In this article, a novel metaheuristic technique named Far and Near Optimization (FNO) is introduced, offering versatile applications across various scientific domains for optimization tasks. The core concept behind FNO lies in integrating global and local search methodologies to update the algorithm population within the problem-solving space based on moving each member to the farthest and nearest member to itself. The paper delineates the theory of FNO, presenting a mathematical model in two phases: (i) exploration based on the simulation of the movement of a population member towards the farthest member from itself and (ii) exploitation based on simulating the movement of a population member towards the nearest member from itself. FNO’s efficacy in tackling optimization challenges is assessed through its handling of the CEC 2017 test suite across problem dimensions of 10, 30, 50, and 100, as well as to address CEC 2020. The optimization results underscore FNO’s adeptness in exploration, exploitation, and maintaining a balance between them throughout the search process to yield viable solutions. Comparative analysis against twelve established metaheuristic algorithms reveals FNO’s superior performance. Simulation findings indicate FNO’s outperformance of competitor algorithms, securing the top rank as the most effective optimizer across a majority of benchmark functions. Moreover, the outcomes derived by employing FNO on twenty-two constrained optimization challenges from the CEC 2011 test suite, alongside four engineering design dilemmas, showcase the effectiveness of the suggested method in tackling real-world scenarios.
Currently, numerous optimization challenges are prevalent in science, engineering, mathematics, and practical applications, requiring suitable methodologies [1,2]. Typically, each optimization problem consists of three basic components: decision variables, constraints, and an objective function. The main goal in optimization is to find the optimal values for the decision variables, ensuring that the objective function is optimized within the constraints [3,4].
Metaheuristic algorithms are recognized as effective problem-solving tools designed to tackle optimization challenges. These algorithms employ random search techniques within the problem domain, along with stochastic operators, to provide feasible solutions for optimization problems [5]. They are valued for their simple conceptualization, ease of implementation, and ability to address various optimization problems, including non-convex, non-linear, non-derivative, discontinuous, and NP-hard problems, as well as navigating through discrete and unexplored search spaces [6]. The operational process of metaheuristic algorithms typically begins with randomly generating a set of feasible solutions, forming the algorithm population. Through iterative refinement processes, these solutions are improved via algorithmic update mechanisms. Upon completion of the algorithm’s execution, the most optimal solution obtained during the iterations is presented as the solution to the problem [7].
For metaheuristic algorithms to conduct an efficient search in the problem-solving space, they need to effectively manage the search process at both global and local levels. Global search, achieved through exploration, allows the algorithm to avoid getting trapped in local optima by thoroughly scanning the problem-solving space to identify the main optimum region. On the other hand, local search, through exploitation, enables the algorithm to converge towards potentially better solutions by closely examining the vicinity of discovered solutions and promising areas within the problem-solving space. In addition to exploration and exploitation, the success of a metaheuristic algorithm in facilitating an efficient search relies on maintaining a fine balance between these two strategies throughout the search process [8].
Because of the random nature of stochastic search methods, metaheuristic algorithms cannot guarantee reaching the global optimum. Therefore, solutions obtained from these algorithms are commonly labeled as quasi-optimal. This built-in uncertainty, along with researchers’ desire to attain quasi-optimal solutions closer to the global optimum, has driven the creation of various metaheuristic algorithms [9].
The main research question revolves around whether the existing metaheuristic algorithms adequately address the challenges in science, or if there’s a necessity for the development of newer ones. The No Free Lunch (NFL) theorem, as posited in reference [10], provides insight into this question by stating that the effectiveness of a metaheuristic algorithm in solving a specific set of optimization problems doesn’t guarantee similar success across all optimization problems. Hence, there’s no assured success or failure of a metaheuristic algorithm on any given optimization problem. According to the NFL theorem, no single metaheuristic algorithm can be deemed as the ultimate optimizer for all optimization tasks. This theorem, by continually stimulating the exploration and advancement of metaheuristic algorithms, encourages researchers to seek more efficient solutions for optimization problems through the design of newer algorithms.
This paper presents a novel contribution by introducing the Far and Near Optimization (FNO) metaheuristic algorithm, aimed at addressing optimization challenges across various scientific disciplines. Literature review shows that many metaheuristic algorithms have been designed so far. In most of these algorithms, researchers have tried to manage the search process by improving the state of exploration and exploitation in such a way as to achieve more effective solutions for optimization problems. In fact, the proper management of local and global search is the main key to the success of metaheuristic algorithms. This challenge of balancing exploration and exploitation has been the main motivation of the authors of this paper to design a dedicated algorithm that specifically deals with the design of the local and global search process. Therefore, it can be stated that the main source of inspiration for the authors in designing the proposed FNO approach was balancing exploration and exploitation.
In this new approach, the authors have used two basic mechanisms to change the position of the population members in the search space.
The first mechanism is the movement of each FNO member towards the farthest member of the population. In this mechanism, with the aim of increasing the discovery power of the algorithm in order to manage the global search, extensive changes are made in the position of each member of the population. These large displacements have a significant impact on the comprehensive scanning of the search space and prevent the algorithm from getting stuck in local optima. The important point in this process is that updating the population based on movement towards the best member is avoided. Because the excessive dependence of the update process based on the position of the best member can lead to the algorithm getting stuck early in inappropriate local optimal solutions.
The second mechanism is the movement of each FNO member towards the nearest member of the population. In this mechanism, small changes are made in the position of each population member with the aim of increasing the ability to exploit FNO in order to manage local search. These small displacements are effective in accurate scanning near the obtained solutions with the aim of obtaining possible better solutions and converging towards the global optimum.
Although based on the best knowledge obtained from the literature review, the originality of the proposed FNO approach is confirmed, however, the authors describe the differences of the proposed approach with one of the recently published algorithms called Giant Armadillo Optimization (GAO) [11]. GAO is a swarm-based algorithm inspired by the natural behavior of the giant armadillo in the wild. The main idea in FNO design is to move giant armadillo towards termite mounds and then dig in that position.
The first difference between proposed FNO approach and GAO is the source of design inspiration. As the title of the paper suggests, FNO is a population-based yet metaphor-free algorithm. But GAO is inspired by nature and wildlife in its design.
The second difference between FNO and GAO is in the design of the exploration phase. In the GAO design, each member of the population is directed to a position in the problem solving space that has a better objective function value compared to the corresponding member. Therefore, in GAO, it does not matter whether the location of the destination member is far or near. Meanwhile, in the design of the exploration phase in FNO, each member of the population moves towards the member farthest from itself. In this mechanism, the criterion for selecting the position of the destination member is the greatest distance to the corresponding member.
The third difference between FNO and GAO is related to the design of the exploitation phase. In GAO, in order to manage the local search, it is assumed that the movement range of each member becomes smaller and more limited during the iterations of the algorithm. This is while in FNO design, local search management is independent of algorithm iterations. The exploitation phase in FNO is designed in such a way that each member of the population moves towards the member closest to itself.
In general, it can be said that FNO is different from GAO from the point of view of source of inspiration in design, theory, mathematical model, exploration phase design, and exploitation phase design.
FNO has several advantages compared to other algorithms, which are described below:
Balanced Exploration and Exploitation
Enhanced Global Search: FNO leverages the exploration phase by moving towards the farthest member, which helps in thoroughly exploring the search space and avoiding local optima.
Improved Local Search: The exploitation phase involves moving towards the nearest member, which refines the search around promising areas to find the optimal solution.
Dynamic Population Update
Adaptive Strategy: FNO dynamically updates population members based on their distances, allowing for a more adaptive and responsive search process.
Efficient Convergence: The dual-phase approach ensures that the algorithm does not prematurely converge and maintains a good balance between exploration and exploitation throughout the optimization process.
Mathematical Rigor
Clear Theoretical Foundation: The FNO algorithm is mathematically modeled and well-defined, providing a clear understanding of its working principles and mechanisms.
Predictable Behavior: The mathematical formulation allows for predictable and consistent behavior of the algorithm across different optimization problems.
Performance on Benchmark Functions
Superior Results: Comparative analyses show that FNO often outperforms other well-known metaheuristic algorithms in solving benchmark optimization problems.
Statistical Significance: The performance improvements offered by FNO are not only observed empirically but are also statistically significant, providing robust evidence of its effectiveness.
Versatility and Applicability
Wide Range of Problems: FNO has demonstrated effectiveness across various problem domains, including both unconstrained and constrained optimization problems.
Real-World Applications: The algorithm has been successfully applied to real-world engineering design problems, showcasing its practical utility.
Scalability
Handling High-Dimensional Problems: FNO is capable of handling optimization problems with a large number of decision variables, making it suitable for complex, high-dimensional tasks.
Scalable Complexity: Despite its quadratic time complexity with respect to the number of population members, the algorithm remains computationally feasible for a wide range of practical applications.
Robustness and Reliability
Consistent Performance: FNO consistently provides high-quality solutions across different types of optimization problems, indicating its robustness.
Reliability: The algorithm’s ability to balance exploration and exploitation ensures that it reliably finds good solutions without getting stuck in suboptimal regions.
The FNO algorithm offers several advantages over other metaheuristic algorithms, including a balanced approach to exploration and exploitation, dynamic and adaptive population updates, strong mathematical foundations, superior performance on benchmark functions, versatility in application, scalability to high-dimensional problems, and overall robustness and reliability. These advantages make FNO a powerful and effective tool for tackling a wide variety of optimization challenges.
The key contributions of this research are outlined as follows:
The main idea in the design of FNO is to effectively update the population member of the algorithm in the problem-solving space by applying the concept of global search based on movement towards the farthest member and local search based on movement towards the nearest member.
The theory of FNO is stated and its implementation steps are mathematically modeled in two phases of exploration and exploitation.
Novel Approach: The paper introduces a new metaheuristic algorithm named Far and Near Optimization, designed to address various optimization challenges in scientific disciplines.
Balance between Exploration and Exploitation: FNO emphasizes a balanced approach to global and local search by employing distinct mechanisms for exploration (movement towards the farthest member) and exploitation (movement towards the nearest member).
Exploration Phase: The algorithm enhances global search capability by moving each population member towards the farthest member, which helps in avoiding local optima and ensures thorough exploration of the search space.
Exploitation Phase: FNO improves local search by moving each population member towards the nearest member, enabling precise scanning near discovered solutions to find potentially better solutions and converge towards the global optimum.
Clear Theoretical Foundation: The FNO algorithm is mathematically modeled, providing a well-defined and understandable framework for its working principles and mechanisms.
Mathematical Rigor: The paper provides a detailed mathematical formulation of the FNO algorithm, ensuring predictable and consistent behavior across different optimization problems.
Consistent Performance: FNO consistently provides high-quality solutions across different types of optimization problems, indicating its robustness.
Reliability: The algorithm’s balanced exploration and exploitation ensure that it reliably finds good solutions without getting stuck in suboptimal regions.
Below outlines the structure of this paper: Section 2 presents the review of relevant literature. In Section 3, we introduce and model the FNO approach. Section 4 covers simulation studies and their results. Section 5 examines the efficacy of FNO in addressing real-world applications. Finally, Section 6 offers conclusions and suggestions for future research.
Literature Review
Metaheuristic algorithms draw inspiration from a range of natural and evolutionary phenomena, including biology, genetics, physics, and human behaviors. As a result, these algorithms are categorized into five groups based on their underlying sources of inspiration: swarm-based, evolutionary-based, physics-based, and human-based.
Swarm-based metaheuristic algorithms take inspiration from the collective behaviors observed in diverse natural swarms, encompassing birds, animals, aquatic creatures, insects, and other organisms within their habitats. Notable examples include Particle Swarm Optimization (PSO) [12], Artificial Bee Colony (ABC) [13], Ant Colony Optimization (ACO) [14], and Firefly Algorithm (FA) [15]. PSO mirrors the coordinated movements of bird and fish flocks during foraging. ABC replicates the collaborative endeavors of bee colonies in locating new food sources. ACO is grounded in the efficient pathfinding capabilities of ants between their nest and food sites. FA is crafted based on the light signaling interactions among fireflies for communication and mate attraction. In the realm of natural behaviors exhibited by wild organisms, strategies like hunting, foraging, migration, chasing, digging, and searching prominently feature and have inspired the design of swarm-based metaheuristic algorithms such as: Arctic Puffin Optimization (APO) [16], Remora Optimization Algorithm (ROA) [17], Sand Cat Swarm Optimization (SCSO) [18,19], Giant Armadillo Optimization (GAO) [11], African Vultures Optimization Algorithm (AVOA) [20], Golden Jackal Optimization (GJO) [21], White Shark Optimizer (WSO) [22], Tunicate Swarm Algorithm (TSA) [23], Whale Optimization Algorithm (WOA) [24], Grey Wolf Optimizer (GWO) [25], Marine Predator Algorithm (MPA) [26], Honey Badger Algorithm (HBA) [27], and Reptile Search Algorithm (RSA) [28].
Evolutionary-based metaheuristic algorithms find their inspiration in principles from biological sciences and genetics, with a particular emphasis on concepts like natural selection, survival of the fittest, and Darwin’s evolutionary theory. Prominent examples include Genetic Algorithm (GA) [29] and Differential Evolution (DE) [30], which are deeply rooted in genetic principles, reproduction processes, and the application of evolutionary operators such as selection, mutation, and crossover. Furthermore, Artificial Immune System (AIS) [31] has been developed, drawing inspiration from the human body’s defense mechanisms against pathogens and diseases.
Physics-based metaheuristic algorithms draw inspiration from a wide array of forces, laws, phenomena, transformations, and other fundamental concepts within physics. Simulated Annealing (SA) [32] is a notable example, inspired by the process of metal annealing, where metals are heated and gradually cooled to achieve an optimal crystal structure. Moreover, algorithms like Spring Search Algorithm (SSA) [33] Momentum Search Algorithm (MSA) [34] and Gravitational Search Algorithm (GSA) [35] have been developed based on the emulation of physical forces. SSA employs spring tensile force modeling and Hooke’s law, MSA is founded on impulse force modeling, and GSA integrates gravitational attraction force modeling. Noteworthy physics-based methods include the Multi-Verse Optimizer (MVO) [36], Archimedes Optimization Algorithm (AOA) [37], Thermal Exchange Optimization (TEO) [38], Electro-Magnetism Optimization (EMO) [39], Water Cycle Algorithm (WCA) [40], Black Hole Algorithm (BHA) [41], Equilibrium Optimizer (EO) [42], and Lichtenberg Algorithm (LA) [43].
Human-based metaheuristic algorithms draw inspiration from diverse aspects of human behavior, encompassing communication, decision-making, thought processes, interactions, and choices observed in personal and social contexts. Teaching-Learning Based Optimization (TLBO) [44] serves as a prime example, inspired by the dynamics of education within classrooms, where knowledge exchange occurs among teachers and students as well as between students themselves. The Mother Optimization Algorithm (MOA) is introduced, drawing inspiration from the care provided by mothers to their children [45]. Poor and Rich Optimization (PRO) [46] is formulated based on economic activities observed among individuals of different socioeconomic backgrounds, with the aim of improving their financial situations. Conversely, Doctor and Patient Optimization (DPO) [47] simulates therapeutic interactions between healthcare providers and patients in hospital settings. Other notable human-based metaheuristic algorithms include War Strategy Optimization (WSO) [48], Ali Baba and the Forty Thieves (AFT) [49], Coronavirus Herd Immunity Optimizer (CHIO) [50], and Gaining Sharing Knowledge based Algorithm (GSK) [51].
In addition to the mentioned groupings, many researchers are interested in combining several algorithms and developing hybrid versions of them to benefit from the advantages of existing algorithms at the same time. Also, designing improved versions of existing algorithms has become a research motivation for researchers. WOA-SCSO is a hybrid metaheuristic algorithm that is derived from the combination of WOA and SCSO [52]. The nonlinear chaotic honey badger algorithm (NCHBA) is an improved version of the honey badger algorithm, which is designed to balance exploration and exploitation in this algorithm [53]. Some other hybrid metaheuristic algorithms are: hybrid PSO-GA [54], hybrid GWO-WOA [55], hybrid GA-PSO-TLBO [56], and hybrid TSA-PSO [57].
Table 1 lists an overview of metaheuristic algorithms and their grouping.
Overview of metaheuristic algorithms
Row
Group
Algorithm
Description and source of inspiration
Year
1
Particle Swarm Optimization (PSO) [12]
Inspired by the social behavior of birds flocking or fish schooling. Particles move through the solution space influenced by their own best position and the best positions of their neighbors.
1995
2
Swarm-based
Ant Colony Optimization (ACO) [14]
Inspired by the foraging behavior of ants. Ants deposit pheromones to mark paths, influencing others to follow and reinforce optimal paths.
1996
3
Artificial Bee Colony (ABC) [13]
Modeled on the foraging behavior of honey bees. The algorithm uses employed bees, onlookers, and scouts to explore and exploit food sources, representing solutions.
2007
4
Firefly Algorithm (FA) [15]
Based on the flashing behavior of fireflies. Attracted to each other based on their brightness, which represents the objective function. Brighter fireflies attract others, guiding the search.
2009
5
Grey Wolf Optimizer (GWO) [25]
Simulates the leadership hierarchy and hunting mechanism of grey wolves. The search is guided by alpha, beta, delta, and omega wolves representing different hierarchical roles.
2014
6
Whale Optimization Algorithm (WOA) [24]
Inspired by the bubble-net hunting strategy of humpback whales. The algorithm mimics the behavior of whales encircling prey and using bubble nets to trap them.
2016
7
Tunicate Swarm Algorithm (TSA) [23]
Modeled on the swarming behavior of tunicates in the ocean. The algorithm uses unique movement patterns and swarming behavior to explore and exploit the search space.
2020
8
Marine Predator Algorithm (MPA) [26]
Based on the behavior of marine predators such as sharks and tuna during the foraging process. It includes strategies like Brownian and Lévy flight motions to enhance exploration and exploitation.
Inspired by the scavenging behavior of African vultures. The algorithm simulates the way vultures search for and locate carrion, representing optimal solutions.
2021
10
Remora Optimization Algorithm (ROA) [17]
Inspired by the symbiotic relationship between remoras and their host fish. The algorithm mimics the behavior of remoras attaching to and detaching from hosts to optimize the search process.
2021
11
Honey Badger Algorithm (HBA) [27]
Mimics the foraging behavior of honey badgers. The algorithm uses a unique approach to balance exploration and exploitation, inspired by the badger’s ability to dig and hunt.
2022
12
Reptile Search Algorithm (RSA) [28]
Based on the hunting strategies of reptiles. It incorporates adaptive strategies to efficiently explore and exploit the search space, inspired by reptiles’ predatory behavior.
2022
13
Golden Jackal Optimization (GJO) [21]
Based on the social hunting behavior of golden jackals. The algorithm mimics the cooperative hunting strategies of jackals to explore and exploit the search space.
2022
14
White Shark Optimizer (WSO) [22]
Inspired by the hunting strategies of white sharks. The algorithm incorporates the predatory behavior of sharks, using strategies like circling and attacking prey to find optimal solutions.
2022
15
Sand Cat Swarm Optimization (SCSO) [18,19]
Inspired by the hunting behavior of sand cats. The algorithm simulates their adaptive strategies for hunting and surviving in harsh desert environments.
2022
16
Crayfish Optimization Algorithm (COA) [16]
Based on the behavior of crayfish. The algorithm uses their movement patterns and social behavior to explore and exploit the solution space.
2023
17
Giant Armadillo Optimization (GAO) [11]
Modeled on the foraging behavior of giant armadillos. The algorithm uses a unique approach to digging and searching for food to optimize solutions.
2023
18
Evolutionary-based
Genetic Algorithm (GA) [29]
Inspired by the process of natural selection and genetics. Uses operators like selection, crossover, and mutation to evolve solutions to optimization problems.
1988
19
Differential Evolution (DE) [30]
Based on the concept of differential mutation, crossover, and selection to optimize a problem. The algorithm relies on the differences between solution vectors to create new candidate solutions.
1997
20
Artificial Immune System (AIS) [31]
Inspired by the human immune system’s ability to recognize and combat pathogens. Uses mechanisms like clonal selection, immune network theory, and negative selection for optimization.
2003
21
Simulated Annealing (SA) [32]
Inspired by the annealing process in metallurgy, which involves heating and controlled cooling of a material to increase the size of its crystals and reduce defects.
1983
22
Gravitational Search Algorithm (GSA) [35]
Based on the law of gravity and mass interactions. Agents are considered as objects, and their performance is measured by their masses. All objects attract each other by the gravity force, and this force causes a global movement of all objects towards the objects with heavier masses.
2009
23
Electro-Magnetism Optimization (EMO) [39]
Inspired by the attraction-repulsion mechanism of electromagnetism theory. Solutions are considered as charged particles that interact with each other based on their charge (quality).
2012
24
Water Cycle Algorithm (WCA) [40]
Inspired by the water cycle process and how rivers and streams flow towards the sea. It simulates the process of evaporation, precipitation, and surface runoff.
2012
25
Black Hole Algorithm (BHA) [41]
Based on the concept of black holes in astronomy. The algorithm uses the attraction of black holes to pull solutions towards them, representing a convergence towards optimal solutions.
2013
26
Physics-based
Multi-Verse Optimizer (MVO) [36]
Inspired by the theory of multiple universes in physics. Solutions are considered as universes, and optimization is achieved by exchanging objects (parameters) between the universes.
2016
27
Thermal Exchange Optimization (TEO) [38]
Inspired by the thermal exchange process, which involves heat transfer principles. The algorithm uses concepts of thermal equilibrium to guide the search for optimal solutions.
2017
28
Equilibrium Optimizer (EO) [42]
Inspired by dynamic and equilibrium states in physics and chemistry. It mimics the process of reaching equilibrium in a system, balancing exploration and exploitation.
2020
29
Spring Search Algorithm (SSA) [33]
Based on the mechanical properties and behaviors of springs. It uses Hooke’s law and the spring force to model the search process.
2020
30
Momentum Search Algorithm (MSA) [34]
Inspired by the momentum in physics. The algorithm uses concepts of velocity and momentum to guide the search process, balancing exploration and exploitation.
2020
31
Lichtenberg Algorithm (LA) [43]
Based on the principles of Lichtenberg figures formed by electrical discharges. The algorithm uses these patterns to explore and exploit the search space.
2021
32
Archimedes Optimization Algorithm (AOA) [37]
Inspired by Archimedes’ principle in fluid mechanics. The algorithm simulates the buoyancy force and how it affects the objects submerged in fluid, guiding the search process.
2021
33
Teaching-Learning Based Optimization (TLBO) [44]
Inspired by the teaching-learning process in a classroom. It mimics the influence of a teacher on learners and the interactions among learners to achieve optimal solutions.
2011
34
Poor and Rich Optimization (PRO) [46]
Inspired by the socio-economic interactions between poor and rich individuals, focusing on wealth redistribution and resource optimization.
2019
35
Gaining Sharing Knowledge Based Algorithm (GSK) [51]
Inspired by the process of gaining and sharing knowledge among individuals, simulating learning and knowledge dissemination to solve optimization problems.
2020
36
Doctor and Patient Optimization (DPO) [47]
Based on the interactions between doctors and patients, aiming to simulate diagnostic and treatment processes to find optimal solutions.
2020
37
Human-based
Coronavirus Herd Immunity Optimizer (CHIO) [50]
Based on the herd immunity concept used in epidemiology, simulating the spread and control of viruses to find optimal solutions.
2021
38
War Strategy Optimization (WSO) [48]
Inspired by military strategies and tactics used in warfare to achieve objectives, focusing on strategic planning and resource management.
2022
39
Ali Baba and the Forty Thieves (AFT) [49]
Inspired by the tale of Ali Baba and the Forty Thieves, using concepts of hidden treasures and strategic planning to solve optimization problems.
2022
40
Mother Optimization Algorithm (MOA) [45]
Inspired by the behavior and strategies of a mother to solve problems and optimize resources for the best outcomes.
2023
41
Hybrid PSO-GA [54]
A combination of Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) to leverage the strengths of both for better optimization.
2016
42
Hybrid GWO-WOA [55]
A hybrid of Grey Wolf Optimizer (GWO) and Whale Optimization Algorithm (WOA) to enhance search capabilities and convergence rates.
2020
43
Hybrid TSA-PSO [57]
Combines Tunicate Swarm Algorithm (TSA) with Particle Swarm Optimization (PSO) to improve exploration and exploitation in the search process.
2022
44
Hybrid GA-PSO-TLBO [56]
An integration of Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Teaching-Learning Based Optimization (TLBO) for robust optimization.
2023
45
WOA-SCSO [52]
A hybrid algorithm combining Whale Optimization Algorithm (WOA) and Sand Cat Swarm Optimization (SCSO) to enhance performance and convergence.
An improved version of the Honey Badger Algorithm (HBA) that incorporates nonlinear chaotic maps to balance exploration and exploitation.
2024
Far and Near Optimization
This section elucidates the theory behind the FNO approach and subsequently delineates its implementation steps through mathematical modeling.
Inspiration of FNO
Metaheuristic algorithms represent stochastic approaches to problem-solving within optimization tasks, primarily relying on random search techniques in the problem-solving domain. Two fundamental principles guiding this search process are exploration and exploitation. In the development of FNO, these principles are harnessed to update the algorithm’s population position within the problem-solving space. Each member of the population undergoes identification of both the farthest and nearest members based on distance calculations. Steering a population member towards the farthest counterpart enhances the algorithm’s exploration capacity for global search within the problem-solving space. Hence, within the FNO framework, a population update phase is dedicated to moving each member towards the farthest counterpart. Conversely, directing a population member towards the nearest counterpart enhances the algorithm’s exploitation capacity for local search within the problem-solving space. Accordingly, within the FNO framework, a separate population update phase is allocated for moving each member towards its closest counterpart.
Algorithm Initialization
The proposed FNO methodology constitutes a population-based metaheuristic algorithm, leveraging the search capabilities of its members within the problem-solving space to attain viable solutions for optimization problems through an iterative process. Each member within the FNO framework derives values for decision variables based on its position in the problem-solving space. Consequently, each FNO member serves as a candidate solution to the problem, represented mathematically as a vector. As a result, the algorithm population, comprising these vectors, can be mathematically represented as a matrix, as indicated by Eq. (1). The initial positions of the population members within the problem-solving space are randomly initialized using Eq. (2).
X=[X1⋮Xi⋮XN]N×m=[x1,1⋯x1,d⋯x1,m⋮⋱⋮⋰⋮xi,1⋯xi,d⋯xi,m⋮⋰⋮⋱⋮xN,1⋯xN,d⋯xN,m]N×mxi,d=lbd+r⋅(ubd−lbd)
Here, X is the FNO population matrix, Xi is the ith FNO member (candidate solution), xi,d is its dth dimension in search space (decision variable), N is the number of population members, m is the number of decision variables, r is a random number in interval [0,1], lbd, and ubd are the lower bound and upper bound of the dth. decision variable, respectively.
In alignment with each FNO member serving as a potential solution to the problem, the objective function of the problem can be assessed. Consequently, the array of evaluated objective function values can be depicted utilizing a matrix, as outlined in Eq. (3).
F=[F1⋮Fi⋮FN]N×1=[F(X1)⋮F(Xi)⋮F(XN)]N×1
Here, F is the vector of evaluated objective function and Fi is the evaluated objective function based on the ith FNO member.
The assessed objective function values provide crucial insights into the quality of population members and potential solutions. The best evaluated objective function value corresponds to the top-performing member within the population, while the worst value corresponds to the least effective member. It should also be explained that since optimization problems fall into two groups of minimization and maximization, the concept of the best and worst value of the objective function is different. In minimization problems, the best value of the objective function corresponds to the lowest evaluated value for the objective function and the worst value of the objective function corresponds to the highest evaluated value for the objective function. On the other hand, in maximization problems, the meaning of the best value of the objective function is the highest value evaluated for the objective function, and the meaning of the worst value of the objective function is the lowest value evaluated for the objective function.
As the position of population members in the problem-solving space is updated in each iteration, new objective function values are computed accordingly. Consequently, throughout each iteration, the top-performing member must also be updated based on comparisons of objective function values. Upon completion of the algorithm’s execution, the position of the top-performing member within the population is presented as the solution to the problem.
Mathematical Modelling of FNO
Within the FNO framework, the adjustment of each population member’s position occurs by moving towards both the farthest and nearest member from itself. The distance between each member and another is computed using Eq. (4).
Di,j=∑d=1m(Xi,d−Xj,d)2,i=1,2,…,N,andi≠j,d=1,2,3,…,m
Here, Di,j is the distance between the ith and jth population members from each other. According to Eq. (4), the highest calculated value for Di corresponds to the farthest member (FMi) and the lowest calculated value for Di corresponds to the nearest member (NMi) for the ith member.
Phase 1: Moving to the Farthest Member (Exploration)
In the initial phase of FNO, the adjustment of population member positions in the problem-solving space is orchestrated by directing each member towards the farthest counterpart. This modeling process instigates significant alterations in the positions of population members within the problem-solving space, thereby enhancing FNO’s capacity for global search and exploration.
The calculation of a new position for each population member is based on the modeling of their movement towards the farthest counterpart, as outlined in Eq. (5). Subsequently, if an improvement in the objective function value is observed at this new position, it supersedes the previous position of the corresponding member, as per Eq. (6).
xi,dP1=xi,d+ri,d⋅(FMi,d−I⋅xi,d),Xi={XiP1,FiP1≤Fi,for minimization problemsXiP1,FiP1≥Fi,for maximization problemsXi,else
Here, FMi is the farthest member for ith population member, FMi,d is the its dth dimension, XiP1 is the new position calculated for the ith population member based on exploration phase of FNO, xi,dP1 is its dth dimension, FiP1 is its objective function value, ri,d are random numbers from the interval [0,1], and I is number which is randomly selected as 1 or 2.
