Comparative analysis of real-coded genetic algorithms for mixture distribution models: Insights from TOPSIS

TL;DR

This study introduces two new crossover operators for genetic algorithms that improve their performance in solving complex optimization problems.

Contribution

The paper proposes two novel parent-centric real-coded crossover operators, MGGX and MRRX, for genetic algorithms.

Findings

MGGX outperformed other operators in 20 out of 36 mean values cases.

MRRX achieved the lowest standard deviation in 21 out of 36 cases.

Quade test, PI, and TOPSIS confirmed the operators' efficiency and robustness.

Abstract

This particular study presents two novel parent-centric real-coded crossover operators, named mixture-based Gumbel crossover (MGGX) and mixture-based Rayleigh crossover (MRRX), to increase the efficiency of genetic algorithms (GAs) in tackling complex optimization problems. Conventional crossover operators often struggle in multimodal and extremely restricted situations and fail to find the ideal balance between exploration and exploitation. Proposed parent-centric real-coded crossover operators increase the precision and robustness of GAs, which is confirmed by empirical results on testing constrained and un-constrained benchmark functions having different complexity levels. MGGX parent-centric real-coded crossover operator performs best in 20 out of 36 mean values cases and achieves the lowest standard deviation values in 21 out of 36 cases. Likewise, to confirm the efficiency, robustness, and reliability of the proposed crossover operator the Quade test, Performance index (PI), and multi-criteria TOPSIS method are utilized.

Figures

Visual parametric framework for all fifteen algorithms.

Visual line representation of algorithms with NUM mutation.

Visual line representation of algorithm with MPTM mutation.

Visual line representation of algorithm with PM mutation.

Visual performance of each algorithm with NUM mutation.

Visual performance of each algorithm with MPTM mutation.

Visual performance of each algorithm with PM mutation.

Visual representation of ranks of algorithms with NUM mutation.

Visual representation of ranks of algorithms with MPTM mutation.

Visual representation of ranks of algorithms with PM mutation.

Equations

Full text

1 Introduction

1.1 Motivation and purpose

Optimized solutions are highly desirable across various domains of today’s rapidly evolving world. The fields of science, particularly computer science, engineering, management science, and applied mathematics are expanding rapidly. We face challenging optimization problems due to the complexity and modality of the problems [1–3]. Problems in science and engineering research are crucial nowadays because technology is changing the world quickly. Many of these problems are complex global optimization problems with high dimensions search spaces [4]. As the problem’s dimension increases, the search space expands exponentially, finding these problems challenging to solve [5]. Classical and gradient-based optimization methods struggle to handle modern complex problems due to the curse of dimensionality. Additionally, gradient-based searches are often ineffective when local optima exist, as they rely heavily on the choice of initial point [6].

Population-based stochastic methods including genetic algorithms (GA), simulated annealing (SA), particle swarm optimization (PSO), and feature selection techniques have been extensively used to overcome these limitations [7,8]. Heuristic and meta-heuristic approaches are crucial in multi-objective hybrid flow shop scheduling problems (HFSP) with machine capability limitations and constrained waiting times [9]. The problem’s complex nature is tackled by developing solution approaches based on GA [10]. Heuristic methods aim to efficiently approximate solutions with minimal computational efforts. The approximate techniques known as “meta-heuristics” are derived from various ideas in classical heuristics, neural systems, biological evolution, artificial intelligence, natural phenomena, and statistical mechanics [11]. Meta-heuristics inspired by nature are devised to address optimization issues that stem from various physical or biological phenomena. The values used by these algorithms are generated at random. The population can be optimized for iterations by adding the best individual value from the current generation to the next generation of individuals. They can manage several local optima simultaneously and effectively handle higher dimensional global optimization problems. These qualities have made these algorithms well-liked and widely utilized in solving global optimization problems. The researchers used all of these in developing optimization strategies that can address optimization issues in the real world. The two stages of the entire search process are called exploration and exploitation. As much random search as possible is preferred during the exploration stage, where the algorithm must explore the search space considering variables. The exploitation stage is determined by evaluating the possible areas discovered during exploration. The exploitation phase is connected to the local search process. Striking a balance between exploration and exploitation remains one of the primary challenges in optimization [12].

