A Cooperation of the Multileader Fruit Fly and Probabilistic Random Walk Strategies with Adaptive Normalization for Solving the Unconstrained Optimization Problems

A swarm-based nature-inspired optimization algorithm, namely, the fruit fly optimization algorithm (FOA), has a simple structure and is easy to implement. However, FOA has a low success rate and a slow convergence, because FOA generates new positions around the best location, using a fixed search radius. Several improved FOAs have been proposed. However, their exploration ability is questionable. To make the search process smooth, transitioning from the exploration phase to the exploitation phase, this paper proposes a new FOA, constructed from a cooperation of the multileader and the probabilistic random walk strategies (CPFOA). This involves two population types working together. CPFOAs performance is evaluated by 18 well-known standard benchmarks. The results showed that CPFOA outperforms both the original FOA and its variants, in terms of convergence speed and performance accuracy. The results show that CPFOA can achieve a very promising accuracy, when compared with the well-known competitive algorithms. CPFOA is applied to optimize two applications: classifying the real datasets with multilayer perceptron and extracting the parameters of a very compact T-S fuzzy system to model the Box and Jenkins gas furnace data set. CPFOA successfully find parameters with a very high quality, compared with the best known competitive algorithms.


Introduction
Over the last few decades, researchers have started to adapt their knowledge of natural phenomena for the development of optimization techniques. These techniques were successfully applied for chemical process applications( [1], [2], [3] and [4]). The main concepts of the aptly named, sources of nature-inspired algorithms, have been observed within the successful biological systems. Accordingly, most nature-inspired algorithms are biologically inspired, or bio-inspired, and mimic specific behavior in nature. Examples of such popular natureinspired algorithms include the particle swarm optimization algorithm(PSO)( [5]),which was inspired by the social behavior of flocking birds, or schooling fish; the ant colony optimization algorithm (ACO)( [6]),which mimics an ant colonys behavior in their search for food; the artificial bee colony algorithm (ABC)( [7]),motivated by the intelligent behavior of a honey bee swarm; the cuckoo search algorithm (CS)( [8]),inspired by the parasitic biointeractions of a cuckoo species, which lays their eggs in the nests of other host birds; and the bat-inspired algorithm (BA)( [9]),which was inspired by the echolocation behavior of bats, to name but a few. However, each natureinspired algorithm has different capabilities when it comes to finding solutions, depending on the personal ability

The Fruit Fly Optimization Algorithm
The drosophila optimized algorithm or fruit fly optimization algorithm (FOA) was developed in 2011 by Pan ( [12]). FOA determines global optimization based on the foraging behavior of fruit flies. Compared to other species, the fruit fly possesses a keener sense of smell and sight in search of food. Their drosophila olfactory organ can detect a food source as far as 40 km away, which triggers a flight reaction toward the target location. For the intelligent sense of fruit flies, the novel FOA optimization algorithm is inspired and established through the simple behavior of fruit flies search for food.
The FOAs process is similar to that of other swarm optimization algorithms. The first phase of the fruit flies quest for food is initiated with a random uniform distribution ( [10], [32], [33], [34], [35], [36]), with no specific 461 position or direction. In the second phase, the fruit fly with the best sense of smell or the best fitness within the group, from the first phase, is determined. The fruit fly with the best sense of smell, represented by X axis t is used as a center for generating new populations in the next generation. The computational steps of the FOA method are summarized in Algorithm 1.
Step 2: Give the random position and fly direction of an individual fruit fly in search of food.
Step 3: Calculate the distance (Dist) to the food's origin, as the exact position of the food's location is not known at this stage.
Step 4: Calculate the smell concentration judgment value(S i ).
Step 5: Calculate the fitness: the smell concentration judgment of the individual fruit fly, obtained from Step 4, is calculated by substituting S i into the smell concentration judgment function(also called the fitness function),in order to find the optimal smell.
Step 6: Determine the fruit fly with the optimal smell concentration judgment among the fruit fly group.
[bestSmell, bestIndex] = f ind the best(Smell) Step 7: Keep the best (x,y) position and the optimal concentration value, and use this position as the flight center towards the next location(in Step 2).
Step 8: Repeat Steps 2-7, and determine whether the smell concentration is better than the previous iterative smell concentration. If yes, go to Step 7. The process will stop if either the smell concentration no longer changes, or the iterative number reaches the maximum iteration number(M ax iter).The outputs are X axis and Y axis.  [15], [16], [18], [30], [31], [37], [38]) and are briefly described as follows. 1. Problems regarding the smell value S i ,according to Step 4, cannot appropriately evaluate the "objective function(S i )" when there are negative numbers in the domain because S i = 1 Disti > 0,so that the function cannot determine S i as a negative ( [15], [18], [37], [38]). 2. The fixed radius, with random uniform distribution,rand(),within the initial process, limited the convergence of FOA in the processes of exploration and exploitation ( [15], [16], [18], [30], [31]).