Phase 2: Moving to the Nearest Member (Exploitation)
In the subsequent phase of FNO, adjustments to population member positions within the problem-solving space are made by guiding each member towards its nearest counterpart. This modeling procedure induces subtle alterations in the positions of population members, thereby enhancing FNO’s capacity for local search management and exploitation.
Utilizing the modeling of each FNO member’s movement towards its nearest counterpart, a fresh position for every population member is computed, as specified in Eq. (7). Subsequently, if an enhancement in the objective function value is observed at this new position, it supplants the previous position of the corresponding member, in accordance with Eq. (8).
xi,dP2=xi,d+ri,d⋅(NMi,d−I⋅xi,d),Xi={XiP2,FiP2≤Fi,for minimization problemsXiP2,FiP2≥Fi,for maximization problemsXi,else
Here, NMi is the nearest member for ith population member, NMi,d is the its dth dimension, XiP2 is the new position calculated for the ith population member based on exploitation phase of FNO, xi,dP2 is its dth dimension, FiP2 is its objective function value, ri,d are random numbers from the interval [0,1], and I is number which is randomly selected as 1 or 2.
Repetition Process, Pseudocode, and Flowchart of FNO
The initial iteration of FNO concludes with the updating of all population members through the first and second phases. Subsequently, based on the updated values for both the algorithm population and the objective function, the algorithm progresses to the next iteration, and the updating process persists using Eqs. (4) to (8) until the final iteration is reached. Throughout each iteration, the best solution attained is updated and preserved. Upon the culmination of the FNO execution, the best solution acquired during the algorithm’s iterations is provided as the solution for the given problem. The procedural steps of FNO are depicted in a flowchart in Fig. 1, while its pseudocode is outlined in Algorithm 1.
Flowchart of FNO
Theoretical Analysis of Time Complexity
To understand the efficiency and scalability of the FNO algorithm, we need to analyze its time complexity. The analysis is broken down into the key steps involved in each iteration of the algorithm.
Initialization Phase
Population Initialization: Generating the initial population of N members, each with m decision variables.
Time Complexity:O(N⋅m)
Distance Calculation Phase
Distance Calculation: For each member, distances to all other N−1 members need to be calculated. Each distance calculation involves m dimensions.
Time Complexity:O(N2.m)
Exploration Phase (Moving to the Farthest Member)
Identify Farthest Member: For each member, determine the farthest member from the distance matrix.
Time Complexity:O(N2)
Update Position: For each member, update its position towards the farthest member, which involves m dimensions.
Time Complexity:O(N⋅m)
Evaluate Objective Function: For each member’s new position, compute the objective function.
Time Complexity:O(N)
Exploitation Phase (Moving to the Nearest Member)
Identify Nearest Member: For each member, determine the nearest member from the distance matrix.
Time Complexity:O(N2)
Update Position: For each member, update its position towards the nearest member, which involves m dimensions.
Time Complexity:O(N⋅m)
Evaluate Objective Function: For each member’s new position, compute the objective function.
Time Complexity:O(N)
Iterative Process
The above phases are repeated for T iterations.
Overall Time Complexity
Combining the complexities of each phase, the total time complexity per iteration is:
O(N.m)+O(N2.m)+O(N2)+O(N.m)+O(N)+O(N2)+O(N.m)+O(N)
Simplifying, we get:
O(N2+m)+O(N2)
Since O(N2+m) dominates O(N2) when m ≥ 1, the overall time complexity per iteration is:
O(N2.m)
For T iterations, the overall time complexity becomes:
O(T.N2.m)
The time complexity of the FNO algorithm per iteration is (N2.m), and for T iterations, it is O(T.N2.m). This analysis indicates that the algorithm scales quadratically with the number of population members N and linearly with the number of decision variables m. The number of iterations T also linearly affects the total computational cost.
Population Diversity, Exploration, and Exploitation Analysis
Population diversity in the FNO pertains to the spread of individuals within the solution space, which is crucial for tracking the algorithm’s search dynamics. This metric reveals whether the algorithm’s focus is on exploring new solutions (exploration) or refining existing ones (exploitation). Evaluating FNO’s population diversity allows for an assessment and adjustment of the algorithm’s balance between exploration and exploitation. Several researchers have proposed different ways to define diversity. For instance, Pant introduced a method to calculate diversity using the following equations [58]:
Diversity=1N∑i=1N∑d=1m(xi,d−x¯d)2,x¯d=1N∑i=1Nxi,dwhere N is the population size, m represents the number of dimensions in the problem space, and xi,d is the mean value of the population in the dth dimension. Consequently, the percentages of exploration and exploitation at each iteration can be determined using the following equations:
Exploration=DiversityDiversitymax,Exploitation=|Diversity−Diversitymax|Diversitymax
This subsection delves into the analysis of population diversity, as well as exploration and exploitation, using twenty-three standard benchmarks. These include seven unimodal functions (F1 to F7) and sixteen multimodal functions (F8 to F23), with detailed descriptions available in [59].
Fig. 2 demonstrates the ratio of exploration to exploitation throughout the iterations of the FNO, providing visual insights into how the algorithm navigates between global and local search strategies. Additionally, Table 2 presents the outcomes of the population diversity analysis, alongside exploration and exploitation metrics.
Exploration and exploitation of the FNO
Population diversity, exploration, and exploitation percentage results
Function name
Exploration
Exploitation
Diversity
First iteration
Last iteration
F1
9.4E-165
1
129.6066
1.2E-162
F2
0
1
17.2539
0
F3
0
1
264.3119
0
F4
0
1
211.8951
0
F5
0
1
39.25608
0
F6
0.01236
0.98764
117.865
1.456773
F7
0.072464
0.927536
1.386376
0.100462
F8
5.96E-10
1
1290.08
1.23E-06
F9
4.04E-10
1
10.44909
4.23E-09
F10
1.70E-17
1
46.04662
7.82E-16
F11
3.61E-11
1
730.849
2.64E-08
F12
0
1
78.17809
0
F13
0
1
83.89034
0
F14
2.32E-09
1
32.43996
7.54E-08
F15
4.37E-11
1
2.875723
1.26E-10
F16
0
1
1.629282
0
F17
1.28E-09
1
3.916857
5.01E-09
F18
2.9E-10
1
0.91886
2.67E-10
F19
0.23545
0.76455
0.378556
0.111708
F20
0.167065
0.832935
0.428116
0.071523
F21
8.75E-11
1
3.661421
3.92E-10
F22
2.79E-10
1
2.763374
7.70E-10
F23
5.79E-11
1
3.303417
2.61E-10
The simulation results highlight that FNO maintains high population diversity at the initial iterations, which gradually decreases as the iterations progress. This pattern indicates a shift from exploration to exploitation over time. Furthermore, the exploration-exploitation ratio for FNO frequently aligns closely with 0.00% exploration and 100% exploitation by the final iterations. These findings validate that the FNO effectively manages the balance between exploration and exploitation through its diverse population, ensuring robust performance across the iterative search process.
Simulation Studies and Results
In this section, the performance of the proposed FNO approach to solve optimization problems is evaluated. For this purpose, CEC 2017 test suite [59] for problem dimensions equal to 10, 30, 50, and 100, as well as to address CEC 2020 [60] have been selected.
Performance Comparison
The performance of FNO has been assessed by comparing its results with those of twelve established metaheuristic algorithms: GA, GSA, MVO, GWO, MPA, TSA, RSA, AVOA, and WSO. Table 3 specifies the parameter settings for each competing metaheuristic algorithm. To tackle the challenges posed by the CEC 2017 test suite, FNO and each competing algorithm were subjected to 51 independent runs, each consisting of 10,000·m (where m represents the number of variables) function evaluations (FEs). The optimization outcomes are evaluated using six statistical measures: mean, best, worst, standard deviation (std), median, and rank. The mean values serve as the basis for ranking the metaheuristic algorithms in their performance across the benchmark functions.
Control parameters values
Algorithm
Parameter
Value
GA
Type
Real coded
Selection
Roulette wheel (Proportionate)
Crossover
Whole arithmetic (Probability = 0.8,α∈[−0.5,1.5])
Mutation
Gaussian (Probability = 0.05)
PSO
Topology
Fully connected
Cognitive and social constant
(C1, C2) =(2,2)
Inertia weight
Linear reduction from 0.9 to 0.1
Velocity limit
10% of dimension range
GSA
Alpha, G0, Rnorm, Rpower
20, 100, 2, 1
TLBO
TF: teaching factor
TF = round [(1+rand)]
random number
rand is a random number between [0−1].
GWO
Convergence parameter (a)
a: Linear reduction from 2 to 0.
MVO
Wormhole existence probability (WEP)
Min(WEP) = 0.2 and Max(WEP) = 1.
Exploitation accuracy over the iterations (p)
p=6.
WOA
Convergence parameter (a)
a: Linear reduction from 2 to 0.
r is a random vector in [0−1].
l is a random number in [−1,1].
TSA
Pmin and Pmax
1, 4
c1, c2, c3
Random numbers lie in the range of [0−1].
MPA
Constant number
P = 0.5
Random vector
R is a vector of uniform random numbers in [0,1].
Fish Aggregating Devices (FADs)
FADs = 0.2
Binary vector
U = 0 or 1
RSA
Sensitive parameter
β=0.01
Sensitive parameter
α=0.1
Evolutionary Sense (ES)
ES: Randomly decreasing values between 2 and −2
AVOA
L1, L2
0.8, 0.2
w
2.5
P1, P2, P3
0.6, 0.4, 0.6
WSO
Fmin and Fmax
0.07, 0.75
τ, ao, a1, a2
4.125, 6.25, 100, 0.0005
Evaluation CEC 2017 Test Suite
In this section, we assess the performance of FNO alongside competitor algorithms in addressing the CEC 2017 test suite across problem dimensions of 10, 30, 50, and 100. This test suite comprises thirty benchmark functions, encompassing unimodal, multimodal, hybrid, and composition functions. Specifically, it includes three unimodal functions (C17-F1 to C17-F3), seven multimodal functions (C17-F4 to C17-F10), ten hybrid functions (C17-F11 to C17-F20), and ten composition functions (C17-F21 to C17-F30). Notably, function C17-F2 is excluded from simulation studies due to its erratic behavior. Detailed information on the CEC 2017 test suite is accessible at [59]. The outcomes of applying metaheuristic algorithms to the CEC 2017 test suite are presented in Tables 4 to 7. Additionally, boxplot diagrams illustrating the performance of metaheuristic algorithms across the CEC 2017 test suite are depicted in Figs. 3 to 6.
Optimization outcomes for the CEC 2017 test suite (dimension = 10)
FNO
WSO
AVOA
RSA
MPA
TSA
WOA
MVO
GWO
TLBO
GSA
PSO
GA
C17-F1
Mean
100
4.76E+09
5.66E+06
8.62E+09
3.54E+07
1.47E+09
1.11E+07
5661835.6
8.01E+07
1.30E+08
5.66E+06
5.66E+06
1.57E+07
Best
100
3.95E+09
4.18E+03
7.47E+09
1.12E+04
3.15E+08
4.92E+06
9201.9453
2.52E+04
5.55E+07
1.88E+03
2.08E+03
7.19E+06
Worst
100
6.09E+09
2.05E+07
1.03E+10
1.29E+08
3.20E+09
2.45E+07
20556443
2.91E+08
3.00E+08
2.05E+07
2.05E+07
3.17E+07
Std
0.00E+00
9.78E+08
1.05E+07
1.33E+09
6.57E+07
1.35E+09
9.49E+06
10491366
1.49E+08
1.20E+08
1.05E+07
1.05E+07
1.17E+07
Median
100
4.49E+09
1.04E+06
8.38E+09
6.49E+06
1.19E+09
7.48E+06
1040849.1
1.47E+07
8.22E+07
1.04E+06
1.04E+06
1.18E+07
Rank
1
12
4
13
8
11
6
5
9
10
2
3
7
C17-F3
Mean
300
7420.6332
479.07361
8363.136
1411.7556
9674.4523
1683.6476
477.52226
2813.1747
837.06161
8877.6904
477.47621
12686.724
Best
300
4047.3872
378.72942
4793.4295
793.1808
4003.6941
660.52489
378.7401
1414.8579
523.17669
5583.4037
378.72941
4074.2349
Worst
300
9754.7015
661.5522
11027.212
2543.4504
13486.237
2935.2008
658.24031
5371.423
1100.3784
11990.252
658.13544
19926.992
Std
0
2621.2477
136.59549
2995.4456
849.45664
4234.1132
1145.8495
135.64703
1920.2358
270.08019
2766.3585
135.60386
8740.8537
Median
300
7940.2221
438.00641
8815.9512
1155.1956
10603.939
1569.4324
436.55432
2233.2089
862.34567
8968.5529
436.51999
13372.835
Rank
1
9
4
10
6
12
7
3
8
5
11
2
13
C17-F4
Mean
400
851.10595
404.76442
1203.5993
406.43193
549.69733
421.98551
403.56824
410.66296
408.49557
404.59703
417.90241
413.17988
Best
400
651.04451
401.44571
777.43767
402.45609
466.11707
407.25809
401.74359
405.53194
408.24557
403.4047
400.47988
411.51274
Worst
400
1032.1503
405.9144
1621.7177
411.42675
646.50645
462.49338
404.5371
425.76359
408.89934
405.87998
459.81981
415.97178
Std
0
182.90496
2.3279243
379.14117
4.4747736
93.308293
28.464626
1.3474457
10.590088
0.3171326
1.3582575
29.663767
2.186983
Median
400
860.61449
405.84879
1207.6209
405.92243
543.0829
409.09529
403.99613
405.67816
408.41869
404.55171
405.65498
412.6175
Rank
1
12
4
13
5
11
10
2
7
6
3
9
8
C17-F5
Mean
501.24636
555.44167
538.5083
563.03261
511.94581
555.82717
535.88604
521.16423
512.06635
529.99139
546.87355
524.747
524.84229
Best
500.99515
542.86299
524.28716
550.49922
508.02119
537.72756
520.86599
509.58873
507.90451
524.99816
542.73113
510.14116
520.74543
Worst
501.9917
563.45867
554.4502
575.74225
516.75578
583.08614
566.41609
533.28796
518.73037
533.45311
556.58047
545.00492
530.20678
Std
0.522698
9.996335
16.711565
14.975468
4.803656
21.239825
22.489086
10.343269
4.900485
3.8382191
6.8854589
16.938782
4.4387094
Median
500.99929
557.7225
537.64792
562.94449
511.50313
551.24749
528.13104
520.89011
510.81526
530.75714
544.0913
521.92097
524.20849
Rank
1
11
9
13
2
12
8
4
3
7
10
5
6
C17-F6
Mean
600
627.83962
614.89965
634.91964
601.09529
621.32902
619.90548
601.91366
601.03807
605.94803
614.80197
606.43361
608.85652
Best
600
624.46953
614.04332
632.135
600.6744
612.9704
606.50871
600.44288
600.54902
604.13958
602.6083
601.27146
605.97698
Worst
600
630.70375
617.04936
638.55334
602.12967
634.7137
638.73196
603.7691
601.58336
608.75983
630.97865
616.56372
612.49364
Std
0
3.0631164
1.51176
3.0217425
0.7279899
9.8689709
14.269137
1.5739663
0.4504082
2.2033685
13.816701
7.3085161
3.0264091
Median
600
628.0926
614.25296
634.4951
600.78854
618.81598
617.19063
601.72132
601.00995
605.44636
612.81047
603.94963
608.47772
Rank
1
12
9
13
3
11
10
4
2
5
8
6
7
C17-F7
Mean
711.12673
790.87231
758.69647
791.91696
723.66381
812.56483
755.72192
729.00651
724.8415
747.1352
717.22623
730.60269
734.13897
Best
710.6726
774.70542
741.26705
781.70142
719.51427
779.35164
745.77109
716.90719
716.97424
744.14442
714.88378
724.08601
724.89382
Worst
711.79949
804.92184
781.92139
802.57737
728.56053
847.89195
780.54267
746.61681
740.9555
753.90822
720.2424
740.31988
738.69955
Std
0.538751
13.3631
19.939207
10.430973
4.0096476
31.385407
17.531888
13.208505
11.581012
4.8395511
2.5162099
7.4693104
6.684949
Median
711.01742
791.93099
755.79872
791.69452
723.29023
811.50785
748.28695
726.25102
720.71813
745.24408
716.88936
729.00244
736.48125
Rank
1
11
10
12
3
13
9
5
4
8
2
6
7
C17-F8
Mean
801.49284
842.78691
827.81151
847.14615
812.00355
842.5128
832.29998
811.28633
814.72846
833.4538
818.16944
820.65722
815.53611
Best
800.99496
835.88776
818.14478
837.51314
808.34501
828.61797
817.35463
807.7985
809.78114
827.82098
811.73209
814.88054
812.40758
Worst
801.99121
849.58647
841.64225
851.2519
813.96612
858.65958
842.72897
815.00262
819.28739
839.91548
824.44168
826.30895
821.82891
Std
0.6047211
6.5728026
10.403119
6.8325399
2.7714905
14.035102
11.481443
3.1089481
4.1863009
6.6741068
5.7336457
6.0656766
4.4817738
Median
801.49259
842.83671
825.72951
849.90977
812.85153
841.38683
834.55815
811.1721
814.92264
833.03937
818.25199
820.71969
813.95398
Rank
1
12
8
13
3
11
9
2
4
10
6
7
5
C17-F9
Mean
900
1348.6519
1147.5446
1386.6599
905.22904
1312.4997
1307.8602
901.46319
910.99909
910.9071
900.77674
904.41039
905.15499
Best
900
1225.8335
946.05487
1303.8944
900.31783
1130.4121
1049.9878
900.06388
900.52832
908.35124
900.03731
900.87844
903.24706
Worst
900
1468.9325
1554.1004
1505.7946
913.58498
1558.6494
1547.8733
903.51813
930.53534
917.19838
902.15635
910.61526
907.81342
Std
0.00E+00
115.18992
296.38668
90.403133
6.3213379
195.12276
220.30744
1.7701066
14.808567
4.4564475
1.0457564
4.4872647
2.0242914
Median
900
1349.9208
1045.0116
1368.4753
903.50668
1280.4687
1316.7899
901.13537
906.46634
909.03938
900.45664
903.07393
904.77974
Rank
1
12
9
13
6
11
10
3
8
7
2
4
5
C17-F10
Mean
1006.179
2153.3678
1706.8564
2386.1904
1485.3041
1923.1247
1916.6759
1709.3747
1662.391
2041.4113
2131.4303
1849.4835
1654.1263
Best
1000.2838
1931.3951
1445.7148
2241.2707
1367.416
1706.2859
1428.3869
1425.3817
1492.6378
1698.2813
1894.2644
1510.9421
1398.9345
Worst
1012.6676
2304.0115
2246.6561
2686.2499
1565.7032
2125.5261
2378.2974
2129.834
1905.419
2302.6484
2216.081
2210.7234
2006.2522
Std
7.0021346
174.80971
393.27276
215.14523
93.386807
237.50991
478.77417
353.81086
184.64503
268.32516
166.46054
302.11784
274.62709
Median
1005.8824
2189.0322
1567.5274
2308.6204
1504.0486
1930.3435
1930.0096
1641.1416
1625.7536
2082.3577
2207.688
1838.1342
1605.6593
Rank
1
12
5
13
2
9
8
6
4
10
11
7
3
C17-F11
Mean
1100
3442.4451
1144.6592
3547.5511
1126.4766
4797.7396
1146.7386
1126.8673
1150.3948
1146.6999
1136.7697
1140.4431
2190.5438
Best
1100
2387.5952
1122.7124
1406.7662
1112.5779
4670.2627
1119.2473
1107.5907
1119.7139
1134.9488
1124.9118
1129.015
1121.0143
Worst
1100
4459.2232
1187.6426
5659.4764
1158.0748
4864.4887
1164.8368
1142.8398
1217.0461
1162.6632
1159.8287
1163.3642
5237.4207
Std
0
980.87612
31.04434
2011.8945
22.555088
91.554661
21.933729
17.382694
47.729054
12.391583
16.515448
16.358967
2137.1669
Median
1100
3461.4811
1134.1409
3561.9809
1117.627
4828.1036
1151.4351
1128.5193
1132.4095
1144.5937
1131.169
1134.6967
1201.8702
Rank
1
11
6
12
2
13
8
3
9
7
4
5
10
C17-F12
Mean
1352.9587
3.00E+08
1.03E+06
5.99E+08
5.74E+05
9.75E+05
2.09E+06
9.66E+05
1.29E+06
4.38E+06
9.58E+05
9.84E+04
6.06E+05
Best
1318.6465
6.73E+07
4.24E+05
1.33E+08
1.99E+04
4.61E+05
2.44E+05
1.06E+05
4.17E+04
1.25E+06
5.25E+05
1.22E+04
2.70E+05
Worst
1438.1762
5.25E+08
1.70E+06
1.05E+09
8.98E+05
1.20E+06
3.46E+06
2.75E+06
2.03E+06
7.74E+06
1.47E+06
1.55E+05
1.01E+06
Std
60.273395
2.43E+08
6.48E+05
4.87E+08
406857.01
3.67E+05
1.54E+06
1.27E+06
9.20E+05
3.58E+06
4.24E+05
6.48E+04
3.31E+05
Median
1327.506
3.05E+08
9.92E+05
6.08E+08
6.89E+05
1.12E+06
2.33E+06
5.04E+05
1.55E+06
4.27E+06
9.20E+05
1.13E+05
5.73E+05
Rank
1
12
8
13
3
7
10
6
9
11
5
2
4
C17-F13
Mean
1305.324
1.46E+07
16351.25
2.92E+07
5393.7543
1.16E+04
7.22E+03
6493.1508
9.53E+03
1.50E+04
9.33E+03
6.40E+03
4.70E+04
Best
1303.1138
1.22E+06
3354.2064
2.43E+06
3693.0197
7.23E+03
3.57E+03
2218.3639
6060.3467
1.39E+04
5.04E+03
3.06E+03
8.04E+03
Worst
1308.5079
4.85E+07
27217.996
9.70E+07
6687.0615
1.77E+04
1.36E+04
11302.106
1.33E+04
1.69E+04
1.31E+04
1.50E+04
1.54E+05
Std
2.3907745
2.38E+07
13090.479
4.76E+07
1442.5993
4.69E+03
4.76E+03
5069.6506
3.11E+03
1.39E+03
3.53E+03
6.06E+03
75055.794
Median
1304.8371
4.36E+06
17416.399
8.72E+06
5597.4679
1.07E+04
5.83E+03
6226.0667
9.39E+03
1.45E+04
9.60E+03
3.78E+03
1.30E+04
Rank
1
12
10
13
2
8
5
4
7
9
6
3
11
C17-F14
Mean
1400.7462
3666.7795
1990.1139
4818.0713
1920.6143
3150.8267
1562.4793
1607.447
2266.1706
1623.5515
5003.9687
2818.3796
11294.412
Best
1400
2897.6488
1694.9945
4194.1896
1434.4845
1480.5258
1473.7195
1425.3931
1457.1239
1504.519
4127.3818
1432.3512
3383.6173
Worst
1400.995
4836.903
2620.3688
6080.2525
2909.0859
4962.7505
1755.1245
2135.2382
4661.1276
1818.4585
6864.1109
6034.232
22182.724
Std
0.5233088
893.59526
448.17093
911.81891
727.89443
1884.0922
138.61727
370.16242
1679.565
142.5087
1351.6216
2291.0016
8348.5021
Median
1400.995
3466.2832
1822.5462
4498.9215
1669.4434
3080.0152
1510.5367
1434.5783
1473.2155
1585.6142
4512.1911
1903.4675
9805.6528
Rank
1
10
6
11
5
9
2
3
7
4
12
8
13
C17-F15
Mean
1500.3314
9.25E+03
5008.9488
1.23E+04
3884.4658
6.46E+03
5791.6279
1814.4186
5.45E+03
1.96E+03
2.08E+04
8.15E+03
4372.7758
Best
1500.0007
3.08E+03
2120.6899
2.85E+03
3099.4047
2.50E+03
2286.5546
1655.8502
3394.4168
1.77E+03
1.01E+04
3015.4991
2165.2967
Worst
1500.5
1.59E+04
11312.08
2.64E+04
4717.945
1.10E+04
1.20E+04
1894.8258
6.44E+03
2.10E+03
3.10E+04
1.31E+04
7339.412
Std
0.2476483
5.65E+03
4465.54
1.08E+04
697.42725
3.86E+03
4479.5662
113.62289
1.47E+03
1.58E+02
1.05E+04
4.43E+03
2669.2889
Median
1500.4125
9.00E+03
3301.5127
9.98E+03
3860.2567
6.16E+03
4443.0871
1853.4991
5.98E+03
1.98E+03
2.11E+04
8232.3278
3993.1973
Rank
1
11
6
12
4
9
8
2
7
3
13
10
5
C17-F16
Mean
1600.7598
1959.1164
1786.5679
1962.4531
1679.9885
1988.7089
1906.3858
1792.2412
1717.6287
1673.7548
2010.7068
1883.5623
1780.4292
Best
1600.3563
1897.2686
1650.5592
1801.1325
1636.6174
1824.2438
1749.9153
1708.7038
1614.5751
1651.5043
1896.2632
1798.6825
1715.5748
Worst
1601.1203
2092.2505
1878.5166
2196.4551
1712.2718
2146.7995
2008.6371
1850.9513
1806.3395
1726.1708
2182.6426
2019.2319
1806.6717
Std
0.3323135
94.264386
101.52865
176.48528
33.687183
154.44341
132.44575
63.003168
83.228217
37.071147
135.40165
105.70158
45.79671
Median
1600.7812
1923.4733
1808.5979
1926.1125
1685.5323
1991.8962
1933.4954
1804.6548
1724.8
1658.672
1981.9607
1858.1674
1799.7351
Rank
1
10
6
11
3
12
9
7
4
2
13
8
5
C17-F17
Mean
1700.0992
1805.0034
1747.8063
1805.1308
1734.8402
1791.3669
1825.1395
1825.8888
1762.734
1754.1053
1829.2845
1748.9843
1752.0637
Best
1700.0202
1795.3463
1730.8768
1797.4
1721.078
1775.6183
1765.2283
1768.3914
1722.4174
1743.4809
1743.2184
1740.5002
1747.5758
Worst
1700.3319
1812.321
1783.2541
1810.9758
1774.8212
1798.8067
1872.1579
1924.1523
1857.0616
1766.7008
1943.4015
1757.3931
1760.0577
Std
0.1632193
7.508255
25.241524
5.9769223
28.040026
11.267494
47.777177
77.082108
66.487125
11.988861
106.39812
7.9036705
5.9769379
Median
1700.0223
1806.1731
1738.5472
1806.0737
1721.7308
1795.5213
1831.5858
1805.5057
1735.7284
1753.1197
1815.2591
1749.0219
1750.3107
Rank
1
9
3
10
2
8
11
12
7
6
13
4
5
C17-F18
Mean
1805.3596
2425001.8
11498.264
4834517.4
10813.884
11670.033
21210.616
19208.262
18325.247
26467.103
9679.6755
19996.51
12310.609
Best
1800.0028
126560.07
6431.6407
241579.24
4092.6721
8655.306
6261.9389
