46E+08 948300.7 2.51E+06 1.81E+07 1004368 1.22E+06 3.62E+07 1.51E+07 1.62E+08 1.29E+08
std 2639.272 1.25E+07 3682210 2.71E+07 2005.676 3.13E+05 5805582 17868.5 1.22E+05 4.66E+06 2.77E+06 1.77E+07 3728471
median 942553.5 6.47E+07 5897864 1.39E+08 945502.6 2.10E+06 9.33E+06 998929.7 9.82E+05 3.56E+07 1.19E+07 1.50E+08 1.24E+08
rank 1 10 6 12 2 5 7 3 4 9 8 13 11
C11-F19 mean 1025341 58527871 7123682 1.25E+08 1366914 2582701 10980935 1514100 1390783 38459399 6702206 1.87E+08 1.24E+08
best 967927.7 49931270 6484134 1.08E+08 1343884 2326479 2150215 1141467 1252651 26910604 2488339 1.70E+08 1.21E+08
worst 1167142 74440929 8651349 1.58E+08 1427450 3068933 19964603 2040777 1575694 47997979 8831409 2.16E+08 1.28E+08
std 94829.24 11061970 1025954 23018090 40535.43 332380.8 8383033 377771.2 134742.5 9135730 2880015 20194234 2785524
median 983146.6 54869643 6679623 1.18E+08 1348161 2467695 10904461 1437079 1367393 39464507 7744538 1.81E+08 1.24E+08
rank 1 10 7 12 2 5 8 4 3 9 6 13 11
C11-F20 mean 941250.4 62243936 6303277 1.36E+08 946579.8 1899989 7807096 975431.8 1003291 37362188 15376888 1.72E+08 1.25E+08
best 936143.2 54753557 5548677 1.19E+08 939384.4 1700681 7351327 964474.4 980067.7 36540674 10188015 1.57E+08 1.19E+08
worst 946866.6 73723256 7110732 1.61E+08 950729.5 2230697 8415523 987887.1 1021436 38250334 23845584 1.87E+08 1.29E+08
std 4769.814 8083207 648345.1 18141817 5030.901 251619 455059.4 10398.75 17836.94 711073.8 5968667 16526880 4479404
median 940995.9 60249467 6276850 1.31E+08 948102.7 1834289 7730768 974682.9 1005831 37328871 13736977 1.73E+08 1.25E+08
rank 1 10 6 12 2 5 7 3 4 9 8 13 11
C11-F21 mean 12.71443 53.37283 22.38723 81.77425 15.62748 31.25063 40.9641 28.77291 23.18251 108.0202 43.03455 113.4746 110.0845
best 9.974206 43.54778 21.15423 60.38387 13.83158 27.68958 37.20046 25.30841 21.3192 51.18422 37.56023 97.80934 62.48432
worst 14.97499 63.92425 24.17974 103.1811 17.7647 33.16732 45.63621 32.03371 25.53745 159.6996 46.07948 126.5846 134.876
std 2.295373 8.790098 1.296205 18.9047 2.066322 2.456868 3.733383 3.680033 1.827882 44.44662 3.850039 14.04357 33.6487
median 12.95425 53.00965 22.10748 81.766 15.45683 32.0728 40.50985 28.87476 22.9367 110.5986 44.24924 114.7523 121.4889
rank 1 9 3 10 2 6 7 5 4 11 8 13 12
C11-F22 mean 16.12513 49.37691 28.39079 67.43929 18.40047 33.47484 48.85642 33.644 25.73388 109.8312 49.24346 114.2011 98.96395
best 11.50133 42.42355 23.0652 48.20173 14.88709 29.46911 42.40217 25.65977 24.30524 70.76134 40.63684 95.90493 98.11324
worst 19.55286 55.13954 33.85428 77.59772 22.55941 35.95157 53.75117 38.90346 26.79215 130.0648 59.08522 125.98 100.4104
std 3.993717 5.454167 5.083363 13.13616 3.347607 2.847801 5.093176 6.037242 1.232421 26.68831 7.570152 13.64839 1.041821
median 16.72317 49.97228 28.32183 71.97886 18.07769 34.23935 49.63617 35.00639 25.91906 119.2493 48.62589 117.4596 98.66611
rank 1 9 4 10 2 5 7 6 3 12 8 13 11
Sum rank 29 190 99 231 45 145 147 118 95 222 158 199 225
Mean rank 1.318182 8.636364 4.5 10.5 2.045455 6.590909 6.681818 5.363636 4.318182 10.09091 7.181818 9.045455 10.22727
Total rank 1 9 4 13 2 6 7 5 3 11 8 10 12
Wilcoxon: p-value 4.8E-12 8.49E-15 1.71E-15 0.001914 5.36E-15 5.76E-15 1.75E-11 2.11E-12 3.66E-15 8.8E-15 1.71E-15 2.5E-15

