Genetic Algorithm (GA) based hybrid approaches have become a cornerstone in modern optimization research, combining GA’s global exploration abilities with various techniques to enhance local refinement. The following highlights some key advancements: 1.

GA-Nelder–Mead (GA-NM): ¹ •

Methodology: Integrates GA for its broad search capabilities with the Nelder–Mead (NM) simplex algorithm, which excels in refining solutions locally.

These developments illustrate the versatility and potential of GA hybrids in addressing a range of optimization challenges while emphasizing the need for careful parameter tuning and scalability enhancements. GANMA builds on this foundation by offering a structured, robust framework that addresses existing limitations.

Despite advancements in hybrid optimization algorithms, several key challenges persist. Many studies lack comprehensive comparisons, failing to evaluate scalability, convergence, and adaptability across diverse tasks. Additionally, the balance between global exploration and local exploitation remains under-explored, limiting efficiency in finding optimal solutions. Scalability issues are prominent, as many hybrids falter in high-dimensional problems, highlighting the need for robust methods capable of maintaining performance in complex spaces. Parameter sensitivity is another hurdle, with insufficient adaptive tuning mechanisms leading to inconsistent results. Furthermore, validation is often restricted to benchmark functions, offering limited insight into real world applicability where constraints and objectives are more complex. These gaps emphasize the need for innovative hybrids that address these issues while ensuring efficiency, scalability, and practical relevance.

Individually, both GA and NM have strengths and limits that make them appropriate for specific optimization scenarios. GA excels in global exploration, utilizing population variety to explore large solution spaces and avoid local optima. On the other hand, NM excels at local refinement, expertly traversing convex and smooth terrain to locate specific optima. The hybridization of GA with NMA addresses the limitation of GA in fine-tuning solutions near optima, at which NMA excels. This synergy improves the algorithm’s convergence speed and solution quality. Other researchers have primarily focused on individual optimization methods or hybridizations excluding GA and NMA, leaving a gap in fully exploiting the complementary strengths of these methods.

The GANMA method effectively addresses these gaps through its innovative design and balanced approach. By seamlessly integrating Genetic Algorithm (GA) and Nelder-Mead Algorithm (NM), GANMA achieves a robust balance between global exploration and local exploitation, enhancing its efficiency in diverse optimization tasks. Its structured framework allows for improved scalability, maintaining performance even in high-dimensional problem spaces. Additionally, GANMA incorporates adaptive mechanisms for parameter tuning, reducing sensitivity and ensuring consistent results across various scenarios. Unlike many existing hybrids, GANMA has been rigorously tested on both benchmark functions and real-world parameter estimation tasks, demonstrating its adaptability and robustness. These features position GANMA as a superior hybrid optimization method, addressing the limitations of existing approaches while offering practical solutions for complex, multidimensional challenges.

Many industries are interested in using the Nelder-Mead Simplex Algorithm (NM) working together with Genetic Algorithms (GA), including bio-informatics, ¹¹ finance, ^{12, 13} and engineering. ^{14, 15} Combining these methods provides a potent method of resolving challenging optimization issues in engineering, where designs are complicated and rules are demanding. Combining GA with NM helps improve portfolio management and risk assessment in the financial industry, where on-time and correct choices are essential. Similarly, hybrid algorithms speed up tasks like genomic analysis and drug discovery ^{16, 17} in bio-informatics, where understanding biology relies on smart computer methods. This article explores how combining NM and GA enhances both, highlighting how they work together to solve real-world issues. This paper has tested the GANMA algorithm on fifteen benchmark problems in three dimensions (10, 20, and 30). According to the results from the experiment, the suggested GANMA algorithm is a promising one that can quickly find the best or almost the best solution for most of the functions examined.

