A Novel Parent Centric Crossover with the Log-Logistic Probabilistic Approach Using Multimodal Test Problems for Real-Coded Genetic Algorithms

ul Haq, Ehtasham; Ahmad, Ishfaq; Almanjahie, Ibrahim M.

doi:https://doi.org/10.1155/2020/2874528

Mathematical Problems in Engineering

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Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Advances in Numerical Optimisation: Theory, Models, and Applications

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Research Article | Open Access

Volume 2020 | Article ID 2874528 | https://doi.org/10.1155/2020/2874528

A Novel Parent Centric Crossover with the Log-Logistic Probabilistic Approach Using Multimodal Test Problems for Real-Coded Genetic Algorithms

Ehtasham ul Haq,¹Ishfaq Ahmad,¹and Ibrahim M. Almanjahie^2,3

Academic Editor: Predrag S. Stanimirović

Received23 Jul 2020

Revised24 Sept 2020

Accepted08 Oct 2020

Published31 Oct 2020

Abstract

In this paper, a comprehensive empirical study is conducted to evaluate the performance of a new real-coded crossover operator called Fisk crossover (FX) operator. The basic aim of the proposed study is to preserve population diversity as well as to avoid local optima. In this context, a new crossover operator is designed and developed which is linked with Log-logistic probability distribution. For its global performance, a realistic comparison is made between FX versus double Pareto crossover (DPX), Laplace crossover (LX), and simulated binary crossover (SBX) operators. Moreover, these crossover operators are also used in conjunction with three mutation operators called power mutation (PM), Makinen, Periaux, and Toivanen mutation (MPTM), and nonuniform mutation (NUM) for inclusive evaluation. The performance of probabilistic-based algorithms is tested on a set of twenty-one well-known nonlinear optimization benchmark functions with diverse features. The empirical results show a substantial dominance of FX over other crossover operators with authentication of performance index (PI). Moreover, we also examined the significance of the proposed crossover scheme by administrating ANOVA and Gabriel pairwise multiple comparison test. Finally, the statistically significant results of the proposed crossover scheme have a definite edge over the other schemes, and it is also expected that FX has a great potential to solve complex optimization problems.

1. Introduction

In many real-life decision-making problems, it is often the best possible solutions are required. These problems may be anything from engineering, science, economics, and finance [1–3]. When the quality of potential solutions can be modeled mathematically, it may be possible to algorithmically find a better and sometimes optimal solution. In this case, decisions are made by developing optimization models that describe the nature of the problem, and then mathematical procedures are applied to solve these models. Hence, the simplest optimization scenario is based on constrained optimization problems, but the present research study focuses only on unconstrained optimization problems. More generally, unconstrained nonlinear optimization problem may be mathematically defined aswhere , and S is a rectangular hypercube with n-dimensions in Rⁿ with limits . These are commonly known as bounds which are based on the decision variables. A point is known as local minima of if , where is the small neighborhood of the point . If , then is said to be the global minima of .

The optimization techniques for unconstrained problems invariably used the gradient information to locate the optima. Hence, proximal gradient descent is also a gradient-based optimization method, which can be used to solve objective functions with nondifferentiable parts. These techniques cannot efficiently handle discrete variables and are most probably stuck at local optima for multimodal objective functions. Hence, the gradient-based methods often ensure that the local optima will be reached at the global optima.

To preserve population diversity and to avoid local optima, there are different types of population-based probabilistic optimization techniques that do not require continuous or differentiable objective functions. Such techniques are genetic algorithms (GAs) [4, 5], particle swarm optimization (PSO) [6], simulated annealing (SA) [7], differential evolution [8], ant colony optimization (ACO) [9], and Tabu search [10, 11]. All these optimization techniques are hence categorized under guided random search methods [12]. The guided random search methods provide a very efficient solution to the combinatorial problems and can be subcategorized into evolutionary techniques. GA is the most efficient procedure to understand and solve problems for which there is limited information. These algorithms can effectively handle both unconstrained and constrained optimization problems depending on a process of natural selection through biological evolution. The working mechanism of GA is linked with a search space that contains all possible solutions. Each part of the search space represents one sufficient solution and its fitness values will be marked by these sufficient solutions. A strong chromosome can survive, and usually, the weak chromosomes are eliminated from the population. GA is working with a search space that contains all feasible solutions. It means that each of the points in the search space is to obtain one feasible solution that will be marked according to its fitness [13]. The main operators of the GA process are selection, crossover, and mutation, and all these operators make the algorithm more unique compared to other conventional algorithms in the optimization scenario. The ideology of the selection process is to select the good chromosomes which are sent to the mating pool for combining with the other chromosomes to reproduce two new offspring. Meanwhile, the mutation process aims to encourage diversity in the new population with minor probability [14].