8172.5536
5929.291
20914.706
6214.6236
4765.8125
5236.974
Worst
1820.4506
7025868
15314.063
1.40E+07
16096.374
14381.1
31617.441
30679.343
30813.877
32092.843
12379.334
35347.946
17762.149
Std
10.585845
3361898.8
3945.5162
6721890.8
5904.2647
2466.1447
12996.961
10652.537
13273.875
5255.1353
2774.0167
17342.485
5546.9078
Median
1800.4924
1273789.7
12123.675
2532128.6
11533.245
11821.863
23481.542
18990.577
18278.91
26430.432
10062.372
19936.141
13121.657
Rank
1
12
4
13
3
5
10
8
7
11
2
9
6
C17-F19
Mean
1900.4447
3.37E+05
6203.3703
5.98E+05
5261.7861
1.07E+05
3.00E+04
2137.1057
5.08E+03
4.50E+03
3.48E+04
2.17E+04
5757.4089
Best
1900.0392
2.16E+04
2144.7479
3.93E+04
2264.3986
2.10E+03
6903.4991
1916.4041
1940.9054
2.02E+03
1.05E+04
2632.5024
2933.2963
Worst
1901.5593
7.11E+05
12289.95
1.28E+06
8278.5987
2.13E+05
5.51E+04
2675.5494
1.28E+04
1.09E+04
5.01E+04
6.55E+04
8674.2095
Std
0.7832733
3.13E+05
5145.6529
5.90E+05
3387.5079
1.27E+05
20825.106
382.35928
5.45E+03
4.50E+03
1.87E+04
3.11E+04
2503.294
Median
1900.0902
3.08E+05
5189.3919
5.34E+05
5252.0736
1.06E+05
2.91E+04
1978.2347
2.80E+03
2.54E+03
3.93E+04
9.28E+03
5711.065
Rank
1
12
7
13
5
11
9
2
4
3
10
8
6
C17-F20
Mean
2000.3122
2193.3805
2155.6474
2200.1488
2089.2879
2186.8782
2186.2131
2129.352
2155.0998
2072.0504
2226.1675
2154.3058
2053.5693
Best
2000.3122
2143.6293
2042.4575
2149.7325
2070.1713
2099.8897
2099.2859
2049.9965
2119.5065
2060.184
2169.4557
2133.0467
2040.5576
Worst
2000.3122
2250.4329
2258.2612
2252.0631
2120.0696
2280.6586
2252.6492
2218.3412
2224.5441
2080.0928
2310.0411
2178.8048
2058.5614
Std
0
46.269775
102.50676
51.845353
22.640443
79.824499
78.07532
72.579052
49.808882
9.9783738
71.823513
22.507088
9.1631705
Median
2000.3122
2189.7299
2160.9355
2199.3999
2083.4553
2183.4823
2196.4586
2124.5352
2138.1743
2073.9624
2212.5866
2152.6858
2057.579
Rank
1
11
8
12
4
10
9
5
7
3
13
6
2
C17-F21
Mean
2200
2286.3188
2219.027
2264.2827
2255.8574
2313.5537
2300.5565
2252.4169
2303.4731
2291.9248
2350.1747
2308.1456
2290.6345
Best
2200
2246.0953
2211.1318
2227.6262
2253.473
2225.2939
2222.8853
2207.0423
2299.6298
2210.4489
2335.0681
2301.0321
2230.1704
Worst
2200
2308.241
2240.4068
2284.8449
2258.331
2353.3982
2337.7968
2298.616
2308.012
2324.4928
2364.8482
2314.5411
2319.771
Std
0
30.60076
15.023608
26.722181
2.1478202
62.994544
55.126044
54.764002
3.6261999
57.518479
13.059611
7.0419485
42.930214
Median
2200
2295.4694
2212.2847
2272.3299
2255.8127
2337.7613
2320.7719
2252.0047
2303.1254
2316.3788
2350.3912
2308.5046
2306.2983
Rank
1
6
2
5
4
12
9
3
10
8
13
11
7
C17-F22
Mean
2300.0725
2671.6169
2308.1915
2823.6641
2304.8121
2652.0979
2320.7772
2288.4799
2307.8667
2317.1867
2300.5656
2311.8331
2315.79
Best
2300
2565.0503
2305.1527
2646.2284
2300.8833
2426.7632
2316.404
2240.147
2301.1577
2311.8731
2300.0818
2300.6237
2312.8473
Worst
2300.2902
2790.6254
2309.5498
2954.7721
2309.3983
2829.5451
2326.7908
2305.0614
2320.4814
2328.0419
2301.4464
2338.7724
2319.5903
Std
0.1526151
109.42018
2.1466184
136.53847
3.7261119
188.88785
4.8595222
33.897008
9.3491972
7.9192899
0.6602797
18.97897
3.0344778
Median
2300
2665.3959
2309.0318
2846.828
2304.4835
2676.0417
2319.957
2304.3556
2304.9138
2314.416
2300.3671
2303.9681
2315.3613
Rank
2
12
6
13
4
11
10
1
5
9
3
7
8
C17-F23
Mean
2600.9194
2683.5874
2636.8034
2686.5847
2613.1558
2706.0115
2642.4583
2618.2085
2612.6594
2637.2078
2764.1977
2638.6902
2648.7811
Best
2600.0029
2648.1609
2626.7669
2661.7054
2610.8654
2630.489
2627.6187
2607.3298
2607.3883
2628.3281
2709.1096
2632.3491
2632.1783
Worst
2602.8695
2704.5253
2652.2916
2721.4998
2615.8269
2743.4006
2659.4206
2628.4298
2618.7466
2644.9101
2881.5475
2648.4222
2655.6101
Std
1.3888855
27.52311
12.486654
29.241572
2.5543836
53.776504
18.013295
9.6077211
6.2223052
7.8882033
85.342398
7.5422556
11.78494
Median
2600.4025
2690.8317
2634.0776
2681.5667
2612.9654
2725.0781
2641.397
2618.5372
2612.2513
2637.7965
2733.0668
2636.9947
2653.668
Rank
1
10
5
11
3
12
8
4
2
6
13
7
9
C17-F24
Mean
2630.4876
2763.2738
2754.8101
2824.8243
2638.2762
2670.3005
2748.8576
2683.0874
2738.7893
2744.7922
2737.6729
2753.0706
2716.9546
Best
2516.677
2707.9507
2720.7595
2796.5348
2615.172
2555.4111
2715.2998
2517.064
2706.9475
2723.2862
2523.3395
2739.3286
2568.7324
Worst
2732.3195
2839.7566
2778.2909
2885.6158
2653.6271
2800.816
2783.5195
2756.1684
2757.6746
2762.2237
2874.7403
2778.9987
2800.0354
Std
122.54978
62.590977
25.634392
43.197622
18.587946
138.58396
30.415546
117.7466
23.162993
20.387004
158.54861
19.519292
106.9475
Median
2636.4769
2752.6939
2760.095
2808.5732
2642.1528
2662.4874
2748.3054
2729.5587
2745.2676
2746.8295
2776.3058
2746.9776
2749.5253
Rank
1
12
11
13
2
3
9
4
7
8
6
10
5
C17-F25
Mean
2932.6395
3126.3222
2916.7491
3225.6256
2920.4965
3103.9039
2911.691
2924.0714
2938.1804
2933.7873
2924.2164
2925.1207
2949.6981
Best
2898.0475
3048.7781
2901.8847
3168.6477
2916.4387
2911.6293
2790.2716
2906.2796
2923.3805
2916.3464
2907.655
2901.5215
2935.1339
Worst
2945.7929
3298.4504
2948.351
3287.6299
2926.5165
3546.8314
2954.8275
2943.8258
2945.6736
2951.007
2943.4285
2945.3138
2960.0259
Std
24.288727
121.91252
22.397416
51.844628
5.0051593
314.00578
85.191096
21.078025
10.699238
19.613827
19.576532
24.450935
11.29632
Median
2943.3588
3079.0301
2908.3804
3223.1123
2919.5154
2978.5775
2950.8324
2923.0901
2941.8338
2933.8979
2922.891
2926.8237
2951.8163
Rank
7
12
2
13
3
11
1
4
9
8
5
6
10
C17-F26
Mean
2900
3519.927
2991.5403
3652.0496
3018.597
3536.7641
3164.3649
2923.7389
3234.3753
3184.4516
3741.6631
2927.0664
2921.2454
Best
2900
3268.7401
2833.0076
3418.0377
2897.7673
3112.2192
2988.2513
2904.639
2963.3632
2922.3718
2833.0075
2885.9911
2740.6481
Worst
2900
3717.1511
3183.5625
3927.2774
3299.8995
4077.8509
3494.9418
2965.1976
3821.8745
3795.145
4197.7742
2997.3981
3091.0648
Std
3.906E-13
233.15046
194.35472
236.07564
198.30374
481.02655
241.66989
29.339685
415.79741
430.48989
651.23366
51.036749
172.70929
Median
2900
3546.9084
2974.7955
3631.4416
2938.3605
3478.4932
3087.1332
2912.5594
3076.1317
3010.1449
3967.9354
2912.4381
2926.6343
Rank
1
10
5
12
6
11
7
3
9
8
13
4
2
C17-F27
Mean
3089.518
3192.3719
3117.1701
3211.6981
3104.1448
3167.807
3180.9036
3093.0319
3113.8587
3112.9921
3207.3632
3130.842
3151.1992
Best
3089.518
3149.4565
3095.0187
3121.9033
3092.1545
3101.0776
3171.3047
3090.0749
3094.0208
3095.0853
3195.6391
3096.2821
3115.2083
Worst
3089.518
3253.6555
3172.9159
3373.951
3132.8375
3202.3755
3189.5088
3095.322
3169.3913
3159.3994
3224.2975
3175.0172
3200.1882
Std
2.762E-13
46.122103
39.183452
116.90737
20.243137
49.442778
7.8477851
2.4854051
38.982802
32.648572
12.707645
34.9409
37.339828
Median
3089.518
3183.1878
3100.373
3175.4691
3095.7935
3183.8875
3181.4005
3093.3653
3096.0114
3098.7419
3204.7582
3126.0344
3144.7
Rank
1
11
6
13
3
9
10
2
5
4
12
7
8
C17-F28
Mean
3100
3560.0545
3231.458
3692.9092
3216.5335
3528.9783
3274.532
3233.6835
3323.9279
3307.0585
3413.8264
3290.5561
3240.1611
Best
3100
3520.6036
3118.7891
3625.9508
3162.9819
3378.5307
3162.9764
3106.2187
3186.5686
3216.9812
3404.952
3184.3081
3144.2455
Worst
3100
3599.0674
3352.7959
3745.7833
3242.0142
3709.7028
3365.906
3365.4723
3385.2633
3365.6769
3432.4598
3352.9785
3470.0257
Std
0
36.051086
109.17755
55.633329
38.366679
181.80825
105.72227
147.77663
97.083903
71.523896
13.223228
82.542029
162.3039
Median
3100
3560.2735
3227.1236
3699.9513
3230.5689
3513.8399
3284.6228
3231.5214
3361.9399
3322.788
3408.9468
3312.4689
3173.1867
Rank
1
12
3
13
2
11
6
4
9
8
10
7
5
C17-F29
Mean
3132.2407
3307.6202
3270.4257
3347.5646
3201.1466
3229.3398
3325.1795
3200.791
3253.967
3209.2618
3322.6187
3254.7089
3230.1883
Best
3130.0764
3288.2063
3202.7142
3282.2068
3164.9039
3164.9358
3230.6385
3144.9381
3185.1782
3171.0395
3228.5479
3166.5459
3183.9606
Worst
3134.8406
3331.411
3334.4362
3406.4848
3244.032
3284.3083
3445.476
3273.8423
3358.7537
3236.1876
3570.3166
3320.4689
3267.328
Std
2.6112316
22.52269
72.885701
69.440415
37.027887
51.594863
93.984831
56.390324
86.896578
29.649665
174.17385
73.331447
37.989204
Median
3132.0229
3305.4317
3272.2761
3350.7835
3197.8253
3234.0576
3312.3017
3192.1919
3235.9681
3214.9101
3245.8052
3265.9104
3234.7323
Rank
1
10
9
13
3
5
12
2
7
4
11
8
6
C17-F30
Mean
3418.7336
1.94E+06
310177.96
3.17E+06
411838.42
5.81E+05
9.01E+05
317081.66
8.53E+05
1.12E+05
7.23E+05
3.89E+05
1.35E+06
Best
3394.6821
1.23E+06
136759.81
7.88E+05
15956.556
1.81E+05
6.98E+04
8764.3179
3.09E+04
2.73E+04
5.97E+05
9.22E+03
5.32E+05
Worst
3442.9073
2.83E+06
716359.94
4.92E+06
604678.05
1.10E+06
3.26E+06
1.07E+06
1.23E+06
1.54E+05
8.49E+05
7.36E+05
2.95E+06
Std
29.212532
6.95E+05
286649.53
1.83E+06
282891.92
4.13E+05
1.66E+06
527102.97
5.95E+05
6.24E+04
1.12E+05
4.12E+05
1198764.2
Median
3418.6725
1.85E+06
193796.04
3.49E+06
513359.55
5.20E+05
1.37E+05
96977.566
1.07E+06
1.33E+05
7.24E+05
4.05E+05
9.68E+05
Rank
1
12
3
13
6
7
10
4
9
2
8
5
11
Sum rank
36
320
178
352
107
285
240
117
189
192
240
184
199
Mean rank
1.2413793
11.034483
6.137931
12.137931
3.6896552
9.8275862
8.2758621
4.0344828
6.5172414
6.6206897
8.2758621
6.3448276
6.862069
Total rank
1
11
4
12
2
10
9
3
6
7
9
5
8
Optimization outcomes for the CEC 2017 test suite (dimension = 30)
FNO
WSO
AVOA
RSA
MPA
TSA
WOA
MVO
GWO
TLBO
GSA
PSO
GA
C17-F1
Mean
100
2.23E+10
1.07E+08
3.48E+10
1.07E+08
1.52E+10
1.54E+09
1.07E+08
1.51E+09
5.31E+09
1.16E+08
1.29E+09
2.57E+08
Best
100
1.91E+10
17600000
3.09E+10
17600000
9.54E+09
1.15E+09
18189683
2.49E+08
3.34E+09
17600000
17700000
1.46E+08
Worst
100
2.78E+10
3.22E+08
4.27E+10
3.22E+08
2.06E+10
2.1E+09
3.22E+08
4.56E+09
7.77E+09
3.26E+08
4.77E+09
4.52E+08
Std
8.63E-15
4.25E+09
1.52E+08
5.66E+09
1.52E+08
5.52E+09
4.75E+08
1.52E+08
2.15E+09
1.93E+09
1.51E+08
2.44E+09
1.41E+08
Median
100
2.11E+10
44100000
3.27E+10
44100000
1.54E+10
1.45E+09
44437917
6.24E+08
5.06E+09
59500000
1.91E+08
2.15E+08
Rank
1
12
2
13
3
11
9
4
8
10
5
7
6
C17-F3
Mean
300
84750.61
40376.27
64739.13
3625.891
42481.21
198183.4
4193.116
37817.86
31930.31
83495.36
29578.43
143656.5
Best
300
77668.66
23140.75
50782.13
3384.862
40702.15
164739.7
3858.621
33041.33
27571.3
72571.02
21862.74
109023.7
Worst
300
92774.38
51735.07
70079.09
3838.437
44279.7
227261.9
4466.153
42232.35
34669.78
91714.8
37534.37
198536.4
Std
0
7700.572
12819.59
9806.415
252.9757
2021.714
27317.34
310.4686
3966.306
3246.36
9119.259
7411.365
44640.38
Median
300
84279.7
43314.64
69047.65
3640.133
42471.5
200366
4223.844
37998.88
32740.08
84847.8
29458.3
133533
Rank
1
11
7
9
2
8
13
3
6
5
10
4
12
C17-F4
Mean
458.5616
5514.996
513.1326
8362.45
494.9801
3911.395
801.7626
498.0427
561.3617
844.6749
580.6721
605.4398
763.8951
Best
458.5616
3132.861
494.7781
5388.965
488.1533
962.903
747.54
493.4245
510.8723
670.7046
564.4267
514.6421
720.829
Worst
458.5616
7439.73
529.0903
11659.91
509.7619
6450.92
872.2656
505.8455
589.8111
1178.504
601.0365
765.3713
784.5743
Std
0
1877.646
14.8523
2737.94
10.57664
2438.42
60.80812
5.884109
36.48582
238.8188
17.8775
122.1118
30.70809
Median
458.5616
5743.697
514.3309
8200.463
491.0026
4115.879
793.6224
496.4504
572.3818
764.7457
578.6126
570.8729
775.0885
Rank
1
12
4
13
2
11
9
3
5
10
6
7
8
C17-F5
Mean
502.4874
799.4111
698.497
832.5222
578.375
756.4103
781.1346
608.9617
611.0662
736.4978
696.3058
620.0834
679.0406
Best
500.995
781.5667
666.3582
809.5298
561.1385
731.2881
758.4144
593.6933
574.1608
719.1954
681.6443
596.7078
634.9207
Worst
503.9798
819.2639
745.4427
863.1647
595.1056
786.6623
790.1124
640.445
637.2454
756.0871
717.5099
660.8697
734.0958
Std
1.351177
16.2677
37.1795
25.80494
15.42662
28.11175
15.98737
22.36284
32.78049
18.73322
17.98576
29.65458
43.17018
Median
502.4874
798.4069
691.0935
828.6972
578.6279
753.8454
788.0058
600.8543
616.4294
735.3544
693.0346
611.3781
673.573
Rank
1
12
8
13
2
10
11
3
4
9
7
5
6
C17-F6
Mean
600
667.9712
640.0658
670.6346
603.5126
665.5659
664.9208
621.2911
610.7685
637.2181
648.316
640.2583
626.1605
Best
600
666.8004
637.9333
666.6184
602.5929
652.1961
655.6832
611.336
604.2382
631.6804
647.8989
630.4598
619.7779
Worst
600
669.1425
642.6248
675.7851
604.7565
673.1688
669.0234
632.4298
617.1937
646.5213
649.2118
648.8411
629.9802
Std
6.9E-14
1.136042
2.034682
4.578038
1.026945
10.45666
6.564253
10.50084
5.609068
6.910974
0.636862
8.645146
4.715387
Median
600
667.971
639.8525
670.0674
603.3504
668.4493
667.4884
620.6993
610.8211
635.3354
648.0766
640.8662
627.4421
Rank
1
12
7
13
2
11
10
4
3
6
9
8
5
C17-F7
Mean
733.478
1219.29
1091.623
1253.658
839.0586
1156.961
1226.86
845.2525
871.9205
1031.863
943.9246
865.7492
940.7737
Best
732.8186
1174.469
994.7341
1241.523
818.8062
1029.695
1191.149
802.7711
808.0542
958.3184
904.9486
850.421
909.8576
Worst
734.5199
1251.416
1230.863
1274.244
885.4101
1284.733
1297.729
903.1913
908.5046
1097.755
998.8954
884.294
987.8676
Std
0.793172
34.9786
109.401
15.50187
33.08332
115.8256
52.06295
45.25291
46.244
78.41539
42.95978
15.65768
35.03757
Median
733.2867
1225.637
1070.447
1249.433
826.009
1156.709
1209.282
837.5238
885.5616
1035.689
935.9273
864.1409
932.6847
Rank
1
11
9
13
2
10
12
3
5
8
7
4
6
C17-F8
Mean
803.3298
1046.685
932.7806
1078.193
883.0652
1024.811
1000.726
885.379
884.2528
994.1707
942.7767
910.855
963.5851
Best
801.2023
1033.646
907.0712
1060.65
877.565
987.1286
953.1646
859.0602
877.9547
977.7503
922.0261
900.5087
950.1541
Worst
804.1574
1063.417
951.4572
1101.524
889.8239
1112.305
1035.907
910.7902
891.4972
1021.979
965.6326
924.7244
980.5287
Std
1.494596
14.36074
21.0321
21.8794
5.319828
61.97885
37.30537
23.7039
6.199516
20.22222
20.10029
11.32729
16.37696
Median
803.9798
1044.838
936.2969
1075.298
882.436
999.9053
1006.917
885.8327
883.7797
988.4766
941.724
909.0934
961.8287
Rank
1
12
6
13
2
11
10
4
3
9
7
5
8
C17-F9
Mean
900
9461.472
4283.897
9175.401
1131.329
9903.382
9513.784
4815.659
1968.546
5083.842
3656.238
3207.913
1307.076
Best
900
8166.16
3239.019
8920.075
1051.039
6168.601
7339.868
3929.968
1479.72
3783.026
3170.149
1967.19
1095.069
Worst
900
10694.1
4818.392
9287.292
1227.411
13267.34
11269.31
7266.008
2683.006
7532.341
4319.993
4827.528
1450.387
Std
6.9E-14
1103.407
747.843
180.2673
84.8897
3087.953
2069.596
1720.807
614.1961
1791.942
519.4671
1271.33
174.0566
Median
900
9492.814
4539.089
9247.118
1123.433
10088.8
9722.979
4033.331
1855.728
4510
3567.405
3018.468
1341.425
Rank
1
11
7
10
2
13
12
8
4
9
6
5
3
C17-F10
Mean
2293.267
6617.502
5124.519
7195.948
3888.845
6060.538
6006.836
4446.157
4563.693
7212.543
4613.82
4776.043
5708.204
Best
1851.756
6108.489
4529.878
6452.183
3597.103
4864.181
5231.715
4208.484
4105.32
6914.032
4395.01
4544.5
5275.655
Worst
2525.027
6861.894
5538.909
7704.967
4250.525
6543.559
7115.713
4800.919
4825.055
7338.563
4939.68
5194.373
6169.657
Std
315.9698
362.4263
515.0489
560.0898
309.6836
842.0577
870.4231
301.3097
335.4876
211.5766
272.3037
303.5602
437.3793
Median
2398.142
6749.814
5214.646
7313.321
3853.877
6417.205
5839.958
4387.612
4662.199
7298.789
4560.295
4682.649
5693.752
Rank
1
11
7
12
2
10
9
3
4
13
5
6
8
C17-F11
Mean
1102.987
6582.375
1304.218
7680.113
1229.656
4576.876
6846.279
1351.868
2096.195
1921.005
2690.562
1297.058
7990.261
Best
1100.995
5419.241
1248.01
6257.126
1149.5
3455.679
5125.362
1266.483
1363.82
1721.828
2096.592
1220.331
3037.177
Worst
1105.977
7458.404
1385.226
8751.453
1395.854
6740.9
9967.603
1524.43
4043.944
2490.803
3395.82
1419.804
14745.55
Std
2.264257
926.6255
65.02055
1173.611
118.4
1588.081
2250.131
126.0088
1367.808
399.7072
628.0357
90.1088
5250.521
Median
1102.487
6725.927
1291.818
7855.936
1186.634
4055.463
6146.076
1308.28
1488.508
1735.694
2634.918
1274.048
7089.158
Rank
1
10
4
12
2
9
11
5
7
6
8
3
13
C17-F12
Mean
1744.553
5.94E+09
21000000
9.23E+09
3400000
4.29E+09
2.13E+08
12900000
47800000
2.59E+08
1.72E+08
5550000
9880000
Best
1721.81
4.91E+09
6710000
8.23E+09
351000
2.21E+09
57800000
7500000
4640000
1.68E+08
32900000
1710000
6370000
Worst
1764.937
7.55E+09
44900000
1.16E+10
7090000
5.61E+09
4.2E+08
23300000
1E+08
4.45E+08
5.4E+08
9820000
14500000
Std
21.19911
1.19E+09
17400000
1.7E+09
3092785
1.55E+09
1.77E+08
7590000
43900000
1.33E+08
2.58E+08
4450000
3600000
Median
1745.733
5.66E+09
16200000
8.53E+09
3070000
4.67E+09
1.87E+08
10300000
43200000
2.11E+08
57300000
5320000
9290000
Rank
1
12
6
13
2
11
9
5
7
10
8
3
4
C17-F13
Mean
1315.791
4.83E+09
175118.5
8.92E+09
50217.03
1.24E+09
813000
125529.5
687000
74600000
79500
76000
10100000
Best
1314.587
2.35E+09
76007.4
4.68E+09
8027.036
16700000
369000
39921.68
83038.86
51800000
32800
24200
2740000
Worst
1318.646
6.77E+09
269199.3
1.1E+10
152150.9
4.29E+09
1280000
184625.3
2130000
1.1E+08
176000
162000
21800000
Std
2.036813
1.92E+09
94504.53
3.01E+09
72389.4
2.16E+09
477000
73902.61
1030000
26500000
68500
63200
8606555
Median
1314.967
5.1E+09
177633.7
1E+10
20345.07
3.19E+08
802000
138785.5
267000
68300000
54700
58900
7970000
Rank
1
12
6
13
2
11
8
5
7
10
4
3
9
C17-F14
Mean
1423.017
1634752
263366.9
1888993
35552.08
1026183
1912000
51507.84
484503.9
152453.4
1000197
50179.14
1729492
Best
1422.014
1060359
37015.7
1005611
3549.286
766299.7
86898.11
24116.25
31353.6
73631.86
681834.3
5015.86
285566.3
Worst
1423.993
2028265
603690.1
2766761
74662.43
1474661
5738512
77656.23
1038274
207296.8
1514900
102374.1
2914484
Std
0.850137
474385.1
255762.2
913836.5
37974.42
351058.9
2729671
27671.05
537723.5
63352.08
415047.3
49656.46
1284368
Median
1423.03
1725192
206381
1891800
31998.3
931886.7
911295.1
52129.44
434194.2
164442.4
902026.1
46663.32
1858958
Rank
1
10
6
12
2
9
13
4
7
5
8
3
11
C17-F15
Mean
1503.129
2.58E+08
1046649
5.05E+08
1016409
13100000
5272313
1051149
14400000
5350000
1030000
1020000
1821548
Best
1502.462
2.26E+08
44622.61
4.39E+08
7813.061
7470000
306514.5
41863.7
89358.33
1130000
24800
8956.889
402511.6
Worst
1504.265
2.85E+08
3827791
5.57E+08
3801484
28400000
17600000
3821003
53800000
12000000
3810000
3800000
3948137
Std
0.899978
31000000
1951161
61300000
1954080
10700000
8725229
1943692
27700000
4870000
1950000
1950000
1635485
Median
1502.893
2.6E+08
157091.8
5.13E+08
128169.8
8380000
1579864
170865.1
1790000
4150000
139000
131371.7
1467773
Rank
1
12
5
13
2
10
8
6
11
9
4
3
7
C17-F16
Mean
1663.469
3961.074
2849.212
4515.955
2027.496
3077.947
3898.56
2500.936
2463.631
3240.441
3410.79
2790.136
2806.118
Best
1614.72
3677.659
2476.504
3855.314
1764.597
2707.522
3265.791
2299.515
2323.967
3064.406
3240.178
2581.133
2501.688
Worst
1744.118
4201.708
3298.531
5115.564
2246.754
3302.559
4621.969
2712.367
2571.478
3443.361
3556.238
3034.575
3106.607
Std
65.1896
247.91
356.8489
702.0488
228.7552
275.4284
588.9608
186.9542
132.7904
175.877
148.535
235.2757
309.1403
Median
1647.519
3982.464
2810.906
4546.472
2049.316
3150.854
3853.24
2495.931
2479.539
3226.999
3423.372
2772.417
2808.088
Rank
1
12
7
13
2
8
11
4
3
9
10
5
6
C17-F17
Mean
1728.099
3163.114
2374.139
3420.292
1857.178
3049.381
2690.766
2042.076
1917.07
2138.156
2416.428
2257.343
2105.831
Best
1718.761
2648.213
2255.375
3092.244
1754.337
2154.186
2279.176
1990.285
1798.503
1949.912
2322.314
2061.615
2065.827
Worst
1733.659
3784.672
2463.238
3995.396
1919.354
5370.143
2957.458
2174.366
2053.107
2380.795
2549.195
2601.013
2170.049
Std
7.056339
507.3959
94.96416
431.1526
75.62203
1630.205
307.3274
93.28991
127.3899
190.9237
117.1204
256.8487
49.57056
Median
1729.987
3109.785
2388.972
3296.763
1877.51
2336.599
2763.215
2001.827
1908.335
2110.958
2397.102
2183.372
2093.725
Rank
1
12
8
13
2
11
10
4
3
6
9
7
5
C17-F18
Mean
1825.696
24010735
2262488
27603268
28667.7
30692338
5006997
567089.3
381073.5
1432871
461586.4
142846.3
3103457
Best
1822.524
6978171
307329.9
8984696
6797.463
1129661
1683388
141343.3
71377.77
721909.7
249063
110706.7
2471012
Worst
1828.42
46583044
4465379
54200000
70864.66
58117699
10284009
1490237
978877.4
1772554
915305.6
184528.4
4537750
Std
2.842212
18427398
2053628
20164677
31714.99
33278329
3879872
653493.7
449261
510714.7
323244
32458.41
1013281
Median
1825.92
21240861
2138621
23623254
18504.34
31760996
4030295
318388.4
237019.5
1618509
340988.5
138075
2702534
Rank
1
11
8
12
2
13
10
6
4
7
5
3
9
C17-F19
Mean
1910.989
4.9E+08
315910.4
8.26E+08
260410
2.49E+08
12300000
1051513
3660000
5110000
328000
296000
1626779
Best
1908.84
3.67E+08
37353.84
5.97E+08
6343.378
3190000
1681847
242120.3
64465.94
2520000
78500
17479.07
545242
Worst
1913.095
6.39E+08
846283.4
1.25E+09
835727.6
6.88E+08
21000000
1870060
11800000
7290000
872000
852000
2538842
Std
2.03232
1.44E+08
380170.1
3.06E+08
406209.7
3.33E+08
9128459
718481.9
5750000
2510000
385000
399000
878084.5
Median
1911.01
4.78E+08
190002.2
7.28E+08
99784.5
1.52E+08
13400000
1046937
1390000
5320000
181000
158000
1711516
Rank
1
12
4
13
2
11
10
6
8
9
5
3
7
C17-F20
Mean
2065.787
2794.033
2570.338
2839.824
2180.062
2752.443
2741.942
2544.446
2348.901
2708.446
2887.979
2495.248
2434.172
Best
2029.521
2724.894
2432.237
2696.952
2067.884
2627.215
2576.548
2334.178
2187.258
2640.185
2567.061
2450.476
2379.227
Worst
2161.126
2877.872
2756.83
2920.153
2263.673
2867.962
2893.318
2897.838
2502.758
2822.006
3303.539
2604.654
2469.3
Std
66.95098
66.35057
147.2205
105.3712
85.94739
105.5067
140.171
257.4623
135.7586
89.57131
323.8441
76.9979
40.56966
Median
2036.25
2786.684
2546.142
2871.095
2194.346
2757.296
2748.951
2472.884
2352.794
2685.796
2840.658
2462.931
2444.081
Rank
1
11
7
12
2
10
9
6
3
8
13
5
4
C17-F21
Mean
2308.456
2581.072
2426.648
2629.866
2363.126
2506.634
2570.745
2396.176
2383.24
2472.976
2536.603
2421.174
2470.527
Best
2304.034
2501.02
2234.454
2563.105
2354.875
2313.541
2502.956
2366.469
2351.179
2462.634
2521.697
2405.49
2443.098
Worst
2312.987
2635.249
2561.544
2707.006
2374.773
2621.898
2627.027
2421.872
2397.203
2483.079
2568.411
2432.981
2511.61
Std
4.690555
65.79854
144.9785
65.94695
9.025478
142.6196
64.13207
24.40785
22.9312
9.764223
22.54543
13.53134
30.59232
Median
2308.402
2594.01
2455.297
2624.676
2361.427
2545.549
2576.499
2398.182
2392.29
2473.095
2528.151
2423.113
2463.7
Rank
1
12
6
13
2
9
11
4
3
8
10
5
7
C17-F22
Mean
2300
7162.287
5280.339
6955.703
2328.61
7834.465
6670.479
3733.822
2669.869
5206.244
5752.875
4525.201
2668.312
Best
2300
6870.709
2320.359
6076.275
2319.631
7629.459
