CLPB: Chaotic Learner Performance based Behaviour

doi:10.21203/rs.3.rs-2963539/v1

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CLPB: Chaotic Learner Performance based Behaviour

https://doi.org/10.21203/rs.3.rs-2963539/v1

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published 25 Apr, 2024

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Learner Performance-based Behavior (LPB) is a recent metaheuristic algorithm that is inspired by the process of accepting students who have recently graduated from high schools into various departments at university according to the student’s GPA (Grade Point Average). Two versions of the algorithm are available; single and multi-objective versions. The exploitation phase, convergence speed, and processing time (PT) of the LPB are not according to the required level.This paper aims to improve the performance of a single objective LPB through modification which helps in reducing its PT, getting closer to the global optima, and bypassing the local optima with the best convergence speed. For that, ten chaos theory maps are used within LPB to propose Chaotic LPB (CLPB) to consider which of the ten maps works well in case of performance. Another modification that has been made to LPB is that the best individuals of a sub-population are forced into the interior crossover to improve the quality of solutions. CLPB is evaluated against multiple well-known test functions such as classical (TF1_TF19) and (CEC_C06 2019). Additionally, the results have been compared to the standard LPB and several well-known metaheuristic algorithms such as DA, GA, and PSO. Finally, the numerical results show that CLPB has been improved with chaotic maps. Furthermore, it is verified that CLPB has a great ability in dealing with large optimization problems compared to LPB, GA, DA, and PSO. Overall, Gauss and Tent maps both have a great impact on improving CLPB.

Artificial Intelligence

Evolutionary Algorithms

Learner Performance-based Behavior

Chaotic Learner Performance-based Behavior

LBP

CLPB

Artificial intelligence (AI) is a discipline that focuses on developing machines that are capable of displaying intelligence (Du and Swamy,2016). The term AI refers to the study and design of systems that are capable of performing actions that are designed to maximize their success (Zhang et al.,2015; Russell and Norvig,2010). Search is a crucial AI concept since it helps all systems and specialties. For that, there are three search mechanisms: enumeration, analytical, and heuristic search techniques (Du and Swamy,2016). The concept of an optimization problem involves searching for the optimal path for solving a particular problem. It can be done by taking into account the objective function of the issue and the various alternatives available. Doing so helps identify the path that will most likely work for the goal (Du and Swamy,2016; Rahman and Rashid,2021). Computational Intelligence (CI) is a fast-evolving research area, it comprises a set of approaches and methodologies that take nature (especially, biology) as an essential source of inspiration to them to solve complex real-world optimization problems in a reliable fast manner with large search spaces of problems. CI initially was created by Bezdek in the early 1990s (Rahman and Rashid,2021). As a term, CI is classified as one of the popular subfields/Branches of AI (Rahman and Rashid,2021; Bezdek,1992). Contrasting AI in that relies upon knowledge resulting from human expertise (Du and Swamy,2016). Subjects that are covered under the umbrella of CI are artificial neural networks (ANN), fuzzy logic, and evolutionary computation (EC) (Zhang et al.,2015). Evolutionary computation [e.g., GA, genetic programming (GP), evolutionary programming (EP), and evolutionary strategy (ES)] (Xing and Gao,2014). EC (metaheuristic, in particular) based on characteristics is divided into two main categories: single solution algorithms and population-based algorithms (Jourdan et al.,2009). population-based algorithm develops a population of individuals by generations, generates offspring by mutation and crossover, and selects the fittest to survive each generation (Du and Swamy,2016).These mechanisms are practical when traditional (exact) methods are no longer useful or effective to be used (Zhang et al.,2015; Rahman and Rashid,2021). In general, Metaheuristic algorithms (MAs)(approximate) techniques use stochastic mechanisms despite they are effective tools for solving many real-life problems, these techniques give an approximate solution and they do not promise optimality. For that reason, using MAs some convergence problems will be encountered (Jourdan et al.,2009).

LPB is a nature-based MA that is based mainly on the notion of GA (Genetic Algorithm). It is inspired by the procedure of accepting the graduated students from high school in different departments at the university. There are two versions available according to the number of objectives they evaluate namely: the single-objective algorithm and the multiobjective algorithm. Exploitation and Exploration are counted as the most important stages of optimization. LPB has proved a great ability in exploration and in preventing local optima also when it encounters a massive number of local optima. In addition, the algorithm has proved its ability in optimizing large-scale optimization problems and its efficiency in solving a real-world NP-hard problem. Moreover, the results were compared to several popular metaheuristic algorithms in the industry, it was proven that the LPB algorithm performed better in most of the cases.

However, the main problems encountered in LPB that will be focused on in this paper are: 1) the processing time (PT) of the LPB algorithm is significantly high. 2) the convergence speed and exploitation of the LPB algorithm are not satisfactory.

