This research presents a novel nature-inspired metaheuristic algorithm called Frilled Lizard Optimization (FLO), which emulates the unique hunting behavior of frilled lizards in their natural habitat. FLO draws its inspiration from the sit-and-wait hunting strategy of these lizards. The algorithm’s core principles are meticulously detailed and mathematically structured into two distinct phases: (i) an exploration phase, which mimics the lizard’s sudden attack on its prey, and (ii) an exploitation phase, which simulates the lizard’s retreat to the treetops after feeding. To assess FLO’s efficacy in addressing optimization problems, its performance is rigorously tested on fifty-two standard benchmark functions. These functions include unimodal, high-dimensional multimodal, and fixed-dimensional multimodal functions, as well as the challenging CEC 2017 test suite. FLO’s performance is benchmarked against twelve established metaheuristic algorithms, providing a comprehensive comparative analysis. The simulation results demonstrate that FLO excels in both exploration and exploitation, effectively balancing these two critical aspects throughout the search process. This balanced approach enables FLO to outperform several competing algorithms in numerous test cases. Additionally, FLO is applied to twenty-two constrained optimization problems from the CEC 2011 test suite and four complex engineering design problems, further validating its robustness and versatility in solving real-world optimization challenges. Overall, the study highlights FLO’s superior performance and its potential as a powerful tool for tackling a wide range of optimization problems.
In optimization, the aim is to determine a best solution from a set of options for a specific problem [1]. Mathematically, optimization problems consist of decision variables, constraints, and one or several objective functions. The objective is to assign appropriate values to the decision variables in order to maximize or minimize the objective function while adhering to the problem’s constraints [2]. Regarding such optimization problems, problem solving techniques can be partitioned into two main groups: deterministic and stochastic approaches [3]. Deterministic methods are particularly useful for solving linear, convex, low-dimensional, continuous, and differentiable problems [4]. However, as problems become more intricate and dimensions increase, deterministic approaches may struggle with being trapped in local optima and providing suboptimal solutions [5]. Conversely, within science, engineering, industry, technology, and practical applications, numerous intricate optimization problems exist that are characterized as non-convex, non-linear, discontinuous, non-differentiable, complex, and high-dimensional. Due to the inefficiencies and challenges associated with deterministic methods in addressing these optimization issues, scientists have turned to developing stochastic approaches [6].
Metaheuristic algorithms stand out as highly effective stochastic methods capable of offering viable solutions for optimization challenges, all without requiring derivative information. They rely on random exploration within the solution space, utilizing random operators and trial-and-error strategies. Their advantages include straightforward concepts, straightforward implementation, proficiency in tackling varied optimization problems, no matter how complex or high-dimensional they may be, as well as adaptability to nonlinear and unfamiliar search spaces. As a result, the popularity and extensive use of metaheuristic algorithms continue to grow [7]. In metaheuristic algorithms, the optimization process begins by randomly generating a set of candidate solutions at the start of the algorithm. These candidate solutions are then enhanced and modified by the algorithm during a certain number of iterations following its implementation steps. Upon completion of the algorithm, the best candidate solution found during its execution is put forward as the proposed solution to the problem [8]. This random search element in metaheuristic algorithms means that achieving a global optimum cannot be guaranteed using these methods. Nonetheless, the solutions derived from these algorithms, being near the global optimum, are deemed acceptable as quasi-optimal solutions [9].
For a metaheuristic algorithm to effectively carry out the optimization process, it needs to thoroughly explore the solution space on both a global and local scale. Global searching, through exploration, allows the algorithm to pinpoint the optimal area by extensively surveying all parts of the search space and avoiding narrow solutions. Local searching, through exploitation, helps the algorithm converge to solutions near a global optimum by carefully examining surrounding areas and promising solutions. Success in the optimization process hinges on striking a balance between exploration and exploitation during the search [10]. Researchers’ desire to improve optimization outcomes has resulted in the development of many metaheuristic algorithms. These metaheuristic algorithms are employed to deal with optimization tasks in various sciences and applications such as: engineering [11], data mining [12], wireless sensor networks [13], internet of things [14], etc.
The key question at hand is whether, based on the available metaheuristic algorithms, there remains a need in scientific research to develop new metaheuristic algorithms. The concept of No Free Lunch (NFL) [15] addresses this by highlighting that while a metaheuristic algorithm may perform well in solving a particular set of optimization problems, it might not guarantee the same solution quality for different optimization problems. The NFL theorem suggests that there is no one-size-fits-all optimal metaheuristic algorithm for all types of optimization problems. It is conceivable that an algorithm may efficiently reach a global optimum for one problem but struggle to do so for another, possibly getting stuck at a local optimum. As a result, the success or failure of employing a metaheuristic algorithm for an optimization problem cannot be definitively assumed.
The novelty of this paper is the introduction of a new innovative bio-metaheuristic algorithm called Frilled Lizard Optimization (FLO) to solve optimization problems in different research fields and real-world applications.
So far, several algorithms inspired by lizards have been introduced and designed. The strategy of Redheaded Agama lizards when hunting their prey has been the main idea of Artificial Lizard Search Optimization (ALSO). The concept originates from a recent study, where researchers observed that the lizards regulate the movement of their tails with precision, redirecting the angular momentum from their bodies to their tails. This action stabilizes their body position in the sagittal plane [16]. The Side-Blotched Lizard Algorithm (SBLA) is an algorithm inspired by lizards, the main idea in its design is derived from the mating process of these lizards as well as imitating their polymorphic population [17]. The Horned Lizard Optimization Algorithm (HLOA) is another algorithm inspired by lizards. The main source of inspiration in the HLOA design is derived from crypsis, skin darkening or lightening, blood-squirting, and move-to-escape defense methods [18]. As it is evident, although from the point of view of the type of living organism, all these algorithms are inspired by lizards, but they have major differences in the details and also in the mathematical model.
The proposed FLO approach is an algorithm derived from the frilled lizard. In order to design FLO, it is inspired by two characteristic behaviors among frilled lizards. The first behavior is related to the smart strategy of frilled lizards during hunting, which is called sit-and-wait hunting strategy. The second behavior is related to the strategy of frilled lizards when climbing trees after feeding. Based on the best knowledge obtained from the literature review, as well as the review of lizard-inspired algorithms, the originality of the proposed FLO approach is confirmed. This means that so far, no metaheuristic algorithm has been designed inspired by these intelligent behaviors of frilled lizards. Overall, it is confirmed that this is the first time that a new metaheuristic algorithm has been designed based on the modeling of frilled lizard’s intelligent behaviors including (i) sit-and-wait hunting strategy and (ii) climbing trees near the hunting site.
The main contributions of this investigation can be summarized as follows:
FLO is based on the imitation of the natural behavior of the frilled lizard in the wild.
The basic inspiration of FLO is taken from (i) the hunting strategy of the frilled lizard and (ii) the retreat of this animal to the top of the tree after feeding.
The concept behind FLO is outlined, and its procedural steps are mathematically formulated into two stages: (i) exploration, which replicates the frilled lizard’s predatory approach, and (ii) exploitation, which emulates the lizard’s withdrawal to safety atop the tree following a meal.
The performance of FLO has been tested on fifty-two standard benchmark functions of various types of unimodal, high-dimensional multimodal, fixed-dimensional multimodal as well as the CEC 2017 test suite.
FLO’s effectiveness has been tested on several practical challenges, including twenty-two constrained optimization issues from the CEC 2011 test suite and four engineering design tasks.
The outcomes produced by FLO are juxtaposed with those of alternative metaheuristic algorithms for performance comparison.
This paper is organized as follows: Section 2 contains a review of the relevant literature. Section 3 describes the proposed Frilled Lizard Optimization (FLO) and gives a mathematical model. Then Section 4 presents the results of our simulation studies. Section 5 investigates the effectiveness of FLO in solving real-world applications, and Section 6 provides some conclusions and suggestions for future research.
Literature Review
Metaheuristic algorithms are devised by drawing inspiration from a diverse array of sources, including natural phenomena, behaviors exhibited by living organisms in their natural habitats, principles from biological sciences, genetic mechanisms, physical laws, human behavior, and various evolutionary processes. These algorithms are typically classified into four distinct groups, each based on the specific inspiration behind its design. These categories include swarm-based approaches, evolutionary-based approaches, physics-based approaches, and human-based approaches, with each group leveraging different principles and mechanisms to guide the optimization process [19].
Swarm-based metaheuristic algorithms derive their principles from the collective behaviors and strategies observed in various natural systems, especially those exhibited by animals, aquatic creatures, and insects in their natural environments. These algorithms aim to mimic the efficient problem-solving strategies seen in nature to address complex optimization challenges. Among the most commonly used swarm-based metaheuristics are Particle Swarm Optimization (PSO) [20], Ant Colony Optimization (ACO) [21], Artificial Bee Colony (ABC) [22], and Firefly Algorithm (FA) [23]. PSO is inspired by the social behavior and collective movement patterns of birds flocking and fish schooling as they search for food. This algorithm simulates how individuals within a group adjust their positions based on personal experience and the success of their neighbors to find optimal solutions. ACO emulates the foraging behavior of ants, particularly how they find the shortest path between their nest and a food source by laying down and following pheromone trails. This method effectively models the way ants collectively solve complex routing problems through simple, local interactions. ABC is modeled after the foraging behavior of honey bees. It simulates how bees search for nectar, share information about food sources, and optimize their foraging strategies to maximize the efficiency of the colony. The Firefly Algorithm draws its inspiration from the bioluminescent communication of fireflies. It uses the concept of light intensity to simulate attraction among fireflies, guiding the search for optimal solutions through simulated social interactions. Furthermore, natural behaviors such as foraging, hunting, migration, digging, and chasing have inspired the development of various other metaheuristic algorithms. Examples include: Greylag Goose Optimization (GGO) [1], African Vultures Optimization Algorithm (AVOA) [24], Marine Predator Algorithm (MPA) [25], Gooseneck Barnacle Optimization Algorithm (GBOA) [26], Grey Wolf Optimizer (GWO) [27], Electric Eel Foraging Optimization (EEFO) [28], White Shark Optimizer (WSO) [29], Crested Porcupine Optimizer (CPO) [30], Tunicate Swarm Algorithm (TSA) [31], Orca Predation Algorithm (OPA) [32], Honey Badger Algorithm (HBA) [33], Reptile Search Algorithm (RSA) [34], Golden Jackal Optimization (GJO) [35], and Whale Optimization Algorithm (WOA) [36].
Evolutionary-based metaheuristic algorithms are inspired by fundamental principles from genetics, biology, natural selection, survival of the fittest, and Darwin’s theory of evolution. These algorithms model natural evolutionary processes to solve complex optimization problems effectively. Among the most notable examples in this category are Genetic Algorithm (GA) [37] and Differential Evolution (DE) [38], which have gained widespread popularity and adoption. Genetic Algorithms (GA) simulate the process of natural selection where the fittest individuals are selected for reproduction in order to produce the next generation. This algorithm involves mechanisms such as selection, crossover (recombination), and mutation to evolve solutions to optimization problems over successive generations. By mimicking biological evolution, GAs can efficiently explore and exploit the search space to find optimal or near-optimal solutions. Differential Evolution (DE) is another powerful evolutionary algorithm that optimizes a problem by iteratively trying to improve candidate solutions with regard to a given measure of quality. DE uses operations like mutation, crossover, and selection, drawing inspiration from the biological evolution and genetic variations observed in nature. The algorithm is particularly effective for continuous optimization problems due to its simple yet robust strategy. Additionally, Artificial Immune Systems (AIS) algorithms are inspired by the human immune system’s ability to defend against pathogens. AIS algorithms mimic the immune response process, learning and adapting to recognize and eliminate foreign elements. This approach provides robust mechanisms for optimization and anomaly detection [39]. Other prominent members of evolutionary-based metaheuristics include Genetic Programming (GP) [40], Cultural Algorithm (CA) [41], and Evolution Strategy (ES) [42]. These evolutionary-based metaheuristics emulate natural processes to harness the power of evolution, providing versatile and powerful tools for solving a wide range of optimization problems in various domains.
Physics-based metaheuristic algorithms are introduced, drawing inspiration from the modeling of forces, laws, phenomena, and other fundamental concepts in physics. Simulated Annealing (SA) [43], a widely employed physics-based metaheuristic algorithm, takes its design cues from the physical phenomenon of metal annealing. This process involves the melting of metals under heat, followed by a gradual cooling and freezing process to attain an ideal crystal structure. Gravitational Search Algorithm (GSA) [44] is crafted by modeling physical gravitational forces and applying Newton’s laws of motion. Concepts derived from cosmology and astronomy serve as the foundation for algorithms like Multi-Verse Optimizer (MVO) [45] and Black Hole Algorithm (BHA) [46]. Some other physics-based metaheuristic algorithms are: Thermal Exchange Optimization (TEO) [47], Prism Refraction Search (PRS) [48], Equilibrium Optimizer (EO) [49], Archimedes Optimization Algorithm (AOA) [50], Lichtenberg Algorithm (LA) [51], Water Cycle Algorithm (WCA) [52], and Henry Gas Optimization (HGO) [53].
Human-based metaheuristic algorithms draw inspiration from human behaviors, decisions, thoughts, and strategies observed in both individual and social contexts. These algorithms leverage the complexities and nuances of human actions to solve optimization problems effectively.
Teaching-Learning Based Optimization (TLBO) [54] is a prominent example of a human-based metaheuristic. TLBO simulates the teaching and learning processes in a classroom setting, modeling the interactions between teachers and students as well as peer learning among students. The algorithm improves solutions by mimicking the educational process where knowledge is imparted from teachers to students and shared among students, leading to enhanced performance and optimization outcomes. The Mother Optimization Algorithm (MOA) is inspired by the nurturing and caring behaviors exhibited by mothers, specifically modeled on Eshrat’s care for her children. This algorithm utilizes the principles of guidance, protection, and nurturing to iteratively improve solutions, reflecting the natural and effective strategies mothers use in raising their offspring [9]. War Strategy Optimization (WSO) takes its cue from the tactical and strategic movements of soldiers during ancient battles. By simulating various military strategies, formations, and maneuvers, WSO effectively explores and exploits the search space, providing robust solutions to complex problems [55]. Poor and Rich Optimization (PRO) [56] is designed based on the socioeconomic dynamics between the rich and the poor in society. This algorithm models the efforts of individuals to improve their financial and economic status, capturing the diverse strategies employed by different socioeconomic groups to achieve better outcomes. Some other human-based metaheuristic algorithms are: Coronavirus Herd Immunity Optimizer (CHIO) [57], Gaining Sharing Knowledge based Algorithm (GSK) [58], and Ali Baba and the Forty Thieves (AFT) [59].
The literature review highlights the absence of any metaheuristic algorithm that simulates the natural behavior of frilled lizards in their habitat. However, the strategic hunting and safety retreats employed by these lizards represent intelligent behaviors that hold potential for inspiring the development of a new optimizer. To fill this void, a novel metaheuristic algorithm has been created, drawing upon mathematical models of two primary behaviors exhibited by frilled lizards: predatory attacks and retreats to elevated positions, such as the top of a tree. This newly devised algorithm will be thoroughly discussed in the subsequent section.