5848.004
2324.847
2551.481
2683.847
3765.296
2477.509
2598.517
Worst
2300
7630.107
6417.946
7853.968
2344.931
7944.492
7393.994
5492.296
2913.245
7991.475
6619.919
6515.202
2734.765
Std
0
343.238
2078.969
802.5975
11.77052
150.2962
674.6946
1735.435
174.0463
3047.671
1402.711
1956.452
65.708
Median
2300
7074.165
6191.526
6946.285
2324.939
7881.954
6719.959
3559.072
2607.376
5074.827
6313.141
4554.047
2669.984
Rank
1
12
8
11
2
13
10
5
4
7
9
6
3
C17-F23
Mean
2655.081
3120.621
2894.091
3167.58
2653.629
3124.79
2996.974
2731.957
2743.71
2874.386
3613.467
2871.495
2935.55
Best
2653.745
3047.576
2800.027
3122.593
2503.876
3023.532
2845.78
2692.6
2725.441
2856.16
3521.287
2842.408
2909.968
Worst
2657.377
3189.22
3042.441
3234.388
2710.94
3292.322
3081.08
2756.299
2762.972
2915.986
3704.465
2915.759
2990.302
Std
1.739034
69.8468
111.8945
51.22572
105.2408
124.4942
110.0882
28.99956
17.02262
29.74436
101.5155
35.06682
38.7881
Median
2654.6
3122.845
2866.949
3156.669
2699.851
3091.652
3030.518
2739.464
2743.214
2862.698
3614.058
2863.906
2920.966
Rank
2
10
7
12
1
11
9
3
4
6
13
5
8
C17-F24
Mean
2831.409
3251.741
3128.122
3337.445
2881.944
3222.413
3081.473
2900.991
2913.982
3017.974
3293.713
3094.043
3175.646
Best
2829.992
3219.322
3009.504
3260.959
2868.63
3129.054
3025.536
2858.848
2902.383
2997.75
3262.042
3028.998
3094.958
Worst
2832.366
3318.283
3261.316
3469.713
2887.389
3266.805
3104.275
2920.647
2920.445
3048.333
3326.556
3192.255
3243.627
Std
1.205041
47.28937
116.2057
101.4474
9.414219
67.32037
39.42918
29.91368
8.562528
22.57761
30.1828
73.57656
72.82889
Median
2831.64
3234.68
3120.834
3309.554
2885.878
3246.897
3098.041
2912.233
2916.551
3012.907
3293.127
3077.459
3181.999
Rank
1
11
8
13
2
10
6
3
4
5
12
7
9
C17-F25
Mean
2886.698
3795.058
2912.454
4330.432
2897.554
3392.39
3060.626
2913.157
2984.862
3054.509
2986.69
2900.698
3082.708
Best
2886.691
3477.227
2897.797
3817.652
2889.3
3066.182
3026.898
2891.875
2949.8
2955.656
2974.308
2891.93
3066.093
Worst
2886.707
4034.019
2943.728
5017.569
2905.01
3726.284
3081.356
2965.62
3050.147
3169.692
2997.536
2915.561
3093.099
Std
0.008001
244.3369
22.28778
526.0797
8.014929
338.2477
25.55259
37.18806
48.94882
109.2882
10.85877
11.19416
12.21118
Median
2886.698
3834.494
2904.146
4243.254
2897.953
3388.547
3067.125
2897.566
2969.751
3046.344
2987.458
2897.651
3085.819
Rank
1
12
4
13
2
11
9
5
6
8
7
3
10
C17-F26
Mean
3578.65
8427.399
6846.745
8929.917
3106.09
8047.021
7741.671
4675.154
4485.424
5646.393
6971.007
4727.61
4339.296
Best
3559.841
8084.62
5740.92
8207.081
3077.579
7520.993
7104.988
4372.919
4120.433
4456.012
6063.07
3654.509
3983.145
Worst
3607.686
9056.172
7534.017
10172.57
3147.082
8396.486
8512.776
5255.964
5049.506
6742.773
7479.708
6046.245
4734.68
Std
23.9591
478.2391
821.2362
959.3764
30.92081
391.5256
610.0976
436.6787
416.9052
1114.625
679.6726
1186.512
326.029
Median
3573.536
8284.402
7056.021
8670.008
3099.849
8135.303
7674.46
4535.866
4385.879
5693.393
7170.626
4604.844
4319.679
Rank
2
12
8
13
1
11
10
5
4
7
9
6
3
C17-F27
Mean
3207.018
3555.358
3336.878
3688.367
3216.307
3438.126
3398.685
3230.689
3246.474
3304.809
4721.009
3271.16
3426.23
Best
3200.749
3505.772
3263.936
3447.52
3205.531
3322.404
3253.301
3214.567
3237.797
3237.528
4335.987
3237.518
3359.991
Worst
3210.656
3640.619
3401.573
3933.77
3232.666
3651.717
3506.54
3253.049
3260.534
3366.91
5001.347
3307.613
3466.384
Std
4.889133
63.51793
76.12302
219.1523
13.37553
154.1578
114.5444
16.96452
10.30028
56.45407
343.1252
32.15411
48.49125
Median
3208.335
3537.521
3341.002
3686.088
3213.514
3389.191
3417.451
3227.571
3243.781
3307.398
4773.351
3269.755
3439.272
Rank
1
11
7
12
2
10
8
3
4
6
13
5
9
C17-F28
Mean
3100
4571.492
3274.378
5350.776
3229.928
4038.758
3422.379
3266.493
3558.309
3620.368
3493.381
3328.645
3546.101
Best
3100
4357.75
3261.3
5073.8
3209.397
3547.01
3357.018
3222.483
3374.319
3478.399
3419.218
3200.17
3505.195
Worst
3100
4783.471
3294.513
5621.595
3248.433
4558.9
3458.389
3296.655
4007.822
3947.304
3610.44
3498.118
3587.261
Std
2.76E-13
198.0358
16.68602
265.2672
21.99204
489.1332
47.05939
34.06323
317.1353
231.7292
87.1701
135.5622
36.94125
Median
3100
4572.373
3270.849
5353.855
3230.942
4024.562
3437.055
3273.417
3425.548
3527.885
3471.933
3308.145
3545.973
Rank
1
12
4
13
2
11
6
3
9
10
7
5
8
C17-F29
Mean
3353.75
5134.859
4219.761
5323.349
3642.492
4999.19
4867.14
3801.405
3756.488
4373.415
4846.548
4080.165
4182.013
Best
3325.385
4749.269
3922.493
4783.376
3505.854
4538.943
4630.085
3697.795
3680.25
4096.975
4607.025
3906.717
3859.084
Worst
3370.797
5542.685
4405.449
6073.321
3759.512
5756.815
5012.457
3900.874
3865.582
4789.895
5069.516
4292.148
4483.304
Std
20.71115
405.6956
224.2443
667.7618
117.3124
600.4622
173.4909
90.23052
86.93062
312.7738
261.4941
167.2987
293.9022
Median
3359.41
5123.742
4275.55
5218.349
3652.3
4850.501
4913.009
3803.476
3740.06
4303.395
4854.825
4060.897
4192.832
Rank
1
12
7
13
2
11
10
4
3
8
9
5
6
C17-F30
Mean
5007.854
1.22E+09
1621953
2.4E+09
418257.8
33000000
33700000
3036637
5820000
32500000
2330000
643000
1010000
Best
4955.449
8.95E+08
671084.5
1.72E+09
98713
12300000
6880000
670571.7
1300000
17300000
1920000
207000
257000
Worst
5086.396
1.34E+09
2235273
2.65E+09
1116520
76400000
53500000
4870000
15700000
67600000
2930000
1120000
1340000
Std
62.03637
2.25E+08
747641.2
4.74E+08
494240.5
30800000
20500000
1835030
7010000
24800000
508000
491000
540838.7
Median
4994.785
1.31E+09
1790728
2.61E+09
228898.9
21700000
37200000
3303337
3140000
22600000
2240000
624000
1210000
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
Optimization outcomes for the CEC 2017 test suite (dimension = 50)
FNO
WSO
AVOA
RSA
MPA
TSA
WOA
MVO
GWO
TLBO
GSA
PSO
GA
C17-F1
Mean
100
5.09E+10
6.06E+08
7.94E+10
6.03E+08
3.26E+10
7.07E+09
6.01E+08
8.47E+09
1.8E+10
1.5E+10
2.73E+09
9.34E+09
Best
100
4.55E+10
4.52E+08
6.95E+10
4.34E+08
3.03E+10
4.25E+09
4.34E+08
6.1E+09
1.24E+10
1.23E+10
1.43E+09
8.76E+09
Worst
100
5.47E+10
8.23E+08
8.69E+10
8.3E+08
3.49E+10
1.05E+10
8.22E+08
1.16E+10
2.44E+10
1.78E+10
3.28E+09
9.98E+09
Std
0
4.2E+09
1.64E+08
7.95E+09
1.74E+08
2E+09
3.07E+09
1.7E+08
2.41E+09
6.08E+09
2.37E+09
9.24E+08
6.14E+08
Median
100
5.17E+10
5.74E+08
8.06E+10
5.73E+08
3.27E+10
6.77E+09
5.74E+08
8.09E+09
1.77E+10
1.5E+10
3.1E+09
9.32E+09
Rank
1
12
4
13
3
11
6
2
7
10
9
5
8
C17-F3
Mean
300
141527.3
131525.3
141034.2
23403.48
100043.7
204985.5
47205.31
117567.3
90951.38
157919.5
129996.3
229748.6
Best
300
121563.3
103028.2
127704
21390.34
87899.5
156596.8
38144.74
103289.3
70765.19
142410.3
99809.81
192833
Worst
300
162532.8
158172.8
153027
27137.16
107129.6
309337.8
57697.29
131966.1
102641.1
177381.3
166822.8
261745.4
Std
0
18102.03
25848.1
11613.55
2714.699
9130.536
75905.38
8548.613
12326.71
15071.96
17773.61
30859.99
29744.5
Median
300
141006.6
132450.2
141703
22543.21
102572.8
177003.7
46489.6
117506.9
95199.63
155943.2
126676.3
232208.1
Rank
1
10
8
9
2
5
12
3
6
4
11
7
13
C17-F4
Mean
470.3679
12542.52
722.9987
20104.5
583.0711
7122.787
1767.146
609.9954
1343.592
2485.619
2711.873
990.6342
1418.679
Best
428.5127
9765.637
684.1463
13308.16
530.698
5751.399
1184.253
551.7917
1007.244
1437.509
2260.271
684.1991
1217.201
Worst
525.7252
14277.93
763.208
24002.36
643.5572
9154.095
2063.632
692.8009
1632.317
4152.242
2895.976
1680.062
1517.505
Std
52.14539
2125.239
37.61654
5136.568
49.2991
1515.247
421.9472
62.38522
296.6327
1247.318
319.3414
486.8769
145.573
Median
463.6168
13063.25
722.3202
21553.75
579.0147
6792.828
1910.35
597.6944
1367.404
2176.363
2845.623
799.1376
1470.005
Rank
1
12
4
13
2
11
8
3
6
9
10
5
7
C17-F5
Mean
504.7261
1018.134
815.083
1042.679
712.9127
1057.679
897.4574
715.0726
703.9083
933.4549
771.6502
757.314
843.6417
Best
503.9798
989.1953
787.7418
1028.692
642.629
936.5436
864.8231
651.5795
679.1571
896.4815
725.7214
709.7572
816.6328
Worst
505.9698
1050.26
851.704
1051.881
768.4815
1151.525
917.0944
811.8154
729.5904
958.033
800.8448
812.7633
860.8587
Std
1.00206
30.61951
29.21637
11.08712
55.65075
110.1758
25.19615
75.57236
28.35438
28.33957
37.77926
44.50367
22.19255
Median
504.4773
1016.541
810.443
1045.07
720.27
1071.323
903.956
698.4477
703.4429
939.6525
780.0173
753.3677
848.5377
Rank
1
11
7
12
3
13
9
4
2
10
6
5
8
C17-F6
Mean
600
681.0686
651.7962
682.8024
610.9442
676.6633
683.342
632.9514
620.5286
655.1734
649.9146
646.2755
642.0595
Best
600
678.1986
647.5107
681.3601
609.2425
659.2651
679.4761
624.1956
615.416
644.5228
646.4357
644.3562
630.9974
Worst
600
684.9664
656.3546
684.8934
613.7656
690.3687
689.8617
652.695
629.4142
662.1647
652.0972
648.8926
653.2569
Std
0
3.289016
4.470527
1.768662
2.102176
14.68595
4.874904
14.18613
6.60734
8.072209
2.579837
2.029196
9.74892
Median
600
680.5547
651.6598
682.478
610.3843
678.5097
682.0152
627.4576
618.6421
657.0031
650.5628
645.9266
641.9919
Rank
1
11
8
12
2
10
13
4
3
9
7
6
5
C17-F7
Mean
756.7298
1644.188
1538.138
1727.768
1004.287
1552.107
1573.357
1025.318
1035.219
1381.144
1324.448
1147.271
1237.482
Best
754.7543
1624.439
1479.328
1661.121
955.0207
1425.341
1521.481
993.0339
1014.131
1274.033
1181.567
1014.313
1172.963
Worst
758.3522
1669.262
1595.085
1815.38
1045.879
1675.914
1645.534
1052.14
1052.398
1431.386
1434.044
1340.702
1278.86
Std
1.633977
19.58956
51.54233
69.9839
46.19779
123.5724
60.54072
25.99127
18.79231
75.88935
119.2501
148.394
49.20122
Median
756.9065
1641.525
1539.07
1717.286
1008.124
1553.587
1563.206
1028.05
1037.173
1409.579
1341.09
1117.033
1249.052
Rank
1
12
9
13
2
10
11
3
4
8
7
5
6
C17-F8
Mean
805.721
1335.022
1087.274
1358.176
991.921
1349.604
1255.265
1001.485
1011.543
1253.122
1100.663
1030.468
1199.121
Best
802.9849
1289.435
1045.995
1333.933
963.6715
1263.856
1145.543
969.2718
980.1505
1204.615
1092.332
992.3313
1165.187
Worst
810.9445
1370.621
1128.433
1374.243
1020.792
1461.103
1348.702
1059.897
1046.142
1301.303
1114.109
1088.202
1217.854
Std
3.761519
39.34634
49.10596
18.02852
30.90425
88.88505
87.91615
42.07854
31.04849
41.79327
10.66891
46.80177
24.52874
Median
804.4773
1340.016
1087.334
1362.263
991.6103
1336.728
1263.407
988.3863
1009.94
1253.285
1098.105
1020.67
1206.722
Rank
1
11
6
13
2
12
10
3
4
9
7
5
8
C17-F9
Mean
900
30772.55
11638.79
30933.8
3307.786
32254.31
28146.79
16978.94
6276.016
20640.78
9452.087
9133.861
11230.66
Best
900
29589.87
11068.87
29155.41
2204.308
29714.04
26184.67
9372.494
5467.319
15959.14
8609.989
8544.643
9392.944
Worst
900
33603.47
12424.93
32478.36
4604.404
35961.33
32879.46
22239.73
7136.336
24255.87
10210.24
10261.73
12857.08
Std
9.76E-14
2011.086
604.8312
1648.716
1038.164
2833.419
3331.318
6400.267
911.7464
3623.59
708.5189
829.514
1956.424
Median
900
29948.43
11530.67
31050.72
3211.217
31670.93
26761.51
18151.76
6250.205
21174.05
9494.061
8864.535
11336.31
Rank
1
11
7
12
2
13
10
8
3
9
5
4
6
C17-F10
Mean
4347.157
11885.23
7971.508
12907.47
6470.665
10860.4
10866.9
7413.879
8256.326
12729.66
8200.247
7522.295
10801.01
Best
3555.132
11265.41
7838.321
12472.74
5600.523
10351.78
9617.516
6113.926
6409.333
11934.6
7802.553
7184.992
10207.11
Worst
5099.795
12500.49
8226.909
13225.36
6977.814
11746.61
11819.14
8210.155
12933.87
13537.33
9113.913
8288.141
11725.77
Std
678.2324
535.8009
189.0794
338.5087
639.5372
695.6393
974.722
1033.641
3307.629
755.9321
645.8184
541.3844
686.608
Median
4366.851
11887.52
7910.4
12965.9
6652.161
10671.6
11015.47
7665.717
6841.048
12723.35
7942.26
7308.023
10635.57
Rank
1
11
5
13
2
9
10
3
7
12
6
4
8
C17-F11
Mean
1128.435
13428.73
1848.699
18128.57
1554.192
11361.14
4780.523
1817.763
5648.534
4793.872
12425.7
1904.268
20669.07
Best
1121.25
12255.22
1591.828
16024.93
1375.253
10122.77
4255.55
1613.982
3434.277
4370.147
11546.64
1653.027
12131.06
Worst
1133.132
13936.53
2065.526
19612.36
1868.384
13401.8
6145.706
2152.267
9729.479
5567.932
13861.21
2040.1
27411.01
Std
5.725574
830.7024
206.4517
1619.792
240.2695
1519.338
961.3885
249.0541
3058.853
580.8578
1052.101
180.4036
6665.257
Median
1129.678
13761.59
1868.721
18438.49
1486.566
10959.99
4360.417
1752.403
4715.189
4618.704
12147.47
1961.973
21567.1
Rank
1
11
4
12
2
9
6
3
8
7
10
5
13
C17-F12
Mean
2905.102
3.68E+10
1.23E+08
5.99E+10
73347319
2.18E+10
1.17E+09
1.28E+08
8.67E+08
4.31E+09
1.89E+09
1.41E+09
2.33E+08
Best
2527.376
3.09E+10
35784212
4.38E+10
21043151
9.19E+09
9.28E+08
78021765
1.36E+08
2.51E+09
6.25E+08
1.24E+08
1.52E+08
Worst
3168.37
4.4E+10
1.93E+08
8.21E+10
1.26E+08
3.67E+10
1.54E+09
1.7E+08
1.61E+09
8.46E+09
3.29E+09
3.91E+09
3.09E+08
Std
287.9189
6.25E+09
69519444
1.86E+10
54844761
1.2E+10
2.75E+08
39799181
7.74E+08
2.96E+09
1.16E+09
1.88E+09
68262316
Median
2962.331
3.61E+10
1.32E+08
5.69E+10
73044386
2.07E+10
1.11E+09
1.32E+08
8.61E+08
3.14E+09
1.82E+09
8.05E+08
2.36E+08
Rank
1
12
3
13
2
11
7
4
6
10
9
8
5
C17-F13
Mean
1340.1
2.07E+10
22967563
3.63E+10
22855892
8.51E+09
1.03E+08
23045160
3.23E+08
5.15E+08
38436442
4.24E+08
57781476
Best
1333.781
1.19E+10
10465749
1.83E+10
10380127
4.52E+09
70415791
10491036
1.47E+08
4.12E+08
10390647
10407244
33139515
Worst
1343.015
2.83E+10
57463352
5.22E+10
57450695
1.32E+10
1.42E+08
57627665
8.13E+08
6.85E+08
67190110
1.03E+09
97872469
Std
4.504617
7.54E+09
24207126
1.49E+10
24276893
3.87E+09
31368815
24268771
3.44E+08
1.27E+08
32865138
5.08E+08
30094534
Median
1341.801
2.13E+10
11970575
3.73E+10
11796374
8.15E+09
99086365
12030970
1.67E+08
4.82E+08
38082505
3.31E+08
50056960
Rank
1
12
3
13
2
11
7
4
8
10
5
9
6
C17-F14
Mean
1429.458
21935048
1103685
40832413
75140.78
2336716
4089133
234560.7
1043637
802696.4
12831098
557162.3
9515152
Best
1425.995
7214597
461328.4
12574637
7187.684
671663.7
3628898
175729.6
81381.62
647883.2
3034735
247531.8
4719792
Worst
1431.939
42938777
2526862
82665781
143642
3662323
4844757
317872.2
2013790
973866.9
21019463
780001
16391720
Std
2.757393
15864301
1009890
31368165
58601.46
1310208
554053.4
67117.14
829888
173927.3
8566000
238718.4
5194623
Median
1429.95
18793408
713275.6
34044616
74866.7
2506438
3941438
222320.6
1039689
794517.8
13635096
600558.1
8474547
Rank
1
12
7
13
2
8
9
3
6
5
11
4
10
C17-F15
Mean
1530.66
2.2E+09
412884.8
3.52E+09
382787.4
1.44E+09
8737447
483112.1
5391658
59814905
1.67E+08
390025.5
7604082
Best
1526.359
1.55E+09
61784.01
2.75E+09
4653.511
4.94E+08
773089.7
155334.7
38503.39
35033078
200784.3
20819.02
3457491
Worst
1532.953
2.88E+09
1029399
4.18E+09
1004987
3.13E+09
15785805
1045281
14200642
77699681
6.45E+08
1005296
15679424
Std
3.086361
6.54E+08
450699.7
6.63E+08
459134.1
1.29E+09
7059025
410424.3
6500456
18795844
3.36E+08
452809.6
5807911
Median
1531.664
2.18E+09
280178.3
3.58E+09
260754.6
1.06E+09
9195447
365916.5
3663743
63263429
10726021
266993.4
5639706
Rank
1
12
4
13
2
11
8
5
6
9
10
3
7
C17-F16
Mean
2062.891
5699.523
4074.397
6795.786
2725.273
4316.641
5033.187
3209.482
3207.036
4234.201
3738.721
3220.766
3703.39
Best
1728.6
4991.884
3783.19
5235.585
2578.603
3828.207
4171.31
3059.139
2858.826
3933.037
3497.091
2857.985
3172.11
Worst
2242.663
7151.479
4473.244
9942.431
2930.036
4561.304
5641.014
3403.622
3747.804
4476.671
4067.87
3579.751
4137.484
Std
245.0055
1056.398
343.4174
2274.21
180.3051
350.3472
673.6919
154.9074
453.2542
237.3669
298.7512
392.506
455.3884
Median
2140.15
5327.365
4020.577
6002.565
2696.226
4438.528
5160.212
3187.583
3110.756
4263.548
3694.961
3222.665
3751.983
Rank
1
12
8
13
2
10
11
4
3
9
7
5
6
C17-F17
Mean
2021.151
6799.16
3376.901
9690.992
2534.684
3713.112
4198.649
2965.845
2879.747
3874.469
3597.558
3202.179
3399.034
Best
1900.43
5247.89
2986.733
7158.665
2460.411
3036.733
3787.688
2486.158
2747.158
3321.536
3202.359
3008.377
3190.795
Worst
2138.267
8241.347
3822.006
12468.86
2608.129
4126.772
4413.815
3372.908
3137.724
4195.892
3855.625
3509.988
3594.281
Std
141.197
1299.605
408.6522
2301.189
65.01964
495.3898
301.7025
387.896
184.7869
412.1108
297.0178
249.3597
193.4599
Median
2022.954
6853.701
3349.433
9568.22
2535.098
3844.472
4296.546
3002.157
2817.053
3990.224
3666.124
3145.176
3405.53
Rank
1
12
6
13
2
9
11
4
3
10
8
5
7
C17-F18
Mean
1830.62
64170678
2398969
95016900
388871.4
29910759
38445004
2592349
5191221
7279008
7454786
1061435
8349872
Best
1822.239
51126603
854818.5
43147219
77122.78
2729594
10386403
1385179
989904.6
4827815
3425281
370055.8
3591932
Worst
1841.673
75968343
3792345
1.31E+08
756875.1
84998459
69519156
4195966
10355161
10199877
13837502
1870582
19261497
Std
8.567484
11186140
1501254
45876193
367878.7
39748350
30518666
1268250
5140465
2331445
4852989
649626.7
7698194
Median
1829.285
64793884
2474356
1.03E+08
360744
15957491
36937229
2394125
4709909
7044171
6278180
1002551
5273029
Rank
1
12
4
13
2
10
11
5
6
7
8
3
9
C17-F19
Mean
1925.185
2.3E+09
293564
3.24E+09
76414.37
2.25E+09
5841277
4394130
1054868
42804093
455626
406423.6
910549.9
Best
1924.437
1.09E+09
141977
2.18E+09
38408.47
8278426
949492.4
3324575
516413.6
36312372
267716.5
84602.72
731142.5
Worst
1926.121
3.83E+09
533634.9
4E+09
116458.8
6.58E+09
13706250
5422001
1621820
54343028
949546.7
865286.8
1168957
Std
0.832428
1.21E+09
184100.1
8.51E+08
34320.55
3.09E+09
5774541
901093.5
486160.9
8436335
346811
404082.5
213076.8
Median
1925.091
2.13E+09
249322.1
3.38E+09
75395.1
1.21E+09
4354684
4414972
1040620
40280485
302620.4
337902.4
871050.2
Rank
1
12
3
13
2
11
9
8
7
10
5
4
6
C17-F20
Mean
2160.172
3596.52
3121.432
3822.334
2610.593
3264.264
3530.367
3133.198
2580.432
3551.812
3776.141
3140.935
3038.794
Best
2104.423
3294.344
2644.519
3567.011
2353.967
2862.637
3260.237
2919.468
2391.942
3451.526
3560.555
2802.434
2964.285
Worst
2323.891
3739.246
3557.368
3991.747
2855.842
3470.266
4010.517
3534.865
2782.628
3679.721
4018.795
3275.322
3132.011
Std
114.8219
219.8205
409.6954
191.3138
227.0213
287.0385
354.336
296.4387
211.1917
109.2904
198.2779
237.9579
75.44452
Median
2106.186
3676.245
3141.92
3865.289
2616.281
3362.078
3425.356
3039.23
2573.579
3538.001
3762.607
3242.991
3029.441
Rank
1
11
5
13
3
8
9
6
2
10
12
7
4
C17-F21
Mean
2314.895
2901.352
2700.793
2933.962
2441.558
2871.823
2864.119
2546.686
2502.41
2756.99
2774.088
2618.818
2695.968
Best
2309.045
2873.079
2593.82
2841.206
2421.936
2783.96
2763.306
2517.667
2452.306
2734.665
2716.293
2554.049
2672.139
Worst
2329.683
2931.93
2860.697
3007.919
2462.981
3011.56
2947.563
2580.988
2541.145
2795.345
2807.956
2710.19
2712.863
Std
10.40007
32.35623
120.2411
82.81497
22.12068
103.152
82.9268
33.343
39.62193
29.74565
43.62064
71.98111
20.23425
Median
2310.426
2900.199
2674.329
2943.362
2440.657
2845.885
2872.803
2544.045
2508.094
2748.976
2786.051
2605.518
2699.436
Rank
1
12
7
13
2
11
10
4
3
8
9
5
6
C17-F22
Mean
3095.169
13517.11
10270.09
14590.28
5375.373
12468.72
12406.76
8454.284
8348.814
14123.76
10516.29
9096.576
8314.903
Best
2300
13223.15
8237.641
14346.73
2766.066
11968.01
11833.33
6990.985
7321.24
13671.59
10201.28
8365.926
4210.276
Worst
5480.678
13859.02
11642.45
15005.66
7931.411
12950.1
12761.38
9504.468
8943.026
14569.17
11046.42
9620.424
12277.72
Std
1672.91
294.0908
1740.035
316.3445
2899.597
455.9694
440.4211
1117.818
748.7427
406.5688
386.128
603.4836
4779.261
Median
2300
13493.14
10600.13
14504.38
5402.007
12478.39
12516.17
8660.842
8565.494
14127.15
10408.73
9199.976
8385.808
Rank
1
11
7
13
2
10
9
5
4
12
8
6
3
C17-F23
Mean
2743.354
3678.351
3227.937
3743.013
2885.902
3612.825
3614.985
2970.319
2996.687
3219.329
4474.671
3300.686
3288.313
Best
2729.988
3608.53
3150.285
3711.87
2875.409
3430.968
3456.415
2930.449
2922.703
3141.649
4302.852
3240.987
3174.615
Worst
2752.657
3771.603
3295.439
3775.913
2901.01
3911.468
3711.137
3030.987
3125.731
3272.81
4630.927
3358.456
3404.945
Std
10.53657
74.72065
74.92236
28.64086
11.36073
240.0266
117.6759
50.85003
93.49217
58.29603
141.2965
62.70506
98.93037
Median
2745.387
3666.635
3233.012
3742.134
2883.595
3554.432
3646.194
2959.919
2969.158
3231.428
4482.452
3301.651
3286.846
Rank
1
11
6
12
2
9
10
3
4
5
13
8
7
C17-F24
Mean
2919.043
4040.878
3444.876
4276.356
3062.307
3865.277
3715.582
3122
3176.363
3389.432
4187.516
3402.075
3574.021
Best
2909.046
3830.395
3348.258
3867.116
3035.341
3776.542
3618.722
3082.915
3085.5
3321.364
4155.978
3263.134
3537.835
Worst
2924.412
4523.088
3597.584
5292.026
3088.615
3984.917
3769.272
3150.67
3295.284
3433.696
4240.84
3536.062
3652.351
Std
7.178381
340.6766
112.5232
719.1763
26.85258
97.6386
71.20958
32.49476
92.03854
55.18308
42.62542
130.3534
55.47785
Median
2921.358
3905.013
3416.831
3973.141
3062.637
3849.824
3737.168
3127.207
3162.334
3401.334
4176.623
3404.552
3552.948
Rank
1
11
7
13
2
10
9
3
4
5
12
6
8
C17-F25
Mean
2983.145
7838.176
3217.399
10680.59
3123.502
5627.323
4048.426
3112.341
3945.876
4235.21
4153.106
3169.032
3958.382
Best
2980.235
6553.487
3197.082
8685.566
3095.221
4678.998
3712.934
3092.796
3765.987
3810.052
3851.107
3125.101
3856.029
Worst
2991.831
8670.519
3246.205
11924.9
3149.163
6551.606
4300.854
3141.903
4132.868
4750.528
4729
3221.047
4076.149
Std
6.091109
983.0776
23.32322
1596.928
26.5223
845.9211
261.7568
22.85313
201.1658
489.5289
431.76
45.2997
94.9917
Median
2980.257
8064.348
3213.155
11055.94
3124.812
5639.344
4089.958
3107.332
3942.324
4190.13
4016.159
3164.989
3950.674
Rank
1
12
5
13
3
11
8
2
6
10
9
4
7
C17-F26
Mean
3776.432
12756.29
10100.84
13596.61
3571.107
11509.16
12526.2
5673.764
6292.395
9031.66
10588.37
7665.832
8404.252
Best
3748.807
12574.2
9638.146
13042.07
3372.993
9721.084
11715.13
5229.332
5932.46
8361.025
10310.84
7155.502
6827.484
Worst
3793.643
12902.04
10559.79
14430.52
3800.907
12574.27
13975.15
5921.823
6629.391
9657.959
10907.47
8167.723
10456.84
Std
20.46024
154.1112
397.8401
627.9618
199.5097
1305.321
1045.028
330.7332
382.6795
568.6506
266.1505
466.3633
1811.848
Median
3781.639
12774.46
10102.72
13456.92
3555.264
11870.64
12207.26
5771.951
6303.864
9053.828
10567.59
7670.051
8166.342
Rank
2
12
8
13
1
10
11
3
4
7
9
5
6
C17-F27
Mean
3251.26
4597.307
3787.875
4758.891
3389.548
4520.769
4308.156
3371.267
3608.452
3771.179
7405.516
3613.129
4294.924
Best
3227.701
4321.079
3743.694
4433.711
3297.904
3912.01
3812.743
3332.574
3565.304
3607.839
7189.611
3383.523
4195.937
Worst
3313.631
4783.86
3845.23
4989.516
3477.239
4945.493
4801.97
3432.577
3652.8
3917.286
7709.005
3823.361
4421.116
Std
43.8751