The main objectives of this work are: 1) To minimize the processing time (PT) of the LPB algorithm, by embedding chaos theory functions into LPB. 2) Speeding up the convergence of the LPB algorithm while reaching the global optima and avoiding trapping in the local optima. 3) Improvement in the exploitation phase of the LPB algorithm by altering the algorithm. 4) Improvement in the accuracy and quality of the solutions. 5)Improvement in the software.6) Evaluating and assessing the performance of the modified version of LPB (CLPB) against multiple popular benchmark functions and well-known optimization techniques in the industry.

The innovative contributions of this paper to overcome the issues that the LPB has are 1) improvement in the performance of single objective LPB through modification. Consequently, chaos theory maps are embedded within LPB to propose Chaotic LPB (CLPB), which helps enhance the global convergence speed and quality of solutions and avoid trapping into local optima. 2) reducing the processing time of the algorithm. 3) the good individuals of the sub-population are forced to crossover between themselves to improve the quality and diversity of the solutions. 4) the importance of CLPB solutions is tested statistically.

The rest sections of the paper are arranged as follows: Section 2 discusses some of the related works in the literature in detail. Section 3 is divided into two sub-sections; in general, both sub-sections outline the background and theoretical approaches followed and used to make the whole work achieved. In addition to that, the single objective version of LPB is discussed in detail also the pseudo-code has been provided by the author of this paper. Finally, the author deliberates on the chaotic maps with their background and characteristics. In section 4 CLPB is described and presented to the readers, and the pseudo-code of the single version beside the flowchart of CLPB is provided by the author. In section 5 the produced experimental results of CLPB are presented, evaluated, and tested using several popular benchmarks and compared against many well-known algorithms in the industry and discussed. Finally, in section 6, a summary of the work is given. Moreover, the significance of future work in the field is provided. The adjustments that might enhance the CLPB work are also stated.

All metaheuristic optimization algorithms (MOA) that existed in the industry; can guarantee to solve a wide range of optimization problems with the promised purpose, to reach either optimality or close to optimality solutions. Despite this, we cannot compliment the fact that MAs could have defects and they may lack some important features that affect the overall performance and quality of promised solutions. For this purpose, various mathematical methods are utilized to improve the performance of MAs, such as the chaos theory. This method is a nonlinear system that has dynamic behavior. It has been used by researchers in various areas because of its effectiveness. There exist numerous applications of chaos theory embedded inside MAs for performance enhancement by avoiding trapping into local optima and speeding up the convergence. The use of chaotic maps in MAs is important. However, different phases of the algorithm utilizing different chaotic maps have different effects on metaheuristic algorithms (Lu et al.,2014).

For instance, earlier chaos theory has been introduced with GA. Since GA has disadvantages such as low convergence speed, so in (Javidi and Hosseinpourfard,2015) the authors offered a new GA based on chaotic maps for that (CGA) was proposed to address this deficiency. CGA allows for avoiding local convergence more often and significantly decreases the number of iterations needed in optimization problems, proving high-performance improvements. Moreover, Chaotic maps have been added to the standard bat algorithm (BA) to create the Chaotic Bat Algorithm (CBA) (Gandomi and Yang,2014). To improve the global search mobility of CBA, four variants of CBA have been presented, and the results show that certain variants of the CBAs can outperform the standard ones. More precisely, CBA-IV is the superior and best CBA. Additionally, in (Kohli and Arora,2018) again the authors utilized the chaotic theory in the GWO algorithm to present the CGWO algorithm that aims to accelerate its global convergence speed, using ten different chaotic maps and 13 standards-constrained benchmark functions. The results showed that the Chebyshev map was the best, and deterministic chaotic signals were used instead of linearly decreasing values. It also performed well compared to other similar algorithms. Also, the authors in (Kaur and Arora,2018) proposed a hybridized version of WOA called the chaotic whale optimization algorithm (CWOA) to address the main problem of slow convergence speed by enhancing global convergence speed and improving performance. Using 20 well-known test functions and Numerous chaotic maps. Thus, the results proved that the chaotic maps (especially the Tent map) can improve the performance of WOA. Similarly, chaotic functions have been embedded into another MA, called fitness dependent optimizer (FDO) to create a chaotic fitness dependent optimizer (CFDO) (Mohammed and Rashid,2021). This work proposes ten chaotic maps inside FDO to improve the performance of the CFDO, with the Tent map being the worst and the Singer map being the best.

