Mixture distribution based real-coded crossover: A hybrid probabilistic approach for global optimization

Abstract

In the present simulation-based study, a novel parent-centric real-coded crossover operator is introduced with a unique probabilistic aspect of the mixture distribution. Moreover, the mixture distribution is a co-integration of double Pareto and Laplace probability distributions with various parameters. The key objective of the newly proposed methodology is to obtained optimal solutions for complex multimodal optimization problems. Hence, for its global comparison, the newly proposed mixture distribution crossover operator (MDX) is compared with double Pareto (DPX), Laplace (LX), and simulated binary (SBX) crossover operators within the conjunction of three mutation operators (MTPM, PM, and NUM). After a descriptive comparison, a Quade multiple comparison test is also administered to examine its statistical significance. Furthermore, the performance of the genetic algorithm (GA) is also examined on a set of twenty-one unconstraint benchmark functions with diverse features. The empirical results of the simulation-based study reveal that the mixture-based crossover operator obtained a substantial dominance over all considered crossover operators in terms of computational complexity, robustness, scalability, and capability of exploration and exploitation. Moreover, the Quade multiple comparison test also showed a significant superiority with graphical authentication of the performance index (PI).

Keywords

Genetic algorithms two-component mixture model real coded crossover Quade test Performance Index

1 Introduction

A general concept of optimization refers to both maximization and minimization of tasks, $min {f (t) : t \in R^{n}},$ where t ∈ T, and T is a rectangular hypercube with n-dimensions in Rⁿ with limits a_i≤ t_i ≤ b_i, i = 1, 2, …, n ., these are commonly known as bounds which are based on the decision variables. A point t⁺ ∈ T is known as local minima of objective function f (t), if f (t⁺)≤ f (t) , ∀ y ∈ N_ɛ (t⁺) ∩ T, where N_ɛ (t⁺) ={ t ∣ ∥ t - t⁺ ∥ < ɛ, ɛ > 0 } is the small neighborhood of the point y⁺. If f (t⁺) ≤ f (t) , ∀ t ∈ T, then t⁺ is said to be the global minima of the objective function f (t).

Several applied nonlinear optimization problems in fields of computing, operations research, engineering, science, and technology can be mathematically formulated with possibly having local or global optimum results. Therefore, most of the researchers are interested in obtaining the global optima, but the determination process is quite complicated to determine global optima instead of local optima. Hence, various optimization approaches are available in the literature which can be classified into two major groups, the first is deterministic and the second is stochastic optimization techniques.

The working framework of deterministic techniques is based on a single solution by using deterministic transition rules that guide the search space to obtain an optimum solution. Naturally, deterministic techniques are mainly focused on the minimization problems due to the search procedure that is based on the guesswork for obtaining optimum solution (random selection of the solution in the search space) otherwise global minimizer could be stuck at the local minimizer. Furthermore, deterministic techniques are not fully capable of solving all sorts of optimization problems that are not naturally generic. More illustratively, conjugate gradient-based optimization methods are good to solve the quadratic problem with mathematical derivation but not capable of handling complex multimodal problems [1]. In the context of the above shortcomings related to deterministic techniques, many stochastic optimization techniques including evolutionary algorithms have been designed and developed. These techniques strongly rely on the computational effort for obtaining the global optimum solutions. The theoretical philosophy of evolutionary algorithms is based on three stochastic-based heuristic methodologies, which are Genetic Algorithms (GAs), Evolutionary Strategies [2], and Evolutionary Programming. Genetic Algorithm (GA) is one of the most influential and efficient algorithmic techniques in the field of evolutionary optimization [3].

The fundamental idea about Natural Selection was originated by the British naturalist Charles Darwin in 1859 [4]. According to natural selection, the theory states that the reproduction of individuals with certain favorable features is more likely to carry on and transfer their prominent attributes to their offsprings. Hence, Individuals with less favorable attributes will gradually depart from the population. According to the natural phenomenon, genetic inheritance is placed in chromosomes that are made of genes [5]. The characteristics of every individual are controlled by the genes, which are transferred to the offsprings when the individuals mate. Once in a while, the structure of the chromosomes changes due to crossover and mutation in the genetic process [6]. The population of individuals will steadily improve on average because the individuals having favorable attributes increase in the natural process of reproduction. The crossover operator incorporates genetic information between chromosomes to explore the new search space, whereas the mutation operator helps to maintain population diversity and avoid premature convergence [7].

In the beginning, the GA uses binary encoding due to the mapping process of discrete values in the search space with the inclusion of the Hamming distance. Hence, the required accuracy of the solution is achieved [8]. Furthermore, binary coding is to be regularly encoded and decoded: which increasing the processing time for possible error conversion. Since the encoding length controls the accuracy of the solution. Binary coding may involve unnecessary lengthy codes, resulting in excessive processing and memory capacity and decreased computational speed for maximum accuracy. In the framework of evolutionary optimization techniques, Lucasius and Kateman introduced an initial concept of real-coded genetic algorithms [9, 10] in 1989. Real-coded genetic algorithms have a loss of befits including sufficient precision, no requirement of encoding, wide random space search, easy computing process, fast convergence with limited chance of stuck at local optima instead of global optima [11, 12].

Generally, GA has extensively applied in various fields such as automatic processing control, machine learning, function optimization, combinatorial optimization, image processing, production scheduling problems, data mining, intelligent manufacturing processes, industrial, system and bioengineering, artificial intelligence, text information filtering, and cooperative multi-objective problem evaluation systems [13 –17]. It is effectively and efficiently handles complex nonlinear optimization problems where the traditional methods fail to solve them [18].

Over the last few decades, a real coded genetic algorithm gathered a lot of attention because of its distinctive and exceptional performance. Therefore, many researchers have done an in-depth analysis and obtained productive results. The crossover along with mutation operators has a significant impact on the GA performance. Therefore, many researchers give a lot of attention to enhance the performance of these operators. Wright [19] introduced a heuristic crossover HX crossover operator, the generation of offspring is located on a straight line connected with parents with the highest fitness value. In earlier 1990s, Eshelman et al. [20] suggested a blend crossover operator (BLX - α) by integrating with the theoretical concept of interval schemata. The generations of offspring are quite similar because of the small difference between the parents, while the difference between parents is large, then offspring are the same as random search space. The parameter of blend crossover is denoted by α, where the suggested value for optimum result is 0.5. Hence, Deb et al. [21] modified single point binary crossover and develop a simulated binary crossover (SBX) in 1995. SBX generates two offspring from both selected parents which are offspring are located on a straight line connected with parents, and n is the SBX distribution index that controls the distance between two offspring. If the value of n is large, the probability of the produced offspring being closer to the two parents is greater; if the value of η is relatively small, the likelihood of the created offspring being farther away from the two parents is greater. The shortcoming of SBX is that: it cannot regulate the size of the parameter n value adaptively and therefore cannot control the distance between two offspring produced through the crossover process. After that, in 1997, Ono et al. [22] introduced a uni-modal normal distribution crossover operator (UNDX) associated with Ellipsoidal probability distribution for the generation of more offspring from three selected parents to boost working capabilities in the genetic process. Again in 2002, Deb et al. [23] proposed another self-adaptive multi-parent crossover (PCX) which uses a large probability, and different vectors are calculated for the generation of N offspring from multiple parents. Deep and Thakur [24] suggested a Laplace crossover (LX) which is linked with Laplace probability distribution and is used to determine the offspring’s location. Hence, two offspring produced by the LX operator are symmetrical concerning their parental position and are not automatically located near the best of both parents. Chuang et al. [25] suggested a direction-based crossover (DBX) that would be able to explore (2ⁿ - 1) alternative search directions. DBX’s search directions are however limited. Though it can generate a route based on the crossover that can guide the chromosomes towards the optimal solution; this is not highly likely. Meanwhile, the null vector solution is likely to be generated when the dimensionalities of the variables are small.

Wang et al. [26] introduced a heuristic-based crossover operator (HNDX) which is linked with the Normal probability distribution. The cross-generated offspring can be assumed to be closer to the best one between the two parents and the search path, to be very similar to the optimal search direction, or to be aligned with the optimal search direction. Nevertheless, HNDX does not determine whether there are stronger individuals in the population than parents, and if so, whether the crossover operator should be treated as such.