Analysis of the FOA Variants
To overcome the disadvantages of the original FOA, researchers have continuously developed new strategies to improve the FOA for solving high-dimensional function optimization problems. The recently proposed FOAs can be grouped in the two categories. In the first category, each of the fruit flies is defined through the random initial base point of X axis, Y axis and the positions of X i and Y i ,as in the original FOA, to update the new generation of populations. The other functions, such as the smell value(S i )and the evaluation of the objective function(S i ),are modified, including the extra mechanisms. In the second category, these the problems are solved by (i) omitting Y i ;(ii) defining each of the fruit flies through X ∈ R N ×D ,where D is the number of decision variables (ordimensions),and N is the population size, i.e., and (iii) removing the distance Dist i and the smell concentration judgment function S i .The fitness value is now calculated by substituting x i into the smell concentration judgment function, with Smell i = objectivef unction(x i ).The position of the fruit fly with the minimal concentration value,X axis,is the base point for flying towards the next location. A brief summary of the two categories of improved FOAs are as follows. The first category is as follows.
• Babalık et al. ( [39]) proposed an improvement of the fruit fly optimization algorithm using sign parameters (SFOA). The algorithm presents the improvement method by using two sign variables, r and q vectors, in order to determine a sign for each decision variable of fruit flies.
• CEFOA ( [31]), proposed by Han et al., revealed that the simple structure of FOA limited the search space and easily trapped the fruit flies in a local minimum. To overcome this drawback, the CEFOA used two mechanisms: The trend search strategy and the co-evolution mechanism. The trend search enhances the local search capability of swarm. The co-evolution mechanism is employed to avoid premature convergence and to improve the global searching ability. However, the key of the CEFOA is the multi-scale equation for updating the fruit fly swarm. To set the variable capacity of each fruit fly, connected with its food quality, the CEFOA used the variance of the multi-level evolutionary operator. The search radius is dynamically adjusted are the multi-scale factors of the i-th fruit fly.
The second category is as follows.
• LGMS-FOA, proposed by Shan et al.( [15]),presented two parameters to tune up the search radius by adding the weight parameter w,when the radius is changed with respect to time. A new fruit fly location is generated as x d i = X axis d i + w × rand[0, 1), w = w 0 × α t ,where w 0 = 1, α = 0.95,t=iteration index, and X axis is the best position obtained during iterations. ,UB is the upper bound and LB is the lower bound of domain problems,λ min = 10 −5 ,t is the iteration index and t max is the maximum iteration number (Max iter). • MFOA, proposed by Yuan et al.( [18]), presented a multi-swarm fruit fly that employed sub-swarm to explore the solutions in the search space simultaneously. Moreover, MFOA shrinks the search radius through where t is the iteration index,UB is the upper bound,LB is the lower bound,G is the number of sub-swarms and G max is the maximum number of sub-swarms. • MSFOA, proposed by Zhang et al.( [30]), presented a strategy to analyze the convergence and showed that the convergence depends on the initial positions of the swarms. MSFOA used the Gaussian mutation operator, rather than the uniform random number (more details of which can be found in ( [30])). For a flying fruit fly, MSFOA used a linear generation mechanism, through the equation,x t i,j = X t j + w × rand(R min , R max ),where w = w 0 × α t ,w 0 = 1,α = 0.95,t is current iteration,w is the search coefficient,α is the initial weight and R min , R max are obtained from the domain boundary of the problem. To recap, the improved FOAs, such as SFOA, CEFOA, LGMS-FOA, IFFO, MFOA and MSFOA, were strategies proposed to enhance the search ability. The search radius was a main point to tackle in several proposed strategies. However, only the dynamic mechanism of the search radius itself might be insufficiently efficient to overcome the lack of diversity and premature convergence, because these FOA variants still use only one leader as the flying base point. The single leader strategy might affect the FOA by easily trapping a local optimum when optimizing a multi-dimensional optimization problem. In this paper, the proposed CPFOA is comprised of two strategies. The first strategy focuses on the enhancement of the search ability based on the multileader fruit fly. The latter is a probabilistic dynamic search radius, with adaptive normalization. These two mechanisms are different from the existing FOA variants. The details of the proposed CPFOA are presented in the next section.