Since the introduction of these methods and techniques of MAs, many researchers have been able to implement these techniques in various fields and a variety of applications. Meanwhile, to find a certain optimal solution for a given problem, there are many different ways to approach the optimization problem. Due to the complexity of the task, it is not feasible to implement a single approach as a universal method. In 1997, Macready and Wolpert stated that if an algorithm called ‘A’ is better than another one called ‘B’ for a certain problem solving, then algorithm ‘B’ will perform better in the other cases of problems. Through the evolution of nature, researchers can now learn how to solve problems efficiently using techniques inspired by nature. Accordingly, this can help them to continue developing new and better algorithms (Yang, 2010). This paper is close to the work of CFDO in terms of embedding ten chaotic map functions inside the standard algorithm and later on using these chaotic functions to initialize the main population of individuals rather than randomly initializing it.

This section is broken down into two sub-sections, which both provide an overview of the theoretical methods used in creating the overall work. The first sub-section delves into the LPB algorithm, which explains how it works and what inspired its design. Additionally, the single objective version of the algorithm is covered in detail, with the steps for its implementation outlined. The algorithm's pseudo-code is also presented. The second sub-section provides general background on the various chaotic maps available in the industry and also discusses the importance and characteristics of the chaotic maps that are used in CLPB.

3.1 The Concept and Implementation of LPB

Every year, groups of high school students apply to universities to obtain a certain degree. The applications of some of these individuals are accepted, while others are rejected. The students are then divided into different groups based on their grade point average (GPA). After graduating from high school, the students (population/groups of individuals) start the process of transferring to a university. The university then start to identify the number of students they want to accept. Also, they set the minimum GPA that the students should have to meet in order to be admitted. The whole process of transferring begins with a group of students (population of individuals) called M. This process is similar to how groups of high school students are arranged into several different groups (different departments) based on their GPA. Then, the departments of the university accept the applications of the students if their GPA is within the required range. There are also several groups composed of students whose GPAs are under the required standard. Those students will not be accepted. In other cases, there are students whose GPA is much higher than the required GPA, hence, these students' applications will be first considered as they have higher priority than others who have lower ranks of GPAs, and so on, until the process reaches that case when the number of students who are accepted is equal to the number of students that the department has identified. However, sometimes, the lack number of students with the required GPA can prevent some departments from having the exact number of students they want. In that case, the university and the departments should decide if they want to accept students with less than the required GPA or not.

According to (Rahman and Rashid, 2021), the LPB algorithm originates from the aforementioned process of transferring students that have recently graduated from high school with their specified GPA to different departments of the universities. In other words, the algorithm is inspired by the process of moving in groups, grouping, and improving. By taking into consideration some important criteria. First, the process of moving/transferring a group of students corresponds to the process of initiating the initial population in the LPB algorithm as it always starts with a group of individuals that have been randomly created. Second, the process of grouping the students according to their GPA which is compared to the pre-identified GPA of each different department corresponds to the process of evaluating the fitness (GPA) of individuals in the initial population of LPB utilizing the fitness function, then, randomly a percentage of the individuals is separated from M called O using a division probability (dp). Thus, this helps in dividing (grouping) the main population into sub-populations (Initial population M, and a separated percentage of the initial population O). Then, according to the fitness function, the fitness value of each individual in the initial population (M) is compared with the two fitness values of individuals of the separated group of population(O) to create three different populations (groups) namely, Bad Population (BP), Good Population (GP), and Perfect Population (PF). Lastly, the process of improving the study behaviors of the students after they got accepted into universities is also another crucial point as the transition from high school to college can be very challenging for new students. To improve their academic performance, new students should start working on their studying behaviors. Doing so will help them improve their scores and in making them good college learners.

Also, the level of metacognition is affecting the studying behavior of the students as well (Qayyum,2018). The LPB corresponds to this by applying the most powerful GA operators to improve the performance (quality, accuracy, convergence speed, and optimality) of the newly created individuals in the population.

In the LPB algorithm, the GA operators that have been used to implement the algorithm are named; selection, crossover, and mutation; they were first implemented in the GA. The operators mimic the approach of the inheritance of genes to produce new individuals in each generation. Then, they make changes to the structure of the individual during the representation (Gen et al.,2008).

The concept of natural selection is the inspiration behind the selection operator of GA. To produce the next generation of a species, the fittest individuals are more likely to mate and reproduce as their genes can contribute to the development of the next generation of organisms. Using various selection operators such as roulette wheel, Boltzmann, Tournament, Rank, Steady-state, Truncation, and Local selection (Mirjalili,2019). The concept of crossover is a fundamental genetic operator that allows two individuals to reproduce by combining their features (Gen et al., 2008). Recombination (crossover) is a process of combining two solutions chosen by the selection operator to create a new generation. It is performed using a crossover operator that is most appropriate to the given problem (Mirjalili, 2019). It can be done in various ways; the most popular and simple one is by Selecting a cut-point stochastically. For instance, one-cut point, two-cut point, multi-cut-point, or uniform crossover (Gen et al., 2008). Mutation involves altering one or more genes to produce random changes in the chromosomes. As it can be employed for the significant purposes of either 1) replacing the genes that were lost during the selection process or 2) providing new genes that were not present in the original population. Many different mutation operators are available for different representations of individuals. the simplest way to achieve mutation would be to alter one or more genes. Others include boundary mutation, replacement mutation, dynamic mutation, and uniform mutation (Gen et al., 2008).