Similarly, the literature reports several other mutation schemes including the non-uniform mutation (NUM) operator which is suggested by Michalewicz [27]. NUM’s global search capability is strong at the beginning as well as the latter part of the simulation process. Hence, the major shortcoming of the NUM operator is the fixation of a maximum number of iterations for all optimization problems. Wang et al. [28] developed a mutation operator, which mutated towards the objective function’s gradient direction. However, the performance of the mutation operator cannot be sufficient if the objective function is non-differentiable. Some real coded mutation operators are proposed by Deb et al. [29], which is known as polynomial mutation with application in diverse fields. The main feature of PM operators is having a strong random search capability for achieving global optimum solutions with slow convergence. Thakur [30] developed a self-adaptive real coded crossover to obtain the global optimum solution of multi-modal complex optimization problems. This crossover operator is associated with double Pareto probability distribution known as double Pareto crossover (DPX). DPX is also co-integrated with three mutation operators and shows a significant dominance over other established crossover operators. Recently, Chuang et al. [25] suggested a mutation operator which is named dynamic random mutation (DRM). Hence, the mathematical formulation of step size opposes its explanation. The mutation operator still requires the maximum number of iterations in advance, which is hard to estimate. In recent past, Rolland and Chandra [31] also introduced PCX to effectively handle the forward kinematics problem (FKP) for parallel manipulators. Ali et al. [32] developed a new Differential Evolution real-coded crossover based on multi-parent crossover ideology. Akopov et al. [33] designed and developed a unique multi-agent real-coded genetic algorithm (MA-RCGA). In recent past, Naqvi et al. [34] suggested a real-coded crossover based on logistic probability distribution. The most recent contribution in the context of this area, a novel Exponentiated Pareto distribution (EPX) based real-coded crossover by Naqvi and Shad [35].

In the context of the co-integration of two real coded crossover operators for the improvement in the GA process through a probabilistic crossover strategy. This novel crossover technique provides an adequate trade-off between exploitation and exploration, which effectively eliminates the impact of selection pressure and maintains a certain genetic diversity within the population. In the present simulation-based research study, an innovative real coded crossover operator called the Mixture distribution crossover (MDX) operator is presented by the co-integration of Double Pareto and Laplace probability distribution. This new crossover operator is introduced by using the concept of two-component mixture probability models in real coded genetic algorithms. The MDX is used to get optimum solutions in combination with three mutation operators are known as Power Mutation (PM), Makinen, Periaux, and Toivanen Mutation (MPTM), and Non-Uniform Mutation (NUM) [30]. To examine the global performance of the mixture distribution crossover, a comparative computational analysis is executed.

This paper is organized in such a way that: the newly suggested two-component mixture distribution crossover is defined in Section 2. In Section 3, some previously defined well-known selection, crossover, and mutation operators are discussed. A set of well-known unconstrainted benchmark functions with diverse features is discussed in Section 4. In Section 5, the experimental setup of the proposed study is presented. The simulated results with statistical analysis are discussed in Section 6 and conclusions are drawn in the final section of the study.

2 The proposed mixture distribution of real coded crossover operator

The mixture densities provide a suitable model for various subsets of the data that exhibit different features and can be better mathematically formulated because they can be more numerically tractable and can be more easily studied. The inclusion of mixture models in evolutionary algorithms have analytical forms with the advantages of generating computationally efficient information. In this regard, standard practice is to use mixture models assuming that the underlying distribution is multi-modal, or data is spread over multiple manifolds and subspace with the compensation of miss-captured parts. This probabilistic procedure is more suitable to tackle certain problems than others, even at the different stages of the genetic process in the same problem. The study of the synergy produced by combining the different styles of the traversal of solution space associated with the different probabilistic functions.

In the current section, a newly developed real-coded crossover operator is based on the unique idea about the hybridization of two well-known existing real-coded crossover operators. This novel technique is designed in such a way that it retains the strength of optimality from which it is accomplished. This newly designed crossover operator is known as a mixture distribution crossover (MDX) by integrating Double-Pareto crossover (DPX) and Laplace crossover (LX) operators for the generation of two offsprings. MDX is robust, accurate, and efficient because the hybrid probabilistic representation is conceptually closest to the real design space. The hybridization ideology enforces the real coded crossover operators to activate the exploitation and exploration mechanism based on the probabilistic approach. In other words, it would be a suitable adjustment between exploitation and exploration that makes the genetic process towards optimality.

Hence, the finite mixture models continued to receive growing interest from both practical and theoretical perspectives over the years in the field of mathematical statistics. There is also a direct implementation of the finite mixture models in many applied fields of science and engineering as mentioned by Sultan and Al-Moisheer [31]. In the present section, we propose a parent-centric crossover operator based on the two-component mixture of Double-Pareto and Laplace probability distributions [30]. The probability density function (pdf) along with the cumulative distribution function (cdf) of the first component (Laplace) is given in Equations 2. The visual representation of density function in Fig. 1. $f_{1} (t) = \frac{1}{2 b} e^{(- \frac{| t - a |}{b}),} - \infty < t < \infty$ (1) $F_{1} (t) = {\begin{matrix} \frac{1}{2} e^{(\frac{| t - a |}{b})} & t \leq a \\ 1 - \frac{1}{2} e^{(\frac{| t - a |}{b})} & t > a \end{matrix}$ (2)

Fig. 1

pdf of Laplace distribution for fix (a).

The pdf and cdf of the second component (Double Pareto) of the mixture function are given in Equations 4. Moreover, the graphical depiction of double Pareto is also shown in Fig. 2. $f_{2} (t) = \frac{1}{2 β} {(1 - \frac{| t |}{α β})}^{- (α + 1)} - \infty < t < \infty$ (3) $F_{2} (t) = {\begin{matrix} \frac{1}{2} {(1 - \frac{t}{α β})}^{- α} & t < β \\ \frac{1}{2} [1 - {(1 - \frac{t}{α β})}^{- α}] & t \geq β \end{matrix}$ (4)

Fig. 2

pdf of double Pareto distribution for fix (α).

Moreover, the pdf and cdf of the two-component mixture probability model are given in Equations 5-8. $f (t) = θ_{1} f_{1} (t) + θ_{2} f_{2} (t)$ (5) where 0 < θ₁, θ₂ < 1, and θ₁ + θ₂ = 1 $f (t) = θ_{1} \frac{1}{2 b} e^{(- \frac{| t - a |}{b})} + θ_{2} \frac{1}{2 β} {(1 - \frac{| t |}{α β})}^{- (α + 1)}$ (6) $F (t) = θ_{1} F_{1} (t) + θ_{2} F_{2} (t)$ (7)

$F (t) = {\begin{matrix} \frac{1}{2} (θ_{1} e^{(\frac{| t - a |}{b})} + θ_{2} {(1 - \frac{t}{α β})}^{- α}) & t \leq a \\ θ_{1} [1 - \frac{1}{2} e^{(\frac{| t - a |}{b})}] + θ_{2} \frac{1}{2} [1 - {(1 - \frac{t}{α β})}^{- α}] & t > a \end{matrix}$ (8) In the Mixture cumulative distribution function, β & b > 0 is defined as scale parameters, and a & α ∈ R known as the location parameters. By using MDX, from a pair of parents $t^{{1}} = (t_{1}^{{1}}, t_{2}^{{1]}, \dots, t_{n}^{{1}})$ and $t^{{2}} = (t_{1}^{{2}}, t_{2}^{{2}}, \dots, t_{n}^{{2}})$ , two offspring u^{1} = $(u_{1}^{{1}}, u_{2}^{{1}}, \dots, u_{n}^{{1}})$ and $u^{{2}} = (u_{1}^{{2}}, u_{2}^{{2}}, \dots, u_{n}^{{2}})$ are obtained through a stepwise procedure.

Step-1: Generate a random number (r_j) from a uniform distribution with a unit range.

Step-2: Obtain the parametric value β_t by generating random numbers from Mixture probability distribution by simply inverting the cumulative distribution function as under in Equation 9.

$β_{t} = {\begin{matrix} θ_{1} α β + θ_{2} a - [θ_{1} α β {(2 r_{j})}^{- \frac{1}{α}} + θ_{2} b {log}_{e} r_{j}], & r_{j} \leq 0.5 \\ θ_{1} α β - θ_{2} + [θ_{1} α β {(1 - 2 r_{j})}^{- \frac{1}{a}} + θ_{2} b {log}_{e} r_{j}], & r_{j} > 0.5 \end{matrix}$ (9) Step-3: Finally, the two off-springs for j = 1, 2, …, n are determined through subsequent Equations 11 respectively. $u_{j}^{(1)} = \frac{(t_{j}^{{1}} + t_{j}^{{2}}) + β_{t} | t_{j}^{{1}} - t_{j}^{{2}} |}{2}$ (10) $u_{j}^{{2}} = \frac{(t_{j}^{{1}} + t_{j}^{{2}}) - β_{t} | t_{j}^{{1}} - t_{j}^{{2}} |}{2}$ (11) If offspring in MDX are obtained outside the variable bounds i.e. $u_{j} < u_{j}^{l}$ or $u_{j} < u_{j}^{r_{j}}$ , then randomly generated values are given to u_j in an interval $[u_{j}^{l}, u_{j}^{r_{j}}]$ .