1.2 Literature review

Normally existing methods fail to strike a balance between exploration and exploitation, especially in multimodal optimization and high-dimensional problems, despite the development of crossover and mutation operators to enhance the efficiency of real-coded GAs [13]. Still, there is a lack of resilient, real-coded crossover operators that can maintain solution stability, avoid premature convergence, and frequently obtain ideal solutions across diverse optimization scenarios. Conventional operators such as double Pareto crossover (DPX), simulated binary crossover (SBX), and Laplace crossover (LX) struggle with population diversity, premature convergence, and adaptation to varying optimization landscapes [14]. To address these challenges, this study introduces two novel parent-centric real-coded crossover operators mixture-based Gumbel crossover and mixture-based Rayleigh crossover (MGGX and MRRX), designed to enhance the balance between exploration and exploitation dynamically. The gaps are filled by this study’s inclusion of two new real-coded crossover operators.

1.3 Contribution and paper organization

We suggest the mixture distribution-based MGGX and MRRX crossover operators to improve balancing between exploration and exploitation.

(i) Theoretically, these operators are developed to adapt dynamically, which reduces the risk of premature convergence while ensuring efficient exploration of complex search spaces. By leveraging the properties of mixture distributions and Gumbel distribution, they enable diverse yet high-quality offspring generation, improving GA robustness across different levels of benchmark functions.(ii) By testing on constrained and unconstrained benchmark functions, we assessed proposed operators and examined that in most cases, the proposed operator especially MGGX outperforms conventional operators in terms of stability and efficiency, attaining the lowest mean and standard deviation. The inclusion of proposed operators within GA provides a practical approach to handling real-world optimization problems requiring strong consistent solutions. Both approaches PI and TOPSIS further validate MGGX’s robustness and scalability, making it an effective option for challenging, high-dimensional optimization tasks.

The remainder of this paper is organized as follows: Section 2 presents previously used operators. Section 3 introduces proposed mixture-based real-coded crossover operators. Section 4 details the benchmark functions. Section 5 discusses computational results, while Section 6 concludes the study with key finding and future research directions.

2 Previously used operators

For comparison, we considered the three real-coded crossover operators previously used in the literature. Laplace crossover operator (LX) was first proposed by Deep and Thakur [15].

It is a Laplace distribution-based crossover operator that is self-parent-centric. The double Pareto probability distribution, a parent-centric operator, is the foundation of DPX (Thakur, 2014). The procedure has three steps for generating two offspring, $[eqn]$ & $[eqn]$ from parents $[eqn]$ and $[eqn]$ . In the first step, it begins the procedure by generating a random number (r_i_) uniformly distributed between zero and one. After a random number is generated in the second step, we calculate $[eqn]$ by simply inverting the Laplace cumulative distribution function in LX and the double Pareto cumulative distribution function in DPX. In the end, offspring are generated with the help of $[eqn]$ :

$$[eqn]$$

and

$$[eqn]$$

Real-coded crossover operator SBX has a special feature with binary transformation to continuous search space, it was first proposed by Deb and Agarwal [16]. Important steps are adopted for generating following offspring, i.e., $[eqn]$ ) by parents $[eqn]$ & $[eqn]$ are these:

(a)  Starting with a uniform distribution, generate a random number r_i_ between zero and one.(b)  In this important step we obtain a parameter $[eqn]$ from mathematical expression.

Thus, from two parents $[eqn]$ $[eqn]$ and an offspring $[eqn]$ is produced in the Equation 3:

$$[eqn]$$

2.1 Mutation operators

Population diversity decreases the chance of premature convergence [17,18]. In literature, several mutation operators are used in GA to increase population diversity. In this study, we used three popular mutation operators. Non-uniform mutation (NUM) is one of the most popular mutation operators, and its working mechanism was introduced by Michalewicz et al. [19]. Makinen et al. proposed the Makinen, Periaux, and Toivanen Mutation (MPTM) [20]. MPTM is mostly used in literature to solve multidisciplinary optimization problems. We also used the third mutation operator named power mutation (PM) proposed by Deep and Thakur [21], which is based on power distribution. In GA the mutation operator helps in searching new search spaces and solving complex optimization problems.