Frilled Lizard Optimization
In this section, the source of inspiration used in the development and theory of Frilled Lizard Optimization (FLO) is stated. Then the corresponding implementation steps are mathematically modeled to be used for the solution of optimization problems.
Inspiration of FLO
The frilled lizard (Chlamydosaurus kingii) is a species of lizard from the family Agamidae, which is native to Southern New Guinea and Northern Australia [60]. The frilled lizard is an arboreal species and diurnal that spends more than 90% of each day up in the trees [61]. During the short time that this animal is on the ground, it is busy with feeding, socializing or traveling to a new tree [60]. A frilled lizard can move bipedally and do this when hunting or escaping from predators. To keep balanced, it leans its head far back enough, so it lines up behind the tail base [60,62]. The total length of the frilled lizard is about 90 centimeters, a head-body length of 27 centimeters, and weighs up to 600 grams [63]. The frilled lizard has a special wide and big head with a long neck to accommodate the frill. It has long legs for running and a tail that makes most of the total length of this animal [64]. The male species is larger than the female species and has proportionally bigger jaw, head, and frill [65]. A picture of a frilled lizard is shown in Fig. 1.
Frilled lizard taken from: free media wikimedia commons
The main diet of the frilled lizard are insects and other invertebrates, although it also rarely feeds on vertebrates. Prominent prey includes centipedes, ants, termites, and moth larvae [66]. The frilled lizard is a sit-and-wait predator that looks for potential prey. After seeing the prey, the frilled lizard runs fast on two legs and attacks the prey to catch it and feed on it. After feeding, the frilled lizard retreats back up a tree [60].
Among the frilled lizard’s natural behaviors, its sit-and-wait hunting strategy to catch prey and retreat to the top of the tree after feeding is much more prominent. These natural behaviors of frilled lizard are intelligent processes that are the fundamental inspiration in designing the proposed FLO approach.
Algorithm Initialization
The proposed FLO method is a metaheuristic algorithm that considers frilled lizards as its members. FLO efficiently discovers near-optimal solutions for optimization challenges by leveraging the search capabilities of its members within the problem-solving space. Each frilled lizard establishes value assignments for the decision variables according to its particular location in the problem-solving space. Consequently, every frilled lizard represents a potential solution that can be interpreted mathematically through a vector. Collectively, the frilled lizards constitute the FLO population, which can be mathematically described as a matrix using Eq. (1). The initial placements of the frilled lizards within the problem-solving space are established by a random initialization 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 denotes the FLO population matrix, Xi represents the ith frilled lizard (candidate solution), xi,d denotes its dth dimension in the search space (decision variable), N gives the number of frilled lizards, m denotes the number of decision variables, r represents a number randomly taken from the interval [0,1], lbd, and ubd are a lower bound as well as an upper bound on the dth decision variable, respectively.
Considering that each frilled lizard represents a candidate solution for the problem, corresponding to each candidate solution, the corresponding objective function value can be calculated for the problem. The set of determined objective function values can be represented mathematically using the vector given in Eq. (3):
F=[F1⋮Fi⋮FN]N×1=[F(X1)⋮F(Xi)⋮F(XN)]N×1
Here F denotes the vector of the calculated objective function values and Fi gives the evaluated objective function value corresponding to the ith frilled lizard.
The determined objective function values are appropriate criteria for measuring the quality of the population individuals (i.e., the candidate solutions). In particular, the best evaluated value for the objective function corresponds to the best individual of the population (i.e., the best candidate solution) and similarly, the worst evaluated value for the objective function corresponds to the worst individual of the population (i.e., the worst candidate solution). Since in each iteration of FLO, the position of the frilled lizards is updated in the solution space, new values are also evaluated for the objective function under consideration. Consequently, in each iteration the position of the best individual (i.e., the best candidate solution) must also be updated. At the end of the implementation of Algorithm FLO, the best candidate solution obtained during the iterations of the algorithm is taken as the solution to the problem.
Mathematical Modelling of FLO
In each iteration of Algorithm FLO, the position of the frilled lizard in the problem-solving space undergoes updating in two distinct phases. Firstly, the exploration phase simulates the frilled lizard’s movement towards prey during hunting, aimed at diversifying the search space and exploring new potential solutions. This phase allows the algorithm to probe different areas of the problem space, facilitating the discovery of novel regions that may contain optimal solutions.
Secondly, the exploitation phase simulates the movement of the frilled lizard towards the top of a tree after feeding. In this phase, the algorithm capitalizes on the knowledge gained during exploration to exploit promising regions identified as potential optimal solutions. By focusing on refining these regions, the exploitation phase aims to improve the quality of solutions and converge towards the global optimum.
Phase 1: Hunting Strategy (Exploration)
One of the most characteristic natural behaviors of the frilled lizard is the hunting strategy of this animal. The frilled lizard is a sit-and-wait predator that attacks its prey after seeing it. The simulation of frilled lizard’s movement towards the prey leads to extensive changes in the position of the population members in the problem-solving space and as a result increases the exploration power of the algorithm for global search. In the first phase of FLO, the position of the population individuals in the solution space of the problem is updated based on the frilled lizard’s hunting strategy. In the design of FLO, for each frilled lizard, the position of other population members who have a better objective function value is considered as the prey position. According to this, the set of candidate preys’ positions for each frilled lizard is determined using Eq. (4):
CPi={Xk:Fk<Fiandk≠i},wherei=1,2,…,Nandk∈{1,2,…,N}
Here CP is the candidate preys set for the ith frilled lizard, Xk is the population member with a better objective function value than the ith frilled lizard, and Fk is its objective function value.
In the FLO design, it is assumed that the frilled lizard randomly chooses one of these candidate preys and attacks it. Based on the modeling of the frilled lizard’s movement towards the chosen prey, a new position for each individual of the population has been calculated using Eq. (5). Then, if the objective function value is better, this new position replaces the previous position of the corresponding individual using Eq. (6):
xi,dP1=xi,d+r⋅(SPi,d−I⋅xi,d),i=1,2,…,N,andd=1,2,…,mXi={XiP1,FiP1<FiXi,else
Here XiP1 denotes the new suggested position of ith frilled lizard based on the first phase of FLO, xi,dP1 represents its dth dimension, FiP1 denotes its objective function value, r is a random number with a normal distribution from the interval [0,1], SPi,d denotes the dth dimension of the selected prey for the ith frilled lizard, I is a number randomly taken from the set {1,2}, N denotes the number of frilled lizards, and m gives the number of decision variables.
Phase 2: Moving Up the Tree (Exploitation)
After feeding, the frilled lizard retreats to the top of a tree near its position. Simulating the movement of the frilled lizard to the top of the tree leads to small changes in the position of the population individuals in the solution space of the problem and as a result, increasing the exploitation power of the algorithm for local search. In the second phase of FLO, the position of the population individuals in the solution space is updated based on the frilled lizard’s strategy when retreating to the top of the tree after feeding.
Based on modeling the movement of the frilled lizard to the top of the nearby tree, a new position for each population individual is calculated using Eq. (7). Then this new position, if it improves the objective function value, replaces the previous position of the corresponding individual using Eq. (8):
xi,dP2=xi,d+(1−2r)⋅(ubd−lbd)t,i=1,2,…,N,d=1,2,…,m,andt=1,2,…,TXi={XiP2,FiP2<FiXi,else
Here XiP2 denotes the new suggested position of the ith frilled lizard based on the second phase of FLO, xi,dP2 represents its dth dimension, FiP2 gives its objective function value, t represents the iteration counter of the algorithm, and T describes the maximum number of iterations of the algorithm.
Repetition Process, Pseudo-Code, and Flowchart of FLO
The initial iteration of the Frilled Lizard Optimization (FLO) algorithm concludes after updating the positions of all frilled lizards within the problem-solving space, following the execution of the first and second phases. Subsequently, armed with the newly updated values, the algorithm proceeds to commence the subsequent iteration, perpetuating the process of updating the frilled lizards’ positions until the algorithm reaches completion, guided by Eqs. (4) to (8). Throughout each iteration, the algorithm also maintains and updates the best candidate solution, storing it based on the comparison of obtained objective function values. Upon the algorithm’s full execution, the best candidate solution acquired throughout its iterations is presented as the ultimate FLO solution for the given problem. The implementation steps of FLO are visually depicted as a flowchart in Fig. 2, providing a comprehensive overview of its execution sequence. Additionally, the algorithm’s pseudocode is detailed in Algorithm 1, offering a structured representation of its operational logic and steps.
Flowchart of FLO
Computational Complexity of FLO
In this subsection, we delve into evaluating the computational complexity of the Frilled Lizard Optimization (FLO) algorithm. The computational complexity of FLO can be broken down into two main aspects: the preparation and initialization steps, and the position update process during each iteration. The preparation and initialization steps of FLO involve setting up the algorithm and initializing the positions of the frilled lizards. This process has a computational complexity denoted as O(Nm), where N represents the number of frilled lizards and m denotes the number of decision variables in the problem. During the execution of FLO, the positions of the frilled lizards are updated in each iteration, incorporating both exploration and exploitation phases. This update process contributes to the overall computational complexity of FLO, which is represented as O(2TNm), where T signifies the maximum number of iterations the algorithm will perform. Combining these aspects, the overall computational complexity of the FLO algorithm can be expressed as O(Nm(1+2T)). This analysis underscores the computational framework within which FLO operates, taking into account factors such as the number of lizards, decision variables, and iterations.
Simulation Studies and Results
In this section, we delve into evaluating the performance of the developed Frilled Lizard Optimization (FLO) algorithm in handling optimization problems. To this end, we utilize a diverse set of fifty-two standard benchmark functions encompassing unimodal, high-dimensional multimodal, fixed-dimensional multimodal types [67], alongside the CEC 2017 test suite [68]. To gauge the efficacy of FLO, we compare its performance against twelve well-known metaheuristic algorithms: GA [37], PSO [20], GSA [44], TLBO [54], MVO [45], GWO [27], WOA [36], MPA [25], TSA [31], RSA [34], AVOA [24], and WSO [29]. Fine-tuning control parameters significantly influences metaheuristic algorithm performance [69]. Therefore, the standard versions of MATLAB codes published by the main researchers are used. In addition, for GA and PSO, the standard MATLAB codes published by Professor Mirjalili are used. The links to the MATLAB codes of the mentioned metaheuristic algorithms are provided in Appendix. Control parameter values for each algorithm are detailed in Table 1.
For optimization of objective functions F1 to F23, FLO and competitive algorithms are executed in 30 independent runs. Each run comprises 30,000 function evaluations (FEs) with a population size of 30. For the CEC 2017 test suite, FLO and competitive algorithms undergo 51 independent runs, with each run incorporating 10,000m FEs (where m denotes the number of variables) and a population size of 30. Simulation results are presented using six statistical indicators: mean, best, worst, standard deviation (std), median, and rank. Ranking of metaheuristic algorithms for each benchmark function is established through comparison of mean index values, providing comprehensive insights into their relative performance.
Unimodal Functions
The unimodal functions F1 to F7, which do not have local optima, provide a suitable benchmark for evaluating the exploitation and local search capabilities of metaheuristic algorithms. These functions are particularly useful because they allow researchers to assess how well an algorithm can focus on finding the global optimum without being distracted by false peaks. The performance of FLO and several competitive algorithms on these functions is detailed in Table 2. The results indicate that FLO excels in exploitation and local search, successfully converging to the global optimum for functions F1 through F6. When it comes to the F7 function, FLO stands out as the leading optimizer, demonstrating its effectiveness. An in-depth comparison of the simulation results reveals that FLO’s high exploitation ability enables it to perform better than the competitive algorithms on the unimodal functions F1 to F7. This superior performance highlights FLO’s capability in focusing its search process effectively, ensuring it finds the optimal solutions for these unimodal benchmark functions.