3 Some recently used real coded GA operators in the literature comparison

In this section, we theoretically describe the basic ideology of some recently considered real coded GA operators in the literature i.e. tournament selection (TS), Laplace crossover (LX), double Pareto crossover (DPX), and simulated-binary crossover (SBX) co-integration with different mutation operators that are Non-uniform mutation (NUM), Makinen, Periaux and Toivanen mutation (MPTM), and Power mutation operator (PM) for empirical comparison.

3.1 Tournament selection (TS)

The tournament selection technique is straightforward to use and easier to apply in the genetic process. The working framework of tournament selection is based on two or more than two chromosomes competing with each other, and the best chromosomes will be picked and selected for the mating pool [5]. The selection of these chromosomes is purely a random process that is based on their fitness values and the winner (higher fitness) goes into the mating pool. Finally, there will be selected chromosomes that have been chosen according to their fitness. This random selection of chromosomes will be compared again and if the chromosomes win, it will have generated offspring for the next crossover process while the weakest chromosomes will not have any copies at all.

3.2 Laplace crossover (LX)

LX real coded crossover operator [24] belongs to a class of self-adaptive crossover operators. This crossover operator is originated from the Laplace probability distribution. Hence, the cdf of the distribution is represented in Equation 12 given below. $F (t) = {\begin{matrix} \frac{1}{2} e^{(\frac{| t - a |}{b})}, & t \leq a \\ (1 - \frac{1}{2} e^{(\frac{| t - a |}{b})}), & t > a . \end{matrix}$ (12) The scale parameter of Laplace probability distribution is b > 0 and a ∈ R defined as the location parameter. Both offspring $u^{{1]} = (u_{1}^{{1}}, u_{2}^{{1}}, \dots, u_{n}^{(1}})$ and u^{2} = $(u_{1}^{(2}}, u_{2}^{{2}}, \dots, u_{n}^{{2}})$ are obtained from a pair of parents $t^{{1}} = (t_{1}^{(1]}, t_{2}^{(1]}, \dots, t_{n}^{{1}})$ and t^{2] = $(t_{1}^{{2}}, t_{2}^{{2}}, \dots, t_{n}^{{2}})$ through the subsequent three steps procedure.

Step-1: Generate a random number (r_j) through a uniform probability distribution with a unity range.

Step-2: Obtain the value of the parameter β_t through random numbers generation process from the Laplace probability distribution by simply inverting the cumulative distribution function as under in Equation 13. $β_{t} = {\begin{matrix} a - b {log}_{e} (r_{j}), & r_{j} \leq \frac{1}{2} \\ a + b {log}_{e} (r_{j}), & r_{j} > \frac{1}{2} \end{matrix}$ (13) Step-3: Hence the offspring are obtained through Equations 15 respectively for j = 1, 2, …, n $u_{j}^{(1)} = \frac{(t_{j}^{{1}} + t_{j}^{{2}}) + β_{t} | t_{j}^{{1}} - t_{j}^{{2}} |}{2}$ (14) $u_{j}^{{2}} = \frac{(t_{j}^{{1}} + t_{j}^{{2}}) - β_{t} | t_{j}^{{1}} - t_{j}^{{2}} |}{2}$ (15)

If offspring in LX are obtained outside the variable limits i.e. $u_{j} < u_{j}^{l}$ or $u_{j} < u_{j}^{rj}$ , then randomly generated values are given to u_j is lies in an interval $[u_{j}^{l}, u_{j}^{r_{j}}]$ .

3.3 Double pareto crossover (DPX)

DPX operator [30] is another type of parent-centric crossover operator that uses the double Pareto probability distribution. The cumulative distribution function of which is expressed as in Equation 16. $F (t) = {\begin{matrix} \frac{1}{2} {(1 - \frac{t}{α β})}^{- a}, & t < β \\ \frac{1}{2} (1 - {(1 - \frac{t}{α β})}^{- a}), & t \geq β, \end{matrix}$ (16) where α ∈ R is represented as the location parameter while β > 0 defined as a scale parameter of the distribution. By using DPX, from a pair of parents $t^{{1}} = (t_{1}^{{1}}, t_{2}^{{1}}, \dots, t_{n}^{{1}})$ and $t^{{2}} = (t_{1}^{{2}}, t_{2}^{(2)}, \dots, t_{n}^{{2}})$ , two offspring $u^{{1}} = (u_{1}^{{1}}, u_{2}^{{1}}, \dots, u_{n}^{{1}})$ and u^{2} = $(u_{1}^{{2}}, u_{2}^{{2}}, \dots, u_{n}^{{2}})$ , are generated through a subsequent step-by-step process.

Step-1: Generate a random number (r_j) from a uniform probability distribution where r_j ∈ [0, 1]

Step-2: Calculate the value of the β_t by generating random numbers from the double Peraeto probability distribution, simply by finding the reverse of the double Peraete distribution cumulative distribution function as given in Equation 17. $β_{t} = {\begin{matrix} α β (1 - {(2 r_{j})}^{- \frac{1}{α_{i}}} & r_{i} \leq 0.5 \\ (α β (1 - {(2 r_{j})}^{- \frac{1}{a_{i}}}) - 1) & r_{i} > 0.5 \end{matrix}$ (17) Step-3: Now the offspring are in Equations 19 for j = 1, 2, …, n.

$u_{j}^{(1)} = \frac{(t_{j}^{{1}} + t_{j}^{{2}}) + β_{t} | t_{j}^{{1}} - t_{j}^{{2}} |}{2},$ (18) $u_{j}^{{2}} = \frac{(t_{j}^{{1}} + t_{j}^{{2}}) - β_{t} | t_{j}^{{1}} - t_{j}^{{2}} |}{2} .$ (19)

If offspring in DPX are obtained outside the variable limits i.e. $u_{j} < u_{j}^{l}$ or $u_{j} < u_{j}^{r_{j}}$ , then randomly generated values are given to u_j in an interval $[u_{j}^{l}, u_{j}^{r_{j}}]$ .

3.4 Simulated binary crossover (SBX)

The SBX was proposed by Deb and Agrawal [21] with a distinct methodology of binary transformation to continuous search space. The initial step of the working framework is the generation of a random number r_j from a uniform probability distribution with limits 0 to 1. Afterward, the value of the parameter β_t is obtained through subsequent mathematical expression in Equation 20. $β_{t} = {\begin{matrix} {(2 r_{j})}^{\frac{1}{(n_{c} + 1)}}; & if r_{j} \leq \frac{1}{2} \\ {(2 - 2 r_{j})}^{\frac{- 1}{(n_{c} + 1)}}; & otherwise . \end{matrix}$ (20) Where the distribution index is defined as n_c ∈ [0, ∞] and an offspring u = (u₁, u₂, …, u_n) is produced from the pair of parents $t^{(1)} = (t_{1}^{{1}}, t_{2}^{(1)}, \dots, t_{n}^{(1}})$ and $t^{(2)} = (t_{1}^{(2)}, t_{2}^{(2}}, \dots, t_{n}^{(2}})$ by using a simulated binary approach in given Equation 21 for j = 1, 2, …, n $u_{j} = \frac{1}{2} [(t_{j}^{{1}} + t_{j}^{{2}}) - β_{t} | t_{j}^{{1}} - t_{j}^{{2}} |] .$ (21)

3.5 Power mutation (PM)

The mutation operator avoids the simulation process stuck at the local optima and keeps a reasonable amount of the population diversity in the genetic process. That means that the genetic process will help to obtain the best-fitted offspring in the population. Goldberg [6] introduced a mutation clock to cater to the shortcomings of the mutation process in terms of computational complexity. He applied the exponential probability distribution to obtain the next location by shuffling the string through location adjustment. The basic aim of the mutation process is to explore the new searching space and to efficiently solve complex optimization problems. Hence, the mutation operator is helpful to prevent local minimum solutions at the cost of exploring more search spaces. Here, we theoretically define the PM mutation operator that was proposed by Deep and Thakur [24]. The PM is linked with power distribution and the pdf and cdf of power distribution are given as under Equations 23, respectively. $f (t) = p t^{p - 1} 0 \leq t \leq 1$ (22) $F (t) = t^{p} 0 \leq t \leq 1,$ (23) where p is known as distribution index and PM is used to generate an offspring u = (u₁, u₂, …, u_n) from single-parent t = (t₁, t₂, …, t_n) through a subsequent step-by-step approach.