The LPB algorithm starts initializing the main population of individuals M (graduated students) randomly to apply for different departments in various universities. Besides, the dp operator is used to randomly choose a percentage of students (individuals) O from M. Later, the fitness value of each individual in O is calculated and then sorted. For that, two populations are created according to the best fitness, the individuals with the highest fitness are grouped into the GG (Good Group) or 1^st group of O, and the rest of the individuals with the lower fitness values are assigned to the BG (Bad Group) or 2^ndgroup of O. Additionally, the fitness value of individuals in M is evaluated. The best individual and worst individual of O is highlighted. In other words, the best individual can be found in the GG, and the worst individual in the BG. Moreover, the defined fitness of individuals in M is filtered by making the comparison with the two fitness values of the GG of individuals in O, this will create three subpopulations. In the end, the Crossover and Mutation operators will be carried out until the condition is met. The pseudo-code for the standard LPB algorithm is demonstrated below. Table 1 shows the definition of symbols in LPB. Also, the most important features of the LPB are:

The initial population is randomly generated.
A percentage of the population is separated.
The population is divided into several sub-populations.
The best fitness in the separated group is used to divide the population into subpopulations.
The sub-populations that contain the best individuals have priority to go through the optimization process first.

Pseudo-Code of Standard LPB

Standard LPB Algorithm
	1. [Initialization]: randomly create a population M
	2. [Specify parameters]: specify the number of required learners N for a department, crossover rate, and mutation rate
	3. [Create Sub-Populations]: Use dp parameter to randomly choose a percentage of individuals O from M Evaluate the fitness of individuals in O Depending on their fitness, sort the individuals in O (descending order), using one of the sorting methods Divide O into two populations, good (individuals with high fitness) and bad (individuals with low fitness)
	While the termination condition is not met
		Use dp parameter to randomly choose a percentage of individuals O from M
		Evaluate the fitness of individuals in O
		Depending on their fitness, sort the individuals in O (descending order), using one of the sorting methods
		Divide O into two populations, good (individuals with high fitness) and bad (individuals with low fitness)
		Find fitness for all individuals in the population M
		Find the highest fitness in good and bad populations
		if an individual from M has fitness <= highest fitness in the bad population
			Move it to the bad population (BP)
		else if an individual from M has fitness <= highest fitness in a good population
			Move it to the good population (GP)
		Else
			Move it to the perfect population (PF)
		end if
		While k <= N
			If PF is not empty
				Select an individual from PF
			else if GP is not empty
				Select an individual from GP
			else
				Select an individual from BP
			end if
		k = k+1;
		end while
	4. Crossover
	5. Mutation
	6. [Termination]: repeat the procedure from step 3 until the termination condition is met.
	end while
	7. [Optimal solution]: select the best solution from the perfect population.

3.2 Chaotic Maps

In 1972, Edward Lorenz introduced the concept of chaos to the world through his concept of the butterfly effect. This theory aims to make a system more predictable. By understanding this concept, we can keep a system's inputs under control. Due to the contributions of scientists and mathematicians to the development of this theory, its applications are widely acknowledged in various scientific fields. For instance, Lorenz developed a mathematical model that was used to study the movement of air in the atmosphere. His model caused significant differences in its results. Through his work, Lorenz was able to gain a deeper understanding of the concept of sensitive dependence on initial conditions (SDIC). This concept became a key component of the chaotic system. The concept of chaos theory can be described as that a small change in the initial conditions can lead to a large change later. This is a field of study in mathematics that focuses on the study of chaotic systems and their applications in various fields such as physics, engineering, economics, and philosophy. Characteristics of a chaotic system: (i) No periodic behavior. (ii) Sensitivity to initial conditions. (iii) Chaotic motion is difficult or impossible to forecast. (iv) The motion looks random. (v) Non-linear (Biswas et al.,2018).

Chaos theory has been used in different areas by researchers Because of its effectiveness in research optimization, synchronization, and chaos control (Mohammed and Rashid,2021). Over the years, chaotic maps have been widely used in the optimization field due to their dynamic behavior. This helps search optimizers explore the entire search space more globally and dynamically. Recently, due to the varying human kingdom, a wide variety of chaotic maps has been created in the optimization field. These maps are commonly used by mathematicians, researchers, and physicians (Kohli and Arora,2018). Most of the available maps are mainly used in the optimization field to make modifications for improving the search efficiency of their algorithms and speeding up convergence (Gandomi and Yang,2014; Kohli and Arora,2018; Kaur and Arora,2018). Table 2 demonstrates the functions and names of the ten common chaotic map functions that were used in this work. Depending on research done previously in this area, there are three most used ways to introduce chaotic maps within the metaheuristic algorithms. Accordingly, in the first method, chaos can be utilized to initialize the initial population, secondly, chaotic can be used as an operator (crossover and/or mutation), and lastly, chaos can be used to initialize any variable that has a possibility of generating random numbers (Mohammed and Rashid,2021; Lu et al.,2014; Lorenz,2017).