Optimization results for the unimodal functions
FLO
AVOA
WSO
RSA
MPA
TSA
WOA
GWO
MVO
TLBO
GSA
PSO
GA
F1
Best
0
4.822882
0
0
3.47E−52
1.32E−50
8.50E−171
0.096099
1.36E−61
5.35E−77
4.88E−17
0.000443
16.32806
Mean
0
60.02965
0
0
1.75E−49
4.24E−47
1.30E−151
0.13629
1.61E−59
2.30E−74
1.21E−16
0.091953
27.78154
Median
0
41.36897
0
0
3.79E−50
3.89E−48
2.00E−159
0.137102
9.80E−60
1.54E−75
1.03E−16
0.008853
25.68391
Worst
0
217.602
0
0
1.51E−48
3.01E−46
2.40E−150
0.183344
7.03E−59
2.36E−73
3.40E−16
1.27308
51.8506
Std
0
49.03061
0
0
3.65E−49
9.30E−47
5.60E−151
0.02579
1.99E−59
5.72E−74
6.65E−17
0.288752
9.721273
Rank
1
11
1
1
5
6
2
9
4
3
7
8
10
F2
Best
0
0.603391
1.20E−306
0
1.68E−29
1.84E−30
7.20E−118
0.145798
4.44E−36
8.04E−40
3.18E−08
0.041243
1.589689
Mean
0
1.948988
9.90E−277
0
6.34E−28
1.92E−28
2.30E−105
0.236058
1.23E−34
6.16E−39
5.00E−08
0.815635
2.539698
Median
0
1.39396
6.00E−290
0
3.20E−28
1.80E−29
3.10E−108
0.244414
5.92E−35
4.53E−39
4.67E−08
0.532063
2.497037
Worst
0
6.781435
2E−275
0
4.29E−27
1.66E−27
2.50E−104
0.332
7.21E−34
2.22E−38
1.12E−07
2.270937
3.46705
Std
0
1.648521
0.00E+00
0
1.02E−27
4.92E−28
6.40E−105
0.058527
1.82E−34
5.19E−39
1.74E−08
0.671498
0.506175
Rank
1
11
2
1
7
6
3
9
5
4
8
10
12
F3
Best
0
947.6498
0
0
5.64E−19
1.25E−21
1880.714
5.44143
2.15E−19
2.00E−29
224.0264
19.82675
1297.164
Mean
0
1626.99
0
0
2.29E−12
1.08E−10
18179.06
14.54867
1.98E−14
3.50E−24
433.0901
353.5142
1975.532
Median
0
1419.306
0
0
1.66E−13
9.79E−14
18511.54
10.81976
4.25E−16
3.68E−26
364.629
266.9079
1913.339
Worst
0
3227.104
0
0
1.31E−11
1.78E−09
31594.58
44.57484
3.69E−13
3.28E−23
1080.509
933.9386
3150.434
Std
0
583.2962
0
0
4.07E−12
4.05E−10
7950.636
10.00187
8.38E−14
1.01E−23
204.6704
267.9873
594.3514
Rank
1
9
1
1
4
5
11
6
3
2
8
7
10
F4
Best
0
10.85214
0.00E+00
0
2.75E−20
0.0000879
0.823893
0.242208
5.97E−16
5.29E−2
9.01E−09
2.085999
2.018782
Mean
0
15.75337
3E−265
0
2.71E−19
4.03E−03
47.1994
0.49832
1.12E−14
1.67E−30
1.13E+00
5.719781
2.577042
Median
0
16.18755
1.80E−282
0
2.36E−19
0.001339
50.48116
0.483681
5.78E−15
5.94E−31
0.826058
5.357815
2.53522
Worst
0
21.70983
4.1E−264
0
8.75E−19
0.032632
83.53016
0.877153
5.23E−14
7.40E−30
4.488194
12.16864
3.636626
Std
0
2.682766
0.00E+00
0
2.13E−19
0.007381
27.51566
0.178584
1.35E−14
2.23E−30
1.288828
2.325016
0.433841
Rank
1
11
2
1
4
6
12
7
5
3
8
10
9
F5
Best
0
1227.144
1.27E−06
7.93E−29
20.77433
23.38145
24.33873
25.16727
23.28628
23.30644
23.57596
23.93699
208.4007
Mean
0
9836.204
1.30E−05
1.18E+01
21.24372
25.93746
24.87395
87.63958
24.21079
24.39873
40.1211
4200.597
542.2831
Median
0
5109.367
8.55E−06
1.12E−28
21.21726
26.2519
24.67096
27.34075
23.89208
23.97967
23.99658
78.41897
433.1568
Worst
0
84446.57
5.38E−05
26.40458
21.90432
26.31483
26.17246
344.199
24.734
26.18823
152.3277
82043.31
2055.752
Std
0
18645.98
1.35E−05
1.37E+01
0.361087
0.732269
0.53677
94.27249
0.489025
0.869977
41.18185
18690.79
394.8645
Rank
1
13
2
3
4
8
7
10
5
6
9
12
11
F6
Best
0
15.44096
6.47E−09
3.336533
7.36E−10
2.325128
0.009582
0.072166
0.224723
0.212329
5.03E−17
0.00000173
14.21997
Mean
0
91.90698
4.53E−08
5.881906
1.64E−09
3.353519
0.074298
0.137535
0.601908
1.1489
9.53E−17
5.78E−02
31.10185
Median
0
63.37101
4.20E−08
6.270887
1.46E−09
3.457431
0.028788
0.145872
0.662447
1.108843
8.63E−17
0.001874
28.85646
Worst
0
348.3797
1.24E−07
6.603377
4.37E−09
4.360664
0.297605
0.227803
1.140588
1.971716
1.65E−16
0.493414
57.16884
Std
0
88.71011
3.05E−08
0.955084
8.69E−10
0.644215
0.094418
0.044022
0.284877
0.461983
3.45E−17
0.138033
12.58959
Rank
1
13
4
11
3
10
6
7
8
9
2
5
12
F7
Best
2.35E−06
1.22E−05
2.30E−06
5.58E−06
0.000102
0.001361
0.0000218
0.003619
0.000166
0.0000829
0.012869
0.062864
0.002764
Mean
2.54E−05
8.43E−05
5.93E−05
2.97E−05
0.0005
0.003958
1.17E−03
0.010581
0.000759
1.40E-03
0.048101
0.16772
0.009646
Median
1.83E−05
0.0000619
0.0000382
0.0000148
0.000488
0.003393
0.000748
0.010309
0.000772
0.001376
0.047213
0.161882
0.009274
Worst
6.89E−05
3.10E−04
2.44E−04
1.23E−04
0.00082
0.009088
0.00492
0.020558
0.001783
0.002684
0.087051
0.374669
0.019985
Std
2.02E−05
8.29E−05
6.81E−05
3.21E−05
0.0002
0.002175
0.001343
0.004677
0.000433
0.000817
0.023188
0.073415
0.004477
Rank
1
4
3
2
5
9
7
11
6
8
12
13
10
Sum rank
7
15
72
20
32
50
48
36
59
35
54
65
74
Mean rank
1
2.142857
10.28571
2.857143
4.571429
7.142857
6.857143
5.142857
8.428571
5
7.714286
9.285714
10.57143
Total rank
1
2
12
3
4
8
7
6
10
5
9
11
13
High-Dimensional Multimodal Functions
The high-dimensional multimodal functions F8 to F13, due to having multiple local optima, are suitable criteria for challenging the ability of the metaheuristic algorithms in exploration and global search. The optimization results for the functions F8 to F13 using FLO and the competitive algorithms are reported in Table 3. Based on the obtained results, FLO with its high ability in exploration has been able to provide a global optimum for these functions by discovering the main optimal region in dealing with the F9 and F11 functions. Also, in order to optimize the functions F8, F10, F12, and F13, FLO is the best optimizer for these functions. The analysis of the simulation results shows that FLO with high capability in exploration and global search in order to cross local optima and discover the main optimal area turned out to be superior in competition with the compared algorithms.
Optimization results for the high-dimensional functions
FLO
AVOA
WSO
RSA
MPA
TSA
WOA
GWO
MVO
TLBO
GSA
PSO
GA
F8
Best
−12622.8
−9319.44
−12574.2
−6277.23
−10663.1
−7789.11
−12572.1
−9494.05
−7348.84
−7525.59
−4717.57
−8584.04
−9939.5
Mean
−12498.6
−7535.73
−12471.8
−6064.7
−9936.78
−6704.97
−11191.6
−8247.67
−6650.74
−6212.41
−3646.54
−7076.8
−8783.73
Median
−12577.8
−7475.79
−12561.2
−6088.96
−9959.64
−6677.51
−12087.2
−8140.19
−6654.03
−6229.64
−3578.49
−7214.2
−8749.63
Worst
−11936.3
−6637.72
−11957.2
−5569.4
−9389.03
−5089.32
−8167.12
−7387.54
−5723.04
−5269.79
−3021.3
−5668.31
−7523.94
Std
194.2272
686.0549
179.0354
209.0513
341.0971
682.0853
1611.369
683.5917
439.8347
568.8056
464.0183
688.4831
597.0402
Rank
1
7
2
12
4
9
3
6
10
11
13
8
5
F9
Best
0
13.31572
0
0
0
81.74052
0
48.07879
0.00E+00
0
12.68706
36.24875
21.1603
Mean
0
22.43339
0
0
0
157.6833
0
89.10431
1.55E−14
0
25.96315
61.67496
49.80422
Median
0
20.66512
0
0
0
151.8098
0
88.42416
0.00E+00
0
24.01479
59.26511
47.92176
Worst
0
41.85228
0
0
0
262.4813
0
135.9663
1.04E−13
0
44.40468
104.3443
70.04209
Std
0
8.007302
0
0
0
47.39195
0
23.41107
3.02E−14
0
8.516422
17.5057
12.82893
Rank
1
3
1
1
1
8
1
7
2
1
4
6
5
F10
Best
8.88E−16
3.081216
8.88E−16
8.88E−16
8.88E−16
7.36E−15
8.88E−16
0.091628
7.36E−15
4.12E−15
4.24E−09
1.542411
2.62492
Mean
8.88E−16
4.819446
8.88E−16
8.88E−16
3.96E−15
1.13E+00
3.80E−15
0.526357
1.53E−14
4.12E−15
7.48E−09
2.483992
3.256237
Median
8.88E−16
4.717522
8.88E−16
8.88E−16
4.12E−15
2.03E−14
4.12E−15
0.176984
1.38E−14
4.12E−15
7.03E−09
2.490083
3.305859
Worst
8.88E−16
7.467465
8.88E−16
8.88E−16
4.12E−15
3.072576
7.36E−15
2.290859
2.03E−14
4.12E−15
1.32E−08
4.606032
4.227951
Std
0.00E+00
1.134893
0.00E+00
0.00E+00
7.38E−16
1.46E+00
2.11E−15
0.629188
3.30E−15
8.26E−31
2.17E−09
0.796999
0.36853
Rank
1
11
1
1
3
8
2
7
5
4
6
9
10
F11
Best
0
1.005423
0
0
0
0
0
0.231481
0
0
2.728462
0.002156
1.173211
Mean
0
1.563093
0
0
0
0.008054
0
0.364028
0.00122
0
6.565133
0.168742
1.342053
Median
0
1.458192
0
0
0
0.008191
0
0.379369
0
0
6.659052
0.111443
1.318588
Worst
0
2.991765
0
0
0
0.018714
0
0.488181
0.017145
0
11.51061
0.797732
1.57193
Std
0
0.504151
0
0
0
0.005847
0
0.076055
0.004166
0
2.528054
0.212292
0.115089
Rank
1
7
1
1
1
3
1
5
2
1
8
4
6
F12
Best
1.57E−32
0.868125
3.67E−10
0.700576
4.73E−11
0.944381
0.001117
0.00091
0.011442
0.021959
4.33E−19
0.0000973
0.055414
Mean
1.57E−32
2.978079
2.35E−09
1.200098
1.85E−10
5.276134
0.018304
0.833065
0.036322
0.064967
1.91E−01
1.37E+00
0.250377
Median
1.57E−32
2.63405
2.18E−09
1.265478
1.87E−10
3.920951
0.005268
0.382795
0.034529
0.062564
0.073046
1.170635
0.24084
Worst
1.57E−32
6.729691
7.13E−09
1.499107
3.47E−10
12.87521
0.124691
3.504838
0.079043
0.123083
0.848666
4.753719
0.592794
Std
2.86E−48
1.699748
1.54E−09
0.28233
8.93E−11
3.605398
0.037167
1.111915
0.01982
0.019465
0.285629
1.194504
0.128821
Rank
1
12
3
10
2
13
4
9
5
6
7
11
8
F13
Best
1.35E−32
12.567
1.04E−09
6.06E−32
9.07E−10
1.832961
0.033885
0.005868
0.0000427
0.536004
4.24E-18
0.008719
1.176729
Mean
1.35E−32
3278.627
9.13E−09
2.86E−31
2.28E−03
2.474572
0.195464
0.029851
4.68E-01
1.00371
5.16E-02
3.285858
2.466324
Median
1.35E−32
40.28554
5.94E−09
3.66E−31
2.57E−09
2.309059
0.15101
0.021526
0.471026
1.015205
1.62E-17
3.010954
2.611495
Worst
1.35E−32
56617.17
3.47E−08
4.96E−31
0.023056
3.382691
0.637881
0.083455
0.865379
1.403745
0.872898
11.46312
3.588802
Std
2.86E−48
12872.05
8.16E−09
2.09E−31
5.89E−03
0.518014
0.170514
0.02303
0.239555
0.214971
1.99E-01
2.816182
0.701
Rank
1
13
3
2
4
11
7
5
8
9
6
12
10
Sum rank
6
11
53
27
15
52
18
32
39
32
44
50
44
Mean rank
1
1.83334
8.83334
4.5
2.5
8.66667
3
5.33334
6.5
5.33334
7.33334
8.33334
7.33334
Total rank
1
2
11
5
3
10
4
6
7
6
8
9
8
Results for Fixed-Dimensional Multimodal Functions
The fixed-dimension multimodal functions F14 to F23, having a smaller number of local optima compared to functions F8 to F13, are appropriate criteria for measuring the ability of the metaheuristic algorithms in balancing exploration and exploitation. The results of employing FLO and the competitive algorithms for the functions F14 to F23 are reported in Table 4. It turned out that FLO is the best optimizer for the functions F14 to F23. In the cases when FLO has the same value for the mean index as some competitive algorithms, it has provided a more effective performance by providing a better value for the std index. The simulation results show that FLO, with an appropriate ability to balance exploration and exploitation, has a better performance by providing superior results for the benchmark functions in comparison with the competitive algorithms.