In previous studies, the representation of chromosomes was in the binary form which contains 0's (absent) and 1’s (present), and initially, it was applied due to its relative simplicity. Binary-coded scheme transforms the continuous search space into discrete nature with grids and the string length depends on the distance between two neighboring grids. The performance of binary encoding is good and also required less precision for the solutions under a limited number of decision variables. On the other way, binary encoding performs unsatisfactorily to solve multidimensional optimization problems where high precision is required. By the use of multiple decision variables in objective functions, the optimization problems may be solved efficiently with the increase in the size of the population. Hence, increment in the string length may be achieved better precision. In earlier optimization studies, Goldberg [12] explored that the increase in string length exponentially increased computation time and with some additional adjustments in the binary encoding will improve the convergence speed in genetic algorithms which were cited in Jin et al. [15].

Generally, GA used binary encoding in the earlier nineteen century, but many researchers [ and 16–22] used real numbers for encoding. Mohamed et al. [23] recently proposed gaining sharing knowledge-based algorithm (GSK) for solving naturally inspired optimization problems over continuous space. The designing and development of a real-encoding scheme were naturally suitable to solve optimization problems with continuous variables. As a binary coding scheme, similar genetic operators such as selection, crossover, and mutation are used in real-coded GA. The major advantage of real encoding over binary encoding is to efficiently handle complex nonlinear optimization problems with the continuous domain. Hence, by using real-encoded schemes, many difficulties such as premature convergence are to be solved, and such situations arise because of low genetic diversity in the population, and also it represents the poor exploration of the search space. If all the chromosomes have nearly the same empirical value, then the resulting process may be trapped at local optima. In the above context, a substantial amount of algorithmic work has been carried out to improve the performance of the GA process by increasing exploitation and exploration potential [24]. The exploration strength of the GA process mainly relies on the use of crossover operators due to the utilization of information about the current population. This is one of the key reasons that the majority of researchers pay more attention to performance improvement of the GA process through efficiently designed and developed real-coded crossover operators.

A dynamic class of crossover operators with real encoding is also known as parent centric crossover operators. Deb et al. [25] have shown that these parent centric crossover operators are quite effective to obtain optimum solutions of real parametric problems. In our current research study, a newly proposed real-coded crossover operator is based on log-logistic probability distribution which is empirically defined. The comparative performance evaluation of newly proposed parent centric crossover operator (FX) is carried out with double Pareto crossover (DPX), Laplace crossover (LX), and simulated binary crossover (SBX) operators. Moreover, these crossover operators are also used in conjunction with three main mutation operators called power mutation (PM), nonuniform mutation (NUM), and Makinen, Periaux, and Toivanen mutation (MPTM) proposed by Deep and Thakur [26], Meittinen et al. [27], Deb [5], and Maaranen et al. [28]. To evaluate the significance of the simulated study, the pairwise comparison is carried out for all algorithms based on statistical measures.

The rest of this paper is organized as follows. Concise detail about some existing crossover operators with real coding is discussed in Section 2. Description with the mathematical formulation of previous crossover schemes along with tournament selection is presented in Section 3. Designing and development of newly proposed crossover operator based on log-logistic probability distribution along with the description of some well-known benchmark functions are presented in Section 4. Experimentation detail with statistical analysis is described in Section 5, while simulated results by using twenty-one well-known benchmark functions are revealed and discussed in Section 6. Performance evaluation for comparative study is presented in Section 7. The conclusions of the study are presented in Section 8 of this paper.