214.7351
47.96221
278.8706
77.51737
473.3302
488.8805
45.81443
46.98394
144.1234
264.5516
209.3332
100.0853
Median
3231.854
4642.145
3781.289
4806.168
3391.525
4612.786
4308.956
3359.958
3607.851
3779.794
7361.724
3622.817
4281.322
Rank
1
11
7
12
3
10
9
2
4
6
13
5
8
C17-F28
Mean
3258.849
7999.693
3618.551
10087.69
3412.849
6741.734
4666.63
3356.222
4309.486
5031.277
4870.076
3856.588
4853.245
Best
3258.849
7286.35
3530.54
8976.483
3363.858
5572.379
4128.84
3324.771
4060.516
4508.556
4819.887
3592.894
4623.979
Worst
3258.849
9829.865
3687.232
12979.58
3473.523
7940.82
4887.833
3380.646
4622.49
5520.56
4954.975
4284.702
5021.755
Std
0
1292.085
82.77575
2032.374
48.77597
1256.929
379.6752
25.40913
275.7401
436.0028
65.74432
313.8239
192.5004
Median
3258.849
7441.278
3628.217
9197.354
3407.007
6726.869
4824.923
3359.736
4277.469
5047.996
4852.721
3774.378
4883.623
Rank
1
12
4
13
3
11
7
2
6
10
9
5
8
C17-F29
Mean
3263.038
12211.53
5280.264
17218.32
4078.035
6473.44
8301.687
4712.762
4744.794
6161.912
7563.355
4715.582
5832.508
Best
3247.132
8272.219
5176.352
9397.87
3762.061
6111.434
5772.278
4350.998
4554.733
5377.936
6337.615
4507.282
5567.211
Worst
3278.787
16541.08
5389.738
26882.5
4272.789
6929.208
10702.86
5211.296
5021.55
7014.496
9721.849
4796.785
6336.51
Std
18.36308
4010.782
92.65268
8198.624
249.787
359.2502
2138.111
379.9115
225.4195
806.0822
1606.279
146.3749
380.5694
Median
3263.116
12016.41
5277.482
16296.46
4138.645
6426.558
8365.808
4644.378
4701.447
6127.608
7096.977
4779.13
5713.156
Rank
1
12
6
13
2
9
11
3
5
8
10
4
7
C17-F30
Mean
623575.2
2.77E+09
27440265
4.64E+09
10394169
1.41E+09
1.43E+08
68465410
1.27E+08
2.62E+08
1.65E+08
13055313
58302603
Best
582411.6
2.14E+09
15652470
2.85E+09
5568300
1.84E+08
96319490
65109024
61308682
1.82E+08
1.33E+08
8841014
45665936
Worst
655637.4
3.76E+09
37658691
7.28E+09
15612508
2.84E+09
1.97E+08
72793179
1.88E+08
3.34E+08
2.16E+08
17182093
75102726
Std
34361.91
7.45E+08
11930648
2.01E+09
5137494
1.44E+09
54229872
3441313
67249195
66105583
38154677
4447187
13355368
Median
628125.9
2.59E+09
28224950
4.22E+09
10197934
1.3E+09
1.39E+08
67979719
1.29E+08
2.67E+08
1.55E+08
13099073
56220875
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
Optimization outcomes for the CEC 2017 test suite (dimension = 100)
FNO
WSO
AVOA
RSA
MPA
TSA
WOA
MVO
GWO
TLBO
GSA
PSO
GA
C17-F1
Mean
100
1.44E+11
6.88E+09
2E+11
4.1E+09
1.1E+11
5.65E+10
3.71E+09
5.18E+10
8.05E+10
1.19E+11
2.05E+10
5.09E+10
Best
100
1.41E+11
4.74E+09
1.96E+11
3.53E+09
9.69E+10
5.41E+10
3.23E+09
4.49E+10
7.7E+10
1.09E+11
1.45E+10
4.79E+10
Worst
100
1.48E+11
8.01E+09
2.02E+11
4.64E+09
1.23E+11
6.23E+10
4.19E+09
5.86E+10
8.86E+10
1.27E+11
2.69E+10
5.76E+10
Std
1.22E-14
3.45E+09
1.54E+09
2.67E+09
5E+08
1.14E+10
4.13E+09
4.75E+08
6.85E+09
5.77E+09
8.04E+09
6.74E+09
4.69E+09
Median
100
1.44E+11
7.39E+09
2.01E+11
4.1E+09
1.1E+11
5.48E+10
3.72E+09
5.18E+10
7.82E+10
1.19E+11
2.04E+10
4.91E+10
Rank
1
12
4
13
3
10
8
2
7
9
11
5
6
C17-F3
Mean
300
384186
298493.9
295264.1
156523.7
329756.2
688135.8
416072.2
333655.9
272979.9
312759.1
479049.1
509734.2
Best
300
350275.4
290066.2
287319.9
123438.6
267026.7
603313.2
347605
305316.2
259758.2
292908.6
366718.8
487879.4
Worst
300
402400.3
306321.3
302732
186111.5
371238
794850.4
495687.3
365383.7
285568.6
342328.3
660960.7
527209.5
Std
0
24852.19
7696.093
7277.639
27938.99
47057.12
86182.54
77637.33
33804.21
11196.94
22074.12
142523.5
18726.87
Median
300
392034.2
298794.1
295502.2
158272.3
340380
677189.7
410498.3
331961.9
273296.4
307899.8
444258.5
511923.9
Rank
1
9
5
4
2
7
13
10
8
3
6
11
12
C17-F4
Mean
602.1722
37839.72
1651.056
63568.58
1200.02
13816.82
9551.933
982.5642
4102.694
9383.969
29054.11
2423.824
8091.054
Best
592.0676
34785.2
1585.253
57587.84
1064.838
9120.618
8121.545
897.6428
3164.425
8892.001
23317.13
1745.081
7624.647
Worst
612.2769
41414.12
1744.126
70750.3
1443.04
18403.11
10414.38
1114.888
6139.211
10264.06
32797.07
2928.695
8719.858
Std
12.27362
3012.543
82.17368
5724.531
179.8372
4030.024
1043.687
101.9348
1441.19
675.2766
4831.003
532.1992
529.5449
Median
602.1722
37579.78
1637.422
62968.09
1146.102
13871.78
9835.905
958.8629
3553.569
9189.906
30051.13
2510.759
8009.856
rank
1
12
4
13
3
10
9
2
6
8
11
5
7
C17-F5
Mean
512.9345
1767.627
1206.576
1742.292
1132.853
1894.7
1640.354
1141.542
1097.76
1669.283
1224.8
1289.127
1428.609
Best
510.9445
1750.961
1192.889
1715.996
1028.606
1876.915
1561.127
1047.197
1046.913
1642.942
1197.928
1207.159
1303.202
Worst
514.9244
1774.985
1215.696
1767.463
1203.278
1920.43
1769.561
1202.99
1141.191
1695.079
1250.498
1433.941
1500.84
Std
1.910853
11.82202
10.56836
29.24215
87.4949
22.32009
95.72828
72.99895
43.28702
22.48711
30.99835
112.5505
94.36175
Median
512.9345
1772.281
1208.859
1742.854
1149.764
1890.728
1615.365
1157.991
1101.468
1669.555
1225.388
1257.704
1455.197
Rank
1
12
5
11
3
13
9
4
2
10
6
7
8
C17-F6
Mean
600
689.6817
653.1467
688.2584
633.0622
693.4257
687.6238
663.6087
635.4128
669.048
654.8867
652.7851
654.1288
Best
600
687.8542
649.6552
684.5988
629.7664
683.6872
679.4285
658.4562
630.9869
661.4931
652.5965
647.2492
647.7613
Worst
600
691.4492
656.7168
690.7339
638.0694
700.095
702.2429
668.6172
641.0052
673.8035
658.7463
657.4664
658.6304
Std
0
1.725645
3.034378
2.803539
4.024104
8.374682
10.74671
4.560116
4.570866
6.124423
2.901033
5.078164
5.657866
Median
600
689.7117
653.1075
688.8504
632.2065
694.9603
684.4119
663.6807
634.8296
670.4477
654.102
653.2124
655.0618
Rank
1
12
5
11
2
13
10
8
3
9
7
4
6
C17-F7
Mean
811.392
3177.953
2742.663
3273.316
1726.896
3034.827
3154.236
1862.1
1874.806
2756.23
2777.075
2246.796
2327.245
Best
810.0205
3114.71
2599.214
3206.928
1666.865
2889.325
3055.234
1730.272
1706.931
2645.585
2657.242
2028.783
2245.133
Worst
813.1726
3250.547
2853.958
3339.395
1796.125
3162.623
3287.682
1951.931
1998.794
2855.117
2958.074
2353.691
2496.857
Std
1.537009
58.61408
134.2186
58.32358
58.13667
129.8823
110.9943
98.81629
128.4496
90.26558
135.1524
159.3284
120.8128
Median
811.1874
3173.278
2758.74
3273.47
1722.298
3043.679
3137.014
1883.098
1896.75
2762.109
2746.493
2302.355
2283.495
Rank
1
12
7
13
2
10
11
3
4
8
9
5
6
C17-F8
Mean
812.437
2169.05
1610.61
2213.877
1360.243
2150.434
2084.906
1379.637
1429.943
2031.524
1682.836
1584.837
1851.062
Best
808.9546
2120.901
1558.193
2187.13
1215.212
2096.215
1915.556
1262.946
1336.635
1986.99
1610.983
1544.083
1804.472
Worst
816.9143
2227.386
1638.175
2227.9
1449.629
2231.086
2220.551
1531.47
1553.933
2068.457
1798.92
1665.095
1902.25
Std
3.57488
48.29645
39.03698
19.22737
107.2002
69.0738
163.5185
117.7724
102.9739
36.9778
89.25118
57.45093
44.60344
Median
811.9395
2163.956
1623.037
2220.238
1388.065
2137.218
2101.759
1362.066
1414.602
2035.325
1660.721
1565.085
1848.763
Rank
1
12
6
13
2
11
10
3
4
9
7
5
8
C17-F9
Mean
900
75606.64
23848.01
65144.33
20550.14
100282.5
64748.58
50474.58
31545.16
62845.3
21473.32
29036.77
39702.93
Best
900
68540.34
19720.36
62200.57
18438.24
83422.25
51420.5
43719.63
19544.48
59438.49
19629.93
25707.2
36736.3
Worst
900
86473.41
27118.34
66555.95
21702.74
124006.2
80493.64
56579.01
42874.22
65006.06
23064.5
31565.89
45234.24
Std
9.76E-14
8136.098
3314.988
2122.257
1619.198
17950.48
15062.24
5658.25
11980.49
2508.21
1703.869
3108.69
4011.37
Median
900
73706.4
24276.66
65910.41
21029.78
96850.83
63540.09
50799.84
31880.97
63468.32
21599.42
29437
38420.59
Rank
1
12
4
11
2
13
10
8
6
9
3
5
7
C17-F10
Mean
11023.04
27081
15321.59
28169.92
13607.5
26334.05
25479.1
16159.95
14682.62
28177.85
16351.9
16226.86
23647.68
Best
9625.608
26811.66
13217.89
27488.87
13078.74
25800.97
24818.57
15684.11
13576.61
26967.2
14905.38
14803.74
23040.65
Worst
11858.81
27403.11
17252.62
28636.67
14375.92
27161.91
26709.72
16706.78
15222.35
29116.84
17249.06
17112.11
24175.73
Std
1019.169
290.9045
1887.648
566.6808
615.592
691.8126
897.8666
489.5861
794.5095
950.4212
1134.619
1047.92
490.1947
Median
11303.87
27054.61
15407.93
28277.08
13487.67
26186.66
25194.05
16124.46
14965.75
28313.69
16626.58
16495.79
23687.17
Rank
1
11
4
12
2
10
9
5
3
13
7
6
8
C17-F11
Mean
1162.329
141439.6
58535.87
176075.5
9554.956
59564.29
177662.8
9387.922
77580.04
64849.26
148239.2
48569.92
120600.9
Best
1139.568
110092
52232.38
135088.3
8926.633
30106.52
106407.9
7967.33
64456.83
55491.73
123527.8
24204.24
93273.55
Worst
1220.662
164393.4
69553.09
249209.4
10644.08
83129.01
282965.5
10241.63
87404.87
80216.59
172330.4
93379.56
164482.5
Std
41.06257
24479.09
8172.572
53862.66
791.8939
23060.55
86798.28
1058.982
10329.7
11254.24
21147.87
32228.44
32704.7
Median
1144.542
145636.4
56179.01
160002.1
9324.554
62510.82
160638.9
9671.365
79229.23
61844.35
148549.2
38347.95
112323.7
Rank
1
10
5
12
3
6
13
2
8
7
11
4
9
C17-F12
Mean
5974.805
8.79E+10
1.26E+09
1.42E+11
9.34E+08
4.77E+10
1.16E+10
9.93E+08
1.02E+10
1.88E+10
5.59E+10
9.06E+09
1.09E+10
Best
5383.905
6.27E+10
1.06E+09
1.07E+11
7.55E+08
2.48E+10
9.59E+09
9.25E+08
7.04E+09
1.5E+10
4.86E+10
1.58E+09
1E+10
Worst
6570.199
9.79E+10
1.45E+09
1.66E+11
1.09E+09
7.87E+10
1.33E+10
1.13E+09
1.21E+10
2.55E+10
6.58E+10
1.66E+10
1.25E+10
Std
520.1462
1.77E+10
1.87E+08
2.82E+10
1.53E+08
2.37E+10
1.64E+09
1.03E+08
2.3E+09
5.01E+09
7.53E+09
7.19E+09
1.18E+09
Median
5972.559
9.54E+10
1.27E+09
1.49E+11
9.42E+08
4.35E+10
1.18E+10
9.56E+08
1.08E+10
1.74E+10
5.46E+10
9.02E+09
1.05E+10
Rank
1
12
4
13
2
10
8
3
6
9
11
5
7
C17-F13
Mean
1407.28
2.31E+10
59560970
3.53E+10
59559855
1.77E+10
4.92E+08
59772432
8.42E+08
2.39E+09
7.28E+09
1.52E+09
2.04E+08
Best
1371.145
2.01E+10
5240419
2.74E+10
5182405
1.27E+10
3.25E+08
5387611
72638228
1.61E+09
4.59E+09
3.18E+08
1.74E+08
Worst
1439.935
2.55E+10
1.57E+08
4E+10
1.57E+08
2.12E+10
6.8E+08
1.58E+08
2.23E+09
2.88E+09
9.28E+09
2.65E+09
2.79E+08
Std
36.55441
2.96E+09
73888065
6.14E+09
73970080
3.77E+09
1.79E+08
73932725
1.05E+09
6.34E+08
2.06E+09
1.24E+09
52968231
Median
1409.02
2.33E+10
37850377
3.7E+10
37813725
1.85E+10
4.8E+08
38065123
5.35E+08
2.53E+09
7.62E+09
1.55E+09
1.81E+08
Rank
1
12
3
13
2
11
6
4
7
9
10
8
5
C17-F14
Mean
1467.509
38201209
6130231
66560108
680421.3
7969907
12650912
3116933
8567929
12120333
10124288
1279978
9300795
Best
1458.803
33118529
3830170
60805305
464925.2
3996189
7585099
1140868
5419411
8962357
7816864
830032.6
5248219
Worst
1472.733
43330494
9554680
72582739
928675.2
14752220
17373173
4364593
12844546
15194947
15183338
1880808
13289419
Std
6.359294
4805838
2618143
6085202
201246.1
4944907
4213852
1468003
3423689
3337191
3579954
502025.8
3537634
Median
1469.25
38177905
5568037
66426194
664042.5
6565611
12822688
3481135
8003879
12162014
8748475
1204536
9332772
Rank
1
12
5
13
2
6
11
4
7
10
9
3
8
C17-F15
Mean
1609.893
1.28E+10
31617831
1.95E+10
31594360
1E+10
89585483
31652734
4.47E+08
1.02E+09
1.06E+09
3.08E+08
42053666
Best
1551.154
1.19E+10
2144784
1.39E+10
2123802
3.02E+08
34393185
2177506
29295145
3.32E+08
4.14E+08
2119821
8842930
Worst
1652.294
1.43E+10
94565759
2.43E+10
94553853
1.87E+10
1.36E+08
94661194
1.34E+09
2.13E+09
1.35E+09
1.18E+09
1.02E+08
Std
46.45507
1.14E+09
44996355
5.43E+09
44998644
8.41E+09
55363513
45033640
6.37E+08
8.25E+08
4.64E+08
6.15E+08
43471904
Median
1618.063
1.25E+10
14880390
1.99E+10
14849892
1.05E+10
94043556
14886117
2.1E+08
8.06E+08
1.24E+09
21955245
28501401
Rank
1
12
3
13
2
11
6
4
8
9
10
7
5
C17-F16
Mean
2711.795
16369.04
6592.343
19438.38
5259.269
12745.91
14130.39
6147.395
5737.624
10203.94
9840.933
6054.665
9415.349
Best
2171.69
15349.77
5682.333
15480.72
5142.58
10630.79
11597.34
5486.457
5194.048
9721.722
8561.5
5892.124
8624.359
Worst
3397.326
16864.19
7158.504
21677.54
5338.165
15225.35
15534.26
6517.712
6317.253
11120.18
11297.05
6252.058
10041.05
Std
536.4345
726.6692
681.9347
2945.635
97.52369
1990.874
1877.149
504.3495
623.8206
692.1885
1298.565
156.3137
695.3848
Median
2639.081
16631.1
6764.267
20297.64
5278.166
12563.74
14694.98
6292.706
5719.598
9986.928
9752.591
6037.238
9497.994
Rank
1
12
6
13
2
10
11
5
3
9
8
4
7
C17-F17
Mean
2716.564
3497941
5481.873
6880833
4490.27
181909.1
14759.07
4755.924
5200.646
7885.493
39106.67
5691.714
6568.89
Best
2275.021
1025540
5231.872
1865463
4229.64
9025.842
9292.5
4370.788
4241.736
7719.051
25995.03
5399.483
6474.75
Worst
3429.127
7957748
5802.97
15832349
4769.554
482199.4
24700.64
5169.876
6710.02
8173.115
63170.41
5826.365
6706.455
Std
541.1628
3437545
281.4699
6912225
279.5708
217461.6
7311.942
397.6002
1155.152
214.537
17350.2
207.4288
103.6696
Median
2581.054
2504239
5446.324
4912761
4480.944
118205.6
12521.58
4741.517
4925.414
7824.903
33630.61
5770.504
6547.178
Rank
1
12
5
13
2
11
9
3
4
8
10
6
7
C17-F18
Mean
1903.746
49098228
3025157
86114835
882863.4
13043036
10639498
4759167
9775293
14122618
10435430
6026153
5695366
Best
1881.15
22599719
1920448
33825726
564261.1
5742485
8069360
3230760
3078416
10657313
5252169
4054730
4767197
Worst
1919.921
87758368
3954017
1.56E+08
1249819
25459826
12901749
7503672
15796532
19655583
21881730
8356217
8361237
Std
20.3854
29199061
1114076
54288953
297587.5
9402383
2338341
1999687
5504577
4130297
8230532
2324072
1870331
Median
1906.955
43017412
3113081
77107955
858686.9
10484916
10793441
4151119
10113112
13088787
7303911
5846833
4826515
Rank
1
12
3
13
2
10
9
4
7
11
8
6
5
C17-F19
Mean
1972.839
1.06E+10
25103411
1.86E+10
22948166
4.21E+09
1.34E+08
36516864
3.22E+08
5.78E+08
1.33E+09
2.46E+08
33331507
Best
1967.139
9.37E+09
2452035
1.36E+10
451315
1.88E+09
45635836
8380169
2554023
2.62E+08
2.37E+08
54035041
7405979
Worst
1977.869
1.24E+10
70713595
2.31E+10
68367939
8.39E+09
2.13E+08
90255473
9.67E+08
1.34E+09
2.47E+09
5.52E+08
73738269
Std
4.772401
1.45E+09
33379074
4.11E+09
33344392
3.03E+09
93655716
40315077
4.74E+08
5.42E+08
1.16E+09
2.52E+08
30165409
Median
1973.174
1.03E+10
13624008
1.88E+10
11486704
3.29E+09
1.38E+08
23715906
1.58E+08
3.52E+08
1.31E+09
1.9E+08
26090890
Rank
1
12
3
13
2
11
6
5
8
9
10
7
4
C17-F20
Mean
3192.04
6786.56
5862.115
6997.391
4461.887
6571.54
6582.012
5558.61
5777.372
6752.353
5982.709
5192.615
5942.198
Best
2806.762
6653.788
5645.851
6832.146
4382.099
6077.651
6298.833
5229.125
4674.732
6105.004
5692.855
4515.708
5459.311
Worst
3662.121
7030.393
6042.001
7154.899
4566.542
7310.59
6840.553
5950.009
6630.726
7070.085
6169.483
6009.876
6286.234
Std
462.1717
178.6222
192.5978
146.4021
80.77073
570.1676
243.4149
312.7201
1013.239
464.0537
232.9712
659.0846
400.3169
Median
3149.639
6731.029
5880.305
7001.259
4449.453
6448.959
6594.331
5527.653
5902.014
6917.161
6034.25
5122.437
6011.624
Rank
1
12
6
13
2
9
10
4
5
11
8
3
7
C17-F21
Mean
2342.155
3993.262
3479.831
4095.943
2788.949
3856.443
3941.788
3123.929
2907.373
3513.908
4347.53
3409.695
3273.544
Best
2338.689
3951.57
3303.936
4035.382
2752.489
3730.444
3691.765
3067
2833.288
3382.693
3878.657
3253.382
3238.239
Worst
2346.015
4053.164
3598.823
4144.363
2818.995
3941.854
4136.567
3227.704
2956.224
3667.875
4721.159
3710.202
3316.122
Std
3.54374
51.52115
132.4952
48.78381
28.92288
108.0884
210.7737
75.48026
54.98993
127.4221
370.0533
219.1612
34.93936
Median
2341.959
3984.157
3508.283
4102.012
2792.156
3876.737
3969.409
3100.505
2919.991
3502.532
4395.153
3337.597
3269.907
Rank
1
11
7
12
2
9
10
4
3
8
13
6
5
C17-F22
Mean
11739
29080.23
19541.57
30460.41
18247.88
28243.97
26884.17
17076.2
22096.66
30358.4
20292.51
20926.96
26602.71
Best
11119.08
28903.56
18102.09
29922.55
16873.58
27921.91
26278.75
15966.08
17858.91
30187.28
19389.43
19449.65
25592.13
Worst
12601.6
29194.49
21769.04
30835.49
19511.99
28931.49
27660.32
17517.06
32076.89
30493.67
21165.68
22120.46
26992.6
Std
686.6098
140.794
1698.94
425.363
1300.083
487.2466
679.3289
780.5536
7110.969
134.9744
763.3099
1318.171
709.8329
Median
11617.67
29111.43
19147.58
30541.79
18302.97
28061.24
26798.8
17410.82
19225.42
30376.33
20307.47
21068.87
26913.04
Rank
1
11
4
13
3
10
9
2
7
12
5
6
8
C17-F23
Mean
2877.697
4978.106
3955.4
4979.918
3275.195
5079.131
4822.038
3431.943
3543.955
4039.899
7116.307
4588.425
4083.7
Best
2872.107
4763.585
3890.138
4751.6
3262.976
4441.438
4699.796
3352.089
3514.192
3992.441
6614.835
4146.553
4030.291
Worst
2884.013
5225.31
4025.858
5160.213
3300.136
5945.907
4941.281
3530.708
3585.504
4108.926
7470.267
4822.508
4136.13
Std
5.486704
216.7082
67.17814
177.8853
17.84688
709.191
121.2573
79.212
34.30244
51.81721
407.8773
319.938
62.56197
Median
2877.334
4961.764
3952.802
5003.929
3268.834
4964.59
4823.538
3422.488
3538.061
4029.115
7190.063
4692.32
4084.189
Rank
1
10
5
11
2
12
9
3
4
6
13
8
7
C17-F24
Mean
3327.407
7762.246
5108.848
9421.599
3715.752
6200.975
5955.771
3928.173
4202.363
4587.434
9685.991
5603.383
5111.337
Best
3295.518
6188.448
4910.294
6507.595
3661.71
5797.666
5587.4
3881.303
3987.354
4402.673
9145.69
5299.36
5021.174
Worst
3357.991
8852.361
5251.505
11357.15
3784.675
6464.177
6511.428
4010.43
4395.986
4794.583
11124.92
5995.987
5267.189
Std
31.15784
1337.708
161.4569
2478.136
55.17383
300.6364
424.3766
62.676
223.1889
169.6631
1010.363
333.603
113.4499
Median
3328.059
8004.088
5136.795
9910.828
3708.311
6271.028
5862.128
3910.479
4213.056
4576.24
9236.68
5559.093
5078.492
Rank
1
11
6
12
2
10
9
3
4
5
13
8
7
C17-F25
Mean
3185.232
13549.67
4195.488
18643.97
3813.876
9521.712
6855.864
3580.533
6122.821
8204.563
9990.768
4196.381
7347.412
Best
3137.371
12906.96
3866.822
17330.56
3687.409
8991.715
6321.705
3548.261
5983.044
7186.91
9291.77
3960.044
6736.166
Worst
3261.571
15027.29
4519.012
21554.91
3914.156
9871.17
7203.615
3630.205
6487.094
9589.049
11270.11
4549.077
7986.061
Std
63.01686
1049.542
282.3588
2090.283
98.70775
420.3253
413.4837
38.62969
256.4221
1160.468
927.654
288.2573
675.5625
Median
3170.992
13132.21
4198.059
17845.2
3826.97
9611.981
6949.068
3571.833
6010.574
8021.147
9700.597
4138.202
7333.711
Rank
1
12
4
13
3
10
7
2
6
9
11
5
8
C17-F26
Mean
5757.621
34824.54
22332.59
39818.21
11406.35
29620
30147.17
11581.86
15804.84
21687.1
30064.86
19032.81
20960.61
Best
5645.905
34335.8
19804.5
37641.14
10904.61
28595.15
27300.88
10409.61
14089.45
17906.8
28925.2
17189.52
19696.16
Worst
5844.642
35125.8
24904.48
41259.34
12080.26
30263.45
32684.34
13477.27
17259.74
26307.69
31482.31
20596.5
21766.88
Std
88.27609
373.8791
2293.116
1767.9
607.5807
753.6511
2825.833
1394.721
1406.436
3653.115
1117.408
1512.922
956.8509
Median
5769.969
34918.28
22310.7
40186.18
11320.27
29810.7
30301.74
11220.27
15935.09
21266.95
29925.98
19172.62
21189.7
Rank
1
12
8
13
2
9
11
3
4
7
10
5
6
C17-F27
Mean
3309.493
8431.117
4079.44
10925.66
3549.154
6137.902
5634.58
3624.834
4011.29
4219.569
12415.63
4004.856
5195.937
Best
3278.01
7193.617
3920.903
8328.655
3526.795
5893.316
5032.975
3578.077
3854.802
3991.86
12117.8
3836.301
4982.324
Worst
3344.5
9697.866
4330.141
13644.12
3566.024
6447.772
6306.475
3699.561
4135.916
4596.338
12658.07
4171.816
5523.266
Std
29.83731
1433.942
184.7193
3016.573
17.79592
256.779
713.5143
55.66379
145.5687
281.6278
251.8467
202.401
245.9069
Median
3307.732
8416.493
4033.359
10864.93
3551.899
6105.259
5599.435
3610.849
4027.222
4145.039
12443.32
4005.654
5139.079
Rank
1
11
6
12
2
10
9
3
5
7
13
4
8
C17-F28
Mean
3322.242
18529.7
4871.811
24712.73
4082.497
14143.31
9651.713
3818.942
8720.66
10325.03
16757.36
7349.623
10589.75
Best
3318.742
17221.14
4746.14
22200.13
3949.672
11194.77
8344.877
3719.588
7427.774
8162.806
14548.96
5240.782
9676.89
Worst
3327.816
20913.76
5033.137
27953.61
4234.788
16439.41
10477.89
3878.271
10565.86
12273.48
18311.19
10788.9
11683.62
Std
4.609511
1757.062
125.1428
2550.493
123.1342
2607.096
968.2252
79.32491
1392.117
1986.002
1679.512
2624.736
1102.466
Median
3321.205
17991.95
4853.983
24348.59
4072.764
14469.52
9892.042
3838.955
8444.501
10431.91
17084.65
6684.406
10499.25
Rank
1
12
4
13
3
10
7
2
6
8
11
5
9
C17-F29
Mean
4450.696
155369.8
9044.079
294907.2
6738.552
16472.6
14872.16
8252.835
7934.814
11403.39
21937.29
8222.392
10904.45
Best
4169.151
88915.29
7974.815
158677.9
6012.683
12863.77
12524.18
7480.357
7756.966
10648.75
18265.13
7675.744
10729.29
Worst
4829.521
211686.9
9659.388
409063.8
7395.178
20638.08
16924.57
8810.071
8207.92
11920.19
28467.68
8950.622
11303.46
Std
297.0014
54987.89
775.3449
112240.6
596.7075
3414.595
2274.751
600.637
206.4733
571.55
4993.753
635.7098
284.6627
Median
4402.056
160438.5
9271.057
305943.5
6773.173
16194.27
15019.95
8360.456
7887.186
11522.31
20508.17
8131.601
10792.53
Rank
1
12
6
13
2
10
9
5
3
8
11
4
7
C17-F30
Mean
5407.166
1.96E+10
1.4E+08
3.18E+10
1.21E+08
1.14E+10
1.38E+09
2.04E+08
1.66E+09
3.3E+09
6.29E+09
6.25E+08
6.76E+08
Best
5337.48
1.72E+10
64490286
2.97E+10
53136534
7E+09
1.08E+09
1.34E+08
6.81E+08
1.35E+09
4.54E+09
1.72E+08
6.06E+08
Worst
5557.155
2.12E+10
1.94E+08
3.43E+10
1.55E+08
1.41E+10
1.83E+09
2.59E+08
2.17E+09
5.94E+09
7.62E+09
1.7E+09
7.52E+08
Std
106.4092
1.77E+09
58117376
2.06E+09
48957562
3.24E+09
3.38E+08
56208885
7.04E+08
2.44E+09
1.37E+09
7.58E+08
73496112
Median
5367.014
1.99E+10
1.52E+08
3.15E+10
1.38E+08
1.22E+10
1.3E+09
2.1E+08
1.89E+09
2.95E+09
6.49E+09
3.14E+08
6.73E+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 diagrams of FNO and competitor algorithms performances on CEC 2017 test suite (dimension = 10)
Boxplot diagrams of FNO and competitor algorithms performances on CEC 2017 test suite (dimension = 30)
Boxplot diagrams of FNO and competitor algorithms performances on CEC 2017 test suite (dimension = 50)
Boxplot diagrams of FNO and competitor algorithms performances on CEC 2017 test suite (dimension = 100)
According to the optimization outcomes, when tackling the CEC 2017 test suite with problem dimensions equal to 10 (m = 10), the proposed FNO approach emerges as the top-performing optimizer for functions C17-F1, C17-F3 to C17-F21, C17-F23, C17-F24, and C17-F26 to C17-F30. In the case of problem dimensions equal to 30 (m = 30), FNO proves to be the foremost optimizer for functions C17-F1, C17-F3 to C17-F22, C17-F24, C17-F25, and C17-F27 to C17-F30. Likewise, for problem dimensions of 50 (m = 50), FNO exhibits superiority as the primary optimizer for functions C17-F1, C17-F3 to C17-F25, and C17-F27 to C17-F30. Lastly, for problem dimensions of 100 (m = 100), FNO stands out as the premier optimizer for functions C17-F1, and C17-F3 to C17-F30.