Optimization results for the fixed-dimensional functions
FLO
RSA
AVOA
MPA
WSO
TSA
TLBO
WOA
GWO
MVO
GA
GSA
PSO
F14
Best
0.998033
0.998004
0.998004
0.998004
0.998004
0.998004
1.903374
0.998004
0.998004
0.998004
0.998004
0.998004
0.998004
Mean
2.920195
0.998004
1.009791
1.089592
1.089412
1.045199
7.965725
3.455667
2.430619
0.999055
0.999056
3.333745
3.365148
Median
2.115668
0.998004
0.998004
0.998004
0.998004
0.998008
10.76083
2.805144
0.998014
0.998004
0.998004
2.722809
1.903377
Worst
1.16E+01
9.98E−01
1.23E+00
1.90E+00
2.81E+00
1.90E+00
1.42E+01
9.89E+00
9.89E+00
1.02E+00
1.02E+00
1.09E+01
1.16E+01
Std
2.84
0
0.0537
0.284
0.412
0.206
4.69
3.47
2.74
0.00479
0.00479
2.56
3.52
Rank
9
1
4
7
6
5
13
12
8
2
3
10
11
F15
Best
0.000772
0.000307
0.000309
0.000308
0.000316
0.000861
0.000317
0.000317
0.000324
0.000317
0.000327
0.000844
0.000308
Mean
0.001135
0.000307
0.001212
0.001349
0.000436
0.014128
0.015074
0.003178
0.000849
0.002523
0.000654
0.002255
0.002388
Median
0.001026
0.000307
0.0016
0.000429
0.000429
0.013087
0.000874
0.000429
0.000704
0.000736
0.000438
0.002087
0.000429
Worst
2.77E−03
3.07E−04
1.67E−03
1.87E−02
6.94E−04
6.11E−02
1.01E−01
1.87E−02
2.20E−03
1.86E−02
1.29E−03
6.49E−03
1.87E−02
Std
0.000435
2.59E−19
0.000552
0.00417
0.0000907
0.0151
0.0279
0.00679
0.000471
0.00562
0.000382
0.00128
0.0057
Rank
5
1
6
7
2
12
13
11
4
10
3
8
9
F16
Best
−1.03161
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
Mean
−1.02928
−1.03163
−1.02916
−1.0313
−1.0313
−1.03129
−1.02986
−1.0313
−1.0313
−1.0313
−1.03129
−1.0313
−1.0313
Median
−1.03119
−1.03163
−1.0316
−1.03163
−1.03163
−1.03162
−1.03163
−1.03163
−1.03163
−1.03163
−1.03162
−1.03163
−1.03163
Worst
−1.00E+00
−1.03E+00
−1.00E+00
−1.03E+00
−1.03E+00
−1.03E+00
−1.00E+00
−1.03E+00
−1.03E+00
−1.03E+00
−1.03E+00
−1.03E+00
−1.03E+00
Std
0.00652
1.87E−16
0.00708
0.000853
0.000853
0.000853
0.00657
0.000853
0.000853
0.000853
0.000853
0.000853
0.000853
Rank
10
1
11
6
2
7
9
4
3
5
8
2
2
F17
Best
0.398697
0.397887
0.397887
0.397887
0.397887
0.397887
0.397893
0.397888
0.397887
0.397887
0.397897
0.397887
0.397887
Mean
0.409491
0.397887
0.398387
0.397919
0.397919
0.459977
0.397952
0.397919
0.397919
0.397919
0.397985
0.397919
0.713742
Median
0.403269
0.397887
0.397974
0.397894
0.397894
0.397943
0.397917
0.397894
0.397894
0.397894
0.397969
0.397894
0.397913
Worst
4.77E−01
3.98E−01
4.01E−01
3.98E−01
3.98E−01
1.63E+00
3.98E−01
3.98E−01
3.98E−01
3.98E−01
3.98E−01
3.98E−01
2.58E+00
Std
0.0181
0
0.000932
0.0000644
0.0000644
0.281
0.0000842
0.0000644
0.0000643
0.0000644
0.0000895
0.0000644
0.659
Rank
10
1
9
4
2
11
7
6
5
3
8
2
12
F18
Best
3.002335
3
3.013933
3.001243
3.001243
3.00321
3.001249
3.001246
3.001243
3.001243
3.001244
3.001243
3.001243
Mean
5.79232
3
6.144686
3.265013
3.265014
7.18414
11.00853
3.265025
3.265037
3.265013
3.265014
3.265013
3.265013
Median
3.08846
3
3.563655
3.035691
3.035691
3.161867
3.099528
3.0357
3.035714
3.035692
3.035692
3.035691
3.035691
Worst
2.88E+01
3.00E+00
3.00E+01
5.41E+00
5.41E+00
3.22E+01
8.41E+01
5.41E+00
5.41E+00
5.41E+00
5.41E+00
5.41E+00
5.41E+00
Std
7.91
1.19E−15
6.49
0.582
0.582
9.71
24.3
0.582
0.582
0.582
0.582
0.582
0.582
Rank
10
1
11
2
6
12
13
8
9
5
7
4
3
F19
Best
−3.85352
−3.86278
−3.86278
−3.86278
−3.86278
−3.86276
−3.86268
−3.86278
−3.86277
−3.86278
−3.86251
−3.86278
−3.86278
Mean
−3.82664
−3.86278
−3.72454
−3.85019
−3.85019
−3.85004
−3.84982
−3.8488
−3.84804
−3.85019
−3.84918
−3.85019
−3.85019
Median
−3.83066
−3.86278
−3.72574
−3.85056
−3.85056
−3.85049
−3.85052
−3.84988
−3.84899
−3.85056
−3.85015
−3.85056
−3.85056
Worst
−3.77E+00
−3.86E+00
−3.29E+00
−3.81E+00
−3.81E+00
−3.81E+00
−3.81E+00
−3.81E+00
−3.81E+00
−3.81E+00
−3.81E+00
−3.81E+00
−3.81E+00
Std
0.0237
2.32E−15
0.14
0.0123
0.0123
0.0124
0.0121
0.0124
0.012
0.0123
0.0117
0.0123
0.0123
Rank
10
1
11
2
3
5
6
8
9
4
7
2
2
F20
Best
−3.0278
−3.322
−3.22483
−3.31333
−3.2804
−3.23904
−3.31126
−3.31333
−3.30816
−3.31333
−3.29698
−3.31333
−3.31333
Mean
−2.74117
−3.322
−2.52925
−3.23202
−3.19953
−3.16292
−3.18729
−3.19091
−3.18259
−3.20485
−3.17608
−3.24826
−3.196
Median
−2.82466
−3.322
−2.58954
−3.24933
−3.19492
−3.17485
−3.17741
−3.19778
−3.18393
−3.22077
−3.17676
−3.25667
−3.2116
Worst
−1.70E+00
−3.32E+00
−1.78E+00
−3.14E+00
−3.09E+00
−2.97E+00
−3.06E+00
−3.00E+00
−3.04E+00
−3.08E+00
−2.92E+00
−3.18E+00
−3.03E+00
Std
0.297
4.53E−16
0.344
0.0502
0.0636
0.0658
0.0699
0.0882
0.0802
0.0703
0.0927
0.0342
0.0841
Rank
12
1
13
3
5
11
8
7
9
4
10
2
6
F21
Best
−5.50974
−10.1532
−10.1515
−10.1437
−10.1531
−9.56481
−10.1221
−10.1529
−10.1524
−10.153
−9.43287
−10.1531
−10.1362
Mean
−5.27848
−10.1532
−7.55875
−8.33089
−9.92179
−6.37604
−6.07089
−9.22698
−9.2225
−8.76716
−6.91569
−7.22664
−5.79638
Median
−5.30903
−10.1532
−7.90122
−9.88105
−9.95235
−7.0501
−5.07671
−9.88043
−9.8783
−9.77271
−7.16601
−9.69851
−5.15726
Worst
−5.06E+00
−1.02E+01
−5.06E+00
−2.89E+00
−9.70E+00
−2.62E+00
−2.83E+00
−5.10E+00
−5.08E+00
−5.06E+00
−3.65E+00
−2.89E+00
−2.87E+00
Std
0.187
2.12E−15
2.09
2.97
0.187
2.63
3.04
1.73
1.75
2.08
1.93
3.26
2.66
Rank
13
1
7
6
2
10
11
3
4
5
9
8
12
F22
Best
−5.56152
−10.4029
−10.4005
−10.4027
−10.4027
−9.99024
−10.3106
−10.4025
−10.3741
−10.376
−9.77163
−10.4027
−10.3804
Mean
−5.35402
−10.4029
−8.0883
−9.84679
−10.1952
−7.43449
−6.9905
−10.1947
−8.10544
−8.40252
−7.96089
−9.94596
−6.53375
Median
−5.44069
−10.4029
−9.04577
−10.1786
−10.2819
−7.85052
−7.67583
−10.2816
−9.92632
−9.98379
−8.27344
−10.211
−5.21856
Worst
−5.09E+00
−1.04E+01
−5.09E+00
−3.41E+00
−9.93E+00
−2.89E+00
−2.12E+00
−9.93E+00
−2.17E+00
−3.29E+00
−4.32E+00
−5.26E+00
−2.97E+00
Std
0.189
3.58E−15
2.13
1.56
0.189
1.86
3.38
0.189
2.82
2.54
1.58
1.14
3.28
Rank
13
1
8
5
2
10
11
3
7
6
9
4
12
F23
Best
−5.60303
−10.5364
−10.4492
−10.5286
−10.5286
−9.73355
−10.4288
−10.5284
−10.5277
−10.5286
−9.73892
−10.5286
−10.5196
Mean
−5.48753
−10.5364
−9.15348
−10.4131
−10.4131
−6.60937
−7.57014
−10.4127
−8.63436
−9.43441
−8.1814
−10.1863
−6.66461
Median
−5.52257
−10.5364
−9.54713
−10.4482
−10.4482
−7.12733
−9.95964
−10.4479
−10.3968
−10.4202
−8.70553
−10.4482
−4.32836
Worst
−5.13E+00
−1.05E+01
−5.13E+00
−1.01E+01
−1.01E+01
−3.04E+00
−3.11E+00
−1.01E+01
−2.33E+00
−5.17E+00
−4.67E+00
−5.88E+00
−2.97E+00
Std
0.134
2.82E−15
1.5
0.134
0.134
2.39
3.18
0.134
3.07
2.08
1.54
1.04
3.57
Rank
13
1
7
2
3
12
10
4
8
6
9
5
11
Sum rank
10
87
33
44
105
101
73
66
66
50
95
47
80
Mean rank
1.00E+00
8.70E+00
3.30E+00
4.40E+00
1.05E+01
1.01E+01
7.30E+00
6.60E+00
6.60E+00
5.00E+00
9.50E+00
4.70E+00
8.00E+00
Total rank
1
9
2
3
12
11
7
6
6
5
10
4
8
The convergence curves resulting from the execution of FLO and the competitive algorithms for the functions F1 to F23 are drawn in Fig. 3.
Convergence curves for FLO and the competitive algorithms on F1 to F23
CEC 2017 Test Suite
In this subsection, we conduct a comprehensive evaluation of the FLO algorithm’s efficiency in addressing the CEC 2017 test suite. The CEC 2017 test suite comprises thirty standard benchmark functions, which are divided into several categories: 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 composite functions (C17-F21 to C17-F30). Due to instability in its behavior, the C17-F2 function was excluded from the simulation studies. For a full description and details of the CEC 2017 test suite, please refer to [68]. The results of optimizing the CEC 2017 test suite using FLO, in comparison with other competitive algorithms, are detailed in Table 5. Furthermore, the performance outcomes of these metaheuristic algorithms are visually represented through boxplot diagrams in Fig. 4. According to the obtained optimization results, FLO excels in several functions, specifically C17-F1, C17-F3 to C17-F21, C17-F23, C17-F24, and C17-F27 to C17-F30.
Optimization results for the CEC 2017 test suite
FLO
WSO
AVOA
RSA
MPA
TSA
WOA
MVO
GWO
TLBO
GSA
PSO
GA
C17-F1
Mean
100
5.47E+09
3736.741
9.92E+09
34277291
1.69E+09
6265768
7309.046
85692339
1.43E+08
728.1107
3057.613
11513604
Best
100
4.53E+09
115.1723
8.57E+09
10886.23
3.62E+08
4562393
4650.116
27005.92
63693665
100.0187
338.6514
5962184
Worst
100
7.01E+09
11575.72
1.18E+10
1.25E+08
3.68E+09
8249654
10768.56
3.11E+08
3.45E+08
1741.869
9048.114
16528771
Std
0
1.13E+09
5637.763
1.54E+09
63646439
1.56E+09
1643381
3018.544
1.59E+08
1.43E+08
748.1019
4247.123
4651212
Median
100
5.16E+09
1628.036
9.64E+09
6282818
1.36E+09
6125512
6908.755
15705576
81669353
535.2778
1421.844
11781730
Rank
1
12
4
13
8
11
6
5
9
10
2
3
7
C17-F3
Mean
300
8293.792
301.8391
9378.914
1375.654
10888.66
1688.689
300.053
2989.135
713.9977
9971.33
300
14356.74
Best
300
4202.111
300
5061.044
777.166
4151.807
610.0958
300.0123
1492.915
466.305
6277.902
300
4233.022
Worst
300
11094.74
303.9338
12545.46
2470.6
15390.93
3243.315
300.1207
5726.5
875.8003
13549.84
300
22687.57
Std
0
3176.215
2.247148
3603.989
822.8102
5026.203
1306.484
0.050178
2057.025
189.0482
3158.628
4.89E−14
10152.4
Median
300
8939.158
301.7113
9954.573
1127.425
12005.95
1450.672
300.0395
2368.563
756.9427
10028.79
300
15253.2
Rank
1
9
4
10
6
12
7
3
8
5
11
2
13
C17-F4
Mean
400
918.5001
404.6184
1324.333
406.5383
571.4825
424.4454
403.2412
411.4095
408.9142
404.4257
419.7445
414.3073
Best
400
686.9377
401.2064
832.4566
402.378
475.6638
406.2617
401.5494
405.9193
408.1513
403.4619
400.1027
411.3519
Worst
400
1127.349
406.3441
1806.129
411.0611
683.3579
471.5001
404.7584
427.5674
409.3958
405.9062
468.4064
417.9233
Std
0
211.487
2.549157
437.6922
4.510676
107.2135
33.13703
1.757159
11.34448
0.561975
1.180512
34.5168
3.028779
Median
400
929.8569
405.4616
1329.372
406.357
563.4541
410.0099
403.3286
406.0757
409.0548
404.1674
405.2343
413.977
Rank
1
12
4
13
5
11
10
2
7
6
3
9
8
C17-F5
Mean
501.2464
562.7628
543.267
571.5024
512.6851
563.2066
540.248
523.2985
512.8239
533.4614
552.8981
527.4234
527.5331
Best
500.9951
548.6366
526.3694
557.1506
508.2448
542.4586
523.0456
510.0618
508.3883
528.0685
548.1185
510.9634
522.9067
Worst
501.9917
572.071
561.7117
586.2257
517.6984
594.6685
575.476
537.3349
519.9718
536.9224
564.4298
550.8372
533.1848
Std
0.523294
11.24763
19.528
17.00721
5.239914
24.39291
25.86418
11.99271
5.257986
4.097238
8.204426
19.37243
4.881442
Median
500.9993
565.1717
542.4935
571.3166
512.3987
557.8496
531.2352
522.8986
511.4678
534.4273
549.5221
523.9465
527.0204
Rank
1
11
9
13
2
12
8
4
3
7
10
5
6
C17-F6
Mean
600
631.9679
617.0699
640.1193
601.1766
624.4721
622.8332
602.1188
601.1108
606.7637
616.9574
607.3227
610.1123
Best
600
628.0964
616.08
636.953
600.7006
614.8572
607.4178
600.4653
600.5875
604.6901
602.8743
601.3351
606.8056
Worst
600
635.2211
619.5846
644.3114
602.3635
639.8378
644.5482
604.2511
601.6942
609.997
635.6217
618.9817
614.2958
Std
0
3.522076
1.770682
3.482394
0.835501
11.34888
16.46555
1.791659
0.482493
2.54797
15.95463
8.429568
3.496574
Median
600
632.277
616.3074
639.6063
600.8212
621.5967
619.6833
601.8795
601.0807
606.1838
614.6668
604.4871
609.6739
Rank
1
12
9
13
3
11
10
4
2
5
8
6
7
C17-F7
Mean
711.1267
801.8243
764.7796
803.027
724.4458
826.7993
761.355
730.597
725.8017
751.4689
717.0341
732.4347
736.5061
Best
710.6726
781.9245
743.4263
789.9792
720.2997
787.2738
750.5297
717.1182
717.3753
746.9989
714.7886
725.3833
726.3134
Worst
711.7995
818.2036
792.1502
815.5044
728.797
867.6759
790.383
749.5856
743.0676
759.4706
720.7105
743.8261
741.0172
Std
0.539366
16.11626
23.59666
12.6169
3.767224
36.78314
20.43924
14.38877
12.42733
5.871453
2.702113
8.863158
7.268468
Median
711.0174
803.5845
761.771