2. Literature Review of Crossover Operators with Real Coding

Different classes of crossover operators with real coding have been proposed in the literature related to the GA process. In the earlier expansion of real coding, Wright [18] and Michalewicz et al. [29] suggested a heuristic crossover approach where the selection of a gene’s position is purely a random process and the generation of two new offspring is resulting by exchanging the position of their respective genes. Radcliffe [30] explained a Flat Crossover where a random generation of offspring is based on uniform distribution. Muhlebein and Schlierkamp-Voosen [31] proposed an extended form of intermediate crossover and line crossover. Both these crossover operators improved their search competencies by allowing population diversity under a specified interval. Eshelman and Schaffer [32] further extended the ideology of Radcliffe [30] by introducing the theoretical approach of interval schemata in the real-coding scheme. A Blend-α crossover is a hybridization of intermediate crossover and line crossover which was introduced by Muhlebein and Schlierkamp-Voosen [31]. With the benefit of the user-defined parameter, this crossover defined a bond in between genes of the parent as well as equally on either side. Moreover, blend crossover can be transformed into an intermediate crossover with α = 0.25.

Michalewicz et al. [29] presented a theory of sandwiched offspring between two parents in an arithmetic crossover operator. Moreover, a random selection of genes for producing offspring which is based on the uniform distribution in single arithmetic crossover and more offspring is generated by duplicating the process of earlier produced offspring through the average measure. The generation process of a single arithmetical crossover is generalized to all arithmetic crossovers. For more logically mathematical extension in the GA process, fuzzy recombination crossover, and fuzzy min-max crossover operators are introduced by Voigt et al. [33, 34]. Herrera and Lozano [21] presented a class of dynamic heuristic real-coded crossover operators that are linked with fuzzy connectives to overcome the shortcomings including premature convergence. These crossover operators focus on maintaining population diversity with the sustainable convergence speed of the genetic process. Another unique idea of a multiparent crossover is presented by Tsutsui et al. [35] in a simplex real-coded crossover operator by using a uniform probability distribution. Deb and Agrawal [19] repeat the development strategy of single-point crossover by using a binary string is continuous while introducing simulated binary crossover. Tutkun [36] is also introduced as a crossover operator linked with Gaussian probability distribution. Kaelo and Ali [37] proposed a hybridization strategy of different crossovers by reviewing the functionality of these crossover operators. Laplace probability distribution is also used in real-coded crossover for producing new offspring by Deep and Thakur [26].

Generally, the crossover scheme creates offspring with the help of those parents who are selected through the selection operator. It combines the parents’ characteristics to procedure new offspring, which may have new sequences compared to those of their parents and play a vital role in the GAs. In the literature, a lot of crossover schemes have been introduced with their significant importance. Chen and Wang [38] proposed a real-coded crossover (UNDX) by using unimodal normal probability distribution where three parents produce multiple offspring through the crossover process. After that Ono and Kobayashi et al. [39] integrated a uniform crossover operator to enhance the working capability of UNDX in the genetic process. A generalized form of UNDX operator having multiple parents was introduced by Kita et al. [40]. Another multiple parent crossover operator, i.e., parent-centric crossover (PCX) operator was reported by Deb et al. [25] with some enhanced features and a modified class of PCX was also introduced by Sinha et al. [41]. Furthermore, Chandra et al. [42] proposed a novel cooperative coevolution working strategy by incorporating PCX with a memetic framework for neural networks without adding to the computational cost in the subpopulations. After the hybridization with the neural network, island-based cooperative coevolutionary algorithms are presented by Bali et al. [43] to solve large-scale fully separable continuous optimization problems under the framework of PCX. Rolland and Chandra [44] also used PCX to efficiently handle the forward kinematics problem (FKP) for parallel manipulators.