The remaining portion of the research study is structured as follows: Section 2 provides the fundamentals of the Genetic Algorithm and the Nelder-Mead simplex search. Section 3 discusses the proposed hybridized method, benchmark functions, and an alternative hybridization approach. The parameter setup for all methods and computational configurations are detailed in Section 4. Section 5 presents the results and discussion for benchmark functions, while Section 6 focuses on the Weibull distribution. Parameter estimation methods are described in Section 7, and Section 8 provides an analysis of Monte Carlo simulations and results. Two real-world wind speed datasets are analyzed in Section 9 to demonstrate the effectiveness of the proposed technique. Finally, Section 10 concludes the study with key observations.

Metaheuristic optimization methods like GA are higher-level frameworks designed to guide heuristic or local search procedures. In contrast, heuristic searches are problem-specific strategies for exploring the solution space. GA leverages metaheuristic principles to perform heuristic searches iteratively, balancing exploration and exploitation. So, we can say that GA is an approach to heuristic search. The ideas of the biological evolution of species serve as its inspiration. In contrast to traditional optimization methods, GA ^{11 , 18} starts with a collection of starting solutions known as chromosomes.

Genetic algorithms (GAs) work by continually improving solutions based on their fitness, which measures how well they solve a problem. Unlike some traditional methods, GAs don’t assume anything about the problem, like whether it’s smooth or has just one best solution. Instead, they explore different possibilities to find good solutions, even in complex situations where there might be many equally good answers. GAs have been used successfully in many difficult optimization problems. They often work better than traditional methods, especially when there are multiple equally good solutions. This flexibility and ability to handle complex situations make GA a valuable tool for solving optimization problems in various fields.

Following is a summary of the GA stages in this study: I.

Initialization: •

First, create a vector of real values for each variable between predefined ranges. This vector will represent the initial population of individuals.

This approach with elitism helps maintain diversity in the population while ensuring that the best individuals are preserved across generations, ultimately leading to the discovery of better solutions in the optimization process. Real-coded genetic algorithms are suitable for optimization problems with continuous decision variables and offer advantages such as direct representation of real-valued solutions, robustness, and ability to handle high-dimensional search spaces.

The simplex search technique has been widely used for basic unconstrained minimization problems, such as nonlinear least squares, nonlinear simultaneous equations, and general function minimization. ¹⁹ Originally proposed by Spendley, Hext, and Himsworth (1962), ²⁰ the method was later refined by Nelder and Mead (1965) ²¹ to improve its efficiency and applicability.

The Nelder-Mead Algorithm (NMA) is selected for its simplicity and effectiveness in local solution refinement, making it a strong complement to the Genetic Algorithm’s (GA) global search capabilities. While a variety of optimization algorithms exist, NMA’s low computational overhead and reliability in small-dimensional spaces make it an efficient and practical choice for hybridization.

However, NMA’s reliance on simplex geometry and localized operations restricts its exploratory capacity, often causing it to converge prematurely to local optima in complex, multi-modal landscapes. Preliminary experiments (to be included) under these limitations highlight the necessity of GA’s global search to overcome such challenges.

The steps of the Nelder-Mead ^{21 , 22} algorithm are summarized in as follows: I.

Initialization: •

A simplex is a collection of n + 1 vertices in a n dimensional space. These vertices can be deliberately selected or created at random.

The algorithm converges when the simplex becomes sufficiently small or when the function values at the vertices are close to each other. The choice of parameters α, γ, and ρ can significantly affect the performance of the algorithm and may need to be tuned based on the problem characteristics.

The combination of Genetic Algorithms (GA) with the Nelder-Mead simplex algorithm (NM) is driven by their supportive characteristics in both global exploration and local exploitation. GA is a population-based technique that effectively explores diverse sections of the search space, although fine-tuning solutions at local optima may provide issues. In contrast, it requires greater capacity for worldwide investigation. Combining both methods intends to take advantage of the characteristics of both algorithms, resulting in a more balanced and efficient optimization process. This hybridization method has the potential to improve convergence rates, solution quality, and robustness, making it a compelling choice for handling complicated optimization problems across several domains.

The suggested algorithm’s (GANMA) stages are summed up as follows: I.

Initialization: •

Generate an initial population of solutions for the GA.

NMA is applied to the best solution after reproduction and mutation in each iteration, not just the final solution. This strategy allows continuous refinement throughout the optimization process. By combining GA with NM in this way, you leverage the GA’s global exploration capability with the NM’s local refinement ability, potentially leading to improved convergence and robustness in optimization tasks.

GANMA stands out as a versatile hybrid algorithm capable of addressing a wide range of optimization problems, transcending the domain-specific focus of many existing hybrids. Its well-balanced framework effectively combines the global search power of Genetic Algorithms (GA) with the local refinement precision of the Nelder-Mead Algorithm (NMA), ensuring scalability, robustness, and efficiency. This synergy enables GANMA to overcome common challenges, such as parameter sensitivity and poor performance in high-dimensional or multimodal landscapes. Furthermore, GANMA’s structured approach is rigorously validated, making it a reliable solution for both theoretical benchmark functions and complex real-world applications.

The pseudo-code for the hybridization of the GA and Nelder-Mead simplex algorithm is presented in Algorithm 1.

The detailed flow diagram of the proposed algorithm is shown in Figure 1.

Here is an algorithm that combines a Genetic Algorithm (GA) with the Nelder-Mead Algorithm (NMA), where the GA first locates the interval containing the global minimum, and NMA refines the solution: I.

Initialize GA Population Generate an initial population of candidate solutions. Define the fitness function for evaluation.

The pseudo-code for the hybridization of the GA and Nelder-Mead simplex algorithm is presented in Algorithm 2.

This study analyzes 15 benchmark test functions for simulation tests to fully investigate the feasibility as well as the effectiveness of GANMA. The 15 benchmark test functions (denoted as f ₁ to f ₁₅), cover different types. The unimodal functions (A function with a single peak or trough, making it straightforward to locate the global optimum) f ₁ through f ₄ are included in the first kind. Multimodal functions (A function with multiple peaks or troughs, presenting challenges in finding the global optimum due to local optima) f ₅ through f ₉ are included in the second category. Shifted unimodal and multimodal functions (Shifted Unimodal Function: an unimodal function whose peak or trough is relocated to a different position in the search space. Shifted Multimodal Function: A multimodal function with its peaks or troughs displaced, adding complexity by altering the relative positions of local and global optima), f ₁₀ - f ₁₅, are included in the third category. Table 1 displays the expressions, ranges, and global minimum values of the 15 test functions. The function’s dimensions (n) are 10, 20, and 30, in that order.

For problem dimensions 10, 20, and 30, the Genetic Algorithm (GA) was executed for 300, 400, and 600 generations, respectively, starting with a population of 100 individuals. Eighty percent of the population had a one-point crossover, which translates to a crossover rate of 0.8. With a mutation rate of 0.05, random mutation was employed. With a tournament size of five, parents were selected by tournament selection, and the top 10 percent of each generation’s top performers were preserved through an elitism technique.

The Nelder-Mead (NM) algorithm was initialized using the solutions provided by the GA. Standard transformation coefficients were applied, including a reflection coefficient ( α) of 1, an expansion coefficient ( γ) of 1.5, and both contraction ( ρ) and shrinkage ( σ) coefficients set to 0.5. The step size was maintained at 1.0. The algorithm’s simplex shrinking process concluded when the convergence tolerance reached 10 ⁻⁶.

The hybrid process iterated through GA and NMA stages for 300, 400, and 600 generations for 10, 20, and 30 dimensions respectively. The stopping criteria were based on either the maximum number of iterations or fitness convergence, defined by a fitness tolerance of ϵ = 10 ⁻⁵, ensuring early detection of optimal solutions. Table 1 displays the expressions, dimensions, ranges, and global minimum values of the fifteen benchmark test functions (denoted as f ₁ - f ₁₅).

The experiments were conducted in a consistent computational environment using Python 3.11. The hybrid GANMA algorithm was implemented from scratch, leveraging key Python libraries. NumPy handled arrays and matrix operations, Matplotlib was used for visualizing convergence and results, and SciPy supported NMA-based optimization. All tests were executed in a Jupyter Notebook environment to allow for easy experimentation and tuning. Each experiment was repeated 50 times to ensure statistical reliability.

Table 2 demonstrates how the performance of the GANMA, GA, and NM algorithms for dimensions (n) 10, 20, and 30 have been evaluated by comparing the mean value (Mean), standard deviation (Std), and best value (Best) of the final solutions for each benchmark function throughout 30 trials. The algorithm achieves the best optimization performance with the least standard deviation, optimal value, and average value closer to the theoretical ideal value. Any value less than 10 ⁻⁶ in terms of mean, standard deviation, and best value will be regarded as zero. The ideal experimental outcomes are truncated.

A versatile continuous probability distribution, the Weibull distribution is frequently used in survival analysis and reliability engineering. It is characterized by its ability to model the distribution of time until an event occurs. Named after Wallodi Weibull, who described it in the 1950s, the distribution is flexible and can take different shapes depending on its parameters. The shape parameter affects the structure of the Weibull distribution curve resulting in whether the distribution appears to be a Rayleigh distribution ( β = 2), an exponential distribution ( β = 1), or another shape. The scale parameter determines the distribution’s scale or size. Together, these factors enable the Weibull distribution to simulate a wide range of events with varying shapes and sizes.

The following is the Weibull two-parameter distribution’s probability density function (PDF): f ( x ; β , η ) = { β η ( x η ) β − 1 e − ( x / η ) β , x ≥ 0 0 , x < 0 (6) where: •

x is the random variable,

The following represents the Weibull distribution’s cumulative distribution function (or CDF): F ( x ; β , η ) = { 1 − e − ( x / η ) β , x ≥ 0 0 , x < 0 (7)

Probability density and cumulative distribution plots for some different parameter values are given in Figure 5.

Two-Parameter Weibull is Commonly applied in reliability engineering for modeling time until the failure of components. Whereas, Three-Parameter Weibull is Useful when considering scenarios where the event initiation may not be at zero, such as analyzing the time until an event occurs after a certain threshold.

The statistical method known as Maximum Likelihood Estimation (MLE) is used to estimate Weibull parameters by maximizing the likelihood function, which determines how well the distribution fits the observed data. MLE is known for its efficiency, but its optimization can be complex due to non-linear equations and numerical stability issues. The PDF of the Weibull distribution is given by Equation (6). Given a sample x ₁, x ₂, … x _n from a Weibull distribution, the likelihood function is given by: L ( β , η ) = ∏ i = 1 n f ( x i ; β , η ) (8) where, f ( x; β, η) is the probability density function of the Weibull distribution.

The Weibull distribution’s log-likelihood function is as follows: LnL ( β , η ) = ∑ i = 1 n [ ln ( β η ) + ( β − 1 ) ln ( x i η ) − ( x i η ) β ] (9) LnL ( β , η ) = nlnβ − nβln η + ( β − 1 ) ∑ i = 1 n ln x i − η − β ∑ i = 1 n x i β (10)

The log-likelihood function is differentiated for β and η, the derivatives are set to zero, and the resultant system of equations is solved to get the MLE. ∂ L ∂ η = − nβ η + β η β + 1 ∑ i = 1 n x i β = 0 (11) ∂ L ∂ β = n β − nlnη − ∑ i = 1 n x i β − ln η ∑ i = 1 n x i β η β + ∑ i = 1 n ln x i = 0 (12)

By eliminating α from the above equations and simplifying the equations we get, η ̂ = ( 1 n ∑ i = 1 n x i β ) 1 β (13) 1 β − ∑ i = 1 n x i β ln x i ∑ i = 1 n x i β + 1 n ∑ i = 1 n ln x i = 0 (14)

Eqn. (13) may be used to calculate the estimate η ̂ . However, because of Eqn. (14) did not give an analytical solution, the estimate β ̂ must be calculated numerically. This is possible by using the optimization strategy. The Nelder-Mead, Newton Rapson, simulated annealing, or GA algorithms can all be used to solve the nonlinear function that the ML estimator of the shape parameter β contains. In this study, the suggested method, GA, and NM were all used to optimize the log-likelihood function. Nelder-Mead is a powerful algorithm that converges quickly, but its performance is dependent on the initial guess. As a result, we took into account the GA while maximizing the Weibull distribution’s loglikelihood function. Eqn. (10) is considered a fitness function for GA and NM methods.