The Chaotic Learner Performance-based-Behavior algorithm is the modified version of LPB as it has been developed by introducing chaos maps in the LPB algorithm itself. It differs from LPB in two main points. First, within the starting step of the algorithm as the CLPB starts with the creation of the initial population by using chaotic map functions instead of generating the population randomly. This improvement increases the performance of the LPB algorithm as it is lacking some vital performance features such as convergence speed, search efficiency, exploitation, accuracy, and PT. Thus, ten chaotic maps (functions) are used in CLPB to create the initial population of individuals using two different benchmark functions (Classical and CEC2019) with different ranges to evaluate the accuracy and quality of the results as these chaotic maps are mainly used to tune the convergency speed and the PT of the CLPB algorithm. The reason for that, is the chaotic maps control the process of choosing the best appropriate random numbers within the range of [0,1] because generating a random number itself cannot always promise to serve in producing optimal solutions as sometimes having a specific random number can be better than other random number and vice versa. Consequently, we need to choose the best random numbers using chaotic map functions that can be valuable for the process of generating the initial population of individuals in CLPB. Second, in CLPB the process of dividing the initial (main) population M into sub-population O; the best individuals (GG) in O crossover between themselves to improve the quality of solutions. Hence, better individuals (parents) are directed to combine their genes to produce offsprings with better results. Therefore, these facts increase the performance of the LPB which will lead to the creation of a set of LPB algorithms, or different variants of the chaotic LPB algorithm. The chaotic maps used in CLPB have different mathematical equations. They are divided into ten maps, which are named Chebyshev, Circle, Gauss/mouse, Iterative, Logistic, Piecewise, Sine, Singer, Sinusoidal, and Tent map. Then, CLPB continues the steps of the algorithm similar to the LPB until it gets closer to the promised area. The pseudo-code for the modified variant of LPB (CLPB) is shown below. Correspondingly, parts 1 and part 2 of Figure 1 shows the enhanced flowchart of the CLPB algorithm.

Pseudo-Code of CLPB

Modified LPB Algorithm
	1. [Initialization]: Use a chaotic map to initialize a population M
	2. [Specify parameters]: specify the number of required learners N for a department, crossover rate, and mutation rate
	3. [Create Sub-Populations]: Use dp parameter to randomly choose a percentage of individuals O from M Evaluate the fitness of individuals in O Depending on their fitness, sort the individuals in O (descending order), using one of the sorting methods Divide O into two populations, good (individuals with high fitness) and bad (individuals with low fitness). Apply Crossover between individuals of the good population of O, for obtaining better results.
	While the termination condition is not met
		Use dp parameter to randomly choose a percentage of individuals O from M
		Evaluate the fitness of individuals in O
		Depending on their fitness, sort the individuals in O (descending order), using one of the sorting methods
		Divide O into two populations, good (individuals with high fitness) and bad (individuals with low fitness)
		Select half of the individuals of a good population of O, using the selection operator
		Apply crossover between half of the selected individuals in the previous step, using the crossover operator
		Find fitness for all individuals in the population M
		Find the highest fitness in good and bad populations
		if an individual from M has fitness <= highest fitness in the bad population
			Move it to the bad population (BP)
		else if an individual from M has fitness <= highest fitness in a good population
			Move it to the good population (GP)
		else
			Move it to the perfect population (PF)
		end if
		While k <= N
			If PF is not empty
				Select an individual from PF
			else if GP is not empty
				Select an individual from GP
			else
				Select an individual from BP
			end if
		k = k+1;
		end while
	4. Crossover
	5. Mutation
	6. [Termination]: repeat the procedure from step 3 until the termination condition is met.
	end while
	7. [Optimal solution]: select the best solution from the perfect population.