By the integration of bound crossover and average crossover, Ling and Leung [45] suggested an average-bound crossover operator where two parents generated four offspring by the selection of two best out of four offspring. It is also noted in the literature that some of the various types of crossover operators including hybrid crossover operators are more beneficial to enhance the performance of the genetic process by introducing a fair amount of population diversity and controlling selection pressure. Hence, Herrera et al. [46] conducted a comprehensive performance evaluation study among real-coded crossover operators to explore the effectiveness of the genetic process. Real-coded crossover operators are parent centric crossover, fuzzy recombination, parent centric blend crossover, simulated binary crossover, and XLM crossover [47], and Laplace real-coded crossover are parent-centric crossovers which by using a unimodal normal probability distribution, blend crossover, and simplex crossover are all based on mean-centric crossover approaches.

3. Some Presently Used GA Operators with Real Coding in the Study

In the current section, we descriptively delineate selection, crossover, and mutation operators that are tournament selection (TS), double Pareto crossover (DPX), Laplace crossover (LX), and simulated-binary crossover (SBX) along with different mutation operators which are power mutation operator (PM), Makinen, Periaux, and Toivanen mutation (MPTM), and nonuniform mutation (NUM). Hence, the empirical findings are to be obtained through the simulated study.

3.1. Selection Operators

3.1.1. Tournament Selection (TS)

TS method is a ranked-based selection procedure that is simple to execute. This selection technique can be classified as binary and large tournament selection. The binary tournament selection is based on the randomized selection of two individuals, conducting a competition to decide which chromosome will win and get selected for the mating pool on the basis of the highest fitness value [14]. On the contrary, there will be no place in the mating pool for the weakest chromosomes in terms of the least fitness value, and then compare it to a predetermined selection probability p_i. Hence, the predetermine selection probability for individual p_i for (t − 1) tournament is given aswhere W is defined as population size and t is the size of the tournament. For the binary tournament t = 2 and large tournament t > 2, the TS can also be further extended by including more than two individuals if desired [48].

3.2. Crossover Operators

3.2.1. Double Pareto Crossover (DPX)

DPX operator [49] is another kind of parent centric crossover operator that uses the double Pareto probability distribution whose cumulative distribution function is expressed aswhere is represented as the location parameter where is known as a scale parameter of the distribution. By using DPX, pair of offspring and are generated from two parents and in a subsequent stepwise procedure: Step 1: generate a random number from a uniform probability distribution, where . Step 2: calculate the value of the parameter by generating random numbers from double Pareto probability distribution by simply finding the inverse of the cumulative distribution function of double Pareto distribution as Step 3: now, the offspring are in equations (5) and (6) for :

If the generation of offspring in DPX is outside the variable bound, i.e., or , then random values are given to in an interval .

3.2.2. Laplace Crossover (LX)

LX crossover operator [26] is a class of self-adaptive crossover operators with real encoding. This crossover is linked with Laplace probability distribution and the cumulative distribution function is expressed subsequently as

In the Laplace probability distribution, is defined as the scale parameter and is known as the location parameter. By using LX, from a pair of parents and , two offspring and are generated through a stepwise approach: Step 1: generate a random number with a range of unity from a uniform distribution. Step 2: obtain the parametric value by generating random numbers from Laplace probability distribution by simply inverting the cumulative distribution function as Step 3: now, the offspring are obtained in the subsequent equations (9) and (10), respectively, for :

If the generation of offspring is outside the variable limits, i.e., or in LX, then random values are given to in certain bound and assigned “0” value to its location parameter.

3.2.3. Simulated Binary Crossover (SBX)

The SBX is an important crossover operator with real coding. This operator was proposed by Deb and Agrawal [19] with a unique feature of binary transformation to continuous search space. At the first step, random number is generated from a uniform distribution with range 0 to 1. After that, the parametric value is obtained from the following mathematical expression aswhere is known as the distribution index.

Hence, from the two parents and , an offspring is produced in the following equation:

3.3. Mutation Operators

3.3.1. Power Mutation (PM)

The mutation process prevents the new population from stopping at the local optima and keeps the fair amount of diversity in the population. In other words, this process will generate a unique and fit offspring in the population. Goldberg [1] proposed a mutation clock to overcome the problem of the computational complexity in the mutation process. He used the exponential distribution to find the next location to change the string by using the first changed string location. The mutation in this process is aiming to explore more searching space while the crossover tries to converge on some point. This is because the role of the mutation is to solve the local minimum problem. Thus, this process is one way to prevent local minimum solutions at the expense of exploring more areas. Here, we subsequently represent the PM mutation operator by Deep and Thakur [26].

The PM is originated from power distribution and p.d.f of power distribution is as with cumulative distribution function in equations (13) and (14), respectively:where p is represented as distribution index and PM is used to obtain an offspring from a parent in the following stepwise procedure: Step 1: obtain a random value from a uniform distribution, where . Step 2: calculate a random value by using power distribution: Now, use the following mathematical expressions to obtain offspring:where and are the lower and upper bound of the ith decision variable. In the context of the above mathematical formulation, it is stated that perturbation in offspring is proportional to the “p” parameter. Hence, the probability of obtaining a mutated offspring is proportional to the distance from parametric bound which always resulted in feasible solution.

3.3.2. Makinen, Periaux, and Toivanen Mutation (MPTM)

The MPTM mutation operator was suggested by Makinen et al. [50], which is used to solve some multidisciplinary shape-related optimization problems in GA especially in the field of electromagnetics and aerodynamics. Meittinen et al. [27] also solved constrained optimization problems under the GA process. Deep and Thakur [22] tested and evaluated its results on multimodal nonlinear optimization problems. From a point , the mutated point is obtained in the following way.

Let a random value be from a uniform distribution, where . Hence, the muted solution is in the following equations:where

Hence, the and are the lower and upper limits of ith decision variable, respectively.

3.3.3. Nonuniform Mutation (NUM)

The most extensively used mutation operators are Michalewicz’s NUM mutation operator in real-coded GAs. The working mechanism of nonuniform mutation could be initiated by Michalewicz et al. [29, 51]. For implementation context, by increasing the number of generations in the simulation process, the strength of the mutation might be reduced so that it can search uniformly for the initial generations of the process while it may search locally for later generations. For a point , a muted point is developed subsequently: Step 1: generate a random value from a uniform probability distribution, . Step 2: obtain a muted solution by following mathematical expression:where “b” is a parametric value that determines the working capability of the mutation operator. P and p represent the number of maximum generations and the number of the current generation. and are the upper and lower limits of the ith decision variable, respectively.

4. The Proposed Fisk Crossover (FX) Operator Based on Log-Logistic Probability Distribution

The log-logistic probability distribution has been used in networking to model the transmission times of data considering both the network and the software, in hydrology to model streamflow and precipitation, and in economics as a simple model of the distribution of wealth or income. This distribution efficiently deals with the modeling of the delays in the transmission of sensory data coming from a networked telerobot, which would allow us to predict the future times of arrival and provide assurance on the time requirements of these systems [52]. Besides, log-logistics distribution is also used to estimate low-flow frequency based on the analysis on extreme low flow event within a specific time interval [53].

So, with diversified application in many optimization related fields, the presently designed novel real-coded crossover scheme is naturally suitable to solve optimization problems with continuous variables. The major advantage of real encoding in GA over binary encoding is to efficiently handle complex nonlinear optimization problems with continuous domain and also overcome the fundamental issues such as exploration (population diversity) and exploitation (selection pressure). Hence, the binary encoding scheme inherited many difficulties such as premature convergence which can be solved by real encoding in the GA process. Hence, we present a Fisk crossover (FX), and this is a parent centric crossover operator that is linked with log-logistic distribution [52].

Figure 1 reveals the working mechanism of a mixture probability distribution-based real-coded crossover operator. Initially, we generate a random population of size “N” chromosomes with predefined encoding followed by evaluating each chromosome according to its fitness value. Afterword, approximately half of the chromosomes is selected through tournament selection. In the next step, recombine new offspring using FX, DPX, LX, and SBX crossover operators within the conjunction of three mutation operators, i.e., MTPM, PM, and NUM with predefined probability for minor changes in the chromosomes. In the final step, the whole algorithmic process will be continued until the required optimum solution is obtained.

The mathematical working principle of Fisk crossover operator is formulated by using the probability density function of log-logistics distribution, which is given below in

The cumulative distribution function is also depicted as subsequent:where is the scale parameter and is the shape parameter.