3 Proposed mixture-based real-coded crossover operators

Several areas of science and engineering directly employ finite mixture models. Indirect applications of mixture models consist of the empirical Bayes method, cluster analysis, latent structure models, modeling of prior densities, and nonparametric (kernel) density estimation [22]. Here in this particular section, we present two real-coded crossover operators (named MGGX and MRRX). The idea is used to generate real-coded crossover operators based on the mixture of Gumbel distribution (MGGX) and then a mixture of Raleigh distribution (MRRX).

3.1 Proposed mixture Gumbel distribution-based real-coded crossover operator

A wide range of scientific disciplines, including geology, insurance, meteorology, hydrology, and finance, have used the Gumbel distribution and its extensions. It is commonly known that the tails of this distribution are heavier than those of the normal distribution. Since the Gumbel distribution has heavy tails, it can be used to model extreme happenings. In a dataset, it captures the likelihood of rare and extreme values. In extreme value theory, the Gumbel distribution is frequently utilized, particularly when modeling the distribution of maxima or minima in a sample [23]. The probability density function (pdf) of the Gumbel distribution is as follows:

$$[eqn]$$

where the cumulative distribution function (cdf) of Gumbel is as shown in equation 5 and μ denotes the location parameter and the scale parameter is denoted by η:

$$[eqn]$$

Pdf and cdf for the proposed two-component mixture probability model using Gumbel distribution can be written as follows:

$$[eqn]$$

where $[eqn]$ , $[eqn]$ and $[eqn]$

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

The cumulative function of the proposed two-component mixture distribution as elaborated in equation 9, has scale parameters as represented by $[eqn]$ and $[eqn]$ , and the location parameters as $[eqn]$ and $[eqn]$ . Using a planned procedure, we get two offspring $[eqn]$ and $[eqn]$ from two-parent $[eqn]$ and $[eqn]$ by employing mixture-based crossover. The three steps procedure is as follows:

i) By utilizing a uniform distribution we generate a random number denoted by $[eqn]$ .ii) The parametric value, shown as $[eqn]$ is then obtained by inverting $[eqn]$ and generating random numbers from a mixture probability distribution:

$$[eqn]$$

iii) In the end, two offspring are generated:

$$[eqn]$$ $$[eqn]$$

where $[eqn]$ and variable bounds are $[eqn]$ or $[eqn]$ , offspring are assigned randomly generated values in the interval $[eqn]$ if they fall outside of these limitations.

3.2 Proposed mixture Rayleigh distribution-based real-coded crossover operator

The Rayleigh distribution is a positively skewed continuous distribution, is also used in wireless communications and signal processing due to its ability to model the magnitude of two-dimensional vectors with independent identical normal components. This distribution was first introduced by Lord Rayleigh [24]. In GA, its adaptive step-size mechanism enhances exploration in the early stages and refines exploitation later, which reduces the chances of pre-mature convergence. This property makes it suitable for high-dimensional and multimodal optimization problems. In equation 13 pdf and in equation 14 cdf of Rayleigh distribution are expressed.

$$[eqn]$$ $$[eqn]$$

where sigma denotes the scale parameter.

The cdf of the two-component mixture probability model for Rayleigh distribution is as follows:

$$[eqn]$$

where $[eqn]$ , $[eqn]$ and $[eqn]$

$$[eqn]$$

A stepwise process is used to obtain two offspring that are $[eqn]$ and $[eqn]$ from two parents $[eqn]$ and $[eqn]$ by using two-component mixture distribution crossover. Where the three-step process is as follows:

a) The uniform distribution is used to generate a random number $[eqn]$ .b) To obtain $[eqn]$ in the second step, the cumulative density function of the Rayleigh distribution is inverted to generate a random number from the two-component mixture distribution.

$$[eqn]$$

c) The last step is the generation of two offspring for $[eqn]$ :

$$[eqn]$$ $$[eqn]$$

where variable boundaries are $[eqn]$ or $[eqn]$ , if offspring are outside these limits then randomly generated values $[eqn]$ are assigned to offspring $[eqn]$ in an interval $[eqn]$ .

4 Benchmark functions

Different types of optimization problems exist, varying in complexity and compatibility. Since there are no set criteria for choosing benchmark functions, there are many different kinds of benchmark functions with distinct characteristics in the literature. For this specific simulated-based study, we choose a set of twelve benchmark functions with various levels of compatibility and complexity. By using the proposed real-coded probabilistic-based algorithms on a set of twelve benchmark functions, we compare their effectiveness with current algorithms. These benchmark functions are extensively employed in numerous comparative analyses to verify the efficacy of algorithms the detail is given in Table 1. We take the first six unconstrained and the remaining six benchmark functions are constrained test problems from CEC 2017.