The Proposed CPFOA
This section presents a cooperation of multileader fruit flies and a probability search of a random walk for FOA to solve the unconstrained optimization problems (CPFOA). The CPFOA consists of (i) the cooperation strategies, called the multileader strategies, and (ii) the probabilistic search by a random walk. The multileader strategy used a main leader and several other leaders as the flying bases. The probabilistic search by a random walk changes the search radius to control the search spaces of the main leader.

Multileader Strategy
Contrary to FOA, which uses a single leader fruit fly, the CPFOA uses multileader fruit flies. Suppose that there is a swarm of fruit flies,X ∈ R N ×D ,where D is the number of decision variables (or dimensions), and N is the number of fruit flies, i.e., The generated multileader strategy has four computational steps, as follows.
Step 1. Sort X in ascending order, based on the individual fitness values, to be: Step 2. DivideẊ into the M disjoint sub-swarms, as in equation (2): where M is the total number of sub-swarms (aka, the number of leaders).
Step 3. Compute leader 1 , . . . , leader M by equation (3): Step 4. Generate M new fruit flies based on leader 1 , . . . , leader M by equation (4): where ⊗ is the Hadamard product. The multileader strategy generates M new fruit flies from M leaders, i.e., a new fruit fly is generated from each leader. These fruit flies are generated from the shared information and might result in the improvement of the exploration ability. Moreover, CPFOA controls the shrinking of the search radius (β) in Step 4 through the probability of P s.There are two types of radius: The normal scope (N scope ) and high scope (H scope ).The generation of β is controlled by P s,as follows: (6) where M ax iter is the maximum number of iterations,t is the iteration index, and 0 ≤ t ≤ M ax iter.
where U b = upper − bound,and Lb = lower − bound of the domain search problems.
Based on Equation (6), the radius of the fruit fly search is probabilistically large,if P s is a small value (or t is at the early phase of the optimization). Otherwise, it is probabilistically a small value, if the iteration is in the latter phase of the optimization.

Probabilistic Search Strategy Based on a Random Walk with Adaptive Normalization
As mentioned in Section 2.1.1, FOA uses the random uniform distribution, with a fixed radius of search. In this paper, CPFOA will employ a random walk generation, which is inspired by the ant lion optimizer ( [40]), and a probabilistic control of the search radius. The random walk is a mathematical equation process, which can provide the series of consecutive random steps ( [41], [42]). The value generated from a random walk at time n > 0 is found by a recursive formula, as follows: where x n is a random value extracted from a random number generator, and R 0 = 0. Equation (9) shows that the changing state of R n is attached to the previous state of R n−1 and every step, obtained from current iteration to the next iteration. The details of the random walk in the CPFOA strategy can be described in this section. 1. The fruit fly with the best fitness (X axis) is determined after the first generation of the evaluation.(X axis) is used as a center for updating the N-M candidate solutions in the next generations. 2. As for the updating step, CPFOA uses the random walk to generate N-M individual fruit flies. The characteristics of the populations of the random walk movements are described as: where ρ is a function that controls the direction of the fruit fly at any changing step, and XR 0 = 0.In the proposed CPFOA,ρ is either-1 or 1 and is determined as: where t denotes the iteration that the random walk came to a halt,r(t) is a stochastic function, and rand is a random uniform point in [0, 1). In order to match the values generated from Equation (10) with the boundaries of the problem and, furthermore, to make the search process smooth, transitioning from the exploration phase to the exploitation phase, equation (10) is adaptively normalized as follows: where a is the minimum of {XR 0 , . . . , XR t },b is the maximum of {XR 0 , . . . , XR t },b,c and d are the minimum and maximum radii at the t-th iteration to control the scope of the search space during the optimization steps, respectively. The value of c and d are determined through the changing values of L,as follows: where L is a special constant parameter determined from the probability of the P s variable. The parameter of L and P s can be calculated as follows: P s = t M ax iter (18) where t is the current iteration,M ax iter is the maximum number of iterations, and L = P s × 10 2 when P s > 0.25, L = P s × 10 3 when P s > 0.5, L = P s × 10 4 when P s > 0.75, L = P s × 10 5 ,when P s > 0.8, andL = P s × 10 6 when P s > 0.9.In the proposed CPFOA,L is used to adjust the accuracy level of exploitation. Equation (14) through Equation (18) perform the probabilistic control of the search radius for CPFOA, which is different from the mechanism in FOA variants. The simulation state of the cooperation of FOA's leader and CPFOA's co-leaders is shown in Figure 1. A graph of the search radius generated during CPFOA, optimized as "Exponential function" (f1 in Table 1), is shown in Figure 2 (a), and the example the graph of the search radius generated by IFFO is shown in Figure 2 (b).We have observed that the search radii in the two figures are very different. The behavior of the search radius in CPFOA is very similar to that of the chaotic gravitational constants in the gravitational search algorithm (GSA)( [43]). This kind of behavior should help CPFOA in smoothly transitioning from the exploration stage to exploitation.