803.3123
724.3433
826.1237
752.2536
727.8421
721.382
749.7031
716.3187
730.2647
739.3469
Rank
1
11
10
12
3
13
9
5
4
8
2
6
7
C17-F8
Mean
801.4928
847.9594
830.7179
852.9783
812.5179
847.6438
835.8856
811.6922
815.6551
837.214
819.6168
822.481
816.585
Best
800.995
840.0504
820.0255
841.9218
808.7429
831.6806
818.3429
807.3408
810.3963
830.393
811.8696
815.4944
812.6473
Worst
801.9912
856.225
846.3057
858.1424
814.642
866.671
847.9269
816.4079
820.5681
845.0906
827.2753
828.8525
824.2671
Std
0.605411
7.772196
11.68425
7.882006
2.862269
16.39951
13.38181
3.92383
4.482335
7.915127
6.904175
6.97152
5.495759
Median
801.4926
847.7811
828.2702
855.9245
813.3434
846.1118
838.6363
811.51
815.8281
836.6862
819.6612
822.7886
814.7128
Rank
1
12
8
13
3
11
9
2
4
10
6
7
5
C17-F9
Mean
900
1415.647
1184.109
1459.407
905.126
1374.025
1368.683
900.7903
911.7692
911.6633
900
904.1835
905.0408
Best
900
1274.159
952.9809
1364.939
900.323
1164.299
1071.705
900.001
900.5653
907.1323
900
900.8869
902.7594
Worst
900
1554.98
1650.596
1594.98
913.158
1658.273
1645.866
903.0715
932.6733
919.7283
900
912.149
908.9528
Std
0
132.6979
340.4279
103.1532
6.084099
225.1024
254.5478
1.602111
15.86347
5.829805
0
5.66366
2.949944
Median
900
1416.725
1066.429
1438.854
903.5116
1336.763
1378.581
900.0444
906.9191
909.8963
900
901.849
904.2254
Rank
1
11
8
12
5
10
9
2
7
6
1
3
4
C17-F10
Mean
1006.179
2273.607
1759.53
2541.66
1504.452
2008.524
2001.099
1762.429
1708.336
2144.709
2248.35
1923.739
1698.821
Best
1000.284
2018.084
1472.325
2374.849
1382.178
1739.505
1438.96
1445.087
1526.348
1763.109
1975.334
1547.422
1405.051
Worst
1012.668
2452.924
2381.05
2893.002
1577.649
2255.004
2513.204
2252.39
1968.77
2426.108
2351.688
2320.273
2084.862
Std
7.010122
207.8178
449.5863
254.1133
96.96531
286.3359
547.0173
411.8901
198.0574
296.8361
192.0592
334.2969
307.0471
Median
1005.882
2311.71
1592.372
2449.394
1528.991
2019.793
2026.115
1676.12
1669.112
2194.81
2333.189
1913.63
1652.685
Rank
1
12
5
13
2
9
8
6
4
10
11
7
3
C17-F11
Mean
1100
3792.805
1147.32
3913.816
1126.386
5353.182
1149.714
1126.836
1153.923
1149.669
1138.236
1142.466
2351.467
Best
1100
2579.105
1116.633
1449.857
1112.878
5208.571
1112.643
1105.411
1121.094
1136.909
1119.165
1131.464
1114.678
Worst
1100
4965.598
1199.302
6347.472
1157.346
5432.526
1171.319
1147.72
1225.241
1170.543
1166.94
1163.436
5860.164
Std
0
1129.665
38.31996
2318.078
22.11378
104.8258
28.5344
22.26206
51.12907
15.28998
21.47371
15.1631
2463.985
Median
1100
3813.26
1136.672
3928.967
1117.659
5385.816
1157.446
1127.106
1134.679
1145.613
1133.42
1137.481
1215.513
Rank
1
11
6
12
2
13
8
3
9
7
4
5
10
C17-F12
Mean
1352.959
3.46E+08
1076623
6.9E+08
555218.2
1017040
2302745
1006704
1384502
4942507
998167
7942.901
591852.2
Best
1318.646
77501553
348274.8
1.53E+08
19458.1
527421.5
168030.9
8666.689
44473.99
1322697
464212.7
2491.975
171450.5
Worst
1438.176
6.04E+08
1952421
1.21E+09
868884.4
1248617
3820158
3162122
2167137
8749709
1688039
13645.63
1044753
Std
60.34215
2.8E+08
790170
5.61E+08
394089.6
358149.2
1787906
1534071
985273.1
4143007
545566.9
5351.454
377639
Median
1327.506
3.51E+08
1002899
7E+08
666265.1
1146061
2611395
428012.7
1663199
4848812
920208
7817.002
575602.7
Rank
1
12
8
13
3
7
10
6
9
11
5
2
4
C17-F13
Mean
1305.324
16818760
17959.57
33627051
5343.993
12487.34
7441.547
6609.75
10102.15
16388.5
9879.403
6504.793
53288.07
Best
1303.114
1403371
2692
2791832
3667.876
7450.198
3237.74
1384.282
6393.425
15476.72
4965.5
2355.43
8385.223
Worst
1308.508
55824623
30752.64
1.12E+08
6529.18
19767.52
14850.39
12137.03
14101.13
18615.71
13903.43
16377.46
176137.9
Std
2.393502
27444488
15277.22
54887056
1437.183
5598.534
5574.972
5865.593
3326.546
1578.577
3978.516
7008.283
86300.99
Median
1304.837
5023524
19196.82
10040363
5589.459
11365.82
5839.029
6458.844
9957.02
15730.79
10324.34
3643.14
14314.57
Rank
1
12
10
13
2
8
5
4
7
9
6
3
11
C17-F14
Mean
1400.746
3939.312
2008.934
5264.816
1928.918
3345.285
1516.59
1568.362
2326.763
1586.904
5478.844
2962.532
12721.15
Best
1400
3117.693
1673.286
4612.291
1434.307
1486.131
1480.161
1422.656
1461.054
1513.755
4535.374
1431.851
3678.382
Worst
1400.995
5351.578
2798.455
6783.07
2872.227
5496.468
1555.51
1981.281
4889.386
1616.566
7425.724
6730.767
25320.97
Std
0.523906
1086.154
558.4401
1073.877
710.2403
2246.597
40.54732
289.9705
1799.198
51.60308
1426.2
2666.975
9655.175
Median
1400.995
3643.988
1781.998
4831.952
1704.569
3199.271
1515.345
1434.756
1478.307
1608.646
4977.139
1843.756
10942.63
Rank
1
10
6
11
5
9
2
3
7
4
12
8
13
C17-F15
Mean
1500.331
10098.58
5218.908
13616.4
3924.269
6887.913
6120.022
1540.983
5724.402
1704.856
23414.1
8840.912
4486.47
Best
1500.001
2973.107
2060.565
2708.398
3187.379
2302.517
2003.782
1525.386
3527.032
1582.367
11023.57
2843.03
1882.431
Worst
1500.5
17679.9
12395.04
29757.67
4821.345
12315.56
13198.16
1552.777
6787.366
1792.663
35125.15
14517.22
7878.356
Std
0.247931
6463.467
5077.324
12437.92
713.8488
4531.581
5138.748
12.60171
1577.67
108.6858
12125.71
5138.289
3139.349
Median
1500.413
9870.658
3210.011
10999.77
3844.176
6466.789
4639.074
1542.885
6291.604
1722.197
23753.83
9001.7
4092.546
Rank
1
11
6
12
4
9
8
2
7
3
13
10
5
C17-F16
Mean
1600.76
2003.841
1805.182
2007.682
1682.475
2037.911
1943.131
1811.714
1725.811
1675.298
2063.238
1916.854
1798.115
Best
1600.356
1932.833
1641.383
1814.741
1640.895
1856.913
1761.641
1723.889
1615.517
1649.88
1939.83
1817.788
1716.236
Worst
1601.12
2155.779
1919.398
2275.751
1712.433
2218.582
2068.663
1872.098
1820.735
1728.436
2253.981
2073.252
1828.527
Std
0.332693
107.9397
123.3381
205.0407
32.40876
172.8169
153.6816
66.01326
89.17165
38.56257
150.4328
124.6204
57.53511
Median
1600.781
1963.376
1829.974
1970.119
1688.287
2038.075
1971.11
1825.435
1733.496
1661.439
2029.57
1888.188
1823.848
Rank
1
10
6
11
3
12
9
7
4
2
13
8
5
C17-F17
Mean
1700.099
1815.785
1749.933
1815.932
1735.005
1800.085
1838.968
1839.831
1767.119
1757.185
1843.74
1751.289
1754.835
Best
1700.02
1806.767
1733.703
1799.367
1721.462
1785.215
1772.092
1776.895
1723.964
1747.255
1746.952
1744.783
1751.768
Worst
1700.332
1820.911
1793.046
1824.963
1773.372
1810.751
1885.438
1945.3
1868.057
1766.927
1967.462
1757.838
1757.219
Std
0.163405
6.604172
30.34816
11.98474
26.95023
11.55193
51.8522
83.97523
71.2283
10.25543
118.4177
5.880496
2.593649
Median
1700.022
1817.731
1736.491
1819.698
1722.593
1802.187
1849.171
1818.564
1738.228
1757.279
1830.274
1751.268
1755.175
Rank
1
9
3
10
2
8
11
12
7
6
13
4
5
C17-F18
Mean
1805.36
2790335
11621.85
5564458
10833.91
11819.61
22803.87
20498.52
19481.89
28855.77
9528.075
21406.05
12557.12
Best
1800.003
143079.7
4773.511
275503.5
4103.869
7333.66
6341.191
8540.919
6218.404
23471.4
6286.716
2855.611
3398.068
Worst
1820.451
8086661
15273.85
16153226
16174.54
15949.14
35793.67
32964.2
32845.25
36080.81
11621.21
39828.48
18092.38
Std
10.59792
3874670
4958.103
7747506
5781.245
3773.409
14945.94
12109.03
14219.8
6108.088
2397.694
20100.54
6759.369
Median
1800.492
1465801
13220.02
2914551
11528.62
11997.83
24540.31
20244.49
19431.95
27935.43
10102.19
21470.05
14369.02
Rank
1
12
4
13
3
5
10
8
7
11
2
9
6
C17-F19
Mean
1900.445
387325.3
6595.99
687462.2
5511.926
122623.9
34035.21
1914.421
5302.419
4631.324
39521.63
24406.65
6082.546
Best
1900.039
24496.92
2170.349
44786.25
2308.105
1948.078
7524.944
1909.202
1943.66
2039.952
10888
2607.662
2205.951
Worst
1901.559
818032
12978.45
1476750
9240.37
244893.5
62270.58
1923.745
13530.05
12241.53
57304.33
75127.73
9695.844
Std
0.784167
360554.3
5534.839
680304
3721.021
146723.9
23668.24
7.235854
5837.141
5343.17
21892
36010.83
3254.361
Median
1900.09
353386.2
5617.582
614156.2
5249.615
121827
33172.67
1912.368
2867.982
2121.906
44947.1
9945.598
6214.195
Rank
1
12
7
13
5
11
9
2
4
3
10
8
6
C17-F20
Mean
2000.312
2210.011
2166.568
2217.803
2090.167
2202.524
2201.759
2136.293
2165.937
2070.321
2247.759
2165.023
2049.043
Best
2000.312
2154.567
2030.604
2160.675
2071.051
2104.208
2096.032
2045.847
2127.852
2059.553
2183.382
2141.464
2034.979
Worst
2000.312
2278.59
2287.603
2271.927
2119.96
2313.389
2281.142
2241.642
2240.244
2080.497
2338.678
2196.123
2056.626
Std
0
53.95395
121.7585
57.65849
22.07053
93.32007
93.18815
84.66573
53.35706
9.247632
79.56738
28.60972
10.51071
Median
2000.312
2203.443
2174.032
2219.306
2084.828
2196.25
2214.931
2128.842
2147.827
2070.617
2234.488
2161.253
2052.284
Rank
1
11
8
12
4
10
9
5
7
3
13
6
2
C17-F21
Mean
2200
2290.968
2213.493
2265.597
2255.897
2322.324
2307.36
2251.936
2310.718
2297.422
2364.486
2316.097
2295.936
Best
2200
2244.697
2204.034
2223.411
2253.464
2220.748
2217.975
2200.007
2306.605
2203.635
2347.406
2308.22
2225.954
Worst
2200
2316.52
2238.126
2289.583
2258.376
2368.215
2350.548
2305.165
2315.574
2335.231
2381.419
2323.478
2329.794
Std
0
35.24304
17.34766
30.82035
2.18896
72.54506
63.5482
63.15665
3.884515
66.31879
14.9697
7.903532
49.76087
Median
2200
2301.327
2205.907
2274.697
2255.874
2350.166
2330.458
2251.285
2310.346
2325.411
2364.56
2316.346
2313.999
Rank
1
6
2
5
4
12
9
3
10
8
13
11
7
C17-F22
Mean
2300.073
2727.205
2308.786
2902.26
2304.896
2704.733
2323.277
2286.092
2308.412
2319.143
2300.006
2312.979
2317.535
Best
2300
2604.512
2304.267
2697.974
2300.923
2445.851
2318.712
2230.996
2301.239
2313.024
2300
2300.624
2314.697
Worst
2300.29
2863.201
2310.901
3052.187
2309.155
2908.01
2330.751
2305.182
2321.915
2330.62
2300.026
2344.466
2321.909
Std
0.152789
125.676
3.213404
157.0784
3.653123
217.2198
5.66243
38.69394
10.01534
8.480752
0.013627
22.1532
3.245621
Median
2300
2720.553
2309.989
2929.44
2304.752
2732.534
2321.822
2304.095
2305.248
2316.463
2300
2303.413
2316.766
Rank
3
12
6
13
4
11
10
1
5
9
2
7
8
C17-F23
Mean
2600.919
2695.142
2641.279
2698.593
2614.053
2720.959
2647.789
2619.87
2613.481
2641.744
2787.95
2643.451
2655.069
Best
2600.003
2654.05
2630.016
2670.241
2611.708
2633.704
2630.257
2607.041
2607.705
2631.074
2724.222
2636.443
2635.506
Worst
2602.87
2718.801
2658.663
2738.344
2616.681
2764.466
2667.61
2631.191
2620.042
2650.904
2923.517
2655.116
2663.229
Std
1.39047
31.84495
14.22616
33.5504
2.494325
62.22928
21.21048
11.06898
6.713412
9.259933
98.62473
8.912385
13.94899
Median
2600.403
2703.859
2638.218
2692.893
2613.911
2742.834
2646.645
2620.625
2613.089
2642.5
2752.031
2641.123
2660.77
Rank
1
10
5
11
3
12
8
4
2
6
13
7
9
C17-F24
Mean
2630.488
2774.562
2764.817
2845.426
2630.649
2667.52
2757.964
2682.241
2746.372
2753.283
2745.087
2762.814
2721.233
Best
2516.677
2721.424
2731.165
2822.869
2614.606
2529.897
2729.886
2501.653
2720.269
2739.08
2503.872
2753.308
2541.917
Worst
2732.32
2855.174
2784.603
2907.973
2639.658
2810.341
2790.428
2758.938
2759.651
2765.909
2894.23
2785.223
2809.443
Std
122.6896
68.4245
26.78397
43.96585
11.87973
158.2364
26.36986
127.588
19.50185
13.53941
177.0559
15.83832
127.6645
Median
2636.477
2760.824
2771.75
2825.431
2634.167
2664.92
2755.771
2734.188
2752.784
2754.072
2791.123
2756.364
2766.787
Rank
1
12
11
13
2
3
9
4
7
8
6
10
5
C17-F25
Mean
2932.639
3155.18
2913.894
3269.51
2918.209
3129.369
2908.071
2922.324
2938.568
2933.51
2922.491
2923.532
2951.829
Best
2898.047
3064.544
2899.073
3202.552
2914.394
2906.642
2768.56
2901.85
2921.811
2915.723
2903.487
2898.655
2937.353
Worst
2945.793
3355.646
2948.92
3343.189
2923.782
3641.612
2957.842
2943.71
2945.838
2952.119
2943.394
2946.537
2962.362
Std
24.31643
142.1273
24.70364
61.2439
4.370192
363.6379
98.02209
24.7485
11.84326
21.01351
23.0419
27.38159