Step-1: Obtain a value (r_j) through the random procedure from a uniform distribution where r_j∈ [0, 1]

Step-2: After obtaining random value, compute s_j by using power distribution in Equation 24. $s_{j} = {(r_{j})}^{\frac{1}{p}}$ (24) Now using the following mathematical expressions to obtain offspring in Equation 25. $u_{j} = {\begin{matrix} t_{j} - s_{j} (t_{j} - t_{j}^{l}), & if \frac{t_{j} - t_{j}^{l}}{t_{j}^{r_{j}} - t_{j}} < r_{j} \\ t_{i} + s_{j} (t_{j} - t_{j}^{l}), & if \frac{t_{j} - t_{j}^{l}}{t_{i}^{ri} - t_{j}} \geq r_{j}, \end{matrix}$ (25) where $t_{j}^{l}$ and $t_{j}^{r_{j}}$ are defined as a lower and upper bound of the j^th decision variable. In this regard, the mathematical formulation stated that perturbation in offspring is in proportion to the parameter “p”. Hence, for the optimum solution, the probabilistic value is obtained from a mutated offspring that is proportional to the distance from the parametric limit.

3.6 Makinen, periaux, and toivanen mutation (MPTM)

Makinen et al. [32] introduced the MPTM mutation operator which is used to solve certain multidisciplinary shape-related optimization problems in GA, particularly in the field of electromagnetics and aerodynamics. Meittinen et al. [33] are also useful in solving constrained optimization problems within the GA method. Deep and Thakur [24] tested and analyzed their findings on multimodal, nonlinear optimization problems. From a point t = (t₁, t₂, …, t_n) the mutated point $\hat{t} = ({\hat{t}}_{1}, {\hat{t}}_{2}, \dots, {\hat{t}}_{n})$ is determined through the following procedure.

Let a random value (r_j) is generated from a uniform distribution where r_j ∈ [0, 1] . Hence, the muted optimal solution is in the following Equations 27. ${\hat{t}}_{i} = (1 - \hat{w}) t_{j}^{l} + (\hat{w}) t_{j}^{rj}$ (26) where ${\hat{w}}_{i} = {\begin{matrix} w_{j} - w_{j} {(\frac{w_{i} - r_{j}}{w_{j}})}^{b} & if r_{j} < w_{j} \\ w_{j} & if r_{j} = w_{j} \\ w_{j} + (1 - w_{j}) {(\frac{w_{i} - r_{j}}{1 - w_{j}})}^{b} & if r_{j} \geq w_{j} \end{matrix}$ (27) and $w = \frac{t - t_{j}^{l}}{t_{j}^{r_{j}} - t} .$ Hence, $t_{j}^{l}$ and $t_{j}^{rj}$ are defined as a lower and upper limit of the $j_{N}^{th}$ decision variable.

3.7 Non-uniform mutation (NUM)

Michalewicz’s NUM mutation operator in real coded GAs is the most widely used mutation procedure. Michalewicz et al. [39, 40] could initiate the working mechanism of Non-uniform mutation. For the implementation context, when increasing the number of generations, the strength of the mutation could be decreased in the simulation process. So that it can search uniformly for the initial generations of the process while it may search locally for later generations. For a point $t^{(1)} = (t_{1}^{(q)}, t_{2}^{{q}}, \dots, t_{n}^{{q}})$ and muted point t^(q+1) = $(t_{1}^{(q + 1)}, t_{2}^{(q + 1)}, \dots, t_{n}^{(q + 1}})$ is developed subsequent procedure.

Step-1: Generate a random value u_j from a uniform probability distribution, u_j ∈ [0, 1].

Step-2: Determine the muted solution through the mathematical expression below Equation 28.

$\begin{matrix} t_{j}^{q + 1} = \\ {\begin{matrix} t_{j}^{q} + (x_{j}^{u} - t_{j}^{q}) {(1 - u_{j}^{(1 - \frac{q}{Q})})}^{b}, & u_{j} \leq 0.5 \\ t_{j}^{q} - (x_{j}^{u} - t_{j}^{q}) {(1 - u_{j}^{(1 - (\frac{q}{Q})})}^{b}, & Otherwise \end{matrix} \end{matrix}$ (28) where “b” is a value of parametric which defines the working strength of the mutation operator. Q and q are known as the number of the maximum and current generation and $x_{j}^{u}$ and $x_{j}^{l}$ are the upper and lower bounds of the j^th decision variable.

4 Test problems with varied characteristics

The optimization method focuses on finding the maximal global value, consequently, the regional optimal areas should be circumvented due to the optimization method could be locally optimized and then called local optima as global optima. To examine the efficiency and sustainability of the proposed real coded crossover operators, twenty-one unimodal, multimodal, separable or non-separable, convex, and continuous benchmarking functions will be used as test problems. The list of benchmark functions [41] is used to determine the efficiency and applicability of proposed evolutionary approaches which are presented in Table 1. In this section, the name of the benchmark function, its fitness function, search limits, and optimum theoretical value are described. These benchmark factions have dynamics features that are most widely found in many comparative studies. The requisite information concerning these test problems is given in Table 1.

Table 1
Detail of test problems for comparison

Benchmarks Fitness function Search limits Optimum value

Ackley’s $f (t) = - 20 exp (- 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} t_{i}^{2}}) - exp (\frac{1}{n} \sum_{i = 1}^{n} cos (2 π t_{i})) + 20 + e$ [-30, 30] 0

Axis parallel ellipsoid $f (t) = \sum_{i = 1}^{n} i t_{i}^{2}$ [-5.12, 5.12] 0

Cigar $f (t) = t_{i}^{2} + 1000000 \sum_{i = 1}^{n} t_{i}^{2}$ [-10, 10] 0

Cosine mixture $f (t) = \sum_{i = 1}^{n} t_{i}^{2} - 0.1 \sum_{i = 1}^{n} cos (5 π t_{i})$ [-1, 1] 0.1n

De-Jong $f (t) = \sum_{i = 1}^{n} (t_{i}^{4} + rand (0, 1))$ [-10, 10] 0

Drop-wave $f (t) = \sum_{i = 1}^{n} {(t_{i} - i)}^{2}$ [-5.12, 5.12] -1

Ellipsoidal $f (t) = \sum_{i = 1}^{n} \frac{1 + cos (12 \sqrt{t_{i}^{2} + t_{i + 1}^{2}})}{0.5 (t_{i}^{2} + t_{i + 1}^{2}) + 2}$ [- n, n] 0

Brown-3 $f (t) = \sum_{i = 1}^{n - 1} (t_{i}^{2})^{(t_{i + 1}^{2} + 1)} + (t_{i + 1}^{2})^{(t_{i}^{2} + 1)}$ [-1, 4] 0

Generalized penalized-1 $f (t) = \frac{π}{n} \times {10 {sin}^{2} (π z_{i}) + \sum_{i = 1}^{n - 1} {(z_{i} - 1)}^{2} [1 + 10 {sin}^{2} (π z_{i + 1})] + {(z_{n} - 1)}^{2}}$ [-50, 50] 0

$+ \sum_{i = 1}^{n} u (t_{i}, 5, 100, 4)$

where: $z_{i} = 1 + \frac{1}{4} (t_{i} + 1), u (t_{i}, 10, 100, 4) = {\begin{matrix} 100 {(t_{i} - 10)}^{4} & if t_{i} > 10 \\ 0 & if - 10 \leq t_{i} \leq 10 \\ 100 {(- t_{i} - 10)}^{4} & if t_{i} < - 10 \end{matrix}$

Generalized penalized-2 $f (t) = 0.1 ({sin}^{2} (3 π t_{i}) + \sum_{i = 1}^{n - 1} {(t_{i} - 1)}^{2} [1 + {sin}^{2} (3 π t_{i + 1})])$ [-50, 50] 0

$+ 0.1 {(t_{n} - 1)}^{2} [1 + {sin}^{2} (2 π t_{n})] + \sum_{i = 1}^{n} u (t_{i}, 5, 100, 4)$

where: $u (t_{i}, 10, 100, 4) = {\begin{matrix} 100 {(t_{i} - 10)}^{4} & if t_{i} > 10 \\ 0 & if - 10 \leq t_{i} \leq 10 \\ 100 {(- t_{i} - 10)}^{4} & if t_{i} < - 10 \end{matrix}$

Levy and Mantalvo-2 $f (t) = 0.1 (({sin}^{2} (3 π t_{i}) + \sum_{i = 0}^{n - 1} {(t_{i} - 1)}^{2} [1 + {sin}^{2} (3 π t_{i})] + {(t_{n} - 1)}^{2} [1 + {sin}^{2} (2 π t_{n})])$ [-5, 5] 0

Matyas $f (t) = \sum_{i = 1}^{n} (0.26 (t_{i}^{2} + t_{i + 1}^{2}) - 0.48 t_{i} t_{i + 1})$ [-10, 10] 0

Neumaier-3 $f (t) = \sum_{i}^{n} {(t_{i} - 1)}^{2} + \sum_{i = 1}^{n - 1} t_{i} t_{i - 1}$ [- n², n²] $\frac{n (n + 4) (n - 1)}{6}$

New function $f (t) = \sum_{i = 0}^{n} [0.2 t_{i}^{2} + 0.1 x_{i}^{2} sin (2 t_{i})]$ [-10, 10] 0