By using FX, a pair of offspring and is generated from a pair of parents and in a subsequent stepwise procedure. Step 1: generate a random number from a uniform probability distribution, where . Step 2: calculate the value of the parameter by generating random numbers from Fisk/Log-logistic probability distribution by simply finding the inverse of the cumulative distribution function of Fisk/Log-logistic distribution as Step 3: finally, the offspring are generated through subsequent equations (24) and (25), respectively, for :

In the context of the two equations (24) and (25), it is visualized that the smaller parametric value of β produces offspring close to the parents while the large parametric value of β generates offspring away from the parents for the fixed parametric value of α in Figure 2. Hence, from Figure 3, it is also observed that, with the smaller parametric value of α which produces offspring which are close to parents conversely for the larger value of α, the generation of offspring is away from the parents for the fixed parametric value of β. There is another matter of fact that the generation of offspring through FX crossover is visually symmetrical about the location of the parents as depicted in Figure 4. Therefore, the above mathematical formulation for and reveals that the distance between the two offspring is proportional to the distance between parents under fixed parametric values of α and β. Hence, proposed FX crossover operator shows self-adaptive behavior with the violation of variable limit constraint, i.e., or , during the generation process of offspring in FX then the random value given to in certain bound and assigns “0” value to its location parameter.

4.1. Benchmark Functions

The optimization process is focused on obtaining the global optimum point; consequently, the regions nearby local optima should be circumvented because the optimization process might be stuck at local optima, and then local optima are considered to be as global optima. To evaluate the performance and sustainability of the proposed real-coded crossover operators, we will use twenty-one unimodal, multimodal, separable or nonseparable, convex, and continuous benchmark functions. Table 1 presents the list of benchmark functions [54] utilized to appraise the efficiency of suggested evolutionary methods. Hence, the benchmark function name, fitness function, search limits, and theoretical optimum value are present in Table 1. These benchmark functions have varying complexities that are most commonly applied in many comparative studies. The necessary details regarding these benchmarks are given in Table 1.

5. Experimental Setup

In the current research study, a newly proposed parent centric crossover (FX) is used to enhance the performance of the genetic process and make a close comparison with existing real-coded crossover operators which comprise of DPX, LX, and SBX. These four crossover operators co-integrated with MPTM, PM, and NUM mutation operators for the evaluation of their global performance. A simulated study of twelve algorithmic combinations along with their respective crossover and mutation probabilities is summarized and final parametric values are represented in Table 2. The suitable adjustment of these parametric values is helpful in obtaining optimum results in the empirical study.

The size of the population for all these algorithms is ten times the number of decision variables and the simulated results are obtained through thirty independent runs for each algorithm. Tournament selection along with Elitism with size one is applied in the whole GA algorithmic process. All experiments are terminated when the number of generations achieved the 500 generations and the optimum results regarding the GA process were obtained through trial run and screening experimentation. To evaluate the efficiency, compatibility, and effectiveness of the simulation process, all algorithms were executed thirty times and the mean and standard deviation along average execution time in seconds are taken as final results. The performance of the newly proposed real-coded crossover scheme linked with log-logistic probability distribution is evaluated on twenty-one benchmark functions by using MATLAB version R2015a.

In the context of probabilistic search algorithms in the GA process, we applied a one-way analysis of variance (ANOVA) as hypothesis testing [55] for comparison of four real-valued crossover operators including the proposed one. The experimental results seem to be statistically significant if it is considered unlikely to have occurred by chance, assuming the significance of the null hypothesis. The statistically significant results justify the rejection of the null hypothesis when a probability ( value) is less than a prespecified threshold level (5% level of significance). The F test is applied to make a comparing between crossover operators through a ratio of between and within variation. Hence, the statistical significance of the crossover schemes is tested through the F test statistic in

On the basis of a significant F test, we have determined that crossover operators’ means are significantly different from each other. Hence, the pairwise multiple comparisons can determine the difference between each pair of means. The Gabriel pairwise comparison test [56] is used, which is based on the studentized maximum modulus and generally more powerful under simulated studies. The test statistic of the Gabriel test is as follows:where = Mean square error from ANOVA, and are two means for comparison, and and are respective sample sizes from population i and j.