11.26241
Median
2943.359
3100.264
2903.792
3266.149
2917.329
2984.611
2952.94
2921.869
2943.312
2933.1
2921.543
2924.469
2953.8
Rank
7
12
2
13
3
11
1
4
9
8
5
6
10
C17-F26
Mean
2900
3586.548
2978.206
3738.663
3009.357
3605.933
3177.182
2900.145
3257.787
3200.309
3841.837
2903.976
2897.274
Best
2900
3249.585
2808.919
3421.475
2892.278
3139.18
2926.653
2900.111
2967.8
2911.806
2808.919
2808.919
2711.383
Worst
2900
3826.851
3151.519
4068.773
3285.46
4241.393
3579.816
2900.189
3886.42
3855.646
4319.2
3006.985
3105.287
Std
3.91E−13
292.2164
205.8885
293.9256
194.7487
567.5965
300.739
0.036902
445.4171
463.1248
736.9779
85.29239
210.1045
Median
2900
3634.878
2976.193
3732.203
2929.845
3521.579
3101.13
2900.14
3088.464
3016.891
4119.614
2900
2886.213
Rank
2
10
5
12
6
11
7
3
9
8
13
4
1
C17-F27
Mean
3089.518
3205.957
3119.375
3228.207
3104.379
3177.675
3192.753
3091.585
3115.563
3114.565
3223.217
3135.116
3158.554
Best
3089.518
3158.16
3095.187
3126.438
3092.187
3102.163
3177.187
3089.706
3094.336
3095.264
3211.305
3096.94
3118.73
Worst
3089.518
3277.829
3179.042
3416.328
3132.899
3219.061
3204.273
3094.852
3174.984
3169.582
3244.326
3181.461
3216.271
Std
2.76E-13
53.52379
42.0126
135.253
20.16865
55.77425
11.90539
2.548962
41.75977
38.63556
15.47514
37.43231
43.43005
Median
3089.518
3193.919
3101.636
3185.032
3096.215
3194.737
3194.776
3090.89
3096.466
3096.708
3218.617
3131.032
3149.607
Rank
1
11
6
13
3
9
10
2
5
4
12
7
8
C17-F28
Mean
3100
3611.463
3233.144
3764.422
3215.961
3575.685
3282.736
3235.706
3339.606
3320.184
3443.108
3301.185
3243.164
Best
3100
3563.288
3100
3683.905
3165.474
3405.693
3151.545
3100.121
3192.63
3211.49
3430.136
3175.433
3143.902
Worst
3100
3652.954
3384.01
3822.542
3240.311
3780.33
3384.51
3384.011
3405.236
3384.247
3461.135
3384.221
3504.385
Std
0
39.54602
132.2986
67.7661
36.47716
204.6273
126.0886
165.1629
103.9998
86.83885
15.12725
99.68882
184.0982
Median
3100
3614.806
3224.283
3775.62
3229.03
3558.358
3297.444
3229.346
3380.28
3342.5
3440.581
3322.542
3162.184
Rank
1
12
3
13
2
11
6
4
9
8
10
7
5
C17-F29
Mean
3132.241
3324.262
3281.439
3370.25
3201.677
3234.136
3344.478
3201.267
3262.49
3211.02
3341.53
3263.344
3235.113
Best
3130.076
3307.388
3208.777
3300.298
3165.245
3165.464
3233.56
3142.258
3188.769
3164.942
3231.698
3167.136
3187.368
Worst
3134.841
3342.951
3360.613
3436.015
3242.35
3302.718
3488.455
3283.301
3374.431
3233.318
3624.637
3344.532
3283.168
Std
2.61421
19.00098
82.36137
73.67541
35.72216
59.15374
112.573
62.88155
93.01789
33.76393
199.5837
84.79596
42.46644
Median
3132.023
3323.354
3278.183
3372.344
3199.556
3234.181
3327.949
3189.755
3243.379
3222.909
3254.892
3270.854
3234.958
Rank
1
10
9
13
3
5
12
2
7
4
11
8
6
C17-F30
Mean
3418.734
2165720
287506.3
3584776
404550
599358.3
967621
295454.7
912690.3
59213.86
763371.5
377738.5
1489498
Best
3394.682
1320162
102168.6
807160.4
15619.96
109607.9
4440.547
7339.401
32840.9
28649.75
586908.4
6321.34
512847.3
Worst
3442.907
3250978
748843.2
5662047
597037.7
1267208
3652633
1126231
1320855
99321.06
974824
748878.9
3393200
Std
29.24586
845640.7
324777.7
2140730
278078.7
518025.2
1887445
583462.4
637318.9
36349.97
169764.2
450721.6
1429827
Median
3418.673
2045870
149506.7
3934949
502771.1
510308.6
106705.4
24124.27
1148533
54442.32
745876.8
377876.9
1025972
Rank
1
12
3
13
6
7
10
4
9
2
8
5
11
Sum rank
38
319
177
351
106
284
239
116
188
191
238
183
197
Mean rank
1.310345
11
6.103448
12.10345
3.655172
9.793103
8.241379
4
6.482759
6.586207
8.206897
6.310345
6.793103
Total rank
1
12
4
13
2
11
10
3
6
7
9
5
8
Boxplot diagrams of FLO and the performance of the competitive algorithms for the CEC 2017 test suite
These optimization outcomes indicate that FLO achieves favorable results for the benchmark functions, which can be attributed to its strong capabilities in both exploration and exploitation, as well as its effectiveness in balancing these two critical aspects throughout the search process. A comparative analysis of the simulation results clearly demonstrates that FLO outperforms the competitive algorithms for most of the benchmark functions. This establishes FLO as the premier optimizer overall, showcasing its superiority in handling the CEC 2017 test suite.
Statistical Analysis
A comprehensive statistical analysis has been undertaken to determine the significance of FLO’s superiority over competitive algorithms from a statistical perspective. To accomplish this, the non-parametric Wilcoxon rank sum test [70] has been employed, a widely recognized method for identifying substantial differences between the averages of two datasets. In the context of the Wilcoxon rank sum test, the primary aim is to assess whether there exists a noteworthy disparity in the performance of two algorithms, as indicated by the calculation of a key metric known as the p-value. The outcomes of conducting the Wilcoxon rank sum test on the performance of FLO compared to each of the competitive algorithms are meticulously presented in Table 6. These results play a pivotal role in elucidating the degree of FLO’s statistical advantage over alternative metaheuristic algorithms. Specifically, instances where the calculated p-value is below the threshold of 0.05 indicate a statistically significant advantage for FLO when pitted against its counterparts. In essence, the extensive statistical analysis underscores FLO’s significant statistical superiority across all benchmark functions examined in the study, reaffirming its efficacy as an optimization algorithm.
Wilcoxon rank sum test results
Compared algorithm
Test functions
F1 to F7
F8 to F13
F14 to F23
CEC 2017
FLO vs. AVOA
2.93E–14
4.68E–08
1.39E–37
3.65E–22
FLO vs. WSO
1.79E–27
1.91E–24
2.02E–37
1.96E–24
FLO vs. RSA
4.12E–10
1.58E–14
1.39E–37
1.91E–24
FLO vs. MPA
9.78E–28
1.01E–17
2.02E–37
1.94E–21
FLO vs. TSA
9.78E–28
1.27E–23
1.39E–37
9.20E–24
FLO vs. WOA
2.36E–27
5.94E–14
1.39E–37
9.20E–24
FLO vs. GWO
9.78E–28
5.17E–19
1.39E–37
5.07E–24
FLO vs. MVO
9.78E–28
1.91E–24
1.39E–37
8.75E–22
FLO vs. TLBO
9.78E–28
6.76E–18
1.39E–37
3.57E–24
FLO vs. GSA
9.78E–28
1.91E–24
1.39E–37
1.55E–21
FLO vs. PSO
9.78E–28
1.91E–24
1.39E–37
1.49E–22
FLO vs. GA
9.78E–28
1.91E–24
1.39E–37
2.63E–22
FLO for Real-World Engineering Applications
This segment delves into the exploration of FLO’s efficiency in tackling real-world optimization dilemmas. To assess its performance, we analyze its application across twenty-two constrained optimization problems sourced from the CEC 2011 test suite, in addition to evaluating its efficacy in solving four distinct engineering design problems.
Evaluation of the CEC 2011 Test Suite
In this subsection, we delve into evaluating the performance of FLO alongside competitive algorithms in addressing optimization tasks within real-world applications, focusing on the CEC 2011 test suite [71]. Comprising twenty-two constrained optimization problems, this test suite encompasses a diverse range of engineering challenges, making it a prime choice for assessing metaheuristic algorithms’ capabilities. Previous studies have often leveraged this suite due to its relevance in simulating real-world scenarios. To gauge FLO’s aptitude in handling such applications, we utilize the standard engineering problems from the CEC 2011 test suite. Employing a population size of 30, both FLO and competitive algorithms undergo rigorous testing across 25 independent runs, each spanning 150,000 function evaluations. The comprehensive results, as depicted in Table 7 and visually presented through boxplot diagrams in Fig. 5, offer insights into their comparative performances. Notably, FLO emerges as the top-performing algorithm across problems C11-F1 to C11-F22. Its superior performance is evident, outshining competitive algorithms in the majority of cases. Statistical analyses, including the p-values derived from the Wilcoxon rank sum test, further underscore FLO’s significant advantage over its counterparts in tackling the challenges posed by the CEC 2011 test suite.
Optimization results for the CEC 2011 test suite
FLO
AVOA
WSO
RSA
MPA
TSA
WOA
GWO
MVO
TLBO
GSA
PSO
GA
C11-F1
Best
2E−10
15.71076
9.066921
20.62179
0.380394
17.94272
8.419541
1.141181
11.67985
17.18251
20.08481
10.70815
22.81227
Mean
5.92E+00
17.99435
13.14302
22.38356
7.610637
18.74465
13.4423
10.99171
14.21507
18.77743
22.09735
18.27145
23.83473
Median
5.687176
17.74842
13.22952
22.04179
8.683689
18.4701
13.92203
12.50987
14.38094
18.7476
22.36096
18.89069
23.29871
Worst
12.30606
20.76982
17.04611
24.82885
12.69478
20.09569
17.50561
17.80591
16.41857
20.43201
23.58269
24.59626
25.92925
Std
7.196379
2.60606
4.637594
2.136403
5.931504
1.059314
4.389729
7.430618
2.408351
1.40313
1.543546
6.81674
1.494991
Rank
1
7
4
12
2
9
5
3
6
10
11
8
13
C11-F2
Best
−27.0676
−15.5758
−21.5126
−11.7957
−25.7104
−14.8607
−21.9772
−24.7158
−10.5976
−11.8759
−20.5101
−23.9972
−15.0976
Mean
−26.3179
−14.211
−20.9668
−11.3516
−25.073
−11.0667
−18.5091
−22.5833
−8.54387
−10.6677
−15.3823
−22.633
−12.7312
Median
−26.3856
−14.1395
−21.0593
−11.3616
−25.4223
−10.2821
−18.8021
−23.3354
−8.29251
−10.5943
−14.8825
−23.1576
−12.4044
Worst
−25.4328
−12.9891
−20.2358
−10.8875
−23.7372
−8.84169
−14.4552
−18.9464
−6.99281
−9.60622
−11.254
−20.2195
−11.0182
Std
0.738935
1.387208
0.593111
0.509167
0.966972
2.990523
4.075175
2.675921
1.64456
0.990412
4.427235
1.741838
2.015308
Rank
1
8
5
10
2
11
6
4
13
12
7
3
9
C11-F4
Best
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
Mean
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
Median
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
Worst
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
2.21E−04
Std
6.01E−21
5.68E−12
3.64E−08
9.26E−12
8.91E−14
9.18E−13
8.24E−18
3.74E−15
9.98E−13
7.86E−14
2.00E−19
6.08E−20
2.77E−18
Rank
1
11
13
12
6
8
4
7
10
9
3
2
5
C11-F4
Best
0
0
0
0
0
0
0
0
0
0
0
0
0
Mean
0
0
0
0
0
0
0
0
0
0
0
0
0
Median
0
0
0
0
0
0
0
0
0
0
0
0
0
Worst
0
0
0
0
0
0
0
0
0
0
0
0
0
Std
0
0
0
0
0
0
0
0
0
0
0
0
0
Rank
1
1
1
1
1
1
1
1
1
1
1
1
1
C11-F5
Best
−34.7494
−25.9018
−29.1581
−22.0228
−33.8571
−31.5428
−27.7524
−34.1779
−31.7223
−12.7456
−31.5218
−11.9996
−10.7279
Mean
−34.1274
−24.7516
−28.0779
−19.864
−33.2723
−27.0943
−27.5969
−31.5621
−26.9534
−10.5939
−27.3127
−8.41031
−9.27774
Median
−34.1871
−24.6441
−27.771
−19.9721
−33.646
−27.5565
−27.7204
−32.2815
−25.8032
−10.3414
−26.7975
−7.48141
−9.39582
Worst
−33.3862
−23.8166
−27.6116
−17.489
−31.9401
−21.7214
−27.1943
−27.5074
−24.4848
−8.94709
−24.1342
−6.67878
−7.59142
Std
0.589989
0.95518
0.770371
2.52328
0.939346
4.252721
0.282677
2.997963
3.555513
1.70007
3.400906
2.637435
1.455951
Rank
1
9
4
10
2
7
5
3
8
11
6
13
12
C11-F6
Best
−27.4298
−14.5571
−20.4029
−13.6442
−25.7465
−16.4981
−22.9899
−22.38
−17.3952
−2.44646
−26.6323
−5.94001
−9.20345
Mean
−24.1119
−13.9676
−19.0042
−12.965
−22.6108
−7.43437
−19.9336
−19.6085
−9.4219
−2.15053
−21.8798
−3.02392
−3.93842
Median
−23.0059
−13.7835
−19.2017
−13.1341
−21.6869
−4.54604
−21.9278
−19.0489
−9.12028
−2.05189
−21.5738
−2.05189
−2.24917
Worst
−23.0059
−13.7463
−17.2104
−11.9475
−21.3227
−4.1473
−12.8889
−17.9562
−2.05189
−2.05189
−17.7395
−2.05189
−2.05189
Std
2.324951
0.414304
1.546268
0.826562
2.226773
6.363543
5.036499
2.223563
8.734288
0.207362
4.026194
2.0434
3.694555
Rank
1
7
6
8
2
10
4
5
9
13
3
12
11
C11-F7
Best
0.582266
1.546644
1.142773
1.679132
0.757596
1.129698
1.628575
0.814227
0.817307
1.528266
0.88491
0.831913
1.350761