Rastrigin $f (t) = 10 n + \sum_{i = 1}^{n} (t_{i}^{2} - 10 \cos (2 π t_{i}))$ [-5.12, 5.12] 0

Rosenbrock $f (t) = \sum_{i = 1}^{n} (100 {(t_{i}^{2} - t_{i + 1})}^{2} + {(1 - t_{i})}^{2})$ [-30, 30] 0

Sum of Power $f (t) = \sum_{i = 1}^{n} {| t_{i} |}^{(n + 1)}$ [-1, 1] 0

Schwefel-1 $f (t) = \sum_{i = 1}^{n} t_{i} sin (\sqrt{| t_{i} |})$ [-500, 500] 0

Schwefel-2 $f (t) = \sum_{i = 1}^{n} | t_{i} | + \prod_{i = 1}^{n} t_{i}$ [-10, 10] 0

Styblinski $f (t) = \frac{1}{2} \sum_{i = 0}^{n} (t_{i}^{4} - 16 t_{i}^{2} + 5 t_{i})$ [-5, 5] -39.16599n

Sphere $f (t) = \sum_{i = 1}^{n} t_{i}^{2}$ [-5.12, 5.12] 0

Benchmarks	Fitness function	Search limits	Optimum value
Ackley’s	$f (t) = - 20 exp (- 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} t_{i}^{2}}) - exp (\frac{1}{n} \sum_{i = 1}^{n} cos (2 π t_{i})) + 20 + e$	[-30, 30]	0
Axis parallel ellipsoid	$f (t) = \sum_{i = 1}^{n} i t_{i}^{2}$	[-5.12, 5.12]	0
Cigar	$f (t) = t_{i}^{2} + 1000000 \sum_{i = 1}^{n} t_{i}^{2}$	[-10, 10]	0
Cosine mixture	$f (t) = \sum_{i = 1}^{n} t_{i}^{2} - 0.1 \sum_{i = 1}^{n} cos (5 π t_{i})$	[-1, 1]	0.1n
De-Jong	$f (t) = \sum_{i = 1}^{n} (t_{i}^{4} + rand (0, 1))$	[-10, 10]	0
Drop-wave	$f (t) = \sum_{i = 1}^{n} {(t_{i} - i)}^{2}$	[-5.12, 5.12]	-1
Ellipsoidal	$f (t) = \sum_{i = 1}^{n} \frac{1 + cos (12 \sqrt{t_{i}^{2} + t_{i + 1}^{2}})}{0.5 (t_{i}^{2} + t_{i + 1}^{2}) + 2}$	[- n, n]	0
Brown-3	$f (t) = \sum_{i = 1}^{n - 1} (t_{i}^{2})^{(t_{i + 1}^{2} + 1)} + (t_{i + 1}^{2})^{(t_{i}^{2} + 1)}$	[-1, 4]	0
Generalized penalized-1	$f (t) = \frac{π}{n} \times {10 {sin}^{2} (π z_{i}) + \sum_{i = 1}^{n - 1} {(z_{i} - 1)}^{2} [1 + 10 {sin}^{2} (π z_{i + 1})] + {(z_{n} - 1)}^{2}}$	[-50, 50]	0
	$+ \sum_{i = 1}^{n} u (t_{i}, 5, 100, 4)$
	where: $z_{i} = 1 + \frac{1}{4} (t_{i} + 1), u (t_{i}, 10, 100, 4) = {\begin{matrix} 100 {(t_{i} - 10)}^{4} & if t_{i} > 10 \\ 0 & if - 10 \leq t_{i} \leq 10 \\ 100 {(- t_{i} - 10)}^{4} & if t_{i} < - 10 \end{matrix}$
Generalized penalized-2	$f (t) = 0.1 ({sin}^{2} (3 π t_{i}) + \sum_{i = 1}^{n - 1} {(t_{i} - 1)}^{2} [1 + {sin}^{2} (3 π t_{i + 1})])$	[-50, 50]	0
	$+ 0.1 {(t_{n} - 1)}^{2} [1 + {sin}^{2} (2 π t_{n})] + \sum_{i = 1}^{n} u (t_{i}, 5, 100, 4)$
	where: $u (t_{i}, 10, 100, 4) = {\begin{matrix} 100 {(t_{i} - 10)}^{4} & if t_{i} > 10 \\ 0 & if - 10 \leq t_{i} \leq 10 \\ 100 {(- t_{i} - 10)}^{4} & if t_{i} < - 10 \end{matrix}$
Levy and Mantalvo-2	$f (t) = 0.1 (({sin}^{2} (3 π t_{i}) + \sum_{i = 0}^{n - 1} {(t_{i} - 1)}^{2} [1 + {sin}^{2} (3 π t_{i})] + {(t_{n} - 1)}^{2} [1 + {sin}^{2} (2 π t_{n})])$	[-5, 5]	0
Matyas	$f (t) = \sum_{i = 1}^{n} (0.26 (t_{i}^{2} + t_{i + 1}^{2}) - 0.48 t_{i} t_{i + 1})$	[-10, 10]	0
Neumaier-3	$f (t) = \sum_{i}^{n} {(t_{i} - 1)}^{2} + \sum_{i = 1}^{n - 1} t_{i} t_{i - 1}$	[- n², n²]	$\frac{n (n + 4) (n - 1)}{6}$
New function	$f (t) = \sum_{i = 0}^{n} [0.2 t_{i}^{2} + 0.1 x_{i}^{2} sin (2 t_{i})]$	[-10, 10]	0
Rastrigin	$f (t) = 10 n + \sum_{i = 1}^{n} (t_{i}^{2} - 10 \cos (2 π t_{i}))$	[-5.12, 5.12]	0
Rosenbrock	$f (t) = \sum_{i = 1}^{n} (100 {(t_{i}^{2} - t_{i + 1})}^{2} + {(1 - t_{i})}^{2})$	[-30, 30]	0
Sum of Power	$f (t) = \sum_{i = 1}^{n} {\| t_{i} \|}^{(n + 1)}$	[-1, 1]	0
Schwefel-1	$f (t) = \sum_{i = 1}^{n} t_{i} sin (\sqrt{\| t_{i} \|})$	[-500, 500]	0
Schwefel-2	$f (t) = \sum_{i = 1}^{n} \| t_{i} \| + \prod_{i = 1}^{n} t_{i}$	[-10, 10]	0
Styblinski	$f (t) = \frac{1}{2} \sum_{i = 0}^{n} (t_{i}^{4} - 16 t_{i}^{2} + 5 t_{i})$	[-5, 5]	-39.16599n
Sphere	$f (t) = \sum_{i = 1}^{n} t_{i}^{2}$	[-5.12, 5.12]	0

5 Algorithmic experimentation framework

A newly proposed parent-centric crossover operator (i.e. MDX) is being used in the current research study to improve the performance of the genetic process and closely compare with existing real coded Crossover operators which comprise DPX, LX, and SBX. To determine their global efficiency, these four crossover operators co-integrated with MPTM, PM, and NUM mutation operators. A simulated study of twelve algorithmic combinations is summarized along with their respective crossover and mutation probabilities. Afterward, the final parametric settings are shown in Table 2. Hence, the appropriate adjustment of these parametric values is more beneficial in achieving optimum results.

Table 2
Parametric framework for all algorithms

Crossover Operators

DPX LX MDX SBX

Mutation Operators

PM MTPM NUM PM MTPM NUM PM MTPM NUM PM MTPM NUM

Crossover Probabilities

0.65 0.7 0.7 0.65 0.7 0.7 0.65 0.7 0.7 0.65 0.7 0.7

Mutation Probabilities

0.005 0.02 0.01 0.005 0.02 0.01 0.005 0.02 0.01 0.005 0.02 0.01

Crossover Operators
DPX	LX	MDX	SBX
Mutation Operators
PM	MTPM	NUM	PM	MTPM	NUM	PM	MTPM	NUM	PM	MTPM	NUM
Crossover Probabilities
0.65	0.7	0.7	0.65	0.7	0.7	0.65	0.7	0.7	0.65	0.7	0.7
Mutation Probabilities
0.005	0.02	0.01	0.005	0.02	0.01	0.005	0.02	0.01	0.005	0.02	0.01

The population size for all these algorithms is ten times the number of decision variables and the simulated results for each algorithm is obtained through thirty independent runs. The tournament selection operator is applied in the whole GA algorithmic process through Elitism with size one. When the number of generations exceeded up to 500 generations, all the simulation process is terminated and optimum results are obtained by the trial run and test experiments involving GA methodology. To obtain the simulation process’s performance, usability, and effectiveness: all algorithmic simulations were executed thirty times and statistical measures including mean, standard deviation, and average execution time in seconds are taken as final results. The performance of the newly introduced real-coded parent-centric crossover scheme associated with the Mixture of Laplace and double Pareto probability distributions is evaluated on twenty-one benchmark functions by using MATLAB version R2015a and R-language version 3.5.2.