Table 1: Detail of benchmark functions comparison.

5 Computational results

5.1 The experimental framework

MGGX and MRRX are two proposed real-coded crossover operators compared to DPX, SBX, and LX three conventional crossover operators in this particular simulation-based study for global optimization tasks. So total of fifteen algorithmic combinations are produced by integrating these five crossover operators with three mutation operators which are NUM, MPTM, and PM. We test them on a set of twelve benchmark functions having both constrained and unconstrained scenarios. This set of benchmark functions was chosen for their diverse levels of modality and complexity. To retain the best individual for each generation, tournament selection with an elitism size of one was used for each of the fifteen algorithmic combinations. The termination criteria for all algorithmic combinations were fixed as five hundred generations, and the population size was also specified as three hundred. In Fig 1 we describe the specific mutation and crossover probabilities for each case.

Visual parametric framework for all fifteen algorithms.

5.2 Results and discussion

The goal is to evaluate the effectiveness of the suggested crossover operators with the simulation results over existing operators. We accomplish this by proposing a pair of new real-coded Probabilistic-based crossover operators one is the mixture-based real-coded crossover operator by using Gumbel distribution, and in the other operator, Rayleigh distribution is used. The following are the proposed mixture-based real-coded crossover operators MGGX-NUM, MGGX-MPTM, MGGX-PM, MRRX-NUM, MRRX-MPTM, and MRRX-PM. Table 2 shows the mean values of twelve benchmark functions having diverse characteristics for fifteen algorithms and similarly, Table 3 indicates the values of standard deviation of different benchmark functions. For multiple mutations (NUM, MPTM, and PM), Tables 2 and 3 demonstrate that the MGGX crossover outperforms all other operators. The computational complexity of the proposed approach is influenced by the mixture-based crossover operators. As shown in Table 4, proposed operators have a higher average execution time due to additional sampling and probabilistic combinations of distributions. The increase in run time is justified by better solution quality, robustness, and a better balance between exploration and exploitation in complex situations.

Table 2: Mean values of different benchmark functions.

Table 3: Stander deviation values of different benchmark functions.

Table 4: Average execution time (in seconds) of different benchmark functions.

5.3 Performance index

It is common practice to compare different heuristic algorithms utilizing the performance index. It examines the behavior of various stochastic search strategies. Performance index (PI) is used here to obtain a more comprehensive understanding of the respective performance of MGGX-NUM, MGGX-MPTM, MGGX-PM, MRRX-NUM, MRRX-MPTM, and MRRX-PM algorithms, this performance index was also used in literature [18]. The least mean and least standard deviation are given prescribed weighted importance by this index. Equation 20 provides a mathematical setup of this PI.

$$[eqn]$$

where

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

and Np denotes the total number of populations, $[eqn]$ denotes the least standard deviation and $[eqn]$ denotes the least value of the coefficient of variation. Where $[eqn]$ , $[eqn]$ , and $[eqn]$ denote weights and the sum of weights equal to one. Following is the mathematical setup of different three cases.

1^st^ case $[eqn]$ , $[eqn]$ 2^nd^ case $[eqn]$ , $[eqn]$ 3^rd^ case $[eqn]$ , $[eqn]$

And for all three cases, weights are between 0 and 1.

We also employed the Quade test, a non-parametric test that adopts ranking methodology [18], to compare the performance of two proposed and three conventional real-coded crossover operators. Here equation number 21 represents the mathematical formulation of the Quade test statistics.

$$[eqn]$$

where,

$[eqn]$ , $[eqn]$ , p = number of crossover operator, b = number of mutation operator, and $[eqn]$ denotes ranked range.

The Quade test statistics value is 5.65 having a p-value of 0.004 concluding a significant difference between crossover operators (MGGX, MRRX, DPX, SBX, and LX). The pairwise comparison is represented in Table 5 below. It demonstrates strong evidence that the proposed operator MGGX outperforms SBX, DPX, MRRX, and LX.

Table 5: Comparison of different crossover operators.