Mean
0.860699
1.607366
1.284792
1.921733
0.929758
1.302666
1.745163
1.067876
0.881104
1.720185
1.079947
1.123948
1.741991
Median
0.91775
1.582628
1.284988
1.950134
0.974943
1.206637
1.716675
1.081482
0.875765
1.744987
1.076935
1.148661
1.835193
Worst
1.025027
1.717562
1.426421
2.107532
1.011549
1.667694
1.918728
1.294312
0.955577
1.862501
1.28101
1.366556
1.946817
Std
0.211503
0.081267
0.161509
0.187417
0.123219
0.258951
0.129822
0.207943
0.071645
0.153083
0.188418
0.290556
0.283994
Rank
1
9
7
13
3
8
12
4
2
10
5
6
11
C11-F8
Best
220
258.6912
223.6432
284.7586
220
220
245.4116
220
220
220
220
248.2351
220
Mean
220
285.3233
240.5843
325.95
222.4592
257.5026
266.3147
227.3776
224.0986
224.0986
246.4917
470.865
222.5047
Median
220
281.17
240.5843
324.1056
222.4592
227.3776
253.6089
227.3776
220
220
236.0957
531.6483
220
Worst
220
320.262
257.5254
370.8302
224.9184
355.2553
312.6294
234.7551
236.3946
236.3946
293.7756
571.9284
230.0189
Std
0
28.33492
15.32562
37.10878
2.984731
68.88765
32.70744
8.954192
8.616175
8.616175
36.77287
161.0301
5.26544
Rank
1
10
6
11
2
8
9
5
4
4
7
12
3
C11-F9
Best
5457.674
374781.4
336665.2
697814.5
11041.59
47806.58
208627.9
18499.88
75972.25
340166.3
708937.6
874293.6
1873402
Mean
8789.286
560697.8
380695.1
1068617
20271.76
66589.3
377005.8
43236.71
134161.7
411177.8
828454.6
1089209
1954852
Median
7828.591
611878.7
388154.1
1161468
20599.99
66996.02
330330.8
39412.95
128749.4
388492.1
856508.3
1074211
1938336
Worst
14042.29
644252.5
409807.3
1253718
28845.46
84558.58
638733.6
75621.04
203175.8
527560.9
891864.2
1334122
2069334
Std
3889.181
133563.1
33735.78
264941.7
8281.714
16458.05
206133.7
25372.04
55157.42
86679.05
85588.09
258343.4
101384.9
Rank
1
9
7
11
2
4
6
3
5
8
10
12
13
C11-F10
Best
−21.8299
−15.1474
−17.0907
−12.6209
−19.405
−18.8457
−13.5067
−14.5551
−21.1801
−11.3313
−13.6317
−11.3867
−11.0857
Mean
−21.4889
−13.9334
−16.8991
−12.2327
−19.0152
−14.3551
−12.8304
−14.0677
−14.6685
−11.236
−13.1122
−11.3363
−11.0392
Median
−21.669
−13.6326
−16.9924
−12.1747
−19.0175
−13.2966
−12.7347
−14.4077
−13.0427
−11.2352
−13.2437
−11.332
−11.0534
Worst
−20.7878
−13.321
−16.521
−11.9607
−18.6207
−11.9813
−12.3453
−12.9002
−11.4084
−11.1423
−12.3297
−11.2945
−10.9644
Std
0.498616
0.873428
0.27595
0.300903
0.421114
3.248101
0.512428
0.827904
4.63444
0.085067
0.667544
0.04007
0.055176
Rank
1
7
3
10
2
5
9
6
4
12
8
11
13
C11-F11
Best
260837.9
5557994
774810
8605193
1549827
4971008
1107924
3655950
612213.8
5205505
1268560
5226548
6108796
Mean
571712.3
5828813
992977
8901363
1664311
5971920
1218465
3849509
1311819
5233232
1415206
5244366
6150774
Median
598725.2
5779629
1011965
8954672
1654404
5848479
1192994
3765694
945640.2
5235642
1400232
5241956
6135506
Worst
828560.9
6198000
1173167
9090917
1798611
7219712
1379948
4210699
2743783
5256141
1591802
5267002
6223290
Std
260922.1
311888.7
182652.4
218241.7
126103.5
976503
121814.2
259907.3
1016990
23271.59
139894.1
21402.87
53468.09
Rank
1
10
2
13
6
11
3
7
4
8
5
9
12
C11-F12
Best
1155937
8077880
3283202
12348593
1198994
4777223
5416874
1260376
1175696
13547393
5521409
2148081
14426897
Mean
1199805
8426155
3386135
13294942
1274918
5048294
5837448
1425143
1328064
14393299
5812159
2319287
14555006
Median
1196965
8445689
3403596
13352032
1273202
5111608
5942811
1438043
1331765
14488324
5853035
2299989
14553136
Worst
1249353
8735363
3454144
14127111
1354273
5192738
6047295
1564112
1473028
15049154
6021158
2529089
14686857
Std
47157.58
286877.7
78506.39
766755.2
71443.68
202702
305338.9
132403.6
127721.8
662190.6
226324.5
165169.6
111676.6
Rank
1
10
6
11
2
7
9
4
3
12
8
5
13
C11-F13
Best
15444.19
15673.41
15447.01
15896.67
15460.95
15480.47
15492.04
15494.46
15487.9
15628.53
93207.96
15473.77
15460.54
Mean
15444.2
15859.93
15448.01
16315.13
15463.27
15490.45
15535.98
15501.42
15508.25
15935.9
128965.6
15491.09
30083.63
Median
15444.2
15727.24
15447.96
16004.54
15462.43
15489.09
15528.33
15498.88
15499.07
15806.47
122594.5
15481.38
15637.37
Worst
15444.21
16311.82
15449.13
17354.77
15467.29
15503.13
15595.2
15513.47
15546.97
16502.13
177465.4
15527.81
73599.23
Std
0.009091
319.7317
0.936378
734.5073
2.951184
11.77287
50.4487
8.855953
28.7804
415.6116
39873.02
26.01087
30492.98
Rank
1
9
2
11
3
4
8
6
7
10
13
5
12
C11-F14
Best
18241.58
85957.61
18404.38
169588
18524.65
19280.26
19076.34
19090.78
19324.11
30320.43
18806.64
18965.34
18831.17
Mean
18295.35
113024.3
18520.66
230315.4
18609.59
19538.42
19231.08
19238.06
19425.84
312498.8
19097.29
19130.04
19117.21
Median
18275.87
104003.8
18529.52
209852.9
18613.95
19391.67
19248.13
19218.59
19435.9
308198.1
19136.96
19138.94
19109.76
Worst
18388.08
158132.1
18619.22
331967.8
18685.8
20090.09
19351.71
19424.28
19507.46
603278.4
19308.62
19276.94
19418.16
Std
71.59938
33932.57
106.2459
76452.06
72.70982
390.523
133.2113
154.6878
81.27908
289123
228.3532
134.2236
252.1723
Rank
1
11
2
12
3
10
7
8
9
13
4
6
5
C11-F15
Best
32782.17
376496.4
43237.1
805730.9
32870.41
33048.29
33010.83
33043.73
33016.97
3251765
266808.2
33282.14
3635414
Mean
32883.58
913742.5
108324.5
1926961
32948.76
54692.41
219807.3
33079.11
33101.38
15519775
301493.4
33290.62
7987262
Median
32897.86
490250.6
104456.3
935843.3
32952.78
33191.73
265754.4
33064.22
33113.92
17841852
306962.8
33289.27
7312427
Worst
32956.46
2297973
181148.2
5030428
33019.06
119337.9
314709.5
33144.26
33160.73
23143632
325240
33301.81
13688781
Std
76.94696
973487
77908.34
2178087
64.00355
45299.42
133672.1
49.07491
66.50997
9507027
28571.3
8.604748
4845152
Rank
1
10
7
11
2
6
8
3
4
13
9
5
12
C11-F16
Best
131374.2
289911.4
133666.6
475920.9
135600.3
142488.4
136342.4
143409.4
133204.9
87188553
9569677
66243039
62144961
Mean
133550
950389
135187.3
1960981
137683.4
145199.6
142217.9
145950.3
141860.1
89472496
18842170
80082023
76892038
Median
133257.5
632638.2
135643.3
1246618
136879.5
145567.6
142484.3
144445.2
141757.3
89326420
15854250
79194589
73536101
Worst
136310.8
2246368
135796
4874767
141374.4
147174.8
147560.5
151501.4
150721
92048591
34090502
95695874
98350989
Std
2392.2
924912.3
1073.35
2079444
2709.018
2414.887
4923.103
3938
7730.783
2140935
11144663
13343828
16165669
Rank
1
8
2
9
3
6
5
7
4
13
10
12
11
C11-F17
Best
1916953
7.69E+09
2.12E+09
1.12E+10
1957612
1.06E+09
6.96E+09
2038930
2299063
2.16E+10
9.93E+09
1.85E+10
2.06E+10
Mean
1926615
9.02E+09
2.33E+09
1.56E+10
2293290
1.29E+09
9.76E+09
3026639
3119727
2.25E+10
1.13E+10
2.10E+10
2.20E+10
Median
1923412
9.20E+09
2.33E+09
1.61E+10
2151018
1.31E+09
9.55E+09
2583866
3212688
2.24E+10
1.16E+10
2.06E+10
2.13E+10
Worst
1942685
1.00E+10
2.55E+09
1.91E+10
2913511
1.47E+09
1.30E+10
4899892
3754468
2.34E+10
1.20E+10
2.42E+10
2.49E+10
Std
12003.53
1.08E+09
2.00E+08
3.55E+09
450632.8
2.22E+08
2.66E+09
1354721
706077.7
7.96E+08
9.67E+08
2.72E+09
2.04E+09
Rank
1
7
6
10
2
5
8
3
4
13
9
11
12
C11-F18
Best
938416.2
38056233
3961154
8.23E+07
949848.4
1798555
4141903
967157.9
964035.1
24725187
8352117
1.14E+08
1.11E+08
Mean
942057.5
55335975
6591790
119000000
971938.3
2057336
9635664
1031700
988411.5
31196699
11194144
1.36E+08
1.15E+08
Median
942553.5
60171362
5547873
1.29E+08
953682.2
2014229
8740133
978717.1
994949.5
33157306
11150777
1.39E+08
1.15E+08
Worst
944706.9
62944940
11310259
136000000
1030540
2402332
16920486
1202207
999712.1
33746997
14122905
151000000
120000000
Std
2774.139
12250909
3597267
2.65E+07
41194.04
305963.2
5671597
119728.3
17307.31
4553484
2710018
1.73E+07
3.64E+06
Rank
1
10
6
12
2
5
7
4
3
9
8
13
11
C11-F19
Best
967927.7
46475624
6108745
1.01E+08
1068411
2231819
2066743
1233805
1129215
25080259
2395060
1.58E+08
1.13E+08
Mean
1025341
54468087
6693128
1.17E+08
1138554
2472748
10278052
1364982
1479593
35816486
6301409
1.74E+08
1.16E+08
Median
983146.6
51071115
6276995
1.10E+08
1096048
2369288
10210584
1339636
1411616
36753610
7266724
1.68E+08
1.15E+08
Worst
1167142
69254493
8109779
147000000
1293711
2920598
18624296
1546849
1965925
44678464
8277126
201000000
119000000
Std
99675.04
10803670
999513.5
2.25E+07
109726.8
322083.5
8192328
137939.2
368407.8
8923056
2806570
1.97E+07
2.73E+06
Rank
1
10
7
12
2
5
8
3
4
9
6
13
11
C11-F20
Best
936143.2
50954706
5223024
1.10E+08
957152.3
1647208
6898584
977723.6
962978.5
34027686
9534816
1.46E+08
1.10E+08
Mean
941250.4
57915854
5924708
1.26E+08
960470.6
1832300
7322355
998911.1
973018.6
34790803
14357706
1.60E+08
1.16E+08
Median
940995.9
56061928
5900417
1.22E+08
961081.7
1770969
7251433
1001253
972340.5
34759730
12833683
1.60E+08
1.17E+08
Worst
946866.6
68584856
6674976
150000000
962566.8
2140056
7887969
1015414
984414.7
35616065
22228640
174000000
120000000
Std
5013.552
7896396
633442.1
1.77E+07
2451.631
245964.6
444598.3
17075.25
9907.006
694445.5
5830855
1.61E+07
4.38E+06
Rank
1
10
6
12
2
5
7
4
3
9
8
13
11
C11-F21
Best
9.974206
41.53049
20.3649
57.17792
13.78391
26.56174
35.6313
20.64111
24.52369
48.57262
35.96567
91.90592
59.07491
Mean
12.71443
50.50227
21.70431
76.8985
15.95743
29.94194
38.96962
22.44344
27.63915
101.2915
40.89389
106.3608
103.21
Median
12.95425
50.18166
21.46124
76.90776
15.89929
30.83809
38.56438
22.17047
27.67676
103.6709
41.89081
107.5927
113.8538
Worst
14.97499
60.11528
23.52986
96.60055
18.24724
31.52984
43.11841
24.79172
30.67941
149.2516
43.82828
118.3517
126.0577
Std
2.412667
8.418648
1.420534
18.29913
2.179408
2.41647
3.47749
1.927552
3.642786
43.35039
3.696548
13.67187
32.75307
Rank
1
9
3
10
2
6
7
4
5
11
8
13
12
C11-F22
Best
11.50133
40.80883
22.24877
46.17906
16.22029
28.20055
40.22051
23.95676
24.842
66.75931
39.13532
89.94588
92.17511
Mean
16.12513
47.02927
27.52481
63.81642
19.10191
32.24992
46.54553
25.05548
32.40714
103.2154
46.90524
107.2767
93.11538
Median
16.72317
47.34708
27.50942
67.79995
19.45731
33.00267
47.31893
25.18537
33.63111
111.9199
46.28905
110.3475
92.79237
Worst
19.55286
52.6141
32.83162
73.4867
21.27275
34.79379
51.32375
25.89443
37.52432
122.2625
55.90755
118.4661
94.70167
Std
4.197797
5.316853
5.254729
12.72251
2.533066
2.999636
5.265485
0.938547
5.947232
26.22423
7.253922
13.5143
1.170916
Rank
1
9
4
10
2
5
7
3
6
12
8
13
11
Sum rank
22
109
191
231
55
146
145
97
118
222
157
198
224
Mean rank
1
4.954545
8.681818
10.5
2.5
6.636364
6.590909
4.409091
5.363636
10.09091
7.136364
9
10.18182
Total rank
1
12
2
4
13
3
11
6
9
7
10
5
8
Wilcoxon: p-value
9.77E−18
1.71E−17
1.71E−18
7.09E−16
9.66E−18
5.69E−16
7.10E−15
3.99E−12
7.15E−14
2.03E−18
9.08E−17
6.54E−18
Boxplot diagrams of FLO and the performance of the competitive algorithms for the CEC 2017 test suite
Pressure Vessel Design Problem
The optimization challenge of the pressure vessel design, illustrated schematically in Fig. 6, revolves around minimizing construction costs while meeting specified design requirements. This problem is encapsulated by a mathematical model outlined as follows [72]:
Schematic representation of the pressure vessel design
The optimization outcomes for pressure vessel design, utilizing FLO alongside competitive algorithms, are detailed in Tables 8 and 9. Additionally, the convergence trajectory of FLO, depicting its journey towards the solution across iterations, is illustrated in Fig. 7. Notably, FLO emerges triumphant, securing the optimal design with design variable values of (0.7780271, 0.3845792, 40.312284, 200) and an associated objective function value of 5882.9013. These results underscore FLO’s superior performance in pressure vessel design optimization, outshining competitive algorithms and delivering superior outcomes.