Since in regard to the GA process associated with mixture distributions probabilistic approach, we applied the Quade test for comparing four real-coded crossover operators including the proposed one with co-integration of three mutation operators. The Quade test is a nonparametric analog of two-way analysis of variance, which may be derived from observations calculated on a higher scale by using the ranking methodology. According to Conover [42], the Quade test is more efficient than the Friedman test when the group size is less than five. The scores of the Quade test are calculated by using the following Equations 30. Hence, the mathematical formulation of test statistics is denoted by Q_t in Equation 31. $S_{ij} = q_{i} (R_{ij} - \frac{(m + 1)}{2})$ (29) and $S_{j} = \sum_{i = 1}^{m} S_{ij}$ (30) $Q_{t} = \frac{(m - 1) Σ_{j = 1}^{c} s_{j}^{2}}{m \sum_{i = 1}^{m} \sum_{j = 1}^{c} s_{ij}^{2} - \sum_{j = 1}^{c} s_{f}^{2}},$ (31)

where R_ij is calculated by ranking within each mutation operator and q_i is the ranked range. Hence, c and m are the numbers of crossover and mutation operators used for comparison. The empirical results of the study tend 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 (p - value) is less than a pre-specified threshold level (5% level of significance). Based on a significant Quade test, we have to further identify that the means of crossover operators differ substantially from each other. In this context, we apply Quade multiple comparison test to determine the significant difference between each pair of crossover means. The mathematical formulation of the Quade multiple comparison test is in Equation 32 which is based on studentized maximum modulus. $| S_{i} - S_{j} | > t_{(1 - \frac{α}{2}) (m - 1, c - 1)} \times A$ (32) where, $A = \sqrt{\frac{2 m (m \sum_{i = 1}^{m} Σ_{j = 1}^{c} s_{ij}^{2} - Σ_{j = 1}^{c} s_{j}^{2})}{(m - 1) (c - 1)}}$ .

After the statistical analysis, we make a comparison of proposed versus considered crossover operators through the performance index (PI) that was used by Haq et al. [5]. The PI was specifically applied to determine the behavior of various controlled stochastic-based search techniques and administered a comparative evaluation of different heuristic algorithms. The theoretical ideology of PI is based on the mean and the least value of different statistical measures. The mathematical formulation of PI is defined in subsequent Equation 33. $PI = \frac{1}{N} \sum_{i = 1}^{N} (l_{1} θ_{1}^{i} + n_{2} θ_{2}^{i} + n_{3} θ_{3}^{l})$ (33) where $\begin{matrix} θ_{1}^{i} & = \frac{{\bar{x}}_{f}^{i}}{{\bar{L X_{f}}}^{i}} \\ θ_{2}^{i} & = \frac{S D_{f}^{i}}{LS D_{f}^{i}} \\ θ_{3}^{i} & = \frac{c v_{f}^{i}}{Lc v_{f}^{i}} for i = 1, 2, \dots, N . \end{matrix}$ Mean, standard deviation, and coefficient of variation with their least values of the objective function are denoted as: ${\bar{X}}_{f}^{i}, {SD}_{f}^{i}, {CV}_{f}^{i}, {\bar{LX}}_{f}^{i}, LS D_{f}^{i}$ , and $LC V_{f}^{i} .$ Where the total population size is defined as N. Hence n₁, n₂ and n₃ are weights assigned to three statistical measures where n₁ + n₂ + n₃ = 1 The mathematical formulations of weights are given below concerning three possible cases:

Case-1:

n₁ = Wt, $n_{2} = n_{3} = \frac{1 - Wt}{2}$

where, 0 ≤ Wt ≤ 1

Case-2:

n₂ = Wt, $n_{l} = n_{3} = \frac{1 - Wt}{2}$

where, 0 ≤ Wt ≤ 1

Case-3:

n₃ = Wt, $n_{l} = n_{2} = \frac{1 - Wt}{2}$

where, 0 ≤ Wt ≤ 1

6 Computational analysis and discussions

Our main contribution is to propose a new real coded crossover operator that is a mixture of double Pareto and Laplace probability distributions and the focus of the study is to assess the performance of the proposed crossover strategies about the simulation results. Thus, to execute a fair performance comparison, MDX outperformed in approximately 52% of benchmark functions, while LX, SBX and DPX showed a limited dominance in 24% , 19 %, and 5% respectively by using the MTPM mutation operator in Table 3. The statistical measures authenticate the least performance of the crossover operator associated with double Pareto and Laplace distributions. The mixture distribution crossover strategy achieved optimum results in generalized penalized Levy and Mantalvo-2, Neumaier-3, Ackley’s Axis, parallel ellipsoid, Ellipsoidal, Rosenbrock, and Styblinski benchmark functions with diverse characteristics of continuous, multimodal, and inseparable. In terms of execution time, the newly proposed crossover technique efficiently handles the problem of selection pressure with sustainable convergence speed. Hence, the results of Table 3 reveal that LX obtained optimum results in Brown-3, Cigar, Sum of Power, Sphere, and New function benchmark functions, while SBX achieved optimized mean values in Rastrigin, Matyas, and Schwefel benchmark functions. On the whole, there is a shred of sufficient evidence that the theoretical optimum values related to benchmark functions are considerably closer to the empirical optimum values obtained by MDX.

Table 3
Computational details regarding Statistical measures of considered real coded crossover operators with Makinen, Periaux, and Toivanen mutation (MTPM) operator

DPX-MTPM MDX-MTPM LX-MTPM SBX-MTPM

Benchmark functions Statistical Measures Average execution time (Sec.) Statistical Measures Average execution time (Sec.) Statistical Measures Average execution time (Sec.) Statistical Measures Average execution time (Sec.)

Generalized-1 Mean 0.231 567 0.002 261 0.003 268 0.004 285

S.D 0.115 0.001 0.005 0.004

Generalized-2 Mean 0.616 409 0.016 225 0.024 241 0.032 231

S.D 0.232 0.028 0.03 0.043

Ackley Mean 8.079 376 1.743 186 1.764 198 2.095 200

S.D 0.947 1.179 1.245 1.018

Axis Mean 45.986 367 0.633 212 0.662 199 1.049 201

S.D 16.578 0.284 0.845 1.509

Brown-3 Mean 0.144 430 0.009 305 0.005 277 0.02 282

S.D 0.064 0.005 0.007 0.018

Cigar Mean 12000000 350 201710 211 102000 182 214000 172

S.D 4570000 134490 129000 232000

Ellipsoidal Mean 963 431 138.236 247 621 178 617 176

S.D 131 33.9036 120 87.7

LevyMont-2 Mean 0.668 371 0.0156 253 0.0243 195 0.0339 196

S.D 0.264 0.0127 0.0307 0.0436

Neumaier-3 Mean 150000 378 1821.4 246 2350 199 3090 204

S.D 58700 1932.5 2760 4280

Powersums Mean 2.5 × 10^-20 377 9.8 × 10^-43 235 2.9 × 10^-4 242 6.4 × 10^-43 203