The CLPB algorithm has utilized ten common chaotic map functions separately in the LPB interiorly to create the initial population of individuals. This essentially helps in calculating the different positions of individuals by choosing any value that can be assumed as the best random value between 0 and 1; as a result, in this work, the initial point of the chaotic maps was set to (0.7). The different functions of the chaotic maps are proposed by (Mohammed and Rashid,2021). As mentioned previously, Table 2 demonstrates the functions and names of the ten chaotic maps. Each individual in the population of the single objective CLPB algorithm has position and cost characteristics. The fitness for the position of each individual is recognized as cost. The cost of each individual in CLPB is calculated by applying either one of the nineteen classical benchmark functions (TF1-TF19) or one of the ten functions of CEC2019 (CEC01-CEC10) and as mentioned above, one of the 10 chaotic maps is used simultaneously at the same time to produce the positions of individuals (initial population) and therefore this will examine CLPB. The results are then evaluated against the standard LPB and three popular algorithms in the literature: DA, PSO, and GA. The results for 19 classical benchmark functions for PSO, DA, and GA are taken from (Mirjalili,2016), and for LPB are taken from (Rahman and Rashid,2021). Also, the results for 10 CEC-C06 2019 test functions for all participated algorithms LPB, PSO, and DA are taken (Rahman and Rashid,2021). In addition to that, the processing time (PT) of the algorithm for the two groups of test functions is computed to examine how fast is the algorithm in finding the optimal results. Moreover, the significance of the results is proved by using the T.TEST. The parameter settings for CLPB are shown in Table 3.

5.1 Benchmark Test Functions

The benchmark is described as a standard, by which things are measured or evaluated accordingly. In this work, the CLPB is subjected to the use of two powerful and common sets of benchmarks to evaluate the overall performance enhancement of the CLPB.

The first set of benchmarks is classical (traditional) benchmark functions that are divided into three groups: unimodal, multi-modal, and composite (Yao et al.,1999; Molga and Smutnicki,2005; Liang et al.,2005; Mirjalili,2016; Price et al.,2018). The properties of each group of test functions are different. For instance, the unimodal test functions are designed to benchmark the convergence and exploitation of an algorithm. This group type of test function has a single optimum. Nevertheless, a multi-modal test function is often referred to as having multi optimum as their name reveals this. It has one global and multi-local optima. One of the most important factors that an algorithm should consider when it comes to approaching the global optimum is avoiding the entire local optimal solution. Thus, this group of test functions can then test the algorithm's ability to avoid local optima and benchmark the exploration ability of an algorithm. Lastly, composite test functions are composed of various versions of the unimodal and multimodal benchmark groups, such as combined, rotated, shifted, and biased versions (Liang et al.,2005). These functions show the difficulties that exist in the search spaces due to the diversity of shapes and local optima. An algorithm should suitably balance the exploration and exploitation of a given test function to achieve the global optimum. This group of test functions can be used to benchmark the combined efforts of both exploration and exploitation (Mirjalili,2016). Tables 4, 5, and 6 demonstrates more information about the test functions with their ranges and dimensions.

The second set of benchmark functions is CEC-C06 2019 test functions. The author of with work has examined this benchmark on the CLPB algorithm against other participating algorithms to show the ability of the CLPB algorithm in solving large-scale optimization problems (Price et al.,2018). In the evolutionary field, various optimization algorithms take advantage of the known properties of the benchmark functions. For instance, they can find optimal values located across the horizontal axes. The CEC-C06 test functions are designed to evaluate the various functions of an algorithm on a horizontal slice of the convergence plot (Liang et al., 2005). These functions are then used to evaluate the algorithm for large-scale optimization problems. the CEC-C06 test functions can examine the challenge known as “The 100-digit challenge” which helps users find the most effective algorithm for their particular situation. The first three functions of the evolutionary field, namely the CEC01, CEC02, and CEC03, have various dimensions. On the other hand, the rest of the test functions CEC04 to CEC10 functions are set as 10-dimensional minimization problems in the range [-100, 100], and they are shifted and rotated. All the CEC-C06 2019 functions are scalable and all global optimums of these functions were united towards point 1(Rahman and Rashid,2021). Table 7 shows the CEC-C06 2019 test function names, dimensions, and ranges.