The Proposed CPFOA
The structure of the CPFOA is similar to that of the IFFO, in that the function Dist i and the smell concentration judgment value (S i ) are eliminated. The pseudo code of CPFOA is presented in Algorithm 2.

The Experiments and Evaluations
There are two groups of competitive algorithms. The first group, shown in Table 2, is the FOA variants, including FOA, LGMS, IFFO, MFOA, MSFOA, and CEFOA. The second group, shown in Table 3, comprises CEFOA and six meta-heuristic algorithms: PSO, DE, GSA, HS, BA, and FA.

Parameters and Settings
There are two experiments, each of which is as follows: 1. In the first experiment, the proposed CPFOA is compared with the competitive algorithms from the first group, i.e., CPFOA and the FOA variants are competing. The comparison is conducted on the 18 scalable functions taken from ( [31]). The dimension of each problem is set to three values: 30, 50, and 1000. 2. In the second experiment, the proposed CPFOA is compared with the competitive algorithms from the second group, i.e., CPFOA and the original version of some state-of-the-art algorithms are competing. The comparison is based on the 18 scalable functions taken from ( [31]). The dimension of each problem is set as those in the first experiment.
The following settings are set to comply with that of the existing CEFOA, the maximum iteration(M ax iter)of each algorithm is fixed to 500, the population size (popsize) is 30, and the average (ave) and the standard deviations  Tables 2 and 3.

Performance Criteria
The criteria for performance evaluation of the competing algorithms are the quality, the robustness, the success rate, and the statistical test, each of which is as follows: 1. The quality of the algorithms is determined by the average value (ave) and standard deviation ( N F E ),where x is a feasible optimal solution of the function, and x * is the best known solution of a specific problem f .A higher average SR ave indicates a better performance. The average success rate (SR ave ) is written as: SR ave = number of successf ul runs total number of runs (19) A statistical test, to investigate the significance of difference between CPFOA's outcome and the competitive algorithm's outcome, the Wilcoxon signed rank test, with the significance level of 0.05, is conducted to judge whether the 50 runs of CPFOA are statistically better than that of its competitors. The h values, signifying the results of the Rank-sum test, are indicated in Tables 6-13 by one of three symbols:"+","-" or "=", where the "+" symbol means that the solutions produced by the competitive algorithm are better than those of CPFOA, the "=" symbol means the outcomes of the competitive algorithm are comparable to or similar to those of CPFOA, and the "-" symbol means that the outcomes of the competitive algorithm are worse than those of CPFOA. To conclude the statistical test, the total h is represented as #1/#2/#3, where #1, #2, and #3 represent the number of wins, ties, and losses of the algorithm, respectively.

Convergence Behavior of CPFOA
This section will provide consistent information about the convergence behavior of CPFOA, when the problem has an unknown number of local optima. The contours of the Egg holder function ( [55]) and Schaffer function ( [56]) are plotted in Figures 3 and 4, respectively. In addition, the artificial fruit flies, appearing at several iterations, are also scattered in the different plots of the two functions. The optimal points of the two functions are located near the top right corner and at the origin. The starting points are far from the optimal points. We have observed, from Figures 3 and 4, that CPFOA has the capability of successfully optimizing the multimodal functions without being trapped in local optima. The experiment has been repeated for 50 runs, and every run reaches the optimal point. Now, we can conclude that CPFOA has a good balance between diversification (exploration) and intensification (exploitation), since it has not been trapped in local optima.