In Figs 2, 3, and 4 line graph shows that MGGX is consistently outperforming all other real-coded operators, showing that the MGGX crossover operator is outperforming the others in the form of a plotted metric line. This growing pattern suggests that the proposed crossover operator (MGGX) is approaching the global solution. As the algorithm develops its search, a more successful global solution exploration is made possible by the innovative crossover operator (MGGX). Visual performance of each algorithm with mutations NUM, MPTM, and PM are expressed in Figs 5, 6, and 7 respectively, MGGX performs outperforming in all three cases.

Visual line representation of algorithms with NUM mutation.

Visual line representation of algorithm with MPTM mutation.

Visual line representation of algorithm with PM mutation.

Visual performance of each algorithm with NUM mutation.

Visual performance of each algorithm with MPTM mutation.

Visual performance of each algorithm with PM mutation.

5.4 TOPSIS multi-criteria decision-making method

One common approach to solving multiple objective decision-making problems is to identify compromise solutions for each of the multiple objective functions. People generally want to make optimal decisions when faced with several options [25]. Developing analytical and numerical techniques that consider multiple alternatives with multiple criteria is the goal, of using scientific terminology. TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) is a numerical method for multi-criteria decision-making [26]. This method is broadly applicable and makes use of a straightforward mathematical framework. It is also an effective strategy that requires technological support. The method has been applied to optimal decisions throughout the last few decades.

Based on the performance score of the TOPSIS, a rank was established for every algorithm as shown in [Tables 6–8](#pone.0324198.t006 pone.0324198.t007 pone.0324198.t008). Consequently, MGGX surpassed the two closest competitors, DPX and LX, to become the most preferred algorithm for every mutation operator among all of these options. Figs 8–10’s visual representation of the ranks for each algorithm further demonstrates how well the recently suggested MGGX algorithm performs.

Table 6: The ranking of algorithms with NUM mutation by TOPSIS.

Table 7: The ranking of algorithms with MPTM mutation by TOPSIS.

Table 8: The ranking of algorithms with PM mutation by TOPSIS.

Visual representation of ranks of algorithms with NUM mutation.

Visual representation of ranks of algorithms with MPTM mutation.

Visual representation of ranks of algorithms with PM mutation.

6 Conclusions and future research

This study introduces a unique approach using the mixture distribution. A pair of new crossover operators are proposed using Gumbel and Rayleigh distributions separately and efficiency is compared with the three operators in the literature. To examine these five crossover operator versions (MGGX, MRRX, DPX, SBX, and LX) with the co-integration of three mutation operators (NUM, MPTM, and PM), twelve widely recognized benchmark functions with diverse characteristics are utilized. Real-coded algorithms of mixture distribution are designed by combining two components of Gumbel distribution with 3 mutations (MGGX-NUM, MGGX-MPTM, and MGGX-PM). Similarly, real-coded algorithms are proposed using Rayleigh distribution (MRRX-NUM, MRRX-MPTM, and MRRX-PM). The strategies are compared concerning the lowest mean objective function values and standard deviation values. Furthermore, this study employs the performance index (PI) and the multi-criteria decision-making technique TOPSIS for comparison. It is confirmed through these techniques (PI & TOPSIS) that proposed algorithms MGGX-NUM, MGGX-MPTM, and MGGX-PM perform significantly better than all the algorithms taken into consideration for comparison in the present case.

The proposed operators have a great deal of managerial potential for resolving challenging optimization problems in real-world situations. The flexibility of the MGGX operator to efficiently handle high-dimensional and multimodal search spaces could be useful for supply chain management, scheduling, and resource allocation applications. Decision-makers in fields like engineering, manufacturing, and logistics might use these insights to develop optimal solutions that are more resilient and need less computing power.

6.1 Limitations

Despite this study evaluating the proposed operators on both constrained and unconstrained benchmark functions, it focused merely at standard benchmark suites and did not evaluate the operators’ effectiveness for use in specific scenarios.

6.2 Future research directions

Future research could explore the application of MGGX and MRRX to practical problems, such as multimodal optimization in resource allocation, supply chain logistics, and scheduling. To enhance generalization, the operators will be tested on benchmark suites beyond CEC 2014/2015 and CEC 2017. Additionally, hybrid strategies incorporating adaptive parameter mechanisms and combining MGGX and MRRX with other meta-heuristics could further improve their performance and adaptability across various optimization scenarios.