6. Results and Discussion

In the current empirical study, our main contribution is introducing a new real-coded crossover operator (FX), and the focus of our study is to evaluate the performance of proposed crossover operators in the context of simulation results. So, according to the results of Table 3, we make a comparison of newly proposed (FX) with Double Pareto crossover (DPX) operator [49], Laplace (LX) crossover operator [26], and simulated binary crossover (SBX) [19] by the co-integration with MTPM mutation operator [50] based on mean, standard deviation (SD), and average execution time. It is observed that the Fisk crossover operator outperformed in fifteen out of twenty-one test problems/benchmark functions under diverse features. In most of the multimodal test problems, the empirical results are considered close to the theoretical optimum value which reflects the improved performance of the newly proposed crossover scheme. The performance of FX in terms of mean values, standard deviation, and average execution time is exceptionally ideal and also helpful to overcome the shortcomings of the GA process including exploitation and exploration. Hence, the smallest mean values with lesser SD along with minimal average execution time reflect better control over the selection pressure and have a sufficient impact on loss of population diversity.

The empirical comparison of novel Fisk distribution- (FX-) based crossover operator with Double Pareto crossover (DPX) operator [49], Laplace (LX) crossover operator [26], and simulated binary crossover (SBX) [19] in Table 4 was administered by incorporating the power mutation proposed by Deep and Thakur [26] in the algorithmic framework. The resulted mean and standard deviation with average executing time are lowest in most of the benchmark functions, i.e., Generalized-2, Ackley, Axis, Brown, Ellipsoidal, Levy-Mont, etc. by using the power mutation operator by [26]. Overall, FX performs best in thirteen out of twenty-one benchmark functions which show a complete dominance over the other three crossover operators.

The co-integration of power mutation (PM) with crossover operators is also helpful to enhance the performance and also have better control over the deficiencies of the GA process. The considerable closeness of empirical results with theoretical optimum value delineates better control on the loss of population diversity and the least execution time demonstrates sustainable selection pressure. In general, FX has a better success rate for obtaining the optimum results.

The empirical results of Table 5 reveal an ample dominance of the FX crossover operator over the Double Pareto crossover (DPX) operator [49], Laplace (LX) crossover operator [26], and simulated binary crossover (SBX) [19] algorithmic approaches with co-integration of nonuniform mutation (NUM) operator [51] for obtaining an optimum value. It is also observed that the FX crossover attained optimum results in thirteen benchmark functions, whereas SBX crossover performs better in 8 benchmark functions regarding optimum mean with fewer SD and lesser average execution time under nonuniform mutation operators. The closeness with theoretical value expresses the superlative performance of the newly proposed real-coded crossover scheme. The empirical results in Table 5 also show sustainable control over the weaknesses of the GA process.

According to the results of Table 6 show that there is a highly significant difference between DPX, FX, LX, and SBX under Ellipsoidal and Rastrigin benchmark functions by using ANOVA. Besides, there is also a significant difference in thirteen benchmark functions out of twenty-one. Gabriel multiple pairwise comparison test is applied for close comparison between real-coded crossover operators. In this regard, there is a highly significant difference between FX and DPX in two benchmark functions while also have a significant impact on 12 benchmark functions at 5% and 10% significance level respectively. When we compare FX with LX then the differences between these real coded crossover operators are statistically significant at 1%, 5%, and 10% under 15 benchmark functions. Gabriel multiple pairwise comparisons show an insignificant difference between FX and SBX under most of the benchmark functions. The overall statistical results represent the numerical uniqueness of FX over DPX, LX, and SBX crossover operators.

The performance of the algorithmic procedure is usually examined through optimum value and execution time required to get an optimal solution. Four real-coded crossover schemes are visually compared with the integration of three mutation operators in Figure 5. Hence, Figure 5 visualized the dominance of FX over other crossover operators. In the MTPM mutation operator, FX obtained the optimum value in 15 benchmark functions while LX and SBX outperformed in three benchmark functions. Similarly, FX distinctly achieved an optimum solution in thirteen benchmark functions on the other hand SBX obtained optimum value in four and three benchmark functions under PM with NUM mutation operators, respectively, while the graphical description also depicted the limited performance of DPX and LX operators in the whole algorithmic procedure.