Performance of the optimization algorithms for the pressure vessel design problem
Algorithm
Values of the variables of the best solution
Cost
Th
Ts
L
R
GA
0.8317876
1.4828762
58.62914
60.436345
11533.208
PSO
0.6521499
1.6439969
31.507574
65.916976
10499.421
WSO
0.3845790
0.7780271
200
40.312281
5882.9011
MVO
0.4202776
0.8412891
159.51516
43.571299
6018.566
AVOA
0.3845812
0.7780312
199.99699
40.3125
5882.9085
GSA
1.2516962
1.1728405
189.66668
44.572154
12726.817
RSA
0.6715044
1.2457527
29.541317
63.011657
7988.1974
MPA
0.3845792
0.7780271
200
40.312284
5882.9013
GWO
0.3859628
0.7785126
199.96009
40.321643
5891.0999
TSA
0.3859699
0.7796786
200
40.39555
5912.596
WOA
0.4591628
0.9277593
125.78933
46.950913
6318.0671
TLBO
0.4930712
1.6576806
115.47982
48.594397
11406.549
FLO
0.3845792
0.7780271
200
40.312284
882.8955
Statistical results of the optimization algorithms for the pressure vessel design problem
Algorithm
Mean
Best
Worst
Std
Median
Rank
FLO
5882.8955
5882.8955
5882.8955
1.92E-12
5882.8955
1
WSO
5892.2389
5882.9011
5975.0303
25.597613
5882.9017
3
AVOA
6260.4989
5882.9085
7187.8793
405.93922
6067.7468
5
RSA
13203.718
7988.1974
21708.462
3602.5764
12075.035
9
MPA
5882.9013
5882.9013
5882.9013
4.24E-06
5882.9013
2
TSA
6318.3698
5912.596
7078.0209
383.81952
6175.3376
6
WOA
8256.0687
6318.0671
13647.68
1937.652
7786.0827
8
MVO
6595.3898
6018.566
7192.4617
369.02769
6656.059
7
GWO
6028.1203
5891.0999
6766.8855
275.78721
5900.4533
4
TLBO
30997.684
11406.549
66934.242
15893.46
27298.577
12
GSA
22439.592
12726.817
35296.066
7732.2649
21527.451
10
PSO
32584.002
10499.421
56166.913
14879.262
35973.439
13
GA
27805.883
11533.208
50355.085
12475.479
24579.373
11
FLO’s performance convergence curve for the pressure vessel design
Speed Reducer Design Problem
The speed reducer design poses an optimization challenge, depicted schematically in Fig. 8, with the primary objective of minimizing the weight of the speed reducer. This mathematical model encapsulates the design as follows [73,74]:
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, and 5≤x7≤5.5.
The outcomes of addressing the speed reducer design using FLO and the competitive algorithms are outlined in Tables 10 and 11. Additionally, the convergence trajectory of FLO is depicted in Fig. 9. Impressively, FLO achieves the best design, with the design variable values of (3.5, 0.7, 17, 7.3, 7.8, 3.3502147, 5.2866832) and an associated objective function value of 2996.3482. These results underscore FLO’s superior performance in speed reducer design optimization, surpassing competitive algorithms and delivering enhanced outcomes.
Performance of the optimization algorithms for the speed reducer design problem
Algorithm
Values of the variables of the best solution
Cost
b
M
p
l1
l2
d1
d2
FLO
3.5
0.7
17
7.3
7.8
3.3502147
5.2866832
2996.3482
WSO
3.5000005
0.7
17
7.3000098
7.8000004
3.3502148
5.2866833
2996.3483
AVOA
3.5
0.7
17
7.3000007
7.8
3.3502147
5.2866832
2996.3482
RSA
3.591081
0.7
17
8.2108102
8.2554051
3.3555991
5.4809743
3180.6287
MPA
3.5
0.7
17
7.3
7.8
3.3502147
5.2866832
2996.3482
TSA
3.512746
0.7
17
7.3
8.2554051
3.3505367
5.2901744
3013.6687
WOA
3.5864383
0.7
17
7.3
8.0068571
3.3614768
5.2867549
3037.755
MVO
3.5022252
0.7
17
7.3
8.0658639
3.3693655
5.2868795
3008.0939
GWO
3.5006336
0.7
17
7.3050825
7.8
3.3637852
5.2887849
3001.4528
TLBO
3.5554343
0.7039501
26.213531
8.0919072
8.1411238
3.6597328
5.3387355
5243.4107
GSA
3.5226393
0.7027207
17.364783
7.8143829
7.8885518
3.4080837
5.3847634
3167.6764
PSO
3.5080871
0.7000711
18.082718
7.3978687
7.867227
3.5925551
5.3433467
3298.9223
GA
3.5770929
0.7054996
17.804212
7.7373556
7.8551847
3.6974126
5.3456293
3.34E+03
Statistical results of the optimization algorithms for the speed reducer design problem
Algorithm
Mean
Best
Worst
Std
Median
Rank
FLO
2996.3482
2996.3482
2996.3482
9.58E–13
2996.3482
1
WSO
2996.6283
2996.3483
2998.7703
0.593604
2996.3642
3
AVOA
3000.8027
2996.3482
3010.9013
4.0273974
3000.7044
4
RSA
3273.4732
3180.6287
3331.0939
58.376714
3288.1754
9
MPA
2996.3482
2996.3482
2996.3482
3.23E–06
2996.3482
2
TSA
3031.7102
3013.6687
3045.2791
10.291508
3033.477
7
WOA
3148.2393
3037.755
3439.8167
107.88934
3115.2947
8
MVO
3029.4277
3008.0939
3069.3067
13.455894
3029.8624
6
GWO
3004.5239
3001.4528
3010.4183
2.5448163
3004.0121
5
TLBO
6.873E+13
5243.4107
4.975E+14
1.175E+14
2.692E+13
12
GSA
3449.2391
3167.6764
4062.9379
266.13006
3321.1203
10
PSO
1.014E+14
3298.9223
5.139E+14
1.258E+14
7.256E+13
13
GA
4.884E+13
3342.7177
3.152E+14
7.902E+13
1.957E+13
11
FLO’s performance convergence curve for the speed reducer design
Welded Beam Design
Welded beam design is an optimization problem of real-world applications with the schematic representation shown in Fig. 10, whose main design goal is the minimization of the fabrication cost of the welded beam. The mathematical model for this problem can be formulated as follows [36]:
The results comparing FLO with competitive algorithms for the welded beam design are summarized in Tables 12 and 13. Additionally, Fig. 11 illustrates the convergence curve of FLO towards the solution. Remarkably, FLO identifies the best design with the design variable values of (0.2057296, 3.4704887, 9.0366239, 0.2057296) and achieves an objective function value of 1.7246798. These findings underscore FLO’s superior performance in tackling the welded beam design optimization problem, showcasing its efficacy over competitive algorithms.
Performance of the optimization algorithms for the welded beam design problem
Algorithm
Values of the variables of the best solution
Cost
h
l
t
b
FLO
0.2057296
3.4704887
9.0366239
0.2057296
1.7248523
WSO
0.2057296
3.4704887
9.0366239
0.2057296
1.7248523
AVOA
0.204974
3.4868751
9.0365185
0.2057344
1.7259067
RSA
0.1968043
3.5338976
9.9140767
0.2176511
1.972399
MPA
0.2057296
3.4704887
9.0366239
0.2057296
1.7248523
TSA
0.2042143
3.4950752
9.0638538
0.2061512
1.7337349
WOA
0.2136308
3.3314508
8.9745837
0.2208114
1.8201429
MVO
0.2059899
3.4648795
9.0445874
0.2060516
1.7283218
GWO
0.2055937
3.4736068
9.0362448
0.2057979
1.7255154
TLBO
0.313913
4.4099301
6.8250934
0.4224048
3.0076755
GSA
0.2927572
2.7308917
7.4410075
0.3066903
2.0800575
PSO
0.3704896
3.4252427
7.3653863
0.5694254
3.9945673
GA
0.2240807
6.8722146
7.7790615
0.3031552
2.7482044
Statistical results for the optimization algorithms for the welded beam design problem
Algorithm
Mean
Best
Worst
Std
Median
Rank
FLO
1.7246798
1.7246798
1.7246798
2.34E−16
1.7246798
1
WSO
1.7248526
1.7248523
1.7248578
1.269E−06
1.7248523
3
AVOA
1.7607647
1.7259067
1.8412254
0.0369935
1.7470447
7
RSA
2.1759749
1.972399
2.519308
0.1462171
2.1512469
8
MPA
1.7248523
1.7248523
1.7248523
3.40E–09
1.7248523
2
TSA
1.7429229
1.7337349
1.7520018
0.0056864
1.743018
6
WOA
2.3034718
1.8201429
4.0174423
0.6509517
2.08131
9
MVO
1.7410203
1.7283218
1.7744279
0.013955
1.7370004
5
GWO
1.7272229
1.7255154
1.7312168
0.0013824
1.7269807
4
TLBO
3.285E+13
3.0076755
3.17E+14
8.229E+13
5.6424231
12
GSA
2.4348133
2.0800575
2.7395728
0.1942722
2.4639914
10
PSO
4.53E+13
3.9945673
2.742E+14
8.886E+13
6.6680784
13
GA
1.112E+13
2.7482044
1.203E+14
3.506E+13
5.609433
11
FLO’s performance convergence curve for the welded beam design
Tension/Compression Spring Design
The optimization task of tension/compression spring design stems from practical applications, featuring a schematic representation in Fig. 12. The primary design objective revolves around minimizing the weight of the tension/compression spring. Formulating this design challenge involves crafting a mathematical model as follows [36]:
FLO’s performance convergence curve for the welded beam design tension/compression spring design
The outcomes of employing FLO alongside competitive algorithms for optimizing the tension/compression spring design are showcased in Tables 14 and 15. Additionally, Fig. 13 illustrates the convergence trajectory of FLO towards the solution. Notably, FLO achieves the optimal design with design variable values of (0.051689105, 0.356717704, 11.2889661) and an associated objective function value of 0.01260189. These findings unequivocally demonstrate FLO’s superior performance over competitive algorithms in addressing the tension/compression spring design problem.
Optimization results for the tension/compression spring design
Algorithm
Values of the variables of the best solution
Cost
P
d
D
GA
2.828481
0.0679933
0.8927686
0.0178071
TSA
12.334688
0.0509975
0.3403048
0.0126818
WOA
12.0511
0.0511727
0.3444345
0.0126706
MVO
13.852933
0.0501506
0.3204164
0.0127487
TLBO
2.828481
0.0675319
0.8850682
0.0174182
GSA
7.8639345
0.055068
0.4400717
0.0130684
RSA
14.669014
0.0501506
0.3146926
0.0131519
MPA
11.286619
0.0516907
0.3567578
0.0126652
PSO
2.828481
0.0674505
0.8819948
0.0173176
AVOA
12.012341
0.0511979
0.3450265
0.0126701
GWO
10.930013
0.0519529
0.3630808
0.0126706
WSO
11.291725
0.0516871
0.3566707
0.0126652
FLO
11.288966
0.0516891
0.3567177
0.0126652
Optimization results for the tension/compression spring design
Algorithm
Best
Std
Mean
Median
Worst
Rank
GA
0.0178071
4.885E+12
1.593E+12
0.0252274
1.647E+13
12
RSA
0.0131519
6.945E−05
0.0132315
0.013211
0.0133718
6
WOA
0.0126706
0.0006048
0.013257
0.0130638
0.0144524
7
MPA
0.0126652
2.85E−09
0.0126652
0.0126652
0.0126652
2
TSA
0.0126818
0.0002418
0.0129549
0.0128831
0.0135043
5
GWO
0.0126706
5.535E−05
0.0127215
0.0127191
0.0129391
4
MVO
0.0127487
0.0016487
0.0163776
0.0172702
0.0177786
9
TLBO
0.0174182
0.0003583
0.0179362
0.0178931
0.0185278
10
GSA
0.0130684
0.0042637
0.0192519
0.0188366
0.0315728
11
AVOA
0.0126701
0.000558
0.0133257
0.0132591
0.014115
8
PSO
0.0173176
8.314E+13
2.039E+13
0.0173176
3.618E+14
13
WSO
0.0126652
3.588E−05
0.0126761
0.0126656
0.012822
3
FLO
0.0126019
7.07E−18
0.0126019
0.0126019
0.0126019
1
FLO’s performance convergence curve for the welded beam design tension/compression spring design
Conclusions and Future Works
This paper introduces Frilled Lizard Optimization (FLO), a novel bio-metaheuristic algorithm inspired by the natural behaviors of frilled lizards. Drawing upon observations of these creatures in the wild, FLO is designed to emulate two key behaviors: the sit-and-wait hunting strategy and the post-feeding retreat behavior. The algorithm is intricately divided into two distinct phases, each aimed at replicating a specific aspect of the lizard’s behavior: exploration and exploitation. Through meticulous mathematical modeling, FLO seeks to capture the essence of these behaviors and apply them in the context of optimization problems. To assess its effectiveness, FLO undergoes rigorous testing on fifty-two standard benchmark functions, spanning a range of complexities and characteristics. The results of these evaluations reveal FLO’s remarkable aptitude in exploration, exploitation, and the delicate balance between these two aspects crucial in problem-solving environments. In comparative analyses against twelve well-established algorithms, FLO consistently emerges as the top-performing optimizer, showcasing its robustness and efficacy across various functions and problem domains. Furthermore, FLO’s capabilities extend beyond theoretical assessments, as it demonstrates remarkable performance when applied to practical scenarios. Testing on twenty-two constrained optimization problems sourced from the CEC 2011 test suite, as well as four engineering design challenges, underscores its versatility and applicability in real-world settings. Notably, FLO outperforms competitive algorithms in handling these challenges, highlighting its potential to address complex optimization tasks in diverse fields.
Despite its strengths, it is important to acknowledge the inherent limitations of FLO, common to many metaheuristic algorithms. The stochastic nature of these algorithms means that there is no guarantee of achieving the global optimum, and the No Free Lunch (NFL) theorem cautions against claims of universal superiority. Additionally, as with any evolving field, there is always the possibility of newer, more advanced algorithms being developed in the future.
Looking ahead, the introduction of FLO presents exciting opportunities for further research and exploration. Future endeavors may include the development of binary and multi-objective variants of the algorithm, as well as its application to a wider array of optimization problems across different disciplines. By continuing to refine and expand upon the principles underlying FLO, researchers can unlock new avenues for innovation and problem-solving in the realm of optimization.
The authors are grateful to editor and reviewers for their valuable comments.
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: M.D., I.A.F., O.A.B., S.A., G.B., S.G.; data collection: F.W., G.B., O.P.M., S.G., I.L., S.A., O.A.B.; analysis and interpretation of results: O.A.B., I.A.F., F.W., O.P.M., S.A.; draft manuscript preparation: I.A.F., F.W., G.B., S.G., M,D., I.L. All authors reviewed the results and approved the final version of the manuscript.
Availability of Data and Materials
The authors confirm that the data supporting the findings of this study are available within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.
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The MATLAB codes of the competitive algorithms used in simulation and comparison studies are available as follows: