1. Introduction
Combinatorial optimization is one of the most widely investigated areas of artificial intelligence. Every year, several research projects focus on issues that arise in this domain. The solution structure, rather than its coding, is utilized by a knowledge-based crossover mechanism for strategic metaheuristic algorithms [
1]. Thus, there are several different approaches that researchers have used in the past to solve single or dual problems. However, in the majority of these kinds of applications, research is limited to solve linear programs, quadratic programs, or, more broadly, convex programming issues. Because the primal optimal solution is closely related to the optimal single or dual solution for convex problems, such a study has assisted investigators in better understanding the relationship [
2]. Therewith, a set of optimization algorithms influenced by natural events and animal intelligence is characterized as evolutionary nature-inspired metaheuristic algorithms. They are, thus, frequently nature-inspired algorithms, and samples of these evolutionary metaheuristic algorithms are Genetic Algorithm (GA) [
3], Artificial Bee Colony [
4], Differential Evolution [
5], and Learner Performance based Behavior algorithm (LPB) [
6]. Therefore, nature-inspired computing is a field of computer science that could be used and shared with an optimization algorithm, computational intelligence, data mining, and machine learning [
7].
A problem with a fluctuating objective function is one of the more difficult metaheuristic optimization methods, but it is much more common in real-world search or self-adaptive optimization. The search technique used to solve such problems for optimality must be flexible enough to adjust to the present function [
8]. A problem with population-based optimizers is that once the investigation has found a locally optimal solution, there may not be enough diversity to move the search forward to a new, better solution. In many of these circumstances, diversity-preserving strategies such as a high amount of crossover or the use of a clustering operator are required. Therefore, most of the algorithms have been improved and enhanced in the field of metaheuristic optimization by these standard operators [
9]. In addition, several effective methods have been developed in this research to improve existing optimizers based on crossing genes between parents. Alternatively, updated standards are always accepted as long as the suggested methods offer novel improvements or comparable results.
When the genetic recombination operator is chosen correctly, crossing genes can match the best-known techniques for a wide range of problems involving restrictions. It would be better to suggest or use the superlative crossover standard to accomplish the goal. The efficiency of this new method could be shown to work for selected features [
10]. The motivation behind this paper is the use of conventional techniques and direct search procedures due to the complexity of variables in the problematic domain. Depending on the situation, fixing the problem might involve modifying the basic algorithm, presenting a systematic and comprehensive overview of the meta-analysis, or utilizing innovative metaheuristics. In addition, this paper proposes a novel standard from several strategies linked to mathematical assessments with originally generated methods.
The fundamental goal of this research is to investigate the effectiveness of experience and understanding methods in GAs. This paper is focused on the generation and standards of crossover and how it affects metaheuristic algorithms. Crossover, also known as recombination, is a genetic operator in the special process which is exploited to connect the genetic codes of two parents to make new offspring (children). Further, crossover techniques can be considered to be extremely useful for generating new solutions from a current population stochastically [
11]. In many studies, crossover and mutation operators have been associated with GA’s success. Some of them conclude that success rests in both, whether the crossover is performed alone or through mutation or both. Crossover operators play a substantial role in balancing exploitation and exploration, allowing for feature extraction from both chromosomes (parents), with the intention that the developing offspring have beneficial qualities from both chromosomes [
12].
Lagrangian Dual Function (LDF) [
13] technique can support to exchange genes between chromosomes by locating the replacement chromosome at the highest or lowest point to produce offspring with improved traits. A proposed standard known as Lagrangian Problem Crossover (LPX) is imperative for generating new operators; the crucial characteristic of LPX is that it allows offspring to inherit some characteristics from their parents for finding a new optimal solution in population solution-based metaheuristics. Likewise, depending on the meaning of the worst chromosome for each problem, this point (its worst chromosome) can be chosen on both chromosomes (parents), or by applying crossover between chromosomes to produce the best novel genes. Furthermore, the new operator is identified automatically based on performance ratings and statistical evidence, so the time spent selecting the best operator is taken into account.
The crossover strategy starts with a low value and adjusts it every generation to avoid premature convergence. Moreover, several crossover functions are employed as a strategy to avoid convergence rates. Population (swarm) algorithms are one of the most successful metaheuristics for managing these kinds of numerous case problems. As a result, population-based methods have become one of the most efficient methods for combinatorial function optimization [
14]. These techniques operate with several populations of solutions that evolve in tandem with algorithm operation. Accordingly, the following summarizes this paper’s main contributions:
There are several standard operators used to illustrate how the implementation was conducted and illustrate the mathematical crossover form using small examples and technique operations;
As a systematic development of previous standards, it has enabled the use of binary form, real-coded form, and ordered-coded form methods;
Based on LDF, a crossover operator has been proposed that can provide a novel optimum solution for population metaheuristic algorithms that use original metaheuristic optimization;
The new anticipated LPX is evaluated by comparing it with selected previous tuning methods, a variation on the traditional GA, as discussed in the next sections;
LPX is compared with other well performing crossover operators using the LPB algorithm as a single objective population-based algorithm and the affected random values and elapsed time are measured;
The proposed standard operator is statistically analyzed and compared, using nonparametric statistical tests.
The next section describes how this paper is organized; this section is dedicated to discussing related work on crossover standards. Several mathematical crossover standards are presented by thinkable pseudocode in the third section of the paper. The fourth segment examines the novel formula which can be developed as a standard crossover in the future metaheuristic algorithm known as LPX. The fifth section is devoted to proving the heuristic and exploitation crossover results by comparing LPX results with specific real-coded crossover standards. At the end of the paper, the conclusion and novel features are discussed.
2. Crossover Standards Overview
As stated, the paper includes two major subjects. The first part includes a systematic review of crossover standards; the second, a proposed a method to generate novel offspring by generating a newly created evolutionary algorithm.
A systematic review has been performed to assess several efficient methods presented in the research. The reviewed articles, as well as standard sources, were examined using search terms including “crossover standard generation”, “generations of crossover standards of parent chromosomes in evolutionary metaheuristic algorithms”, or “Genetic Algorithm Based on crossover standard generation”. These queries are searched in ScienceDirect, PubMed, and Google Search Engine. These articles and sources were culled to address only crossover standards, evolutionary algorithms, Gas, and crossover operators in GAs. Search dates include up to March 2022. A huge number of articles was retrieved, but in the end, the standard crossover operators were approved by 41 eligible papers.
The second part is the proposed LPX standard which is used as the superlative crossover standard to accomplish the target; the efficiency of this new method might be proved for the feature selection optimization problems. The standard is generated from the mathematical model of the LDF theorem.
Many forms of the crossover have been produced over the years, and comparisons between various types have been proposed. These started with one-point crossover and evolved into a range of ways to cover a variety of conditions, including uniform crossover [
15]. To generate improved self-adaptive combinational optimization, a set of assumptions (rules) have been developed to simplify natural/biological events, and these include a list of control parameters to define intensification and diversification rules [
16]. The best solution is identified, and other solutions advance toward the most optimal solution according to the given rules. In the stochastic mode, the location of a few solutions can be altered and controlled, such as crossover or mutation operations in several metaheuristic algorithms, which is illustrated in a simple flowchart in
Figure 1.
Likewise, several standards for permutation applications, such as the Traveling Salesman Problem (TSP), were defined. There are several ways to approach the TSP using evolutionary algorithms, including binary, route, closeness, ordinal, and vector representations. To reduce the overall distance, the researchers presented a novel crossover operator for the TSP [
17]. Another research study confirmed that sequential constructive crossover (SCX) fixed the TSP in 2010. The primary idea behind this strategy is to choose a random point, termed the crossover point; then, before the crossing point, use an SCX technique with improved edges. After the crossover site, the remaining chromosomes are swapped between parents to generate two children; if a chromosome currently exists, it is replaced with an unoccupied chromosome [
18].
Ring Crossover was offered as a solution to the recombination problem. Parents were grouped in the design of a ring of this type, and then a cut point was chosen. Parents were grouped in the design of this circle procedure, and then a slice point was selected randomly. The other location was the length of the chromosome; the first offspring develops clockwise from the line (the original cut), and the second offspring evolves counter clockwise. They employed this type of crossover for the aspects mentioned and it outperformed the other types of assessed crossover [
19]. Despite that, to prevent creating erroneous solutions, evolutionary algorithms that optimize the ordering of a very large series require specific crossover operators. It is difficult to list all of them. Thus, several standard crossovers have already been documented in
Table 1 and each of them has been generated for a specific global solution. However, some of them produced offspring from parents based on real code, whereas others relied on the binary-coded crossover started in the next section.
In application, crossover standards have typically been classified based on the representation of the gene; genetic sequence has been stored in a chromosome represented by a bit matrix or real code in the different algorithms. Crossover strategies for both techniques are popular, and illustrative instances or classes are genetic recombinations, which are thoroughly explained in the following sections. Several current methods ensure that these techniques can be applied to global numerical optimization and current practical problems as a recently proposed meta-heuristic; for example: Slime Mold Algorithm [
40], Moth Search Algorithm [
41], Hunger Games Search [
42], Harris Hawks Optimization [
43], and Colony Predation Algorithm [
44]. As a result, novel standards should be proposed for evolving evolutionary algorithms.
The capacity of solutions to learn from one another, however, is what gives rise to the intriguing behavior of GAs. Solutions can combine to form offspring for the next generation. Occasionally they will share their poor information, but if we use crossover in conjunction with a harsh selection method, better solutions will emerge; there are many details to crossover with permutations, as in previous population-based algorithms, especially these algorithms based on GA, such as the Quantum-based Avian Navigation Optimizer Algorithm [
45]. Thus, we will cover the basic crossover techniques, known as “modification genes”, by Lagrange method techniques. Accordingly,
Table 2 highlights the weak and strong points of crossover operators in metaheuristic population-based algorithms.
3. Mathematical Crossover Standards
In metaheuristic algorithms, the exploration of the optimal solution is based on the generation of new members from existing members. The crossover process facilitates the interchange of genetic code between parents, which results in a higher probability of genes being exceptional to the parents. However, there are numerous crossover techniques recorded in the cited study. Researchers should focus on finding and tackling the question of whether the most effective standard strategy has been improved and adopted. As mentioned, the crossover operator is comparable to multiplication and biological recombination. The data suggest that more than one genome must be selected and that children are produced using the genetic codes (genomes as blue balls on the parents’ chromosomes [
50]) and then two more children using two new offspring genes (pink balls). This probabilistic scale is illustrated in
Figure 2. Then, the graphic depicts the range of potential offspring in two-dimensional constrained real space between x and y dimensions by generating a box crossover between genes and new offspring.
Typically, the crossover is used in metaheuristic algorithms with a significant probability, particularly in a GA, as challenging in real-coded crossovers [
51]. Consequently, the goal of establishing crossover likelihood is to prevent gene loss from the parents, even if the offspring are not better than the parents. According to the distribution of crossover standards,
Section 3.1 has determined the forms of binary crossover. The
Section 3.2 has classified the categories of real-coded or floating-point crossover. In
Section 3.3, the form of order-coded crossover is distributed.
3.1. Binary Form Crossover
This section provides a broader collection of crossover operators used in binary representation for metaheuristic algorithms. Improvements to previous results demonstrate that, according to the current challenges, most of these results are effective crossovers. How to implement some types of crossover standards and point out some interesting comparisons between others could also be shown [
52]. Traditionally, genetic material has been stored in a gene, which is represented as a bit collection in various techniques. Crossover procedures for bit-order are prominent, and specific examples or categories include genetic manipulation, as explained in the points below.
Binary Single-point crossover [
53]: A crossing point in the parent entity string is picked. Apart from that point in the biological sequence, all data transfer between two units, including biological parents and situational bias, is conducted through strings. As indicated in
Figure 3, Genes with sub-blocks that include three bits to the right of that point are transferred correspondingly between the two parents. As illustrated by pink and blue color bites, it has generated two new offspring.
Double-point and n-point crossover [
53]: two randomly generated locations (strings) or n-point locations on the individual chromosomes are chosen, and the gene code is switched at these locations. As seen in
Figure 4, two equally spaced points on the right and left sides are selected on parent chromosomes, then pink color and blue color bits are swapped to perform two single-point crossovers.
Uniform crossover [
49]
and half-Uniform crossover [
54]: such as in the coin-throwing approach, each gene (bit) is drawn randomly from one of the comparable genes in the selected parents. Each genome is addressed separately rather than being separated into segments. In this situation, we just flip a coin to see if each genome is present in the child. The idea may be tossed about to support one parent having more genetic information in their newborn.
Figure 5 shows two chromosomes arrayed as a two-dimensional array with bits exchanged in a light blue color and a pink color.
Uniform Crossover with Crossover Mask (UCM) [
27]: The matrices are separated into several non-overlapping zones, and the matrix created by the logical operator is known as the crossover mask (CM) generated by pseudocode control. In
Figure 6, we present an example and pseudocode to display how the new offspring has spawned between chromosomes and CMs based on these conditions.
Shuffle Crossover (SHX) [
55,
56]: Initially, we choose a crossover point at random, such as the highlighted line in
Figure 7, then mix the gene code of both parents. It should be emphasized that Shuffle chromosomes for the right and left sites are handled independently. A single point of crossing is chosen, which splits the chromosome into two sections, known as schema. Chromosomes are scrambled in each schema by both parents. To produce offspring, schemas are transferred (as in a single crossover).
Three-Parent Crossover (TPX) [
57]: According to the prior solution approach, in this kind of operator, there are numerous probability rate algorithms with which to create innovative offspring from three parent genes. In the elucidation example,
Figure 8 highlights the problems involved in calculating future generations based on deliberate offspring generated by swapping genes according to the general pseudocode.
3.2. Real-Coded (Floating Point) Form Crossover
The genes are real-valued without encoding or decoding into binary, and the digits are left alone to speed up the process. Nevertheless, it is less logical than binary representation since crossover has demonstrated that floating-point format can function as well as, if not better than, ordinary binary strings. Therefore, there is no reason to be concerned about algorithm efficiency if the floating-point encoding is utilized [
51]. Several crossover techniques for real-coded crossovers were developed. The method is based on effectively adjusted real-coded crossover operations that use the likelihood function to create very distinct sequences that may be candidates for alternative solutions [
58]. The crossover techniques are described mathematically in the next points.
Real Single-point Crossover (RSPX) [
59]: It is marginally comparable with a binary single-point crossover; it could be combined with two chromosomes and use a real number for each gene at the crossover point. It can also be generated for two-point, three-point, and n-point crossovers. As shown in
Figure 9, two genes were crossed and real numbers were swapped between them, resulting in two new offspring.
Single arithmetic crossover (SAX) [
57]: This standard should be derived from a single genome (
), randomly chosen from both chromosomes (
). For instance, in
Figure 10,
= 2 and then we define a random parameter (α = 0.5). We modify the
kth gene of chromosome 1 and chromosome 2 to generate offspring by the selected asthmatic formula that is calculated as Equation (1) [
60,
61].
Whole Arithmetic Crossover (WAX) and Linear Crossover (LX) [
62]: The calculations for the whole arithmetic (linear) crossover have been handled for all genes on the chromosome (
) with a single similar arithmetic crossover. The calculation is shown in Equation (2) [
58], and an example of how to create new offspring can be seen in
Figure 11. Despite this, the probability rates are higher since the number of offspring produced will equal the number of chromosomes (genes =
). It should be defined as a random parameter (
and
) and as (
,
) and (
,
) and, for this example, when the numbers of genes (k) equals three. This example calculated and produced three offspring when
; in consequence, it can be calculated for k numbers. However,
Figure 11 just generates three offspring, the generation could be developed by other children according to the
numbers.