S.D 1.2 × 10^-19 5.3 × 10^-42 1.6 × 10^-47 2.9 × 10^-42

Rastrigin Mean 200 497 49.4144 271 85.4 223 22.5 183

S.D 16.4 23.6542 60.2 5.47

Rosenbrok Mean 148000 352 29.7324 256 42.2 185 184 174

S.D 94000 237.8369 40.7 322

Comix Mean -3.8 360 -2.9946 279 -2.98 192 -2.97 188

S.D 0.242 0.0172 0.0229 0.042

Dejong Mean 37.5 380 14.4971 250 16.3 202 14.5 208

S.D 17.3 8.1997 8.29 8.84

Dropwave Mean -0.338 393 -0.7545 278 -0.864 204 -0.887 201

S.D 0.0825 0.0784 0.106 0.0925

Matyas Mean -1.2 × 10⁴⁸ 414 -6.4 × 10⁴⁸ 403 -7.1 × 10⁵³ 299 -8.8 × 10⁵³ 280

S.D 6.4 × 10⁴⁸ 3.4 × 10⁴⁹ 2.1 × 10⁵⁴ 2.0 × 10⁵⁴

Schwefel-1 Mean -20400 384 -95628 379 -69300 291 -71700 302

S.D 2420 14171 10300 9780

Schwefel-2 Mean -1, 2 × 10⁴⁰ 413 -9.9 × 10⁴¹ 380 -2.3 × 10⁴⁶ 299 -8.5 × 10⁴⁸ 264

S.D 4.3 × 10⁴⁰ 4.7 × 10⁴² 1.2 × 10⁴⁷ 1.8 × 10⁴⁹

Sphere Mean 3.14 416 0.0581 226 0.0299 208 0.0989 195

S.D 1.03 0.0369 0.0316 0.0833

New function Mean 2.56 376 0.0381 254 0.0242 216 0.0584 217

S.D 0.768 0.0198 0.044 0.0827

Styblin-30 Mean -1170 464 -1174.6 314 -1170 284 -1170 234

S.D 2.77 0.257 0.337 0.634

	DPX-MTPM	MDX-MTPM	LX-MTPM	SBX-MTPM
Benchmark functions	Statistical Measures	Average execution time (Sec.)	Statistical Measures	Average execution time (Sec.)	Statistical Measures	Average execution time (Sec.)	Statistical Measures	Average execution time (Sec.)
Generalized-1	Mean	0.231	567	0.002	261	0.003	268	0.004	285
	S.D	0.115		0.001		0.005		0.004
Generalized-2	Mean	0.616	409	0.016	225	0.024	241	0.032	231
	S.D	0.232		0.028		0.03		0.043
Ackley	Mean	8.079	376	1.743	186	1.764	198	2.095	200
	S.D	0.947		1.179		1.245		1.018
Axis	Mean	45.986	367	0.633	212	0.662	199	1.049	201
	S.D	16.578		0.284		0.845		1.509
Brown-3	Mean	0.144	430	0.009	305	0.005	277	0.02	282
	S.D	0.064		0.005		0.007		0.018
Cigar	Mean	12000000	350	201710	211	102000	182	214000	172
	S.D	4570000		134490		129000		232000
Ellipsoidal	Mean	963	431	138.236	247	621	178	617	176
	S.D	131		33.9036		120		87.7
LevyMont-2	Mean	0.668	371	0.0156	253	0.0243	195	0.0339	196
	S.D	0.264		0.0127		0.0307		0.0436
Neumaier-3	Mean	150000	378	1821.4	246	2350	199	3090	204
	S.D	58700		1932.5		2760		4280
Powersums	Mean	2.5 × 10^-20	377	9.8 × 10^-43	235	2.9 × 10^-4	242	6.4 × 10^-43	203
	S.D	1.2 × 10^-19		5.3 × 10^-42		1.6 × 10^-47		2.9 × 10^-42
Rastrigin	Mean	200	497	49.4144	271	85.4	223	22.5	183
	S.D	16.4		23.6542		60.2		5.47
Rosenbrok	Mean	148000	352	29.7324	256	42.2	185	184	174
	S.D	94000		237.8369		40.7		322
Comix	Mean	-3.8	360	-2.9946	279	-2.98	192	-2.97	188
	S.D	0.242		0.0172		0.0229		0.042
Dejong	Mean	37.5	380	14.4971	250	16.3	202	14.5	208
	S.D	17.3		8.1997		8.29		8.84
Dropwave	Mean	-0.338	393	-0.7545	278	-0.864	204	-0.887	201
	S.D	0.0825		0.0784		0.106		0.0925
Matyas	Mean	-1.2 × 10⁴⁸	414	-6.4 × 10⁴⁸	403	-7.1 × 10⁵³	299	-8.8 × 10⁵³	280
	S.D	6.4 × 10⁴⁸		3.4 × 10⁴⁹		2.1 × 10⁵⁴		2.0 × 10⁵⁴
Schwefel-1	Mean	-20400	384	-95628	379	-69300	291	-71700	302
	S.D	2420		14171		10300		9780
Schwefel-2	Mean	-1, 2 × 10⁴⁰	413	-9.9 × 10⁴¹	380	-2.3 × 10⁴⁶	299	-8.5 × 10⁴⁸	264
	S.D	4.3 × 10⁴⁰		4.7 × 10⁴²		1.2 × 10⁴⁷		1.8 × 10⁴⁹
Sphere	Mean	3.14	416	0.0581	226	0.0299	208	0.0989	195
	S.D	1.03		0.0369		0.0316		0.0833
New function	Mean	2.56	376	0.0381	254	0.0242	216	0.0584	217
	S.D	0.768		0.0198		0.044		0.0827
Styblin-30	Mean	-1170	464	-1174.6	314	-1170	284	-1170	234
	S.D	2.77		0.257		0.337		0.634

After evaluating the performance impact of the crossover operator on MTPM mutation operator, now we examine the empirical results of different benchmark functions under power mutation (PM) in Table 4. Therefore, according to the results in Table 4, shows a substantial dominance of newly introduced real coded crossover (MDX) over all other considered crossover operators including LX, DPX, and SBX. MDX achieved optimality in 52% of benchmark functions while LX and SBX attained optimum value in the remaining 48% benchmark function. Whereas DPX has no contribution in achieving optimality under PM. The statistical measures along with average execution time reveal that the proposed strategy has better control over selection pressure, and it is more helpful to preserve population diversity.

Table 4

Computational details regarding Statistical measures of considered real coded crossover operators with Power Mutation (PM) operator

		DPX-MTPM		MDX-MTPM		LX-MTPM		SBX-MTPM
Benchmark functions		Statistical Measures	Average execution time (Sec.)	Statistical Measures	Average execution time (Sec.)	Statistical Measures	Average execution time (Sec.)	Statistical Measures	Average execution time (Sec.)
Generalized-1	Mean	0.029	519	0	232	0.001	259	0.001	246
	S.D	0.011		0		0.002		0.001
Generalized-2	Mean	0.636	385	0.043	302	0.016	253	0.036	268
	S.D	0.207		0.023		0.022		0.037
Ackley	Mean	7.467	479	1.563	249	1.836	243	2.312	227
	S.D	1.187		0.464		0.934		1.196
Axis	Mean	45.807	425	2.287	257	0.896	211	1.153	237
	S.D	16.464		1.325		1.323		1.441
Brown-3	Mean	0.188	441	0.006	355	0.008	306	0.011	262
	S.D	0.076		0.027		0.008		0.004
Cigar	Mean	11200000	516	499680	248	176000	219	263000	211
	S.D	234000		219890		306000		109
Ellipsoidal	Mean	878	432	49.96	271	612	347	601	292
	S.D	0.25		20.05		0.02		0.04
LevyMont-2	Mean	0.59	466	0.02	273	0.02	261	0.03	299
	S.D	46000		0.03		2650		3400
Neumaier-3	Mean	123000	450	14294	272	3180	277	4070	261
	S.D	4532		9874		142		256
Powersums	Mean	4.56 × 10^-19	477	6.73 × 10^-23	283	3.57 × 10^-42	291	8.68 × 10^-47	277
	S.D	9.65 × 10^-20		6.22 × 10^-25		6.52 × 10^-43		1.59 × 10^-47
Rastrigin	Mean	193	412	33.04	315	111	329	24	310
	S.D	87200		14.12		40.9		508
Rosenbrok	Mean	129000	365	4337.7	227	47.7	220	199	285
	S.D	0.3		2576.9		0.41		0.1
Comix	Mean	-1.83	389	-2.9272	246	-0.253	211	-2.61	219
	S.D	16.2		0.068		355		8.73
Dejong	Mean	47	370	16.0505	273	727	228	23.7	238
	S.D	0.071		10.8424		0.169		0.116
Dropwave	Mean	-0.316	390	-0.9324	261	-0.799	212	-0.821	209
	S.D	0.218		0.0861		0.187		0.0454
Matyas	Mean	-7.58 × 10⁴⁸	466	-4.78 × 10²⁹	382	-3.24 × 10⁵⁴	317	-2.76 × 10⁵⁴	308
	S.D	6.56 × 10⁴⁷		3.55 × 10²⁷		1.78 × 10²⁷		1.82 × 10⁵⁴
Schwefel-1	Mean	-19800	447	-92501	306	-67200	360	-78800	425
	S.D	23.4577		47.0278		35.6812		42.6824
Schwefel-2	Mean	-2.69 × 10⁴²	466	-9.95 × 10²⁹	428	-8.29 × 10⁴⁶	326	-2.01 × 10⁵⁰	404
	S.D	1.48 × 10⁴³		6.09 × 10²⁷		3.78 × 10⁴⁷		9.23 × 10⁵⁰
Sphere	Mean	4.91	346	0.1816	234	19.3	213	1.46	210
	S.D	1.21		0.0843		3.69		0.516
New function	Mean	2.35	479	0.1901	256	0.0432	238	0.0431	242
	S.D	0.803		0.1316		0.0776		0.0445
Styblin-30	Mean	-1160	510	-1173.2	251	-1170	317	-1170	332
	S.D	4.65		1.0359		0.44		0.389

With the inclusion of mixture distribution, ideology is helpful to solve multidimensional complex optimization problems. The results of Table 5 describe that the MDX shows a sufficient improvement for obtaining an optimal solution by using non-uniform mutation (NUM). Moreover, MDX attains optimum results in sixteen out of twenty-one benchmark functions, which is around 76% of the total considered test problems. The MDX outperformed most of the benchmark functions except for Rastrigin, Dropwave, Matyas, Schwefel, and Sphere. The considerable closeness of simulated optimum results with the theoretical value represent sufficient control over the reduction in population diversity and the sustainable execution time indicates adequate selection pressure.