The optimization outcomes highlight FNO’s exceptional ability in exploration, exploitation, and maintain a harmonious balance between these strategies throughout the search procedure, resulting in the discovery of viable solutions for the benchmark functions. Examination of the simulation results reveals FNO’s superior performance in tackling the challenges presented by the CEC 2017 test suite when compared to its competitors. FNO consistently outperforms other algorithms across a significant portion of the benchmark functions, earning the top rank as the most effective optimizer.
In addition to comparing the proposed approach of FNO and competing algorithms using statistical indicators and boxplot diagrams, it is valuable to present a comparison analysis of the computational cost and convergence speed between FNO and other competing metaheuristic algorithms. For this purpose, the results obtained from the analysis of the computational cost between FNO and competing algorithms in handling the CEC 2017 test suite for dimensions 10, 30, 50, and 100 are reported in Table 8. In order to analyze the computational cost of each algorithm in handling each benchmark function, the average execution time of each algorithm (in seconds) in different implementations on a benchmark function has been used. It should be mentioned that the experiments have been implemented on the software MATLAB R2022a using a 64-bit Core i7 processor with 3.20 GHz and 16 GB main memory. In fact, the results reported in Table 8 show that each algorithm needs a few seconds on average to run an implementation on each of the benchmark functions. The findings show that FNO has a lower computational cost compared to competing algorithms in most of the benchmark functions.
Computational cost (in seconds) results of CEC 2017 test suite
F
D
FNO
AVOA
WSO
RSA
MPA
TSA
WOA
MVO
GWO
TLBO
GSA
PSO
GA
C17-F1
10
0.943638
1.583096
4.024191
6.405592
3.911916
1.303047
1.162363
2.05708
1.372409
4.741238
3.213548
1.517339
2.037542
30
1.265738
1.413261
1.846767
16.48146
4.009595
1.943186
1.343995
2.564344
2.286843
5.168074
9.150027
1.627347
1.886391
50
1.99737
2.089604
2.211443
28.36328
5.273381
2.952156
2.034547
3.944988
3.213452
6.750656
6.569285
2.657902
2.457211
100
4.51712
4.579711
4.271255
56.39666
11.53558
6.761015
4.556632
8.365783
7.24913
14.96532
13.36212
4.896176
5.327995
C17-F3
10
0.929089
1.04286
1.383004
6.16697
2.937675
1.256226
1.088733
1.522373
1.285341
4.677143
2.703576
1.259734
1.515914
30
1.225901
1.391617
1.902824
16.514
3.922917
1.97258
1.305722
2.236516
2.229595
5.213546
4.488007
1.593242
1.849118
50
1.859628
1.994077
2.301813
30.99464
5.280301
2.970793
1.870235
3.278145
3.204619
6.799276
6.592748
2.194544
2.317156
100
4.414052
4.482771
4.211579
54.33139
11.43952
6.747196
4.550615
7.193599
7.37274
14.72922
13.19979
4.901213
5.146623
C17-F4
10
0.915769
1.032237
1.384822
6.208042
2.976414
1.237309
1.08464
1.630819
1.291011
4.61799
2.657587
1.327702
1.472259
30
1.185222
1.349146
1.858214
16.34037
3.842284
1.959314
1.291446
2.585777
2.137474
5.276378
4.353586
1.637337
1.748408
50
1.870192
2.035837
2.467055
27.98853
5.349814
3.049928
1.863984
3.835268
3.194857
6.76349
6.626663
2.163599
2.298814
100
4.405351
4.519008
4.428645
55.26566
11.48673
6.761333
4.611475
8.24082
7.379116
14.90168
13.33273
5.161936
5.048013
C17-F5
10
0.933385
1.04602
1.381088
6.273573
3.015628
1.323984
1.152157
1.683115
1.379372
4.862399
2.760354
1.345287
1.582547
30
1.377501
1.523034
1.934754
16.30239
4.219738
2.135077
1.482399
2.743176
2.414575
5.709299
4.668204
1.765746
1.928986
50
2.301316
2.413232
2.5762
27.62056
5.81124
3.37138
2.246185
4.144463
3.478511
7.676686
6.844373
2.4271
2.607201
100
5.030091
5.138763
4.951171
56.21442
12.58825
7.288918
5.108913
8.77272
7.909036
16.22265
14.20052
5.458182
5.584455
C17-F6
10
1.126398
1.294872
1.829421
6.466014
3.375569
1.507921
1.328946
1.897401
1.59733
5.499674
2.886424
1.536365
1.738574
30
1.912489
2.10214
2.624145
16.83683
5.491943
2.717874
2.070499
3.427208
2.927015
7.549411
5.089443
2.314395
2.519413
50
3.288581
3.346758
3.172633
28.89353
7.869561
4.273647
3.156906
5.138662
4.504524
10.65626
7.780686
3.491476
3.640336
100
6.786919
6.796686
5.996134
58.8152
16.60051
9.297162
7.13046
10.77563
9.977606
22.16249
15.7976
7.380006
7.599076
C17-F7
10
1.012659
1.156141
1.604793
6.439517
3.212544
1.371907
1.213753
1.751752
1.444604
5.031541
2.8185
1.333314
1.606219
30
1.525872
1.707508
2.245283
16.51654
4.44808
2.201025
1.568279
3.016487
2.40867
5.808502
4.540416
1.904321
1.982735
50
2.335053
2.437017
2.555998
27.61007
6.153847
3.377765
2.333907
4.45428
3.612121
7.827493
6.877402
2.499854
2.656767
100
5.194727
5.224492
4.700902
57.56033
12.73284
7.407609
5.195627
8.873078
7.999521
16.81509
13.96065
5.501317
5.710446
C17-F8
10
0.992334
1.20789
1.947352
6.283318
3.087241
1.317752
1.171157
1.698311
1.397969
5.10731
2.722496
1.318552
1.641561
30
1.430777
1.705661
2.628195
16.44981
4.313289
2.171037
1.536237
2.78404
2.318408
5.805171
4.630729
1.834799
1.966238
50
2.220374
2.331793
2.502783
27.35975
6.141061
3.30834
2.329043
4.238619
3.583135
7.823168
6.965961
2.488757
2.776774
100
5.140905
5.165027
4.625524
57.47095
12.79873
7.492479
5.23961
8.872122
7.956285
16.68273
13.92364
5.600977
5.804097
C17-F9
10
1.031931
1.161275
1.550988
6.311677
3.158561
1.38957
1.236032
1.957807
1.41982
4.984022
2.833075
1.323421
1.669605
30
1.39125
1.73815
2.953635
16.50098
4.362066
2.203161
1.566925
2.798778
2.39086
5.963327
4.527136
1.796131
1.967704
50
2.147766
2.310723
2.696847
27.50515
6.138456
3.380792
2.297848
4.362373
3.646226
7.813568
6.872742
2.473953
2.66579
100
4.996202
5.033284
4.563522
58.32623
12.75466
7.426297
5.195485
8.842833
7.982558
16.55273
14.04435
5.491215
5.636551
C17-F10
10
0.961296
1.08633
1.467542
6.264981
3.145646
1.375752
1.164882
2.280881
1.429785
5.111629
2.787736
1.357167
1.617746
30
1.393677
1.614085
2.323308
16.43479
4.532878
2.317967
1.623748
2.898713
2.461401
6.234178
4.604056
2.097938
2.083909
50
2.190106
2.33686
2.652936
27.66271
6.422074
3.509917
2.410516
4.683054
3.761256
8.420045
7.178027
2.707651
2.9727
100
4.637676
4.834292
5.047025
61.24723
13.7027
7.979256
5.674736
9.529146
8.458111
18.24622
14.36243
5.994234
6.183812
C17-F11
10
0.952038
1.066553
1.406832
6.37464
3.175761
1.320965
1.136778
1.697331
1.360972
4.933614
2.734688
1.363847
1.632728
30
1.31211
1.529566
2.237064
16.33456
4.090852
2.100505
1.409705
2.371073
2.316734
5.643277
4.387027
1.736832
1.899494
50
1.94532
2.083396
2.395045
27.85837
5.617448
3.130142
2.038194
3.732305
3.37602
7.292931
6.67387
2.368867
2.68521
100
4.482023
4.58193
4.427082
55.82035
12.13102
7.270974
4.877049
7.426876
7.748179
15.63841
13.65734
5.176525
5.608402
C17-F12
10
1.030992
1.123171
1.364338
6.209154
3.06695
1.315664
1.125769
1.800704
1.385309
4.994331
2.748337
1.39018
1.621577
30
1.364377
1.602179
2.384595
16.48504
4.331025
2.173524
1.515228
2.748809
2.311146
5.874035
4.615434
1.906525
1.960998
50
2.194549
2.32554
2.578012
27.51985
6.062683
3.312017
2.223965
4.210764
3.558431
7.861787
6.9194
2.654364
2.802153
100
4.967953
5.028629
4.656741
58.48422
12.85632
7.538511
5.260312
9.031602
8.065777
16.70683
13.88842
5.618708
5.96717
C17-F13
10
1.014724
1.13616
1.49637
6.281446
3.215371
1.383178
1.193222
1.937149
1.418381
5.075544
2.789648
1.404257
1.615838
30
1.270571
1.435383
1.937448
16.42227
4.225391
2.156006
1.479912
2.704063
2.392655
5.71325
4.523365
1.798766
1.922749
50
2.375259
2.449219
2.451216
27.28862
5.763
3.217029
2.114405
4.242541
3.474091
7.502669
6.748075
2.515707
2.58898
100
4.511082
4.617708
4.486141
56.90503
12.27883
7.24645
5.015775
8.769536
7.846623
15.83409
13.85038
5.246276
5.386116
C17-F14
10
1.073627
1.182944
1.487391
6.304518
3.135322
1.391854
1.196807
1.848184
1.409921
5.105715
2.801147
1.424332
1.705614
30
1.519454
1.63361
1.902259
16.5747
4.695981
2.306656
1.653657
2.725056
2.794987
6.228647
4.771402
2.025846
2.116047
50
2.215228
2.361414
2.672111
27.36955
6.409105
3.549084
2.426521
4.2535
3.838332
8.41478
7.047098
2.838305
2.888709
100
4.993744
5.133095
5.072713
57.31802
13.72988
7.933819
5.692025
9.126907
8.546136
18.14142
14.38219
6.094356
6.209456
C17-F15
10
0.96446
1.061595
1.330819
6.31015
3.058498
1.313433
1.155879
2.205131
1.345113
4.878095
2.754639
1.35315
1.533785
30
1.336478
1.448251
1.730002
16.25109
4.082402
2.056317
1.513005
2.611331
2.442912
5.436479
4.393152
1.74878
1.871535
50
1.94656
2.083208
2.388985
26.89541
5.60423
3.125149
2.026338
4.000995
3.399974
7.227762
6.804952
2.329487
2.568064
100
4.367917
4.495948
4.467709
56.88235
11.92176
7.097925
4.797472
8.441369
7.671604
15.40191
13.51557
5.141028
5.269202
C17-F16
10
0.982363
1.092563
1.411833
6.354393
3.09145
1.345011
1.152327
2.155294
1.374953
4.979615
2.775434
1.379423
1.580402
30
1.429814
1.534563
1.776673
16.5913
4.26574
2.150607
1.613688
2.761878
2.422901
5.790602
4.47891
1.835975
1.976617
50
2.102965
2.229841
2.477186
27.28086
5.838524
3.253143
2.164184
4.341446
3.481773
8.050005
7.06325
2.472514
2.643974
100
4.820672
4.908906
4.665472
55.32919
12.51733
7.465189
5.14283
8.809536
7.929018
16.17491
13.82993
5.445556
5.693101
C17-F17
10
1.221679
1.317902
1.551661
6.445206
3.482107
1.566034
1.415488
2.318922
1.587765
5.741628
2.99679
1.547552
1.794696
30
2.015923
2.107086
2.222342
17.03146
5.538777
2.733683
2.095749
3.3172
2.935717
7.503896
5.352274
2.447915
2.534663
50
3.181131
3.264275
3.203312
28.97495
7.712769
4.210812
3.104564
5.18683
4.411553
10.48736
8.238313
3.405478
3.57624
100
6.367798
6.423244
5.85726
58.48177
16.31319
10.68463
6.987761
10.45312
9.780839
21.86651
15.70102
7.306115
7.456471
C17-F18
10
0.990391
1.14438
1.637812
6.31286
3.039392
1.37345
1.18141
2.114435
1.432993
5.049462
2.774735
1.400416
1.592656
30
1.344953
1.476479
1.836197
16.50794
4.439156
2.169175
1.520467
2.629998
2.305029
5.727457
4.481437
1.855193
1.974733
50
1.98622
2.101119
2.314997
27.16928
5.922221
3.258925
2.147991
4.263039
3.466858
7.574409
6.982974
2.565446
2.65009
100
4.623392
4.726155
4.565243
56.72284
12.53637
7.645209
5.255334
8.743703
7.952184
16.20433
13.72077
5.467938
5.664581
C17-F19
10
1.955417
2.095637
2.414612
7.333061
5.116793
2.408426
2.185875
3.148839
2.444147
8.19639
3.898278
2.445585
2.6079
30
4.239491
4.27608
3.897961
19.42097
10.65424
5.221357
4.589672
5.773174
5.404539
14.96195
7.703112
4.994248
5.057154
50
6.900162
6.896193
6.026687
32.21721
16.04018
8.341345
7.271696
9.803923
8.6811
24.11799
12.13466
7.579072
7.912256
100
13.34602
13.34456
11.68769
68.29124
33.09738
17.48625
15.39419
18.96095
18.26364
46.70063
24.32194
15.61414
15.88095
C17-F20
10
1.213736
1.37911
1.890453
6.551243
3.476617
1.576621
1.443784
2.250428
1.671454
5.755097
3.013205
1.594801
1.821807
30
1.943917
2.067675
2.322222
17.23006
5.746742
2.850805
2.23519
3.468045
3.209074
7.935644
5.209856
2.463929
2.661997
50
3.122607
3.330494
3.775742
28.25861
8.188441
4.448716
3.327005
5.605411
4.676373
11.69909
8.00799
3.614669
3.802783
100
6.496966
6.646131
6.439043
60.20271
17.2582
9.685805
7.542347
11.13922
10.36337
23.18623
16.36276
7.950868
7.962873
C17-F21
10
1.246397
1.365075
1.685593
6.509065
3.67821
1.571497
1.383807
2.096905
1.619051
5.661371
3.025062
1.52408
1.808687
30
2.207122
2.293717
2.367049
17.31182
6.164841
3.042045
2.413057
3.652132
3.205279
8.44079
5.405171
2.760043
2.857065
50
4.045295
4.134449
3.990613
29.33839
10.19363
5.345634
4.241401
6.332394
5.69715
14.44177
8.949361
4.481806
4.821397
100
11.43088
11.40177
9.871171
65.44038
28.04148
15.01686
12.88325
16.61197
15.65352
39.34907
21.54734
13.16289
13.4443
C17-F22
10
1.358797
1.488432
1.838874
6.626836
3.684655
1.772597
1.497466
2.13106
1.726702
5.981163
3.148835
1.693244
1.901028
30
2.465815
2.544686
2.555118
17.58437
6.58743
3.251819
2.626755
3.880485
3.404116
9.152478
5.615678
2.885135
3.054332
50
4.276386
4.348363
4.107236
29.66646
10.83081
5.685521
4.609728
6.547461
5.968059
15.27601
9.208929
4.874647
5.142972
100
12.5568
12.44629
10.45083
65.52973
29.34387
15.71686
13.49964
17.35543
16.31821
41.40921
22.23516
13.77283
14.15693
C17-F23
10
1.310664
1.440466
1.797534
6.619369
3.832611
1.771668
1.532794
2.16684
1.771407
6.15411
3.16642
1.730208
1.93059
30
2.542504
2.626794
2.649416
17.69085
6.897547
3.458376
2.866457
4.104512
3.657416
9.707524
5.821798
3.089022
3.298424
50
4.547107
4.772215
5.110118
30.57443
12.09766
6.401262
5.302998
7.469741
6.647262
17.54353
10.04646
5.525315
5.679863
100
10.41426
11.07139
12.41156
66.17142
35.51952
18.66019
16.57562
20.25858
19.37997
1326.56
25.21985
16.77714
16.84864
C17-F24
10
1.362291
1.490608
1.835344
6.672468
4.01277
1.750173
1.593779
2.191942
1.941247
6.276043
3.1959
1.831297
2.006232
30
2.725426
2.825318
2.887719
17.86842
7.378802
3.654539
3.042872
4.275243
3.846216
10.42713
6.222771
3.2371
3.457234
50
5.076273
5.339674
5.765289
30.62493
12.74456
6.690299
5.620654
7.720118
6.944134
19.02458
10.27779
5.857397
6.289918
100
12.99194
13.39079
13.37895
67.70093
36.73724
19.35246
17.22645
20.81343
19.80846
52.53937
26.01052
17.42242
17.61044
C17-F25
10
1.257575
1.39261
1.777173
6.583291
3.712224
1.751549
1.459254
2.094637
1.714968
5.942937
3.088336
1.63839
1.884882
30
2.578471
2.660071
2.667484
17.86822
6.815261
3.390773
2.765187
4.046097
3.565367
9.740856
5.808113
2.964355
3.242644
50
5.230465
5.347034
5.16624
30.53703
12.60585
6.584492
5.493316
7.485713
6.778405
18.16804
10.14696
5.779912
5.918255
100
17.85481
17.54259
14.08484
68.49214
39.34139
20.6831
18.54472
22.25741
21.42674
56.27984
27.35005
18.87771
18.95531
C17-F26
10
1.442191
1.562987
1.867755
6.79877
4.273244
1.919745
1.651185
2.237936
1.87562
6.472393
3.240714
1.812288
2.125065
30
3.086272
3.199675
3.271482
18.20593
7.909905
3.931616
3.485475
4.563493
4.192108
11.37455
6.425178
3.491959
4.0234
50
6.212109
6.188404
5.325075
31.61121
14.37638
7.502224
6.384111
8.41198
7.73557
20.81697
11.0707
6.623204
6.873124
100
19.1232
18.78656
15.07423
74.59587
42.9371
22.54353
20.36853
24.20819
23.06341
62.13359
31.59323
1266.628
20.82038
C17-F27
10
1.392044
1.528941
1.904317
6.868196
4.134728
1.867463
1.694939
2.283099
1.930563
6.715525
3.329278
1.842092
2.078829
30
3.035675
3.149578
3.229646
18.85547
8.60177
4.290813
3.762545
4.927789
4.498036
12.8161
6.693306
3.975765
4.528151
50
6.226264
6.56681
7.158735
751.4049
16.45292
8.501632
7.367071
9.311285
8.67136
23.61473
12.10706
7.552317
8.20048
100
19.43779
19.48405
17.26439
75.07449
50.08541
25.96096
23.95681
27.61783
26.66497
73.20217
35.49329
25.63387
24.56765
C17-F28
10
1.318753
1.467862
1.901153
6.806764
3.979335
1.768222
1.598092
2.209172
1.823445
6.356397
3.247905
1.729504
2.013281
30
2.974985
3.0557
3.010519
18.28915
7.850114
3.871084
3.309712
4.519823
4.079317
11.3554
6.226844
3.509667
3.948564
50
6.361014
6.528713
6.412581
31.50602
462.7972
7.808279
6.632258
8.691254
7.967651
21.51879
11.34388
6.840787
7.841928
100
21.30956
20.8315
16.28302
74.30412
47.22546
24.64309
22.39027
26.21548
25.22822
68.47812
33.33656
22.66888
22.7536
C17-F29
10
1.492716
1.690406
2.296503
6.831717
4.04011
1.837098
1.635945
2.236849
1.847688
6.525595
3.281418
1.820307
2.044941
30
2.734513
2.843583
2.941571
17.98335
7.133749
3.572183
3.105965
4.303773
3.728723
10.63919
6.304657
3.10057
3.456765
50
4.591509
4.765972
4.895802
30.34651
14.81977
6.187883
5.028518
7.096739
6.355616
16.67771
9.757751
5.312877
5.610104
100
14.01545
13.7303
10.85583
65.29449
30.96446
16.63042
14.33984
17.74328
17.11014
43.88943
24.59769
14.58347
14.78207
C17-F30
10
2.028156
2.207273
2.672835
7.659937
5.684242
2.689453
2.51453
3.216611
2.727053
9.068879
4.098566
2.715899
2.887235
30
4.705207
4.82594
4.726783
20.51795
12.15729
6.131017
5.462438
6.672822
6.23661
18.04703
8.628704
5.622371
6.123689
50
7.736195
8.591965
11.05799
34.26105
19.98942
10.3383
9.198505
11.26854
10.54938
29.16824
14.0752
9.443989
9.82425
100
19.25983
19.20305
16.59325
73.79778
47.71889
24.95313
22.64243
26.40534
25.54835
68.85316
33.08901
22.9091
23.04624
Also, with the aim of analyzing the convergence speed between FNO and competing algorithms, the convergence curves obtained from the performance of metaheuristic algorithms are drawn in Figs. 7 to 10. The findings show that FNO has different mechanisms to achieve the optimal solution during algorithm iterations. What is evident is that FNO, by balancing exploration and exploitation, has been able to handle most of the CEC 2017 test suite benchmark functions in a reasonable number of iterations.
Convergence curves of FNO and competitor algorithms performances on CEC 2017 test suite (dimension = 10)
Convergence curves of FNO and competitor algorithms performances on CEC 2017 test suite (dimension = 30)
Convergence curves of FNO and competitor algorithms performances on CEC 2017 test suite (dimension = 50)
Convergence curves of FNO and competitor algorithms performances on CEC 2017 test suite (dimension = 100)
Evaluation of CEC 2020 Test Suite
In this sub-section, the effectiveness of the proposed approach to address CEC 2020 is challenged. This test suite consists of ten bound-constrained numerical optimization benchmark functions. Among these, C20-F1 is unimodal, C20-F2 to C20-F4 is basic, C20-F5 to C20-F7 is hybrid, and C20-F8 to C20-F10 is composition. Complete information of CEC 2020 is provided in detail on [60].
The implementation results of the proposed FNO approach and competing algorithms are reported in Table 9. Boxplot diagrams obtained from metaheuristic algorithms are drawn in Fig. 11. Based on the obtained results, FNO has been the first best optimizer to handle all 10 functions C20-F1 to C20-F10. The simulation findings show that FNO has provided superior performance for handling CEC 2020 by providing better results in each benchmark function compared to competing algorithms.