Blended Crossover (BX) [
50,
63]: One of the effective crossovers that improved several algorithms. If the two-parameter values in a pair of chromosomes are
and
,
is thus smaller than
, and the blend crossover method produces an offspring option in the range
. Where
is a constant to be established, the solutions of the offspring do not exceed the scope of the single variable [
50,
64].
The example is stated in
Figure 12 which indicated
number is equal to 2,
= 0.13 <
= 0.94, so we calculate the range by
; when
= 0.5 the range is [1.345, 0.535] so, we randomly select
and
between the range.
If the process was evaluated in the previous range, it could not find a global solution, as documented in several improvement algorithms. According to earlier researchers [
50], the condition might be calculated by the updated blend formula. Thus, a novel technique has been developed for the BX standard. The parameter
must be determined using
and a random integer
in the range limitation between (0.0, 1.0), both of which are excluded from the Equation (3) [
65].
and
are the offspring solutions which are regulated by the parents as Equations (4) and (5) [
65], consecutively:
Figure 13 pointed out the example when
(gene) = 2, then identified randomly
and
, so parameter
is calculated by Equation (3).
Simulated Binary Crossover (SBX) [
66,
67]: It is preferable and common to implement a standard crossover operation across all standards. Due to these reasons, SBX was applied to the real-coded parameter without any mutation operator, and SBX was also developed based on the single-point crossover [
66]. This approach focuses on the probability distribution of obtainable offspring (Gene) by the specified parents (Genes), as shown in Equation (11) [
68]. SBX initially computes the number of children using Formulas (6) and (7) [
68], or by using Formulas (8) and (9), that enhance the last two formulas by Azevedo [
66], which are the most commonly used. The steps to calculate the float number resulting from the crossover are started by fixing a random number µ ~ (0, 1) at first; then, calculate the
and generate offspring by using
.
As a function of (9), Alpha (
) [
63] must be calculated as a function of the two previous offspring. H is the index of a user-defined distribution (not negative), which means η is the number of parameters chosen by the user.
Utilize the probability distributions to compute the function of Alpha (
).
When selecting the second gene as a parent 1 and 2 from
Figure 14 will produce two new offspring genes, we need to find (
) if
, and the user chooses two parameters. The calculation is executed as follows:
However, this technique has many advantages, such as a wider range of offspring explored, and results are reliable and often reach global optima. In
Figure 15, it is illustrated that sometimes the results of the offspring gene are out of range based on the probability distributions. This is because the new gene’s impact should be bigger than the old gene, but in this case, the updated gene is smaller than the old gene.
3.3. Order-Coded Problem Methods Crossover
Several different crossover operators have been proposed to reproduce either the relative order or the precise organization of chromosomes from the homologous chromosome. The majority of standard-matched crossovers are operators that maintain exact locations [
69]. As a result, the following sections focus on the most basic forms of order-coded crossovers.
Partially Mapped Crossover (PMX) [
70]: The procedure produces offspring solutions by transferring sub-orders from one genome to the other while maintaining the original sequence (order) with possible several points, as shown in
Figure 16. To initiate, we choose a random range of crossovers and produce children by swapping genes. Unselected substrings are then used to identify the mapping relationship at the time of legalization of the resulting offspring [
71].
The Cycle Crossover Operator (CX) [
17,
72]: Determines the number of cycles that exist between two-parent chromosomes. It may be used with numerical strings in which each component appears only once. This assures that each index point in the resultant offspring is filled with a value from one of his parents [
73].
Figure 17 shows that the first offspring world is generated by relying on the pseudocode when the random cycle includes {2, 5, 7, 6, 11}.