Below the proposed method on MLE of Weibull Distribution has been described briefly.

7.1.1 Proposed method

(Genetic and Nelder -Mead Algorithm (GANMA))

To improve the precision and reliability of parameter estimation, we proposed a hybrid approach GANMA that integrates the GA and the NM method with MLE for two-parameter Weibull distributions. The GA aids in exploring the parameter space globally, generating diverse candidate solutions, while the NM fine-tunes these solutions through local search, aiming for optimal parameter estimates. To the best of our knowledge, this is the first instance where the GANMA is being utilized to estimate the Weibull distribution’s parameters.

The steps of the proposed method in this study are summarized as follows:

Step 1: Problem Formulation - We aim to find the MLE parameters β (shape) and η (scale) for a Weibull distribution.

Step 2: Genetic Algorithm (GA) Phase - •

Generate an initial population (P) of possible solutions. For the Weibull distribution, each solution indicates a collection of parameters ( β, η).

Step 3: Nelder-Mead Algorithm (NM) Phase - •

Take the best individual from the final population of the GA as an initial guess for the parameters ( β1, η1).

Step 4: Repeat the selection, crossover, and mutation steps for several generations until convergence is met (i.e. end of GA phase).

Step 5: Apply the NM method to the best GA solution once again after the GA phase.

Step 6: Result - The final parameters ( β ̂ , η ̂ ) obtained from the Nelder-Mead optimization represent the Maximum Likelihood Estimates (MLE) for the Weibull distribution.

Figures 6- 8 illustrate the outcome across various shape parameters while keeping the scale parameter constant—as well as various data sizes by plotting the convergence graph of the PDF of Weibull parameters and the PDF of MLE of parameters using NM, GA, and GANMA. The solid black line depicts the PDF of parameters ( β, η), whilst the usual genetic algorithm is illustrated by the solid green line, the yellow solid line shows the Weibull PDF using NM, and the suggested method GANMA is shown by the solid red line. It has been found that parameter estimation using the suggested technique converges with the original PDF as the shape parameter and data size increase. GANMA, the suggested algorithm, performs better than GA and NM in all types of situations.

Based on MAE and bias criteria, the simulation results demonstrate that the GANMA technique outperformed NM and GA in the estimation of shape and scale parameters. In each simulated scenario, the GANMA technique yielded the best shape parameter efficiency in terms of bias and MAE for sample sizes of 20, 100, and 500 respectively.

Throughout almost every simulated scenario, GANMA achieved the maximum efficiency in the estimate of scale parameters for sample sizes of 20, 100, and 500, based on at least one decision criterion. By analyzing MAE and bias for each simulation scenario, GANMA proved to be the most effective approach for the data size 20. For small, moderate, and high sample sizes, GANMA is a fairly effective strategy overall. Additionally shown in Figures 9- 12 are the absolute values of the biases and the MAE.

The MAE values for the shape parameter β are shown in Figure 9. In every simulated scenario, GANMA outperformed NM and GA in terms of efficiency. The second-best approach is NM. An increase in sample size resulted in lower MAE values. On the other hand, MAE values increased along with an increase in the form parameter value.

The scale parameter η’s MAE values are displayed in Figure 10. For sample sizes of 20, 100, and 500, GANMA proved to be the most effective approach. When the shape parameter is set to a higher value, the MAE values drop. Likewise, as the sample size is raised, the MAE values drop.

The shape parameter β’s absolute bias value is displayed in Figure 11. The most efficient results were obtained using GANMA. NM outperformed GA on some occasions. As with MAE values, larger sample sizes resulted in lower absolute bias levels. Increasing the parameter value resulted in higher absolute bias levels.