7. Performance Index (PI)

After statistically examining the performance of FX real-coded crossover operator with others, we make a comparison between GAs’ real-coded crossover schemes based on the performance index (PI) that was used by HaqHussainn and Ahmad [14]. This performance index was precisely applied to examine the behavior of various controlled stochastic search methods. This index is a widely used procedure for making a comparison between different heuristic algorithms [50, 57]. The mathematical derivation of PI is given below in the following equation:where MFⁱ: mean of the objective function for the ith optimization problem LMFⁱ: least mean value of objective function obtained by all algorithms for the ith optimization problem SFⁱ: standard deviation of the objective function for the ith optimization problem LSFⁱ: least standard deviation value of objective function obtained by all algorithms for the ith optimization problem CVFⁱ: the value of the coefficient of variation linked with the objective function for the ith optimization problem LCVFⁱ: least coefficient of variation value of objective function obtained by all algorithms for the ith optimization problem N_p: the total population to be analyzed

The , , and ( + + = 1 and 0 ≤ , , ≤ 1) are weights assigned to three statistics were considered, respectively.

In regards of the above description, the performance index is a mathematical formulation of , , and , respectively. Hence, + + = 1 and one of γ_i, for (i = 1, 2, 3), could be excluded by reducing the number of dependent variables from the mathematical formulation of the performance index. However, it is still difficult to visually evaluate the trend of all GAs’ real-coded crossover schemes because of the overlapping of the surface plot of PI. So, we adopt the modified process by assigning the same weights to any two terms in PI. Hence, the PI becomes a function of a single variable. Now, resultant cases are below:

The visual representation for cases (1–3) in Figures 6–8 reveals that the horizontal axis represents weights (wt) and scaled value of performance index (PI) defined on the vertical axis. The PI of FX is outperformed in all figures instead of other parent-centric crossover operators which shows a substantial dominance towards perfection. More specifically, the graphical depiction of PI endorses the performance improvement in the FX crossover operator.

8. Conclusions

In this paper, an improved real-coded crossover operator called the Fisk crossover operator (FX) is introduced to enhance the performance of the GA process with a fine tradeoff between selection pressure and population diversity. This newly proposed parent centric crossover operator has a characteristic of self-adaption and the development ideology of FX is linked with Log-logistic probability distribution. FX and three crossover operators (DPX, LX, and SBX) are attached with three well-known mutation operators (MPTM, PM, and NUM), which are compared, and evaluated their algorithmic performance under twenty-one benchmark functions with diverse features. All considered test problems are scalable with the varied number of decision variables.

The tournament selection is applied in the reproduction phase for the newly proposed algorithm and performance is examined under an identical simulation strategy. Two different strategies are used to evaluate the performance of GA. The first strategy is to make a comparison between FX and three other crossover operators for obtaining optimal solutions on the basis of mean, SD, and average execution time. The empirical results show a complete dominance of FX over other crossover operators. Hence, FX outperformed in between thirteen to fifteen benchmark functions under MTPM, PM, and NUM mutation operators, respectively. Furthermore, relevant statistical techniques including ANOVA and Gabriel pairwise multiple comparison test are also administrated which indicate the significance of the proposed crossover scheme.

The second strategy is to evaluate the performance by comparing all algorithms through performance index (PI) with the use of three statistical measures. The graphical representation of PI reveals the optimal performance of FX instead of other crossover operators. Finally, the statistically significant results of the proposed crossover technique have a definite edge over the others and have great potential to solve more complex optimization problems.

Data Availability

The data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The authors are very grateful to the deanship of scientific research at King Khalid University, Abha, Kingdom of Saudi Arabia, for funding this work through the Research Groups Program under the project number R.G.P-2/67/41.

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Copyright © 2020 Ehtasham ul Haq et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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