4. Lagrangian Problem Crossover
Two types of crossover operators have been used in population-based algorithms. The first type is operated with binary numbers, while the other is operated with real numbers. By using real code, the crossover operator is vindicated by most of the algorithms rather faster than by binary code [
74,
75]. As discussed in the private table, crossover standards have several weak and strong points. As a result, the major motivation for proving the crossover standards is to propose and demonstrate a distinctive crossover technique. Using novel generated algorithms, crossover operators will take on those responsibilities with varying degrees of precision to find global convergence quickly. The suggested technique is based on LDF for gene crossings. Thus, some following points highlight the novel properties of LPX that should be obvious.
- ▪
When there are limitations on the input values that can be used, the Lagrange multiplier technique can be utilized. This technique is used to determine the maximum or minimum of a multivariable function [
76,
77];
- ▪
The main goal is to find locations where the contour lines of the multivariable function and the input space are adjacent to one another [
76];
- ▪
The constrained population-based optimization problem is transformed into an unconstrained problem using the Lagrange multiplier technique [
77,
78], which provides the optimum solution when used as a point in the crossover standard;
- ▪
Optimization with the Lagrangian method explores the application of Lagrange multiplier methods to find local and global convergence for Lagrangian methods under constraint minimization and maximization [
79];
- ▪
Based on LDF, LPX attempts to calculate the most appropriate offspring, which often involves taking a fairly significant step away from each parent.
The event points show that the Lagrange multiplier method is used for determining a function of local maxima and local minima with equality constraints or requirements. This issue contains the requirement that one or more equations be exactly solved by the selected variable values [
80]. The correlation between the gradient (slope) of the function and the gradients (slopes) of the constraints leads to a natural form of formulation of the global problem, characterized as the Lagrangian Function [
81,
82]. These points around or near the slopes may have a function to generate new genes within the specific chromosome.
According to
Figure 18, there may be an objective function, signified as
which must be optimized while it is subject to the restriction
. Let us put this problem statement into perspective before delving further into its discussion. The Gradient
is a vector which, at every position
, indicates the direction in which
should be increased as effectively as possible.
will rise as long as the stationary point continues to move in that direction, which is the steepest curve in this direction. Any function’s gradient that is calculated at a specific position
will always result in a vector that is perpendicular to the contour line that passes through that point. It is worth noting the exploring point must always remain on the constraint curve
when ascending this top of the climax point (at Gradient Vector), such as the global point. In other words, the only directions the solution can travel are tangents to this constraint curve. These tangents’ values remain constant throughout the constraint curve
because they are orthogonal to the gradient of the limitation function
. Then Gradient Vector must track the optimizer’s motion on the surface of
as the optimizer moves along the constraint curve
. The solution point should continue to raise
, even if it is shifted in a direction along the Gradient
that has a non-trivial component. A gradient can only be shifted orthogonally to the gradient
once if it moves only in the direction orthogonal to the gradient. In this case, the solution has reached a local maximum. The gradients of
and
are now pointing in the same general direction.
Thus, as illustrated in
Figure 18, the restriction
appears as a red curve. The blue curves are characteristics
. Because
, the point where the red constraint tangentially contacts a blue curve is the maximum
, which means tangential to
in the sideways constraint. Moreover, it shows that the assumption that line graphs are tangential has no bearing on the size of any of these gradient vectors, but it is well. When two vectors have the same orientation, we may multiply one by a constant to obtain the other [
83]. The Lagrange multiplier works by assuming that the local minimums and maximums along the constraint occur when the constraint is parallel to the contours. This is stated in
.
In each of the above scenarios, assuming the point is on the contour line,
Figure 18 shows how to calculate the stationary point of
given a function
and the Lagrange multiplier
. Thus, it uses a general method, called the Lagrange multiplier method, as formulated in Equation (12), for solving constrained optimization problems [
84,
85]. Equation (12) is explained in the following evaluation.
From these points, it is proved that the Lagrange multiplier
λ in Equation (12) is improved by maximizing (or minimizing).
Gradient vectors indicate that this model should be fixed to compute many examples and to determine the optimum point. It will, for example, assist with physical routing. It will be used to select the smallest point to find the shortest physical path. However, the Lagrange dual function may be convenient for identifying several global solutions [
86]. As such, the LDF theorem depends on real equation samples [
87]. To identify several local points, the theorem would be formulated using a novel crossover operator. Thus, each station should generate a gene for offspring from parent chromosome(s). The developed alternative to the Conic Duality theory is called LDF theory. The Lagrangian Duality Problem theory is more applicable to generic nonlinear limitations [
88]. From the stationary point in Equation (12), an offspring is derived in Equation (13) using the LDF theorem.
Consequently, it is possible to generate Offspring1 and Offspring2 at the stationary point by including Equation (13) if the Lagrangian Multiplayer has produced a random value for
in the specified range according to the population-based generations crossover rate. Therefore, we developed Equations (14) and (15).
In this case, the proposed standard for this crossover standard (LPX) is the same as the real-coded crossover. It creates offspring solutions by inserting a sub-sequence from one of the genomes into the parent, while the initial order would be as many point states as reasonable, such as in the example calculated by Equations (14) and (15) to generate new offspring, which is illustrated in
Figure 19 as a modification sample crossover. While
is indicated as Gene two
in chromosome one,
is indicated as Gene two
in chromosome two when the stationary multiplier is defined randomly as (
). In the next section, the comparison results are improved heuristically and statistically.
6. Conclusions
In conclusion, the most evolutionary metaheuristic algorithm is based on efficient computation techniques that are applied effectively to different problems. Encoding techniques and standard operators, particularly crossover operators for enhanced metaheuristic optimization, have been responsible for determining their outcomes. In this research, several crossover standards have been collected to assist researchers in obtaining an effective crossover operator and selecting a global solution to the problem. The majority of them were easy to calculate; thus, the calculation was faster, as illustrated. In addition, they allowed for the production of a wide range of offspring from two parameters as determined by two parent values. In addition, the study provided an improved standard option crossover operator to assist in the enhancement of novel mathematical evolutionary algorithms.
In this study, standard crossover operators such as binary-coded crossover, real-coded crossover (floating point), and order-coded problem crossover were mathematically and systematically reviewed. Then, we recommend the optimum crossover standard operator, and LPX has been discovered to be a novel mathematical approach to crossover standards. The LPX is derived from the LDF theorem, which is based on the stationary Lagrange multiplier. Additionally, the capability of the technique was evaluated heuristically for the generation of parent chromosomes and compared with BX and SBX. LPX has the most impressive performance in terms of rate of exploitation and convergence fitness for selected random values. In this manner, these random values assist in selecting the most appropriate range when generating newly developed population-based populations. Moreover, we evaluated LPX results by LPB as a metaheuristic algorithm when LPX, SBX, BX, and Qubit-X were determined by all algorithms. Most test functions have reasonable convergence in the exploitation evaluation for selected random values. Finally, most statistical results for LPX with other standards have proved the significant hypothesis.
In future works, the researcher can evaluate LPX by comparing it with other standardized crossover techniques based on binary form, real-code form, or order-coded problem methods for crossover, and it will be proved by comparison with more test functions such as two-dimensional test functions. LPX may be determined with functional tests that illustrate multimodal test functions and composite test functions. From another perspective, the researchers improved a novel evolutionary metaheuristic algorithm based on populations through single-objective optimization or multi-objective optimization [
98,
99]. Moreover, LPX equations will be improved based on algorithms such as frequency-modulated synthesis with multi-parent crossover [
100] in the Ant Nesting Algorithm [
101], Child Drawing Development Optimization [
102], and Capuchin Search Algorithm [
103]. Thus, LPX asserts that it can deliver the most effective fitness solution. Several research studies by heuristic suggest that large-scale structural optimization, modified to evaluate several strategies pooled with programming techniques, can be used for quantitative appraisal in compliance with enforcement learning [
104,
105,
106]. As a result, LPX might be used for evaluating time-scales in heuristic game problems such as A* or (BA*) search algorithms [
107].