Optimization results of CEC 2020 test suite
FNO
AVOA
WSO
RSA
MPA
TSA
WOA
MVO
GWO
TLBO
GSA
PSO
GA
C20-F1
Mean
100
5612.07
254.0577
254.0577
254.0578
646429.2
585.798
431.1935
1609.957
6189.223
329.3244
1083.229
586.4009
Best
100
208.0469
100
100
100
100
542.7263
233.3933
100
100
165.4577
384.7248
165.4577
Worst
100
15672.09
542.7263
542.7263
542.7263
2579671
611.8795
655.0074
4439.268
23903.82
686.4838
2994.488
974.7303
Std
9.99E-07
7020.009
197.5098
197.5098
197.5098
1288829
32.63734
200.4796
1935.841
11811.23
240.2066
1276.046
331.5199
Median
100
3284.07
186.7523
186.7523
186.7523
2972.937
594.2932
418.1867
950.2805
376.5343
232.6781
476.8524
602.7078
Rank
1
10
2
2
3
12
6
5
9
11
4
8
7
C20-F2
Mean
1109.997
2434.894
1794.357
1706.476
1882.635
1970.708
2783.519
2643.909
7042.562
116943.3
3463.76
2292.66
2603.58
Best
1100
1605.608
1256.933
1100.166
1255.67
1574.792
1803.602
1729.792
1100.166
1803.602
1100.166
1500.962
2192.343
Worst
1120.638
3453.718
2192.343
2192.343
2192.343
2576.384
3851.913
3668.555
11806.32
207485.4
4904.348
3162.643
3069.593
Std
8.462024
910.8041
398.1137
452.2742
426.1382
440.6826
1020.862
969.4117
4433.318
93546.47
1712.811
681.7148
445.9755
Median
1109.674
2340.125
1864.076
1766.697
2041.263
1865.827
2739.28
2588.644
7631.881
129242.1
3925.263
2253.518
2576.193
Rank
1
7
3
2
4
5
10
9
12
13
11
6
8
C20-F3
Mean
710.7399
956.2997
1059.434
925.9339
934.5255
2414.686
1364.274
845.1602
1141.895
829.5544
898.7981
1781.918
913.868
Best
700.0001
701.2474
703.7808
792.4092
700
792.4092
700
700.0213
700
700
732.22
704.0253
784.6324
Worst
714.3198
1215.565
1638.705
1035.392
1044.754
3525.084
2109.735
1024.31
1603.359
1037.98
1235.284
3447.834
1103.032
Std
7.159875
292.9431
424.3349
107.5292
161.3755
1173.015
592.2339
167.8835
490.0681
145.7985
228.8461
1300.955
134.5699
Median
714.3198
954.1933
947.6255
937.967
996.6739
2670.625
1323.681
828.1546
1132.111
790.1189
813.8445
1487.906
883.9038
Rank
1
8
9
6
7
13
11
3
10
2
4
12
5
C20-F4
Mean
1941.536
2457.537
6964.756
2579.226
2770.2
2508.438
3397.925
2233.294
3248.065
2178.674
2558.146
3255.209
2343.38
Best
1900
1936.524
1936.524
1936.524
1900
1936.524
1936.524
2084.561
1900
1940.98
1900
2011.54
1900
Worst
1983.073
2967.722
11009.4
3163.728
3559.686
2995.979
5056.833
2425.333
5856.74
2440.504
3370.068
5855.145
3148.67
Std
41.6773
439.4324
3769.793
502.3301
832.6108
565.4491
1423.251
160.5537
1797.781
233.5194
610.9464
1775.042
566.0155
Median
1941.536
2462.952
7456.547
2608.327
2810.557
2550.625
3299.172
2211.64
2617.759
2166.606
2481.257
2577.076
2162.425
Rank
1
5
13
8
9
6
12
3
10
2
7
11
4
C20-F5
Mean
1702.285
2337.247
1871.272
2245.405
1786.11
3520.324
2051.627
1930.957
2148.966
1913.67
1777.165
1788.926
2131.297
Best
1700
1813.561
1793.517
1765.604
1700
1700
1704.217
1704.217
1700
1700
1736.904
1707.399
1704.217
Worst
1704.57
3070.717
2004.301
2781.858
1894.087
7174.555
2288.053
2390.347
2812.285
2059.271
1846.726
1840.713
2431.711
Std
2.410464
554.9085
95.00707
416.2292
95.59094
2578.037
258.8121
310.6519
530.7737
174.17
47.98654
57.62389
309.1618
Median
1702.285
2232.355
1843.636
2217.079
1775.176
2603.369
2107.118
1814.632
2041.79
1947.704
1762.515
1803.796
2194.629
Rank
1
12
5
11
3
13
8
7
10
6
2
4
9
C20-F6
Mean
1603.727
2615.996
1953.889
1827.439
1820.596
2864.624
1951.399
3820.267
3117.815
2143.747
6701.285
3699.206
2371.762
Best
1600
1741.01
1741.01
1690.778
1600
1741.01
1808.386
1870.293
1600
1741.01
1870.293
1741.01
1741.01
Worst
1609.322
3070.568
2305.198
1934.076
1972.651
3570.869
2131.085
5414.023
4248.29
2353.274
10489.21
6143.423
2875.183
Std
4.558769
613.3696
249.9763
104.593
182.5059
806.9465
165.1641
1519.262
1136.337
273.7522
3676.626
1877.921
499.5352
Median
1602.793
2826.203
1884.674
1842.451
1854.868
3073.308
1933.063
3998.376
3311.486
2240.352
7222.817
3456.197
2435.427
Rank
1
8
5
3
2
9
4
12
10
6
13
11
7
C20-F7
Mean
2206.269
3982.005
2828.481
2366.614
4058.79
2883.029
3342.977
3039.795
2926.368
3404.023
2678.63
3572.791
3348.542
Best
2100
2102.328
2165.527
2165.527
2165.527
2305.424
2305.424
2238.232
2102.328
2171.14
2305.424
2171.14
2177.255
Worst
2327.209
6110.685
4014.801
2585.222
6309.339
3367.293
5045.988
5057.327
4075.858
4225.813
3290.587
4832.517
4651.728
Std
95.35991
1723.317
869.6533
182.0925
1736.29
444.4175
1183.745
1352.058
827.495
979.0758
442.0911
1209.711
1289.525
Median
2198.933
3857.505
2566.798
2357.854
3880.147
2929.699
3010.248
2431.81
2763.643
3609.569
2559.253
3643.754
3282.592
Rank
1
12
4
2
13
5
8
7
6
10
3
11
9
C20-F8
Mean
2218.874
2586.823
2500.863
2353.317
2566.249
2554.43
2335.656
2504.404
2526.738
2351.101
2322.496
2339.469
2387.394
Best
2200
2200.701
2253.626
2200.701
2230.552
2253.626
2200.701
2230.552
2200.701
2200.701
2253.626
2253.626
2253.626
Worst
2237.305
2761.507
2678.263
2421.571
2858.097
2931.105
2430.407
2751.89
2724.962
2619.704
2386.932
2470.734
2634.236
Std
19.65987
259.4729
204.0465
104.4793
336.9903
285.6439
103.0891
254.8663
238.3239
184.5883
65.90148
93.6301
169.462
Median
2219.095
2692.542
2535.782
2395.499
2588.173
2516.495
2355.759
2517.586
2590.646
2291.999
2324.713
2316.758
2330.858
Rank
1
13
8
6
12
11
3
9
10
5
2
4
7
C20-F9
Mean
2404.914
2511.727
3028.135
3030.942
2497.218
2680.029
2745.351
2458.897
2503.327
2529.488
2435.639
2612.916
2500.01
Best
2400
2400.11
2401.226
2425.972
2415.193
2415.193
2415.193
2415.193
2400.11
2401.226
2417.49
2415.193
2401.226
Worst
2411.095
2641.865
3457.152
3529.858
2596.745
3040.057
3214.858
2498.374
2654.112
2643.281
2467.442
2738.017
2572.971
Std
4.761701
111.6805
448.8483
468.9578
75.74189
282.4686
383.5691
35.93406
121.3041
128.5102
21.98644
143.527
80.96721
Median
2404.28
2502.467
3127.081
3083.968
2488.468
2632.433
2675.676
2461.011
2479.543
2536.723
2428.811
2649.227
2512.923
Rank
1
7
12
13
4
10
11
3
6
8
2
9
5
C20-F10
Mean
2500.005
6167.644
2507.443
2510.208
2500.733
2506.725
2517.898
2502.757
2506.417
2503.958
35525.22
2504.073
2501.125
Best
2500
2500
2501.329
2500.539
2500.019
2500
2500.751
2500.539
2500
2500.539
2501.329
2500.027
2500.539
Worst
2500.017
7436.058
2521.801
2517.744
2501.329
2520.226
2557.207
2503.921
2513.03
2511.398
46553.02
2514.363
2501.839
Std
0.007955
2445.313
9.622326
7.156732
0.539975
9.173942
26.40593
1.517281
5.387741
5.021045
22015.94
6.882091
0.548087
Median
2500.001
7367.259
2503.32
2511.274
2500.791
2503.336
2506.817
2503.283
2506.32
2501.947
46523.27
2500.951
2501.06
Rank
1
12
9
10
2
8
11
4
7
5
13
6
3
Sum rank
10
94
70
63
59
92
84
62
90
68
61
82
64
Mean rank
1
9.4
7
6.3
5.9
9.2
8.4
6.2
9
6.8
6.1
8.2
6.4
Total rank
1
13
8
5
2
12
10
4
11
7
3
9
6
Boxplot diagrams of FNO and competitor algorithms performances on CEC 2020 test suite
Statistical Analysis
In this section, a statistical analysis is conducted to determine whether FNO exhibits a significant advantage over competitor algorithms. To address this inquiry, the Wilcoxon rank sum test [61] is utilized, a non-parametric test renowned for discerning significant differences between the means of two data samples. In the Wilcoxon rank sum test, the presence or absence of a noteworthy discrepancy in performance between two metaheuristic algorithms is gauged using a parameter known as the p-value. The results of applying the Wilcoxon rank sum test to evaluate FNO’s performance against each competitor algorithm are documented in Table 10.
Wilcoxon rank sum test results
Compared algorithm
Objective function type
CEC 2017
CEC 2020
D = 10
D = 30
D = 50
D = 100
FNO vs. WSO
6.32E-25
6.32E-25
8.84E-25
c
3.50E-07
FNO vs. AVOA
1.69E-22
1.36E-24
8.84E-25
8.84E-25
1.18E-07
FNO vs. RSA
8.84E-25
6.32E-25
6.32E-25
8.84E-25
1.71E-07
FNO vs. MPA
8.99E-22
6.99E-20
2.97E-21
8.84E-25
1.10E-07
FNO vs. TSA
4.26E-24
8.84E-25
8.84E-25
8.84E-25
2.28E-07
FNO vs. WOA
4.26E-24
8.84E-25
8.84E-25
8.84E-25
2.45E-07
FNO vs. MVO
4.05E-22
9.51E-25
8.84E-25
8.84E-25
1.27E-07
FNO vs. GWO
2.34E-24
8.84E-25
8.84E-25
8.84E-25
8.10E-06
FNO vs. TLBO
1.66E-24
8.84E-25
8.84E-25
8.84E-25
8.68E-07
FNO vs. GSA
7.19E-22
9.06E-25
8.84E-25
8.84E-25
7.59E-08
FNO vs. PSO
6.94E-23
1.06E-24
8.84E-25
8.84E-25
7.59E-08
FNO vs. GA
1.21E-22
8.84E-25
8.84E-25
8.84E-25
1.27E-07
Based on the outcomes of the statistical analysis, instances where the p-value falls below 0.05 indicate that FNO demonstrates a statistically significant superiority over the corresponding competitor algorithm. Consequently, it is evident that FNO outperforms all twelve competitor algorithms significantly across problem dimensions equal to 10, 30, 50, and 100 when handling the CEC 2017 test suite. Also, the findings indicate that FNO has a significant statistical advantage compared to competing algorithms for handling the CEC 2020 test suite.
FNO for Real-World Applications
In this section, we examine the efficacy of the proposed FNO methodology in tackling real-world optimization challenges. To achieve this, we have selected a subset of twenty-two constrained optimization problems from the CEC 2011 test suite, along with four engineering design problems, for assessment. As in the previous section, here the simulation results are reported using six statistical indicators: mean, best, worst, std, median, and rank. It should be noted that the values of the “mean” index are the criteria used to rank the algorithms in handling each of the CEC 2011 test suite problems as well as each of the engineering design problems.
CEC 2011 test suite has been used in many algorithms recently published in similar articles, in order to judge the efficiency of designed metaheuristic algorithms. This test suite consists of twenty-two constrained optimization problems from real world applications. For this reason, in this paper, CEC 2011 test suite has been chosen to evaluate the effectiveness of the proposed FNO approach in handling optimization problems in real world applications.
In order to adapt the proposed approach of FNO to deal with constrained optimization problems, there are different solutions. In this study, the method of penalty coefficient is used. In this case, for a candidate FNO solution, a penalty value is added to the objective function for each of the constraints that are not satisfied. As a result, automatically this inappropriate corresponding solution will not be placed as a solution in the output. However, it is possible that during the update process in the next iteration, a new solution will be produced that satisfies the constraints.
Evaluation of CEC 2011 Test Suite
In this section, we assess the performance of FNO and competitor algorithms in tackling the CEC 2011 test suite, comprising twenty-two constrained optimization problems derived from real-world applications. A comprehensive description and detailed information on the CEC 2011 test suite can be found in [62]. The proposed FNO approach and each of the competitor algorithms is implemented on the CEC 2011 functions in twenty-five independent implementations where each implementation contains 150,000 FEs. The implementation outcomes of FNO and competitor algorithms on the CEC 2011 test suite are documented in Table 11, while boxplot diagrams illustrating the performance of metaheuristic algorithms are presented in Fig. 12.
Optimization results of CEC 2011 test suite
FNO
AVOA
WSO
RSA
MPA
TSA
WOA
MVO
GWO
TLBO
GSA
PSO
GA
C11-F1
Mean
5.920103
12.52841
16.74213
20.55445
7.723161
17.39382
12.78836
13.45957
10.65985
17.42229
20.30587
16.98281
21.8149
Best
2E-10
8.402211
13.72119
17.98675
0.405713
16.51963
7.388265
11.2761
1.066509
16.51282
17.52035
10.47591
20.46457
Worst
12.30606
16.34896
19.54639
23.07193
12.72065
18.96086
16.7112
15.47524
16.6408
18.92175
21.98956
23.05791
24.2157
Std
7.196379
4.623049
2.891741
2.46555
5.891302
1.155486
4.518685
2.427897
7.078568
1.177854
2.032557
5.830417
1.750385
Median
5.687176
12.68124
16.85048
20.57957
8.883141
17.04739
13.527
13.54346
12.46605
17.12729
20.85678
17.1987
21.28967
Rank
1
4
7
12
2
9
5
6
3
10
11
8
13
C11-F2
Mean
−26.3179
−21.4236
−15.5557
−13.0722
−24.9902
−12.8247
−19.289
−10.6335
−22.8276
−12.4781
−16.5731
−22.8708
−14.2704
Best
−27.0676
−21.9622
−16.824
−13.4768
−25.6486
−16.2028
−22.3841
−12.5221
−24.7628
−13.5921
−21.0914
−24.1606
−16.4087
Worst
−25.4328
−20.8936
−14.5993
−12.7739
−23.5778
−10.997
−15.8323
−9.03411
−19.4166
−11.639
−12.7353
−20.839
−12.8069
Std
0.738935
0.556485
1.134413
0.347572
1.012612
2.57848
3.448741
1.558767
2.495606
0.902004
3.950344
1.504127
1.850662
Median
−26.3856
−21.4193
−15.3998
−13.019
−25.3672
−12.0494
−19.4698
−10.4888
−23.5656
−12.3407
−16.2328
−23.2418
−13.933
Rank
1
5
8
10
2
11
6
13
4
12
7
3
9
C11-F3
Mean
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
Best
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
Worst
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
Std
2E-19
2.21E-09
1.93E-11
4.35E-11
1.33E-15
2.06E-14
2.47E-16
8.67E-13
3.49E-15
6.84E-14
2.47E-16
2.46E-16
2.46E-16
Median
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
1.16E-05
Rank
1
13
11
12
6
8
4
10
7
9
3
2
5
C11-F4
Mean
13.68174
15.43223
14.39174
17.84035
15.43663
14.88466
15.08699
17.05755
14.48897
19.65085
15.96374
15.73561
13.78715
Best
13.61536
13.67784
13.72765
13.66863
13.72765
13.72765
13.72765
13.72596
13.64036
13.72596
13.64036
13.72765
13.63191
Worst
13.73188
16.52614
15.29246
19.85408
16.73196
15.57639
16.10149
19.20792
15.20547
22.70577
18.55033
17.02715
13.91219
Std
0.058052
1.323076
0.756844
2.966604
1.48494
0.907301
1.108411
2.666134
0.821883
4.247393
2.113695
1.595204
0.12558
Median
13.68986
15.76248
14.27343
18.91935
15.64345
15.1173
15.2594
17.64815
14.55502
21.08584
15.83213
16.09382
13.80225
Rank
1
7
3
12
8
5
6
11
4
13
10
9
2
C11-F5
Mean
−34.1274
−28.7037
−25.8146
−21.5693
−33.2154
−27.8494
−28.2859
−27.727
−31.73
−13.5176
−28.0391
−11.621
−12.3745
Best
−34.7494
−29.7163
−26.8808
−23.5116
−33.7977
−31.8374
−28.4882
−31.6034
−34.1261
−15.5107
−31.4293
−14.4729
−13.3683
Worst
−33.3862
−28.033
−24.7368
−19.2408
−31.7926
−23.257
−28.0604
−25.707
−27.9425
−12.1616
−25.4024
−10.1843
−10.9842
Std
0.589989
0.755729
0.964252
2.29988
1.000596
3.70703
0.230197
2.909826
2.788723
1.499243
2.784274
2.122759
1.13996
Median
−34.1871
−28.5328
−25.8204
−21.7625
−33.6357
−28.1515
−28.2975
−26.7987
−32.4256
−13.199
−27.6623
−10.9135
−12.5727
Rank
1
4
9
10
2
7
5
8
3
11
6
13
12
C11-F6
Mean
−24.1119
−19.3783
−15.0036
−14.1328
−22.5108
−9.32911
−20.1856
−11.0554
−19.9032
−4.73973
−21.876
−5.49833
−6.29263
Best
−27.4298
−20.9932
−15.3343
−14.6856
−25.6345
−17.0202
−22.803
−17.9435
−22.7104
−5.39677
−25.8224
−8.43116
−10.6843
Worst
−23.0059
−17.783
−14.63
−13.0677
−21.2106
−6.29264
−13.8854
−4.47264
−18.2866
−4.47264
−18.6799
−4.47264
−4.47264
Std
2.324951
1.497394
0.345464
0.761753
2.222427
5.422546
4.458259
7.468412
2.212179
0.465924
3.29799
2.056378
3.106794
Median
−23.0059
−19.3684
−15.0251
−14.389
−21.5991
−7.00181
−22.0269
−10.9027
−19.3078
−4.54476
−21.5009
−4.54476
−5.00682
Rank
1
6
7
8
2
10
4
9
5
13
3
12
11
C11-F7
Mean
0.860699
1.242726
1.522903
1.795953
0.934354
1.258251
1.64259
0.892095
1.054319
1.620895
1.064804
1.103022
1.639835
Best
0.582266
1.1199
1.454609
1.592827
0.769266
1.092462
1.555277
0.841362
0.82802
1.461789
0.879847
0.85696
1.28447
Worst
1.025027
1.37333
1.612629
1.951345
1.012985
1.582892
1.787355
0.970732
1.264946
1.738519
1.253393
1.327696
1.82533
Std
0.211503
0.141593
0.070239
0.156563
0.119362
0.231495
0.105518
0.064912
0.187998
0.131701
0.177573
0.251075
0.25626
Median
0.91775
1.238838
1.512188
1.81982
0.977583
1.178825
1.613864
0.878143
1.062156
1.641636
1.062989
1.113716
1.72477
Rank
1
7
9
13
3
8
12
2
4
10
5
6
11
C11-F8
Mean
220
238.3658
277.2247
312.5117
222.6229
253.0605
260.7144
224.0469
226.8948
224.0469
243.4968
438.3804
222.6624
Best
220
224.1382
254.5798
277.2211
220
220
242.0717
220
220
220
220
245.4979
220
Worst
220
253.5672
308.0582
351.0064
225.2457
338.4523
301.4288
234.2398
233.7896
235.2136
285.053
526.6478
229.6759
Std
0
13.32447
24.72279
31.85048
3.183398
60.22576
28.78087
7.158832
8.368285
7.83979
32.45774
139.596
4.938214
Median
220
237.8789
273.1304
310.9098
222.6229
226.8948
249.6785
220.9738
226.8948
220.4869
234.4671
490.6879
220.4869
Rank
1
6
10
11
2
8
9
4
5
4
7
12
3
C11-F9
Mean
8789.286
334088.5
490433.2
931595.7
21036.05
61266.01
330884.1
119957.3
40982.68
360564.8
722998.3
949481.7
1701352
Best
5457.674
297764.7
329399.1
609975.6
11289.81
46871.22
182907.1
68778.55
17767.84
298249.2
619636.7
761083.4
1631054
Worst
14042.29
357645.2
564925.4
1094288
30402.18
76236.36
557575.5
180347.9
71029.98
462098.6
776344.7
1162652
1800150
Std
3889.181
27700.03
115985.9
230040.8
8813.927
13560.96
179294.8
48505.11
23657.09
75392.44
73781.12
225753.1
86839.5
Median
7828.591
340472.1
533704.1
1011060
21226.1
60978.23
291526.9
115351.4
37566.45
340955.7
748005.9
937095.6
1687102
Rank
1
7
9
11
2
4
6
5
3
8
10
12
13
C11-F10
Mean
−21.4889
−17.0126
−14.4367
−12.9595
−18.8505
−14.8029
−13.4786
−15.0751
−14.5533
−12.0938
−13.7234
−12.1809
−11.9229
Best
−21.8299
−17.2235
−15.4235
−13.3512
−19.2436
−18.6357
−14.0346
−20.7755
−15.0311
−12.2211
−14.2247
−12.2692
−12.0177
Worst
−20.7878
−16.6527
−13.9493
−12.7677
−18.4764
−12.7956
−13.1018
−12.176
−13.4717
−11.981
−12.9762
−12.1131
−11.8264
Std
0.498616
0.281136
0.705055
0.282262
0.424606
2.765963
0.415821
4.063559
0.772761
0.127544
0.641605
0.083879
0.092564
Median
−21.669
−17.0871
−14.1869
−12.8596
−18.8411
−13.8901
−13.389
−13.6745
−14.8551
−12.0866
−13.8464
−12.1707
−11.9237
Rank
1
3
7
10
2
5
9
4
6
12
8
11
13
C11-F11
Mean
571712.3
1153936
5354192
8022913
1737036
5478490
1349788
1430873
3635030
4836889
1520671
4846559
5633837
Best
260837.9
967939
5101288
7769164
1619920
4591451
1236097
827905.4
3449232
4813404
1375620
4813404
5600872
Worst
828560.9
1319938
5679544
8197048
1863179
6566972
1499541
2656953
3952245
4866281
1678745
4870909
5679144
Std
260922.1
163646.1
279127.2
190070.7
119477.7
856487.1
116048.5
871229.6
230816.3
27507.81
130353.3
28465.49
36072.03
Median
598725.2
1163934
5317967
8062720
1732523
5377768
1331757
1119316
3569322
4833935
1514160
4850962
5627666
Rank
1
2
10
13
6
11
3
4
7
8
5
9
12
C11-F12
Mean
1199805
3113654
7491258
11720134
1279917
4557354
5242788
1326078
1410399
12674132
5220823
2187024
12814586
Best
1155937
3020903
7175822
10885229
1201037
4334324
4881360
1197606
1254352
11951819
4972156
2050733
12699967
Worst
1249353
3185139
7772240
12455343
1361256
4669878
5412120
1448642
1543516
13230852
5389417
2365904
12932976
Std
47157.58
75270.06
259442.9
676945.3
73312.95
165786.6
259966.3
108704.4
126165.3
564168.1
189579.1
137821
101134.8
Median
1196965
3124288
7508485
11769981
1278688
4612607
5338837
1329032
1421863
12756929
5260860
2165729
12812701
Rank
1
6
10
11
2
7
9
3
4
12
8
5
13
C11-F13
Mean
15444.2
15451.29
15809.07
16204.44
15464.54
15488.14
15527.69
15503.61
15497.68
15875.05
114049.1
15488.7
28163.32
Best
15444.19
15450.37
15646.6
15840.7
15462.07
15479.02
15490.32
15486.73
15491.17
15607.62
82990.68
15473.2
15461.89
Worst
15444.21
15451.82
16202.36
17108.23
15468.82
15499.11
15578.85
15536.95
15508.94
16366.82
156175.3
15521.39
65959.56
Std
0.009091
0.717424
278.2839
638.5421
3.147698
10.09012
43.54899
24.66549
8.276395
360.9329
34633.02
23.16551
26485.21
Median
15444.2
15451.49
15693.65
15934.42
15463.64
15487.22
15520.79
15495.38
15495.3
15762.88
108515.2
15480.11
15615.92
Rank
1
2
9
11
3
4
8
7
6
10
13
5
12
C11-F14
Mean
18295.35
18553.26
100636.2
202511.6
18630.5
19437.26
19170.31
19339.48
19176.37
273893.4
19054.11
19082.55
19071.41
Best
18241.58
18461.39
77122.08
149760.8
18543.49
19208.23
19022.67
19237.88
19035.22
28811.09
18796.85
18934.7
18818.16
Worst
18388.08
18647.79
139802.1
290790.4
18705.61
19925.54
19284.21
19419.49
19347.24
526464.6
19224.42
19219.06
19341.93
Std
71.59938
93.57007
29465.91
66396.98
74.25737
345.7248
127.1688
81.53748
144.974
251121.5
197.5448
122.9683
225.598
Median
18275.87
18551.94
92810.28
184747.5
18636.45
19307.63
19187.18
19350.27
19161.52
270149
19097.58
19088.23
19062.78
Rank
1
2
11
12
3
10
7
9
8
13
4
6
5
C11-F15
Mean
32883.58
98422.07
797983
1678033
32953.1
51838.95
195252.6
33085.66
33066.31
13484325
266202.6
33250.03
6941821
Best
32782.17
41898.3
331339
704158.7
32876.29
33048.61
32998.24
33003.58
33029.58
2828717
236067.3
33239.7
3161932
Worst
32956.46
161675.2
2000291
4373616
33023.23
107987
277680.7
33137.91
33131.97
20106167
286828.8
33268.81
11893981
Std
76.94696
67668.53
845546.8
1891824
63.23147
39344.94
116109.2
61.75003
49.5816
8257502
24822.65
13.9656
4208351
Median
32897.86
95057.39
430151
817178.8
32956.44
33160.1
235165.7
33100.58
33051.85
15501209
270957.1
33245.8
6355685
Rank
1
7
10
11
2
6
8
4
3
13
9
5
12
C11-F16
Mean
133550
135790.5
843849.2
1721618
137958.6
144486.8
141897
141586.3
145138.9
77731378
16384089
69575107
66804386
Best
131374.2
134202.1
270072.4
431634.4
135881.6
141864.4
136687.2
133962.1
142825.4
75747443
8330119
57554826
53995364
Worst
136310.8
136628.2
1970043
4252988
141711.4
146029.6
147084.5
149014.9
150507.4
79968786
29628046
83137356
85442852
Std
2392.2
1136.541
803717.8
1806507
2739.482
2070.447
4549.789
6581.747
3796.672
1859489
9679765
11590320
14041017
Median
133257.5
136165.9
567640.5
1100924
137120.6
145026.7
141908.2
141684.1
143611.4
77604641
13789096
68804123
63889663
Rank
1
2
8
9
3
6
5
4
7
13
10
12
11
C11-F17
Mean
1926615
2.02E+09
7.83E+09
1.36E+10
2317696
1.12E+09
8.48E+09
3035513
2954660
1.95E+10
9.8E+09
1.82E+10
1.91E+10
Best
1916953
1.84E+09
6.68E+09
9.74E+09
1960318
9.24E+08
6.05E+09
2309786
2030949
1.88E+10
8.63E+09
1.61E+10
1.79E+10
Worst
1942685
2.22E+09
8.69E+09
1.66E+10
2979619
1.28E+09
1.13E+10
3542354
4704927
2.04E+10
1.04E+10
2.1E+10
2.16E+10
Std
12003.53
1.74E+08
9.35E+08
3.08E+09
480715.1
1.93E+08
2.31E+09
576727.1
1265977
6.92E+08
8.4E+08
2.36E+09
1.78E+09
Median
1923412
2.02E+09
7.99E+09
1.4E+10
2165424
1.14E+09
8.29E+09
3144956
2541382
1.94E+10
1.01E+10
1.79E+10
1.85E+10
Rank
1
6
7
10
2
5
8
4
3
13
9
11
12
C11-F18
Mean
942057.5
5855154
48192826
1.04E+08
973927.2
1916670
8498968
988235.3
1025834
27226207
9852615
1.18E+08
1E+08
Best
938416.2
3566023
33180053
71603383
950609.3
1687657
3723128
978489.2
965531.5
21601124
7381068
99118324
96585265
Worst
944706.9
9964895
54798683
1.18E+08
1036253
2212191
14837763
993919.4
1185358
29452742
12407872
1.31E+08
1.04E+08
Std
2774.139
3132056
10642560
22977772
43805.75
261885.6
4933562
7290.961
112018
3958415
2359599
15025192
3163274
Median
942553.5
4944849
52396283
1.12E+08
954422.9
1883415
7717491
990266.4
976223.6
28925481
9810760
1.21E+08
1E+08
Rank
1
6
10
12
2
5
7
3
4
9
8
13
11
C11-F19
Mean
1025341
5970619
47466451
1.02E+08
1146090
2304929
9084371
1442306
1342758
31266255
5630384
1.51E+08
1.01E+08
Best
967927.7
5484323
40513198
87698402
1073916
2084415
1946393
1132085
1217571
21929855
2258733
1.37E+08
98112621
Worst
1167142
7196938
60305322
1.28E+08
1302136
2688025
16329519
1860579
1522004
38959353
7342291
1.75E+08
1.04E+08
Std
99675.04
863968.5
9380732
19523560
110512.2
277372.4
7123152
319822.8
134824.1
7750765
2423854
17128570
2383291
Median
983146.6
5600607
44523642
95411344
1104154
2223638
9030786
1388279
1315728
32087906
6460255
1.46E+08
1E+08
Rank
1
7
10
12
2
5
8
4
3
9
6
13
11
C11-F20
Mean
941250.4
5273531
50431412
1.1E+08
961749.9
1718994
6487484
972648.7
995138.1
30345716
12598168
1.39E+08
1.01E+08
Best
936143.2
4664825
44384145
95906503
957836.9
1558156
6119757
963856.1
975704.5
29681866
8409912
1.27E+08
96060897
Worst
946866.6
5925538
59698100
1.3E+08
964325.5
1985270
6978686
982895
1010227
31062861
19434543
1.51E+08
1.05E+08
Std
5013.552
549976.2
6858771
15393470
2927.238
213153.7
386197.8
9087.642
15634.21
603666.9
5064395
14022604
3800527
Median
940995.9
5251881
48821700
1.06E+08
962418.6
1666275
6425745
971921.8
997310.5
30319069
11274109
1.39E+08
1.01E+08
Rank
1
6
10
12
2
5
7
3
4
9
8
13
11
C11-F21
Mean
12.71443
21.16485
46.17786
69.10479
16.17329
28.31979
36.16095
26.31967
21.80684
90.29177
37.83232
94.6948
91.95816
Best
9.974206
19.75353
38.68815
52.27901
14.03749
25.1995
33.5643
23.56607
20.05703
44.63123
33.85471
82.26917
53.75319
Worst
14.97499
23.05336
54.27946
85.96943
18.46504
29.56548
39.51651
29.26324
24.14938
131.7641
40.51045
104.8618
111.555
Std
2.412667
1.496981
7.047461
15.62425
2.166017
2.198137
2.760114
3.244344
1.936904
37.53317
3.07227
11.86593
28.27838
Median
12.95425
20.92626
45.87191
69.08536
16.09532
29.25709
35.7815
26.22468
21.51047
92.3859
38.48206
95.82412
101.2622
Rank
1
3
9
10
2
6
7
5
4
11
8
13
12
C11-F22
Mean
16.12513
26.61591
43.55688
58.13767
19.30005
30.72
43.13672
30.85655
24.47113
92.35837
43.44915
95.88593
83.58583
Best
11.50133
21.77054
38.35566
43.02007
16.53439
26.94007
37.38023
24.20705
23.65659
60.61512
36.84021
80.57012
82.50637
Worst
19.55286
31.36502
48.54746
66.67674
21.38723
33.13118
47.4267
35.44095
25.12117
109.1037
51.18962
105.8063
85.16528
Std
4.197797
4.774252
4.606404
10.95576
2.422862
2.794893
4.789876
5.183775
0.654518
22.87719
6.228956
11.89961
1.183616
Median
16.72317
26.66405
43.6622
61.42693
19.6393
31.40438
43.86997
31.8891
24.55339
99.8573
42.88339
98.58364
83.33583
Rank
1
4
9
10
2
5
7
6
3
12
8
13
11
Sum rank
22
197
111
242
62
150
150
128
100
234
166
206
225
Mean rank
1.00E+00
8.95E+00
5.05E+00
1.10E+01
2.82E+00
6.82E+00
6.82E+00
5.82E+00
4.55E+00
1.06E+01
7.55E+00
9.36E+00
1.02E+01
Total rank
1
2
12
4
13
3
11
9
6
7
10
5
8
Wilcoxon: p-value
3.20E-18
1.80E-15
6.43E-19
7.23E-07
2.01E-18
2.17E-18
6.61E-15
7.92E-16
1.38E-18
3.31E-18
6.43E-19
9.48E-19
Boxplot diagrams of FNO and competitor algorithms performances on CEC 2011 test suite
Based on the optimization results, FNO emerges as the top-performing optimizer for optimization problems C11-F1 to C11-F22, demonstrating its adeptness in exploration, exploitation, and maintaining a balance between the two throughout the search process. The simulation results indicate FNO’s superior performance in handling the CEC 2011 test suite, achieving superior results across most optimization problems and securing the top rank as the first best optimizer compared to competitor algorithms. Additionally, the p-value results obtained using the Wilcoxon rank sum test affirm that FNO exhibits a statistically significant superiority over all twelve competitor algorithms in handling the CEC 2011 test suite.