5.1.1 Results of the Classical Benchmark Test Functions

Tables 8 to 14 show the rank results of the comparison between the CLPB algorithm and other popular participating algorithms in the industry GA, PSO, DA, and LPB to demonstrate the differences and optimal results. The parameters for GA, PSO, and DA are given in reference (Mirjalili,2016) and the parameters for standard LPB are discussed in reference (Rahman and Rashid,2021). The data for the test functions (TF1-TF19) for LPB, DA, PSO, and GA is from (Mirjalili,2016; Rahman and Rashid,2021). Each participating algorithm's traditional benchmark test functions are solved 30 times using 80 search agents over a total of 500 iterations. The standard deviation and average of the results are then calculated. Also, the PT has been restored 30 times. The contributed algorithms have been tested by the author of this work using the rest of the test functions. The dp for all the test functions is set to 0.5. The standard deviation and the average of the optimal solution are calculated in the last iteration and the PT as well. For each TF the results of CLPB1 to CLPB10 have been averaged to one solution to be able to compare it with other contributed algorithms. In Table 8, the overall average performance of the CLPB is 2.47 on a scale of 1 to 5, with 1 being the best and 5 the worst. This ranking takes into consideration that CLPB ranked first three times, second seven times, third four times, fourth two times, and fifth once. See Table 9 for more details. Additionally, the ranking of CLPB by the type of the benchmark function (Unimodal test functions) is 3.1428; (Multi-modal test functions): 2.1667; (Composite test functions): 1.75. Table 10, shows that, on a scale of 1 to 5, where 1 is the best and 5 is the worst, the global standard deviation performance of the CLPB is 2.294. This ranking takes into consideration that CLPB ranked first four times, second six times, third five times, fourth two times, and fifth zero times. See Table 11 for more details. Generally, the processing time of the CLPB is 1.117 on a scale of 1–5 where being 1 the best and 5 being the worst. Table 12 shows that. This ranking takes into consideration that CLPB ranked first fifteen times, second two times, and ranked zero times as third, fourth, and fifth. See Table 13 for more details. Additionally, the ranking of CLPB by the type of the benchmark function is (Unimodal test functions): 1.1428; (Multi-modal test functions): 1; (Composite test functions): 1.25. Table 14, proves the superiority of the CLPB algorithm against all the participated algorithms in terms of PT in seconds. The PT for the CLPB for optimizing all the functions is much smaller. This is noticeable as the CLPB algorithm has ranked 1^st (superior) in 15 out of 17 applicable functions and the difference is significantly large. This has two possible reasons, first, As discussed previously, a subset of the population is chosen in the first step of the CLPB, and other subpopulations are constructed based on this smaller group. Priority is given to optimizing the perfect subpopulation, followed by the good and bad subpopulations. The subpopulations are substantially smaller than the overall population, making it quicker to look for answers there. As a result, this increases randomization also reducing overall optimization time. Second, using chaotic map functions helped in choosing the best random values out of a range of values to create the initial populations of individuals rather than creating them randomly and consuming time until reaching the aimed solution. Therefore, this will speed up the processing time which will lead to speed up the convergence to reach the global optimum and choose the best solution in the shortest possible time. In addition, The convergence curve for some of the classical benchmark test functions for the CLPB algorithm is presented in Figure 2. The shown data of the CLPB in the aforementioned tables is proved in the Figure.

5.1.2 Results of the CEC-C06 2019 Test Functions

The results of the CEC-C06 2019 test functions of the CLPB algorithm with the participated algorithms (LPB, DA, GA, and PSO) are shown in Table 15. In addition to the CLPB algorithm, the author of the work evaluated LPB, DA, GA, and PSO against the benchmarks, to compare them to the CLPB. For each test function in Table 15, Bold results indicate superior outcomes. Over 500 iterations, 30 times are used to solve the test functions using 80 search agents. From the last iteration; the average, standard deviation, and processing time are computed. Also, the value of the metric standard deviation for the CLPB algorithm in almost all the CEC-C06 2019 test functions is smaller than DA, PSO, GA, and LPB. In contrast, according to the value of the metric, average the LPB showed its superiority in almost all of the functions compared to the participated algorithms. However, CLBP in CEC03 has the same average and standard deviation compared to LPB but with a significant difference in processing time (2.101408) seconds for CLPB while LPB is (144.194876) seconds. In addition to that, CLPB has proved its enormous superiority in the processing time of all the test functions of CEC-C06 2019 in comparison to all the contributed algorithms (LPB, DA, PSO, and GA). Also, the results of the CLPB and LPB for optimizing CEC08 and CEC10 are comparative. On the other hand, CLPB in CEC02, CEC03, CEC06, and CEC10 test functions scored superior results compared to DA, PSO, and GA.

The aforementioned comparison can be noticeable/ proved in rank tables 16, 17, 18, 19, 20, and 21. which shows that, generally, the global average performance of the CLPB is 2.4 on a scale of 1–5 where being 1 the best and 5 is the worst. This ranking takes into consideration that CLPB ranked first two times, second four times, third two times, fourth two times, and fifth zero times. See Table 17 for more details. Also, the standard deviation performance of the CLPB is 1.9 on a scale of 1–5 while LPB ranked 2.3. According to the processing time, CLPB ranked 1.1 on a scale of 1–4 while LPB ranked 2.9. In general, the results of the CEC-C06 2019 benchmark functions revealed that for large-scale optimization problems, CLPB provides better results compared to the DA, PSO, GA, and LPB. The convergence curve for some of the CEC-C06 2019 test functions for the CLPB algorithm is presented in Figure 3. The shown data of the CLPB in Table 15 is proved in the Figure. Precisely, the processing time and name of the function with the chaotic map that has been used are highlighted in the figures.