The absolute bias for the scale parameter η is shown in Figure 12. Most of the time, GANMA outperformed other methods in terms of efficiency. The second-best approach is NM. Increasing the shape parameter and sample size leads to lower absolute bias levels.

In this challenge, two real-world data sets have been used to examine wind-speed analysis. The very first set of data came from the seas surrounding the Maluku Islands and Sulawesi. The data under analysis were gathered by the satellite Quikscat, which measured the ocean wind 10 meters above sea level using a scatterometer. The measurement’s horizontal and vertical spatial resolution is 0.25°earth grid. The information from the January measurement point at latitude 116° and longitude 85.5° is included in the accessible data. ⁴⁸

Tarama Island and Iriomote Island, which are close to northern Taiwan, had their wind speeds recorded in the second data set. At Iriomotejima Meteorological Station, the maximum daily wind speed and direction were recorded in March 2012. ⁴⁹

The Kolmogorov-Smirnov (K-S) test is a nonparametric statistical test used to compare two distributions. The K-S test calculates the maximum absolute difference between the empirical cumulative distribution functions (ECDFs) of the distributions being compared, providing a test statistic (D). A p-value derived from this statistic indicates the significance of the difference, helping in goodness-of-fit testing, comparing sample distributions, and model validation without assuming any specific distribution for the data.

The statistical confirmation that the monthly data sets come from the Weibull distribution can be obtained by doing the K-S test separately for each data set. The most significant difference between the theoretical distribution, S _N ( x), and the observed distribution, F ₀( x), is the K-S test statistic. ⁵⁰ D = max | F o ( x ) − S N ( x ) | (21)

Monthly distributions from the Weibull distribution are selected for further investigation following the K-S test ( p − value ≥ 0.05 ), which indicates the probability of observing a discrepancy as large as the one computed if the two distributions were the same.

Results across shape and scale parameters were obtained by plotting the convergence graph between the PDF and CDF of MLE of parameters using NM, GA, and GANMA, as shown in Figures 13 and 14. The solid green line and dotted green line represent the PDF and CDF of the standard genetic algorithm, the yellow solid line, and dotted yellow line represent the Weibull PDF and CDF using NM, and the solid red line and dotted red line represent the suggested method for both the PDF and CDF, respectively. Figure 13 illustrates that the PDF and CDF for both GANMA and NM convergence are on the same line.

Tables 6 and 7 present the shape and scale parameters, k-s value, p-value, and power density for the first and second data sets, respectively, for all three estimation techniques. The greatest p-value and the lowest k-s statistic for both data sets are produced by the suggested approach (GANMA) out of the three estimation techniques. The Weibull distribution and the actual wind speed data seem similar, as indicated by the p-value exceeding the selected significance threshold (e.g., 0.05). In other words, the data is well-fitted by the Weibull distribution. The parameters estimated using GANMA are considered the best fit for describing the wind speed data, based on the K-S test findings. The observed wind speed data and the predicted Weibull distribution with these parameters were well recognized, as evidenced by the low K-S statistic and high p-value.

The maximum power density is demonstrated by the parameters estimated through MLE implementing NM, as shown in Table 6. This suggests that the parameters possess greater absolute performance in terms of power generation. Despite the slightly lower power density value of the parameters estimated by MLE using GANMA compared to NM, they are nevertheless selected as the best fit since they have the greatest p-value and the least k-s statistic. This suggests that for wind speed data set 1, parameters calculated by MLE using GANMA offer the best match.

The parameters that are estimated by MLE using GANMA are found to provide the best fit in Table 7, as shown by their lowest K-S statistic and highest p-value. Additionally, superior performance in terms of power generation is indicated by the higher power density value associated with these parameters.

We certify that the submitted manuscript is our original work that is not currently being considered elsewhere. The paper is an unfunded independent piece of labor.

This article does not include any research that any of the authors conducted using humans or animals.