Computational Time of CPFOA
The optimization problem should be solved in a short time. Therefore, a good meta-heuristic optimization should have a short computational time. The average computational time consumed by eights algorithms, when they optimized 18 of the scalable functions taken from ( [31]), is shown in Figure 5. From the figure, we found that CPFOA consumes quite a short computational time, compared to the other algorithms, especially when the dimension of the problem is 1000. Therefore, CPFOA can be used effectively in optimizing a largescale problem.

The First Experiment: Comparison of CPFOA and Six FOA Variants
The average (ave) and standard deviation ( As can be seen from Tables 5 and 6 Tables 5 and 6 show that CPFOA has the highest average SR value (0.94).
The results in relation to a large-scale problem are shown in Table 7. In addition, from the last row of Table 7, it can be found that CPFOA has the highest average SR value (0.83).
To confirm the efficiency of the CPFOA, the convergence graphs of all seven algorithms, when they are optimizing 18 benchmark test functions, with dimension = 1000, are shown in Figure 6, where the red lines represent the convergence graph of CPFOA. The graph is plotted in the log − log scale, where the x − axis is the number of iterations, and the y − axis is the average fitness values, obtained at the corresponding iterations, of the algorithms. From the graphs, CPFOA can reach the best solution faster than the other six FOA variants. NA mean that CPFOA was not compared itself. The total h is represented as #1/#2/#3, where #1, #2, and #3 represent the number of wins, ties, and losses of the algorithm, respectively. For example, the total h in the last row of Table 8, there is only the h value "+" of CEFOA = 1. It meant that from 18 benchmark functions, CEFOA win CPFOA an only 1 function. Moreover, the h value "-","=" of CEFOA = 11 and 6. It meant that from 18 benchmark functions, CEFOA losses CPFOA 11 and ties CPFOA 6 respectively. Hence, CPFOA outperforms CEFOA. The SR is the success rate. For each function, the lowest of ave and std values, and the highest of SR value, are highlighted in boldface.

The Second Experiment: Comparison of the CPFOA and Meta-Heuristics Algorithms
As can be seen from Tables 8 and 9 Tables 8 and 9 show that CPFOA has the highest average SR value (0.94). The results regarding a large-scale problem are shown in Table 10.  Table 8, it can be found that CPFOA has the highest average SR value (0.83).

Applications
In this section, the proposed CPFOA is applied for (i) training the Multi-Layer Perceptron (MLP) to classify five datasets, and (ii) estimating the T-S fuzzy system parameters. MLPs and T-S fuzzy system method have been proposed as useful tools to model complex systems for process in chemical applications ( [57], [58], [59]).

Bio-Medical Real-Life Classification Problems
The datasets are a synthesis dataset, 3-bits XOR, a small dataset, Iris, and three bio-medical datasets: Balloon, Breast cancer and Heart. The details of these datasets and parameter settings, including the MLP structure for solving these datasets, are taken from the literature ( [60], [61]). Brief details of the dataset, for the implementation and performance comparison of the algorithms, are presented in Table 11. CPFOA is compared with six meta-heuristic algorithms in Table 4: JADE, BLPSO, CLPSO, GWO, MGWO and HAGWO ( [61]). HAGWO was successfully tested using these five datasets. It is a hybrid nature-inspired optimization technique that has been constructed using a hybridization of the Mean Grey Wolf Optimizer (MGWO) and Whale Optimizer Algorithm (WOA). The parameters of GWO and HAGWO are set as in ( [60], [61]). Each algorithm is coded and run in MATLAB environment. The convergent graphs, based on five datasets, are shown in Figure 7, where the red line represents the graph of CPFOA. From Figure 7, it can be found that CPFOA can produce the lowest MSE in every dataset. To confirm our claim, the statistical results, in which the minimum objective function and maximum objective function values are extracted, are shown in Table 12. A lower objective function value is better. In addition, the average and the standard deviation of the classification rate are shown in Table 12. A higher classification rate, with a lower standard deviation, is better.
The results from Table 12 are as follows: (i) The XOR dataset contains 8 training and 8 testing samples. Each sample has 3 input attributes and 1 output. The outputs of the XOR dataset are the same as those of the input values. After encoding, the dimension of a fruit fly is 36. As can be seen in (v) The Heart dataset, is the hardest problem among the four classification problems, containing 80 training and 187 testing samples, each of which has 22 input attributes and 2 classes. After encoding, the dimension of a fruit fly is 1081. It is a large-scale problem. As can be seen from To summarize, CPFOA has the ability and is suitable for training the Multi-Layer Perceptron (MLP) in order to solve real-life classification problems.