Boxplot of OOA and competitor algorithms in optimization of the CEC-2011 test suite.

 6 Conclusion and future works

This paper introduced a new metaheuristic algorithm named the Osprey Optimization Algorithm (OOA) to solve real-world optimization problems. The real inspiration in the proposed OOA approach is the ospreys’ strategies when hunting fish from the sea during the steps of identifying the prey, attacking the prey in the sea, and transporting the prey to a suitable place. The proposed OOA approach theory was explained, and its implementation steps in two phases of exploration and exploitation were mathematically modeled. The effectiveness of OOA in solving optimization problems was evaluated on twenty-nine standard benchmark functions from the CEC 2017 test suite. The quality of the proposed approach was compared with the performance of twelve well-known metaheuristic algorithms. The simulation results showed that OOA had achieved better results in most of the benchmark functions by balancing exploration and exploitation during the search process, and compared to competitor algorithms, it has superior performance in optimization tasks. Also, the employment of OOA in dealing with twenty-two up-to-date real-world constrained optimization problems from the CEC 2011 test suite showed the adequate performance of the proposed approach in solving optimization problems in real-world applications.

Following the introduction of the proposed OOA approach, several research directions are activated for future studies. For example, designing binary and multi-objective versions for the proposed OOA approach is one of the central potentials of this study for further work. In addition, the employment of OOA in optimization problems in various science and real-world applications is another research proposal for further work in the future.

 Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

 Author contributions

Conceptualization, PT; methodology, PT; software, MD; validation, PT and MD; formal analysis, MD; investigation, PT; resources, PT.; data curation, PT and MD; writing—original draft preparation, PT and MD; writing—review and editing, PT and MD; visualization, PT; supervision, PT; project administration, MD; funding acquisition, PT.

 Funding

The research was supported by the Project of Excellence of Faculty of Science No. 2210/2023–2024, University of Hradec Králové, Czech Republic.

The authors thank Dušan Bednařík from the University of Hradec Kralove for our fruitful and informative discussions.

 Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

 Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

 References Abdollahzadeh B. Farhad S. G. Mirjalili S. (2021). African vultures optimization algorithm: A new nature-inspired metaheuristic algorithm for global optimization problems. Comput. Industrial Eng. 158, 107408. 10.1016/j.cie.2021.107408 Abualigah L. Abd Elaziz M. Sumari P. Geem Z. W. Gandomi A. H. (2022). Reptile search algorithm (RSA): A nature-inspired meta-heuristic optimizer. Expert Syst. Appl. 191, 116158. 10.1016/j.eswa.2021.116158 Al-Betar M. A. Zaid A. A. A. Awadallah M. A. Abu Doush I. (2021). Coronavirus herd immunity optimizer (CHIO). Neural Comput. Appl. 33 (10), 5011–5042. 10.1007/s00521-020-05296-6 Assiri A. S. Hussien A. G. Amin M. (2020). Ant lion optimization: Variants, hybrids, and applications. IEEE Access 8, 77746–77764. 10.1109/access.2020.2990338 Awad N. H. Ali M. Liang J. Qu B. Suganthan P. Problem Definitions (2016). Evaluation criteria for the CEC 2017 special session and competition on single objective real-parameter numerical optimization. Technology Report. Ayyarao S. V. Rama Krishna N. S. S. Madurai Elavarasam R. Polumahanthi N. Rambabu M. Saini G. (2022). War strategy optimization algorithm: A new effective metaheuristic algorithm for global optimization. IEEE Access 10, 25073–25105. 10.1109/access.2022.3153493 Beyer H. G. Schwefel H. P. (2002). Evolution strategies–a comprehensive introduction. Nat. Comput. 1 (1), 3–52. 10.1023/a:1015059928466 Braik M. Hammouri A. Atwan J. Al-Betar M. A. Awadallah M. A. (2022a). White Shark optimizer: A novel bio-inspired meta-heuristic algorithm for global optimization problems. Knowledge-Based Syst. 243, 108457. 10.1016/j.knosys.2022.108457 Braik M. Hashem Ryalat M. Al-Zoubi H. (2022b). A novel meta-heuristic algorithm for solving numerical optimization problems: Ali Baba and the forty thieves. Neural Comput. Appl. 34 (1), 409–455. 10.1007/s00521-021-06392-x Brunetti G. Stumpp C. Šimůnek J. (2022). Balancing exploitation and exploration: A novel hybrid global-local optimization strategy for hydrological model calibration. Environ. Model. Softw. 150, 105341. 10.1016/j.envsoft.2022.105341 Cavazzuti M. (2013). “Deterministic optimization,” in Optimization methods: From theory to design scientific and technological aspects in mechanics (Berlin, Heidelberg: Springer Berlin Heidelberg), 77–102. Chopra N. Ansari M. M. (2022). Golden jackal optimization: A novel nature-inspired optimizer for engineering applications. Expert Syst. Appl. 198, 116924. 10.1016/j.eswa.2022.116924 Cuevas E. Oliva D. Zaldivar D. Pérez-Cisneros M. Sossa H. (2012). Circle detection using electro-magnetism optimization. Inf. Sci. 182 (1), 40–55. 10.1016/j.ins.2010.12.024 Das S. Suganthan P. N. (2010). Problem definitions and evaluation criteria for CEC 2011 competition on testing evolutionary algorithms on real world optimization problems. Kolkata: Jadavpur University, Nanyang Technological University, 341–359. De Castro L. N. Timmis J. I. (2003). Artificial immune systems as a novel soft computing paradigm. Soft Comput. 7 (8), 526–544. 10.1007/s00500-002-0237-z Dehghani M. Mardaneh M. Guerrero J. M. Malik O. P. Kumar V. (2020). Football game based optimization: An application to solve energy commitment problem. Int. J. Intelligent Eng. Syst. 13, 514–523. 10.22266/ijies2020.1031.45 Dehghani M. Montazeri Z. Ali D. Seifi A. R. (2017). “Spring search algorithm: A new meta-heuristic optimization algorithm inspired by Hooke's law,” in 2017 IEEE 4th International Conference on Knowledge-Based Engineering and Innovation (KBEI), Tehran, Iran, 22-22 December 2017 (IEEE). 10.1109/KBEI.2017.8324975 Dehghani M. Montazeri Z. Trojovská E. Trojovský P. (2023). Coati optimization algorithm: A new bio-inspired metaheuristic algorithm for solving optimization problems. Knowledge-Based Syst. 259, 110011. 10.1016/j.knosys.2022.110011 Dehghani M. Samet H. (2020). Momentum search algorithm: A new meta-heuristic optimization algorithm inspired by momentum conservation law. SN Appl. Sci. 2 (10), 1720–1815. 10.1007/s42452-020-03511-6 Dehghani M. Trojovská E. Trojovský P. (2022). A new human-based metaheuristic algorithm for solving optimization problems on the base of simulation of driving training process. Sci. Rep. 12 (1), 9924. 10.1038/s41598-022-14225-7 Dehghani M. Trojovský P. (2021). Teamwork optimization algorithm: A new optimization approach for function minimization/maximization. Sensors 21 (13), 4567. 10.3390/s21134567 Dorigo M. Maniezzo V. Colorni A. (1996). Ant system: Optimization by a colony of cooperating agents. IEEE Trans. Syst. Man, Cybern. Part B Cybern. 26 (1), 29–41. 10.1109/3477.484436 Eskandar H. Ali S. Bahreininejad A. Hamdi M. (2012). Water cycle algorithm–A novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput. Struct. 110, 151–166. 10.1016/j.compstruc.2012.07.010 Faramarzi A. Heidarinejad M. Mirjalili S. Gandomi A. H. (2020a). Marine predators algorithm: A nature-inspired metaheuristic. Expert Syst. Appl. 152, 113377. 10.1016/j.eswa.2020.113377 Faramarzi A. Heidarinejad M. Stephens B. Mirjalili S. (2020b). Equilibrium optimizer: A novel optimization algorithm. Knowledge-Based Syst. 191, 105190. 10.1016/j.knosys.2019.105190 Goldberg D. E. Holland J. H. (1988). Genetic algorithms and machine learning. Mach. Learn. 3 (2), 95–99. 10.1023/A:1022602019183 Grove R. A. Henny C. J. Kaiser J. L. (2009). Osprey: Worldwide sentinel species for assessing and monitoring environmental contamination in rivers, lakes, reservoirs, and estuaries. J. Toxicol. Environ. Health, Part B 12 (1), 25–44. 10.1080/10937400802545078 Hashim F. A. Houssein E. H. Hussain K. Mabrouk M. S. Al-Atabany W. (2022). Honey Badger Algorithm: New metaheuristic algorithm for solving optimization problems. Math. Comput. Simul. 192, 84–110. 10.1016/j.matcom.2021.08.013 Hashim F. A. Hussain K. Houssein E. H. Mabrouk M. S. Al-Atabany W. (2021). Archimedes optimization algorithm: A new metaheuristic algorithm for solving optimization problems. Appl. Intell. 51 (3), 1531–1551. 10.1007/s10489-020-01893-z Iba K. (1994). Reactive power optimization by genetic algorithm. IEEE Trans. power Syst. 9 (2), 685–692. 10.1109/59.317674 Karaboga D. Basturk B. (2007). “Artificial bee colony (ABC) optimization algorithm for solving constrained optimization problems,” in International fuzzy systems association world congress, (Berlin, Heidelberg: Springer) Kaur S. Awasthi L. K. Sangal A. L. Gaurav D. (2020). Tunicate swarm algorithm: A new bio-inspired based metaheuristic paradigm for global optimization. Eng. Appl. Artif. Intell. 90, 103541. 10.1016/j.engappai.2020.103541 Kennedy J. Eberhart R. (1995). “Particle swarm optimization,” in Proceedings of ICNN'95 - International Conference on Neural Networks, Perth, WA, Australia, 27 Nov.-1 Dec. 1995. Kirkpatrick S. Gelatt C. D. Vecchi M. P. (1983). Optimization by simulated annealing. Science 220 (4598), 671–680. 10.1126/science.220.4598.671 Koza J. R. Koza J. R. (1992). Genetic programming: On the programming of computers by means of natural selection. Vol. 1. Cambridge, MA: MIT press. Mirjalili S. Gandomi A. H. Mirjalili S. Z. Saremi S. Faris H. Mirjalili S. M. (2017). Salp swarm algorithm: A bio-inspired optimizer for engineering design problems. Adv. Eng. Softw. 114, 163–191. 10.1016/j.advengsoft.2017.07.002 Mirjalili S. Lewis A. (2016). The whale optimization algorithm. Adv. Eng. Softw. 95, 51–67. 10.1016/j.advengsoft.2016.01.008 Mirjalili S. Mirjalili S. M. Hatamlou A. (2016). Multi-verse optimizer: A nature-inspired algorithm for global optimization. Neural Comput. Appl. 27 (2), 495–513. 10.1007/s00521-015-1870-7 Mirjalili S. Mirjalili S. M. Lewis A. (2014). Grey Wolf optimizer. Adv. Eng. Softw. 69, 46–61. 10.1016/j.advengsoft.2013.12.007 Mirjalili S. (2015). The ant lion optimizer. Adv. Eng. Softw. 83, 80–98. 10.1016/j.advengsoft.2015.01.010 Moghdani R. Salimifard K. (2018). Volleyball premier league algorithm. Appl. Soft Comput. 64, 161–185. 10.1016/j.asoc.2017.11.043 Mohamed A. W. Hadi A. A. Mohamed A. K. (2020). Gaining-sharing knowledge based algorithm for solving optimization problems: A novel nature-inspired algorithm. Int. J. Mach. Learn. Cybern. 11 (7), 1501–1529. 10.1007/s13042-019-01053-x Mohar S. S. Goyal S. Kaur R. (2022). Localization of sensor nodes in wireless sensor networks using bat optimization algorithm with enhanced exploration and exploitation characteristics. J. Supercomput. 78 (9), 11975–12023. 10.1007/s11227-022-04320-x Moosavi S. H. S. Bardsiri V. K. (2019). Poor and rich optimization algorithm: A new human-based and multi populations algorithm. Eng. Appl. Artif. Intell. 86, 165–181. 10.1016/j.engappai.2019.08.025 Pereira J. L. J. Francisco M. B. Diniz C. A. Oliver G. A. Cunha S. S. Jr Ferreira Gomes G. (2021). Lichtenberg algorithm: A novel hybrid physics-based meta-heuristic for global optimization. Expert Syst. Appl. 170, 114522. 10.1016/j.eswa.2020.114522 Poole A. F. Bierregaard R. O. Martell M. S. (2002). Osprey: Pandion Haliaetus. Philadelphia, PA: Birds of North America, Incorporated, No. 683. Rao R. V. Savsani V. J. Vakharia D. P. (2011). Teaching–learning-based optimization: A novel method for constrained mechanical design optimization problems. Computer-Aided Des. 43 (3), 303–315. 10.1016/j.cad.2010.12.015 Rashedi E. Nezamabadi-Pour H. Saryazdi S. (2009). GSA: A gravitational search algorithm. Inf. Sci. 179 (13), 2232–2248. 10.1016/j.ins.2009.03.004 Reynolds R. G. (1994). “An introduction to cultural algorithms,” in Proceedings of the third annual conference on evolutionary programming, San Diego, California, USA, 24-26 Feb 94. Storn R. Price K. (1997). Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11 (4), 341–359. 10.1023/a:1008202821328 Strandberg R. (2013). Ageing, sexing and subspecific identification of Osprey, and two WP records of American Osprey. Dutch Bird. 35 (2), 69–87. Szaro R. C. (1978). Reproductive success and foraging behavior of the osprey at seahorse key, Florida. Wilson Bull. 1978, 112–118. Trojovský P. Dehghani M. (2022). Pelican optimization algorithm: A novel nature-inspired algorithm for engineering applications. Sensors 22 (3), 855. 10.3390/s22030855 Wei Z. Huang C. Wang X. Han T. Li Y. (2019). Nuclear reaction optimization: A novel and powerful physics-based algorithm for global optimization. IEEE Access 7, 66084–66109. 10.1109/access.2019.2918406 Wilcoxon F. (1992). “Individual comparisons by ranking methods,” in Breakthroughs in statistics (Berlin, Germany: Springer), 196–202. Wolpert D. H. Macready W. G. (1997). No free lunch theorems for optimization. IEEE Trans. Evol. Comput. 1 (1), 67–82. 10.1109/4235.585893 Xian H. Yang C. Wang H. Yang X. (2021). A modified sine cosine algorithm with teacher supervision learning for global optimization. IEEE Access 9, 17744–17766. 10.1109/access.2021.3054053 Xue J. Shen B. (2020). A novel swarm intelligence optimization approach: Sparrow search algorithm. Syst. Sci. Control Eng. 8 (1), 22–34. 10.1080/21642583.2019.1708830