5.2 Statistical Tests

Assessing the performance is not accurate enough since it only relies on the standard deviation, average, and PT. To evaluate the CLPB chaotic algorithm, the statistical test is used to verify the importance of the results statistically. The result of the test is then compared with other metaheuristic algorithms to evaluate the mean and make sure that the result is significant or not. An independent T.TEST is a statistical test (hypothesis test) that measures the difference between the mean values of two unrelated groups of populations. It can also be used to determine if there is a significant difference between the two groups of populations (Microsoft). The p-value of the CLPB algorithm is computed using the T.TEST function. It should be less than 0.05, at that time the results are significant. Therefore, the results are underlined as it is shown in Tables 22 and 23. Figure 4 shows the chaotic map/maps that are the best/worst in improving the performance of CLPB. The Circles are indications of the number of times a specific chaotic map has scored significant results in the performance improvement of CLPB.

It can be seen that the Gauss/mouse map and Tent map both show the best significant results in 9 out of 17 functions while the Chebyshev map, Logistic map, Sine map, and Singer map provide the least impact on improving CLPB in 6 out 17 functions. The iterative map, Piecewise map, and Sinusoidal map have the second-best significant results in 8 out of 17 functions, and the Circle map has the third-best significant results in 7 out of 17 functions. The p values reported in Tables 22 and 23 for classical benchmark test functions prove that almost all the ten chaotic maps could enhance the performance of CLPB. Precisely, Gauss/mouse map and Tent map. This is because these chaotic maps can improve the exploitation and exploration capability of CLPB and speed up the convergency which will lead find the best global optima among various solutions. Therefore, avoid local optima. In the case of (Rahman and Rashid,2021), it was proved that the results of the LPB are statistically significant compared to DA, PSO, and GA. This means there is no need to compare the CLPB algorithm with DA, PSO, and GA statistically since the CLPB algorithm proved its superiority against the LPB in most of the test functions and with the use of ten chaotic maps.

In this paper, the author aims to modify the LPB algorithm for better performance. The modification is taking place by introducing 10 chaotic map functions inside LPB to propose chaotic LPB (CLPB), Also, the good individuals of the population are forced to local crossover. Then the proposed algorithm is tested on TF1_TF19 and CEC-C06 2019 benchmarks. The results of these benchmarks are compared with the standard LPB and other participating MAs like PSO, DA, and GA. The use of chaotic maps helped in improving CLPB. The rank results of the classical benchmarks and CEC-C06 show that the global average performance of CLPB is (2.4 and 2.47) also the standard deviation performance is (2.294 and 1.9) on a scale of 1–5 being 1 the best and 5 being the worst. In terms of the PT, CLPB outperforms the LPB in all cases of both types of test functions. Also, CLPB in almost all cases has superior PT in comparison with the DA, GA, and PSO. Outcomes show that the chaotic maps helped in improving the exploitation phase of LPB since LPB was not up to the required level. CLPB has a brilliant ability in terms of exploration and in avoiding many local optima compared to GA, DA, and PSO while, CLPB showed a slight difference compared to LPB. In general, according to the findings of the CEC-C06 2019 benchmark functions results, CLPB outperforms DA, PSO, GA, and LPB in terms of solving large-scale optimization issues. Moreover, the significance of the results of the proposed work is proved statistically.

For future works, several research directions can be recommended. First of all, the author will focus on utilizing the CLPB algorithm to optimize a real-world optimization problem and compare the results with other metaheuristic techniques. Moreover, embedding/introducing chaos to initialize any variable that has a possibility of generating random numbers in the CLPB algorithm is another area that the author is planning to implement in the future. Using chaos as operators crossover and/or mutation in CLPB is another research direction. Additionally, developing the multi-objective version of CLPB to solve complex optimization problems that are associated with multiple objectives of the problem can be another research topic. Besides, another future work is evaluating CLPB using different benchmark test functions and comparing the results with other metaheuristic techniques.

Acknowledgments:

The authors would like to thank the University of Kurdistan-Hewler for providing facilities for this research work.

Funding: This study was not funded.

Compliance with Ethical Standards Conflict of Interest: The authors declare that they have no conflict of interest.

Ethical Approval: This article does not contain any studies with human participants or animals performed by any of the authors.

Data Availability Statement: The datasets generated during and/or analysed during the current study are not publicly available but are available from the corresponding author on reasonable request.

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Tables 1 to 23 are available in the Supplementary Files section

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CLPB: Chaotic Learner Performance based Behaviour

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Abstract

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1. Introduction

2. Literature Work

3. Background and Theory

3.2 Chaotic Maps

4. Chaotic Learner-Performance Based-Behavior CLPB

5. Experimental work of CLPB

5.1 Benchmark Test Functions

5.1.1 Results of the Classical Benchmark Test Functions

5.1.2 Results of the CEC-C06 2019 Test Functions

5.2 Statistical Tests

6 Conclusion

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References

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