Pressure Vessel Design Problem
Designing pressure vessels is a pertinent optimization task in real-world applications, as illustrated in Fig. 13, with the objective of minimizing construction costs. The mathematical model for this design is outlined as follows [63]:
The implementation outcomes of FNO and rival algorithms in addressing pressure vessel design are detailed in Tables 12 and 13. Fig. 14 illustrates the convergence curve of FNO during the attainment of the optimal design for this problem.
Performance of optimization algorithms on speed reducer design problem
Algorithm
Optimum variables
Optimum cost
Ts
Th
R
L
FNO
0.7781686
0.3846492
40.319619
200
5885.3269
WSO
0.7782309
0.38468
40.322845
200
5885.3325
AVOA
0.7782526
0.3846907
40.323967
199.94929
5885.34
RSA
0.8539516
0.4168657
40.388059
200
8086.5619
MPA
0.7782309
0.38468
40.322845
200
5885.3325
TSA
0.7798201
0.3858965
40.399775
200
5916.3803
WOA
0.8129108
0.5410562
40.39966
198.94944
6340.3278
MVO
0.8182678
0.4062317
42.356098
173.54905
6027.1791
GWO
0.7785163
0.3856561
40.33039
199.9589
5893.9046
TLBO
1.1979804
1.2640954
61.061039
91.748927
11660.68
GSA
0.9570947
0.4737652
49.585704
145.01146
13041.109
PSO
1.2768702
2.3223385
50.651074
110.16225
10712.216
GA
1.1435231
0.780001
54.789155
96.522722
11793.11
Statistical results of optimization algorithms on speed reducer design problem
Algorithm
Mean
Best
Worst
Std
Median
Rank
FNO
5885.3269
5885.3269
5885.3269
2.06E-12
5885.3269
1
WSO
5895.0956
5885.3325
5981.6596
28.716785
5885.3329
3
AVOA
6280.1363
5885.34
7249.7752
455.40456
6078.6009
5
RSA
13539.742
8086.5619
22432.026
4041.5649
12359.627
9
MPA
5885.3325
5885.3325
5885.3325
4.76E-06
5885.3325
2
TSA
6340.6443
5916.3803
7134.9108
430.58947
6191.0944
6
WOA
8366.6396
6340.3278
14003.934
2173.7628
7875.2374
8
MVO
6630.2874
6027.1791
7254.5665
413.99521
6693.7212
7
GWO
6037.1687
5893.9046
6809.5978
309.39299
5903.6843
4
TLBO
32144.537
11660.68
69718.639
17830.143
28276.87
12
GSA
23196.473
13041.109
36638.786
8674.4726
22242.768
10
PSO
33803.139
10712.216
58460.668
16692.36
37347.025
13
GA
28807.292
11793.11
52384.008
13995.666
25433.758
11
FNO’s performance convergence curve on pressure vessel design
According to the findings, FNO has delivered the optimal design with design variable values of (0.7781686, 0.3846492, 40.319619, 200), and an objective function value of 5885.3269. Examination of the simulation outcomes highlights the superior performance of FNO in handling pressure vessel design compared to rival algorithms.
Speed Reducer Design Problem
The optimization task of speed reducer design finds practical application in real-world scenarios, as depicted in Fig. 15, with the objective of minimizing the weight of the speed reducer. The mathematical model for this design is outlined as follows [64,65]:
With
2.6≤x1≤3.6,0.7≤x2≤0.8,17≤x3≤28,7.3≤x4≤8.3,7.8≤x5≤8.3,2.9≤x6≤3.9,and5≤x7≤5.5.
Schematic of speed reducer design
The outcomes of employing FNO and rival algorithms to optimize speed reducer design are presented in Tables 14 and 15. Fig. 16 illustrates the convergence curve of FNO during the attainment of the optimal design for this problem.
Performance of optimization algorithms on speed reducer design problem
Algorithm
Optimum variables
Optimum cost
b
M
p
l1
l2
d1
d2
FNO
3.5
0.7
17
7.3
7.8
3.3502147
5.2866832
2996.348165
WSO
3.5002244
0.7
17.000002
7.4136937
7.8000061
3.3518435
5.2873567
2998.2849
AVOA
3.5000092
0.7
17.000629
7.4084741
7.8136361
3.3523231
5.2868134
2998.3407
RSA
3.5531625
0.7
17
7.9230907
8.076683
3.3562084
5.3978997
3104.2319
MPA
3.5002366
0.7
17
7.3018979
7.8
3.3552821
5.287468
2998.2543
TSA
3.5089329
0.7
17
7.3851797
8.0837291
3.3509261
5.2895882
3008.8748
WOA
3.5500856
0.7
17.002304
7.3248187
7.9504512
3.3629435
5.2871019
3023.4616
MVO
3.5015097
0.7
17
7.3018979
7.9521085
3.3662389
5.2875803
3004.9744
GWO
3.5005991
0.7
17
7.3048057
7.8
3.3630462
5.2886704
3001.1749
TLBO
3.532811
0.70226
22.271331
7.9247095
8.0775428
3.5287298
5.3165585
4286.1391
GSA
3.5131765
0.7015566
17.208703
7.7076307
7.8506631
3.3849498
5.3434699
3096.3026
PSO
3.5047988
0.7000407
17.619454
7.3913565
7.9493891
3.4917496
5.3195215
3173.2805
GA
3.5441317
0.7031465
17.460113
7.6348954
7.8478162
3.5515621
5.3215997
3197.0824
Statistical results of optimization algorithms on speed reducer design problem
Algorithm
Mean
Best
Worst
Std
Median
Rank
FNO
2996.348165
2996.348165
2996.348165
1.03E-12
2996.348165
1
WSO
2999.5614
2998.2849
3001.6023
1.01E+00
2999.5448
3
AVOA
3001.9497
2998.3407
3007.4644
2.8969711
3001.046
4
RSA
3157.9524
3104.2319
3192.6152
36.072081
3166.2611
9
MPA
2999.4011
2998.2543
3001.6022
1.0196094
2999.21
2
TSA
3019.6328
3008.8748
3026.7945
6.23825
3020.9355
7
WOA
3086.3025
3023.4616
3252.0021
65.814052
3067.2525
8
MVO
3018.3269
3004.9744
3040.5666
8.312502
3018.4292
6
GWO
3004.0787
3001.1749
3009.6522
2.5818026
3003.5947
5
TLBO
3.93E+13
4286.1391
2.85E+14
7.21E+13
1.54E+13
12
GSA
3258.5133
3096.3026
3611.8294
163.63956
3185.717
10
PSO
5.80E+13
3173.2805
2.94E+14
7.72E+13
4.15E+13
13
GA
2.79E+13
3197.0824
1.80E+14
4.85E+13
1.12E+13
11
FNO’s performance convergence curve on speed reducer design
According to the findings, FNO has yielded the optimal design with design variable values of (3.5, 0.7, 17, 7.3, 7.8, 3.3502147, 5.2866832), and an objective function value of 2996.348165. Analysis of the simulation results underscores the superior performance of FNO in addressing speed reducer design when compared to rival algorithms.
Welded Beam Design
The optimization challenge of welded beam design holds significance in real-world applications, as depicted in Fig. 17, with the aim of minimizing the fabrication cost of the welded beam. The mathematical model for this design is articulated as follows [24]:
Tables 16 and 17 present the outcomes of addressing welded beam design using FNO alongside rival algorithms. Fig. 18 illustrates the convergence curve of FNO as it achieves the optimal design for this problem.
Performance of optimization algorithms on welded beam design problem
Algorithm
Optimum variables
Optimum cost
h
l
t
b
FNO
0.2057296
3.4704887
9.0366239
0.2057296
1.7246798
WSO
0.2056788
3.4716602
9.0364826
0.2057551
1.7251006
AVOA
0.2049682
3.4877457
9.0372012
0.2057455
1.7262173
RSA
0.2004819
3.5100311
9.5392895
0.2125585
1.8668773
MPA
0.2056789
3.471653
9.0364823
0.2057551
1.7250999
TSA
0.2047591
3.4897943
9.0521996
0.2059867
1.7306038
WOA
0.2097327
3.4067416
9.0014555
0.2143872
1.7809749
MVO
0.2050577
3.4851199
9.0428067
0.2059109
1.7282199
GWO
0.2056011
3.473437
9.0362654
0.2057942
1.7254793
TLBO
0.2675139
4.0119186
7.7707622
0.3297252
2.4592782
GSA
0.2554716
3.0475935
8.1263568
0.2634856
1.9284377
PSO
0.2996899
3.4546043
8.0800161
0.413847
3.0245026
GA
0.215722
5.4287681
8.3186692
0.2614805
2.3115269
Statistical results of optimization algorithms on welded beam design problem
Algorithm
Mean
Best
Worst
Std
Median
Rank
FNO
1.7246798
1.7246798
1.7246798
2.51E-16
1.7246798
1
WSO
1.7257138
1.7250766
1.7272081
5.54E-04
1.7256231
3
AVOA
1.7462601
1.7261934
1.791771
0.0226569
1.7383159
7
RSA
1.983814
1.8668534
2.1817364
0.0900392
1.9701757
8
MPA
1.7257136
1.725076
1.727205
0.0005539
1.7256231
2
TSA
1.7360523
1.7305798
1.7423464
0.0037193
1.736282
6
WOA
2.0567587
1.7809509
3.0388629
0.3998829
1.9294051
9
MVO
1.7349638
1.7281959
1.7555686
0.0088849
1.7323746
5
GWO
1.7270698
1.7254554
1.7308463
0.0014025
1.7268408
4
TLBO
1.88E+13
2.4592542
1.81E+14
5.05E+13
3.966695
12
GSA
2.1319031
1.9284138
2.3060492
0.1190564
2.1487315
10
PSO
2.59E+13
3.0244787
1.57E+14
5.45E+13
4.55E+00
13
GA
6.36E+12
2.311503
6.89E+13
2.15E+13
3.9480164
11
FNO’s performance convergence curve on welded beam design
As per the results obtained, FNO has delivered the optimal design with design variable values of (0.2057296, 3.4704887, 9.0366239, 0.2057296), and an objective function value of 1.7246798. Analysis of the simulation outcomes reveals the superior performance of FNO in optimizing welded beam design compared to rival algorithms.
Tension/Compression Spring Design
The optimization challenge of tension/compression spring design holds practical relevance in real-world scenarios, as depicted in Fig. 19, with the goal of minimizing the weight of the tension/compression spring. The mathematical model for this design is articulated as follows [24]:
Tables 18 and 19 depict the outcomes of employing FNO and competitor algorithms to address tension/compression spring design. Fig. 20 showcases the convergence curve of FNO as it attains the optimal design for this problem.
Schematic of tension/compression spring design
Performance of optimization algorithms on tension/compression spring design problem
Algorithm
Optimum variables
Optimum cost
d
D
P
FNO
0.0516891
0.3567177
11.288966
0.0126019
WSO
0.0517981
0.3593483
11.140038
0.0126672
AVOA
0.050839
0.336765
12.627845
0.0126879
RSA
0.0508249
0.3330204
13.213587
0.012955
MPA
0.0517857
0.3590499
11.157505
0.0126672
TSA
0.0507372
0.3343296
12.794302
0.0126975
WOA
0.0508192
0.3363009
12.661049
0.0126892
MVO
0.0502892
0.3238205
13.591003
0.0127312
GWO
0.0519385
0.3627343
10.94956
0.0126703
TLBO
0.0601787
0.6456368
7.3809506
0.0154035
GSA
0.0530532
0.3911432
10.25442
0.0129158
PSO
0.0608052
0.6596202
6.3144359
0.015329
GA
0.0624287
0.7002184
4.9706342
0.0157093
Statistical results of optimization algorithms on tension/compression spring design problem
Algorithm
Mean
Best
Worst
Std
Median
Rank
FNO
0.0126019
0.0126019
0.0126019
7.58E-18
0.0126019
1
WSO
0.0126837
0.0126585
0.012766
3.05E-05
0.0126775
3
AVOA
0.0130553
0.0126792
0.013507
0.0003454
0.0130109
8
RSA
0.0130014
0.0129462
0.0130806
4.525E-05
0.0129959
6
MPA
0.0126775
0.0126585
0.0127587
2.218E-05
0.0126765
2
TSA
0.0128432
0.0126887
0.0131486
0.0001479
0.0128097
5
WOA
0.013016
0.0126804
0.0137001
0.0003669
0.0129008
7
MVO
0.0148014
0.0127224
0.0155865
0.0010085
0.0153132
9
GWO
0.0127096
0.0126615
0.0129154
5.616E-05
0.0127073
4
TLBO
1.57E-02
0.0153948
1.61E-02
2.26E-04
0.0156604
10
GSA
0.0164459
0.012907
0.0234955
0.0026141
0.0162001
11
PSO
1.17E+13
0.0153202
2.07E+14
5.10E+13
1.53E-02
13
GA
9.11E+11
0.0157005
9.43E+12
3.00E+12
0.0198638
12
FNO’s performance convergence curve on tension/compression spring
According to the findings, FNO has achieved the optimal design with design variable values of (0.0516891, 0.3567177, 11.288966), and an objective function value of 0.0126019. Analysis of the simulation results underscores the superior performance of FNO in handling tension/compression spring design compared to competitor algorithms.
Conclusions and Future Works
This paper introduced a novel metaheuristic algorithm called Far and Near Optimization, tailored for addressing optimization challenges across various scientific domains. FNO’s core concept drew inspiration from global and local search strategies, dynamically updating population members within the problem-solving space by navigating towards both the farthest and nearest members, respectively. The theoretical underpinnings of FNO were elucidated and mathematically formulated in two distinct phases: (i) exploration based on the simulation of the movement of the population member towards the farthest member from itself and (ii) exploitation based on simulating the movement of the population member towards nearest member from itself. FNO’s performance was rigorously assessed on the CEC 2017 test suite across problem dimensions of 10, 30, 50, and 100, as well as to address CEC 2020. The optimization results underscored FNO’s adeptness in exploration, exploitation, and maintaining a delicate balance between the two throughout the search process, yielding effective solutions across various benchmark functions. The effectiveness of FNO in addressing optimization problems was assessed by comparing it with twelve widely recognized metaheuristic algorithms. The results of the simulation and statistical analysis indicated that FNO consistently outperformed most competitor algorithms across various benchmark functions, securing the top rank as the premier optimizer. This superiority is statistically significant and underscores FNO’s exceptional performance. Furthermore, FNO’s efficacy in real-world applications was evaluated through the resolution of twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems. The results demonstrated FNO’s effectiveness in tackling optimization tasks in practical scenarios.
While the FNO algorithm has shown promise in balancing exploration and exploitation, it has several shortcomings, including computational complexity, parameter sensitivity, risk of premature convergence, complexity in mathematical modeling, and limited experimental validation. Potential improvements include enhancing computational efficiency, adopting adaptive parameter tuning, mitigating premature convergence through hybridization and diversification strategies, simplifying the mathematical modeling, and conducting extensive experimental validation. These enhancements can help make FNO a more robust, efficient, and widely applicable optimization algorithm.
The introduction of the proposed FNO approach presents numerous avenues for future research endeavors. One particularly noteworthy proposal is the development of binary and multi-objective versions of FNO. Additionally, exploring the utilization of FNO for solving optimization problems across various scientific disciplines and real-world applications stands as another promising research direction.
None.
Funding Statement
The authors received no specific funding for this study.
Author Contributions
The authors confirm contribution to the paper as follows: study conception and design: Tareq Hamadneh, Khalid Kaabneh, Kei Eguchi; data collection: Tareq Hamadneh, Zeinab Monrazeri, Omar Alssayed, Kei Eguchi, Mohammad Dehghani; analysis and interpretation of results: Khalid Kaabneh, Omar Alssayed, Zeinab Monrazeri, Tareq Hamadneh, Mohammad Dehghani; draft manuscript preparation: Omar Alssayed, Zeinab Monrazeri, Kei Eguchi, Mohammad Dehghani. All authors reviewed the results and approved the final version of the manuscript.
Availability of Data and Materials
All data are available in the article and their references are cited.
Ethics Approval
Not applicable.
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.
ReferencesAl-NanaAA, BatihaIM, MomaniS. A numerical approach for dealing with fractional boundary value problems. Mathematics. 2023;11(19):4082. doi:10.3390/math11194082.HamadnehT, AthanasopoulosN, AliM. Minimization and positivity of the tensorial rational Bernstein form. In: 2019 IEEE Jordan International Joint Conference on Electrical Engineering and Information Technology (JEEIT), 2019; Amman, Jordan; IEEE.SergeyevYD, KvasovD, MukhametzhanovM. On the efficiency of nature-inspired metaheuristics in expensive global optimization with limited budget. Sci Rep. 2018;8(1):1–9. doi:10.1038/s41598-017-18940-4; 29323223QawaqnehH. New contraction embedded with simulation function and cyclic (α, β)-admissible in metric-like spaces. Int J Math Comput Sci. 2020;15(4):1029–44.AlshantiWG, BatihaIM, HammadMMA, KhalilR. A novel analytical approach for solving partial differential equations via a tensor product theory of Banach spaces. Partial Differ Equ Appl Math. 2023;8(2):100531. doi:10.1016/j.padiff.2023.100531.LiñánDA, Contreras-ZarazúaG, Sánchez-RamírezE, Segovia-HernándezJG, Ricardez-SandovalLA. A hybrid deterministic-stochastic algorithm for the optimal design of process flowsheets with ordered discrete decisions. Comput Chem Eng. 2024;180:108501. doi:10.1016/j.compchemeng.2023.108501.de ArmasJ, Lalla-RuizE, TilahunSL, VoßS. Similarity in metaheuristics: a gentle step towards a comparison methodology. Nat Comput. 2022;21(2):265–87. doi:10.1007/s11047-020-09837-9.ZhaoW, WangL, ZhangZ, FanH, ZhangJ, MirjaliliS, et al. Electric eel foraging optimization: a new bio-inspired optimizer for engineering applications. Expert Syst Appl. 2024;238:122200. doi:10.1016/j.eswa.2023.122200.KumarG, SahaR, ContiM, DevgunT, ThomasR. GREPHRO: nature-inspired optimization duo for Internet-of-Things. Int Things. 2024;25:101067. doi:10.1016/j.iot.2024.101067.WolpertDH, MacreadyWG. No free lunch theorems for optimization. IEEE Trans Evol Comput. 1997;1(1):67–82.AlsayyedO, HamadnehT, Al-TarawnehH, AlqudahM, GochhaitS, LeonovaI, et al. Giant armadillo optimization: a new bio-inspired metaheuristic algorithm for solving optimization problems. Biomimetics. 2023;8(8):619. doi:10.3390/biomimetics8080619; 38132558KennedyJ, EberhartR. Particle swarm optimization. In: Proceedings of ICNN’95-International Conference on Neural Networks, 1995 Nov 27–Dec 1; Perth, WA, Australia; IEEE.KarabogaD, BasturkB. Artificial bee colony (ABC) optimization algorithm for solving constrained optimization problems. In: International Fuzzy Systems Association World Congress, 2007; Berlin, Heidelberg; Springer.DorigoM, ManiezzoV, ColorniA. Ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern B Cybern. 1996;26(1):29–41. doi:10.1109/3477.484436; 18263004YangXSed. Firefly algorithms for multimodal optimization. In: International Symposium on Stochastic Algorithms, 2009; Berlin, Heidelberg; Springer.WangWC, TianWC, XuDM, ZangHF. Arctic puffin optimization: a bio-inspired metaheuristic algorithm for solving engineering design optimization. Adv Eng Softw. 2024;195:103694. doi:10.1016/j.advengsoft.2024.103694.JiaH, PengX, LangC. Remora optimization algorithm. Expert Syst Appl. 2021;185:115665. doi:10.1016/j.eswa.2021.115665.WuD, RaoH, WenC, JiaH, LiuQ, AbualigahL. Modified sand cat swarm optimization algorithm for solving constrained engineering optimization problems. Math. 2022;10(22):4350. doi:10.3390/math10224350.SeyyedabbasiA, KianiF. Sand cat swarm optimization: a nature-inspired algorithm to solve global optimization problems. Eng Comput. 2023;39(4):2627–51. doi:10.1007/s00366-022-01604-x.AbdollahzadehB, GharehchopoghFS, MirjaliliS. African vultures optimization algorithm: a new nature-inspired metaheuristic algorithm for global optimization problems. Comput Ind Eng. 2021;158:107408. doi:10.1016/j.cie.2021.107408.ChopraN, AnsariMM. Golden jackal optimization: a novel nature-inspired optimizer for engineering applications. Expert Syst Appl. 2022;198:116924. doi:10.1016/j.eswa.2022.116924.BraikM, HammouriA, AtwanJ, Al-BetarMA, AwadallahMA. White Shark Optimizer: a novel bio-inspired meta-heuristic algorithm for global optimization problems. Knowl-Based Syst. 2022;243:108457. doi:10.1016/j.knosys.2022.108457.KaurS, AwasthiLK, SangalAL, DhimanG. Tunicate Swarm Algorithm: a new bio-inspired based metaheuristic paradigm for global optimization. Eng Appl Artif Intell. 2020;90:103541. doi:10.1016/j.engappai.2020.103541.MirjaliliS, LewisA. The whale optimization algorithm. Adv Eng Softw. 2016;95:51–67.MirjaliliS, MirjaliliSM, LewisA. Grey wolf optimizer. Adv Eng Softw. 2014;69:46–61. doi:10.1016/j.advengsoft.2016.01.008.FaramarziA, HeidarinejadM, MirjaliliS, GandomiAH. Marine Predators Algorithm: a nature-inspired metaheuristic. Expert Syst Appl. 2020;152:113377. doi:10.1016/j.eswa.2020.113377.HashimFA, HousseinEH, HussainK, MabroukMS, Al-AtabanyW. Honey Badger Algorithm: new metaheuristic algorithm for solving optimization problems. Math Comput Simul. 2022;192:84–110. doi:10.1016/j.matcom.2021.08.013.AbualigahL, Abd ElazizM, SumariP, GeemZW, GandomiAH. Reptile Search Algorithm (RSA): a nature-inspired meta-heuristic optimizer. Expert Syst Appl. 2022;191:116158. doi:10.1016/j.eswa.2021.116158.GoldbergDE, HollandJH. Genetic algorithms and machine learning. Mach Learn. 1988;3(2):95–9. doi:10.1023/A:1022602019183.StornR, PriceK. Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim. 1997;11(4):341–59. doi:10.1023/A:1008202821328.De CastroLN, TimmisJI. Artificial immune systems as a novel soft computing paradigm. Soft Comput. 2003;7(8):526–44. doi:10.1007/s00500-002-0237-z.KirkpatrickS, GelattCD, VecchiMP. Optimization by simulated annealing. Science. 1983;220(4598):671–80. doi:10.1126/science.220.4598.671; 17813860DehghaniM, MontazeriZ, DhimanG, MalikO, Morales-MenendezR, Ramirez-MendozaRA, et al. A spring search algorithm applied to engineering optimization problems. Appl Sci. 2020;10(18):6173. doi:10.3390/app10186173.DehghaniM, SametH. Momentum search algorithm: a new meta-heuristic optimization algorithm inspired by momentum conservation law. SN Appl Sci. 2020;2(10):1–15. doi:10.1007/s42452-020-03511-6.RashediE, Nezamabadi-PourH, SaryazdiS. GSA: a gravitational search algorithm. Inf Sci. 2009;179(13):2232–48. doi:10.1016/j.ins.2009.03.004.MirjaliliS, MirjaliliSM, HatamlouA. Multi-verse optimizer: a nature-inspired algorithm for global optimization. Neural Comput Appl. 2016;27(2):495–513. doi:10.1007/s00521-015-1870-7.HashimFA, HussainK, HousseinEH, MabroukMS, Al-AtabanyW. Archimedes optimization algorithm: a new metaheuristic algorithm for solving optimization problems. Appl Intell. 2021;51(3):1531–51. doi:10.1007/s10489-020-01893-z.KavehA, DadrasA. A novel meta-heuristic optimization algorithm: thermal exchange optimization. Adv Eng Softw. 2017;110:69–84. doi:10.1016/j.advengsoft.2017.03.014.CuevasE, OlivaD, ZaldivarD, Pérez-CisnerosM, SossaH. Circle detection using electro-magnetism optimization. Inf Sci. 2012;182(1):40–55. doi:10.1016/j.ins.2010.12.024.EskandarH, SadollahA, BahreininejadA, HamdiM. Water cycle algorithm–a novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput Struct. 2012;110:151–66. doi:10.1016/j.compstruc.2012.07.010.HatamlouA. Black hole: a new heuristic optimization approach for data clustering. Inf Sci. 2013;222:175–84. doi:10.1016/j.ins.2012.08.023.FaramarziA, HeidarinejadM, StephensB, MirjaliliS. Equilibrium optimizer: a novel optimization algorithm. Knowl-Based Syst. 2020;191:105190. doi:10.1016/j.knosys.2019.105190.PereiraJLJ, FranciscoMB, PereiraHD, AbrahamA. Elephant search algorithm: a bio-inspired metaheuristic algorithm for optimization problems. Memet Comput. 2020;12(4):375–88. doi:10.1109/ICDIM.2015.7381893.BouchekaraHREH, BelazzougM, KouzouA, TlemçaniC. HHO–A novel Harris Hawks Optimization algorithm for solving engineering optimization problems. Appl Intell. 2022;52(3):3484–520.Santos CoelhoL, MarianiVC. Firefly algorithm approach based on chaotic Tinkerbell map applied to multivariable PID controller tuning. Comput Math Appl. 2013;64(8):2371–82. doi:10.1016/j.camwa.2012.05.007.LinY, GenM. Auto-tuning strategy for parameter settings of meta-heuristics based on ordinal transformation and scale transformation. Expert Syst Appl. 2009;36(4):7461–71.BlumC, RoliA. Metaheuristics in combinatorial optimization: overview and conceptual comparison. ACM Comput Surv. 2003;35(3):268–308.NguyenTT, MemarmoghaddamH, AbrahamA, ZhangM. Metaheuristics: advances and trends. J Ambient Intell Humaniz Comput. 2020;11:853–5.BraikM, RyalatMH, Al-ZoubiH. A novel meta-heuristic algorithm for solving numerical optimization problems: Ali Baba and the forty thieves. Neural Comput Appl. 2022;34(1):409–55. doi:10.1007/s00521-021-06392-x.GloverF. Tabu search—part I. ORSA J Comput. 1989;1(3):190–206. doi:10.1287/ijoc.1.3.190.Salcedo-SanzS, Del SerJ, Landa-TorresI, Gil-LopezS, Portilla-FiguerasA. The coral reefs optimization algorithm: a novel metaheuristic for efficiently solving optimization problems. Sci World J. 2014;2014:1–15. doi:10.1155/2014/739768; 25147860Coello CoelloCA. Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput Methods Appl Mech Eng. 2002;191(11–12):1245–87. doi:10.1016/S0045-7825(01)00323-1.AliMZ, AwadNH, SuganthanPN. Problem definitions and evaluation criteria for the CEC 2017 special session and competition on real-parameter optimization. J Simul. 2017;32(1):1–35. doi:10.1007/s11548-006-0027-7.GloverF. Heuristics for integer programming using surrogate constraints. Decis Sci. 1977;8(1):156–66. doi:10.1111/j.1540-5915.1977.tb01074.x.PriceKV, StornRM, LampinenJA. Differential evolution: a practical approach to global optimization. Springer Berlin, Heidelberg: Springer; 2006.GaoW, LiuS. A modified artificial bee colony algorithm. Comput Oper Res. 2011;39(3):687–97. doi:10.1016/j.cor.2011.06.007.LukeS. Essentials of metaheuristics. 2nd ed. San Francisco, CA, USA: Lulu; 2013.BäckT, FogelDB, MichalewiczZ. Handbook of evolutionary computation. Boca Raton: IOP Publishing Ltd.; 1997.SimonD. Evolutionary optimization algorithms: biologically-inspired and population-based approaches to computer intelligence. Hoboken, New Jersey: Wiley; 2013.GoldbergDE. Genetic algorithms in search, optimization, and machine learning. Boston, USA: Addison-Wesley; 1989.YaoX. Progress in evolutionary computation. Springer Berlin, Heidelberg: Springer; 1995.BonabeauE, DorigoM, TheraulazG. Swarm intelligence: from natural to artificial systems. New York, USA: Oxford University Press; 1999.KannanB, KramerSN. An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design. J Mech Des. 1994;116(2):405–11. doi:10.1115/1.2919393.WolpertDH, MacreadyWG. Coevolutionary free lunches. IEEE Trans Evol Comput. 2005;9(6):721–35. doi:10.1109/TEVC.2005.856205.HansenN, OstermeierA. Completely derandomized self-adaptation in evolution strategies. Evol Comput. 2001;9(2):159–95. doi:10.1162/106365601750190398; 11382355