Table 5

Computational details regarding Statistical measures of considered real coded crossover operators with Non-uniform (NUM) operator\label tab5

		DPX-MTPM		MDX-MTPM		LX-MTPM		SBX-MTPM
Benchmark functions		Statistical Measures	Average execution time (Sec.)	Statistical Measures	Average execution time (Sec.)	Statistical Measures	Average execution time (Sec.)	Statistical Measures	Average execution time (Sec.)
Generalized-1	Mean	0.523	479	0.003	288	1.45	280	0.049	256
	S.D	0.252		0.002		0.545		0.023
Generalized-2	Mean	0.629	391	0.05	279	1.484	243	0.181	246
	S.D	0.151		0.034		0.442		0.047
Ackley	Mean	9.125	414	3.56	247	12.37	266	5.412	248
	S.D	0.887		0.533		1.462		0.559
Axis	Mean	64.426	537	1.81	231	65.839	355	18.276	210
	S.D	20.08		0.778		21.632		6.269
Brown-3	Mean	1.492	457	0.057	372	4.369	448	0.411	277
	S.D	0.378		0.021		1.777		0.125
Cigar	Mean	17600000	343	275580	294	16100000	217	4510000	209
	S.D	4030000		275580		4570000		1500000
Ellipsoidal	Mean	950	368	46.0313	271	455	227	582	209
	S.D	127		13.2687		117		95.1
LevyMont-2	Mean	0.644	346	0.0475	261	1.49	211	0.175	211
	S.D	0.175		0.0398		0.544		0.0647
Neumaier-3	Mean	134000	340	14496	251	250000	215	49800	212
	S.D	40500		5790.5		54400		16300
Powersums	Mean	3.01×10^-17	374	1.19×10^-26	280	2.56×10^-14	238	2.33×10^-24	236
	S.D	1.65×10^-16		5.46×10^-26		4.24×10^-14		1.12×10^-23
Rastrigin	Mean	203	371	30.6708	256	128	221	23.2	219
	S.D	17.5		14.3423		2.07		5.21
Rosenbrok	Mean	263000	353	6033.1	213	385000	243	43100	218
	S.D	180000		5020.7		201000		23600
Comix	Mean	-1.44	359	-2.9601	292	-0.735	216	-2.63	215
	S.D	0.26		0.0281		0.341		0.103
Dejong	Mean	48	377	13.1831	341	64.3	238	19.3	245
	S.D	19.1		8.2814		24.1		7.77
Dropwave	Mean	-0.9261	368	-0.5628	321	-0.113	202	-0.467	209
	S.D	0.0447		0.0673		0.0248		0.0693
Matyas	Mean	-2.36 × 10²⁹	520	-4.61 × 10²⁹	315	-4.80 × 10²⁹	248	-4.65 × 10²⁹	223
	S.D	1.33 × 10²⁹		1.64 × 10²⁹		2.64 × 10²⁹		2.13 × 10²⁹
Schwefel-1	Mean	-10900	394	-12500	258	-12100	253	-12400	243
	S.D	880		28.2484		583		32.4
Schwefel-2	Mean	-1.74 × 10²⁸	407	-1.12 × 10²⁹	432	-2.03 × 10²⁸	203	-4.73 × 10²⁹	238
	S.D	9.04 × 10²⁸		6.22 × 10²⁸		5.81 × 10²⁸		1.74 × 10²⁹
Sphere	Mean	0.126	347	0.184	298	4.1	213	1.27	206
	S.D	1.19		0.073		1.44		0.409
New function	Mean	3.8	405	0.1724	260	8.14	257	0.899	218
	S.D	3.8		0.0987		3.62		0.303
Styblin-30	Mean	-1120	443	-1173.5	281	-830	218	-1170	213
	S.D	17.7		0.7644		41.2		2.92

For statistical comparison, the test statistics of the Quade test (Q_t = 3.2192) along with its probability value (p - value = 0.1038) concludes the insignificant difference between four consider crossover (DPX, MDX, LX, and SBX) operators. However, the results of pairwise comparisons using the Quade post-hoc test represented in probabilities matrix (P_M) reveal that MDX shows significantly different from DPX, LX, and SBX with respective P values are 0.034, 0.039, and 0.102 while DPX, LX, and SBX are insignificantly different from each other.

$P_{M} = [\begin{matrix} DPX & MDX & LX \\ MDX & 0.034 & 1.000 & 0.039 \\ LX & 0.923 & 0.039 & 1.000 \\ SBX & 0.448 & 0.102 & 0.505 \end{matrix}]$

Usually, the performance of the algorithmic procedure is examined by the optimum value and the execution time. Hence, for the further graphical appraisal (Fig. 3), four parent-centric crossover operators (DPX, MDX, LX, and SBX) are visually compared with three mutation operators (MTPM, PM, and NUM) in the context of obtained optimum values in all benchmark functions. The proposed crossover operator MDX shows considerable dominance with 76% in NUM and 52% in each PM and MTPM mutation operator over other crossover operators, while DPX has limited performance for obtaining the optimum solution in all twenty-one benchmarks with 5% to 10%

Fig. 3

The visual depiction of crossover with mutation operators about benchmark functions.

overall, the variability of results reveal that the proposed scheme had better control over selection pressure and loss of population diversity. Hence, the numerical outcomes of proposed technique is very close to theoretical optimum value which is an evidence of the best-performing mixture distribution based real coded crossover technique with authentication of performance index (PI). Moreover, hybrid probabilistic technique might be a good candidate to get accurate convergent results and show efficiently increased the search-ability of the algorithm by producing multiple possible solutions. The graphical depiction for cases (1-3) in Fig. (4-6) represents that the horizontal axis defines weights (wt) and scaled values of PI specify the vertical axis. According to the Fig. 4 and 5 reveal that the values of PI are lies in between 0.7 to 0.8 for MDX, which is relatively higher than LX, DPX and SBX that show considerable dominance. Moreover, there is also a significant difference in the numerical results of PI at initial stage but latterly, the results are merged with other considered crossovers operators in Fig. 6. Hence, the overall visual performance of PI describes the substantial dominance of MDX over other real-coded crossover operators. The graphical representation of PI endorses the efficiency enhancement in the MDX crossover operator.

Fig. 4

Graphical representation of PI for case-1.

Fig. 5

Graphical representation of PI for case-2.

Fig. 6

Graphical representation of PI for case-3.

7 Conclusions

The foremost ideology of our simulation-based experimental study is to evaluate the performance of mixture distribution crossover (MDX) for solving complex unconstraint optimization problems. Therefore, four different versions of crossover operators (MDX, DPX, LX, and SBX) with co-integration of three mutation operators (MTPM, PM, and NUM) are examined on twenty-one well-known benchmark functions with varied characteristics. Furthermore, some statistical measures along with average execution time are to be analyzed in the context of efficiency, complexity, robustness, and accuracy for algorithmic comparison.

For its global comparison, three different natures of evaluation strategies is executed. Initially, a descriptive comparison is conducted between the proposed MDX and other considered crossover operators in terms of mean, standard deviation, and average execution time. Whereas a mixture distribution strategy outperformed in 52%-76% benchmark functions, while the performance of other crossover operators (DPX, LX, and SBX) is a very limited due to the less percentage (5%-25%) of obtaining optimum solutions. Therefore, MDX has better control over the selection pressure and loss of population diversity. Hence, MDX found a suitable adjustment between exploitation and exploration because of the mixture distribution probabilistic ideology.

In the second phase of evaluation, a Quade multiple comparison test shows a fairly significant difference between MDX versus DPX, LX, and SBX for real coded crossover operators. The performance is visually evaluated by making a comparison of all algorithms through performance index (PI) with the use of statistical measures in the third phase of evaluation. The visual depiction of PI shows a considerable dominance rather than other crossover operators. Ultimately, the statistically significant results of the mixture distribution probabilistic crossover strategy have a definite edge over the other operators and have a great potential for addressing more complex problems of optimization.

Finally, another potential avenue for future research is to examine the performance of parent centric real-coded crossover with the other crossover methods like modified extended line crossover, fuzzy connectives based crossover (FCB), and direction-based exponential crossover (DBXO). Moreover, mixture distribution-based real coded crossover can also be further extended by introducing multiple changes in its parameters and components for achieving global optimization.

Data availability

The data used to support the findings of this manuscript are taken from the website (https://www.sfu.ca/ssurjano/optimization.html).

Footnotes

Acknowledgment

The authors are very grateful to the deanship of scientific research at King Khalid University, Abha, Kingdom of Saudi Arabia, for their funding of this work through Research Groups Program under the project number RGP-2/82/42.

Disclosure statement

All authors of this article declare that there is no conflict of interest regarding the publication of this article.

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