T-S Fuzzy System Parameter Extraction
This T-S fuzzy system parameter estimation has been conducted in several literary works, and seven of the existing works compare the other algorithms with CPFOA. The dataset is a Box and Jenkins gas furnace data set ( [62]). It consists of 296 input and output measurements of a gas-furnace process. It is the collection of recorded data from a combustion process of a methane-air mixture. At each sampling time k, the input x(k) is the gas flow rate, and the output y(k) is the output CO 2 concentration.
To compare CPFOA with the other novel algorithms, we borrow a procedure from the literature, namely, the "parameter estimation of Takagi-Sugeno's fuzzy system using a heterogeneous cuckoo search algorithm" ( [63]) by choosing u(k), u(k − 1), y(k − 1) and u(k − 2), as the input variables of the model. To prepare the data, the first 148 input-output data were utilized as training data, and the last 148 were the testing data. The individual parameter settings of the simulation and the best competitor, HeCoS, are set as in ( [63]), which is as follows: P a = 0.15, α = 1.3, β = 1.3, δ = 0.9, the maximum iteration (Max iter) = 2000, and the number of agents (popsize) = 40. CPFOA solve a very compact model as the number of rules in the model is 2-this is the smallest number of rules. After the encoding, the total number of parameters for a T-S fuzzy system model is 26.
The training results of CPFOA are visualized in Figure 8, and the testing results are shown in Figure 9. The blue line is the real data, and the red dashed line is the output of the model, trained by CPFOA. From those two figures, we have observed that the real data and the output from the model are not very different.
Based on the study in ( [63]), the HeCoS optimizing iTaSuM claimed that it is the best model, with training and testing MSE values of 0.0271 and 0.138, respectively. We also re-run the HeCoS, and there is no better solution found. These two training and testing MSE values are not the lowest values, but the number of rules of the T-S fuzzy system is less than or equal to that of the other six approaches (model no. 1 through no. 6). Regarding CPFOA, from Table 13, it can be found that the training and testing MSEs of CPFOA do not obviously outperform that of HeCoS, but CPFOA can optimize a more compact model than HeCoS as the number of rules of the T-S fuzzy system trained by CPFOA is less than that of HeCoS. The parameters of the identified model are listed in Table 14.

Conclusions
This paper addresses the problem of the low exploration ability of FOA and its variants. The cooperation of the multileader and the probabilistic random walk strategies forms the proposed CPFOA algorithm. CPFOA can smoothly transition from the exploration stage to the intensification stage. The experimental results show that the performance of CPFOA is improved, compared with the original FOA and the FOA variants, in finding the optimal solution.

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The main characteristics of CPFOA in finding the optimal solution can be summarized as follows: • The CPFOA uses the probabilistic random walk algorithm, with adaptive normalization, as the main procedure. • The CPFOA uses the multileader strategy to further enhance the exploration ability. • The population with two types of behavior can prevent the search from becoming trapped in a local optimum, whereas only one population behavior in the existing FOA variants can lead the algorithms to be easily trapped. • The CPFOA demonstrated its promising performance in solving unconstrained function optimization problems, especially when the dimension of the problem is high.
We evaluated the CPFOA's performance in 18 well-known standard benchmark functions. The experimental results from the benchmark functions clearly illustrated that the CPFOA outperforms both the original FOA and the FOA variants, in terms of the convergence speed, the success rate, and the solution accuracy, in finding the optimal solution.
CPFOA is applied for training the MLPs in relation to classified real-life datasets, and the results demonstrate that it achieves a higher level of accuracy in the classification of the proposed CPFOA trainer than the competitive algorithms.
Moreover, CPFOA is applied for the T-S fuzzy system parameter extraction of a Box and Jenkins gas furnace data set. The results demonstrate that CPFOA can achieve a very promising accuracy of the T-S fuzzy system in modeling, when compared with the best known competitive algorithms.
Future work will involve applying CPFOA to optimize the multi-objective function problems, which are very challenging problems.
Author Contributions : The authors have contributed equally to this research and to the writing of this paper. W.A. designed the experiments, conceptualization, investigation, methodology, software and wrote the original draft of this paper. K.S. was responsible for supervision, performing the numerical experiments, conceptualization, investigation, methodology, software and the writing of the original draft of this paper. S.C. analyzed the numerical results and was responsible for conceptualization, investigation, methodology, software and the writing of the original draft of this paper. All authors have read and approved the final version of the manuscript.