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 ^{6 , 7} 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 GAs 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 addresses basic unconstrained minimization cases like Olsson & Nelson, ⁸ nonlinear least squares, nonlinear simultaneous equations, and function minimization. Spendley, Hext, and Himsworth ⁹ initially suggested the simplex search technique, which was later improved upon by Nelder and Mead. ¹⁰

The steps of the Nelder-Mead ^{10 , 11} 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.

By combining GA with NM in this way, you leverage the global exploration capability of the GA with the local refinement ability of the NM algorithm, potentially leading to improved convergence and robustness in optimization tasks.

The pseudo-code for the hybridization of the GA and Nelder-Mead simplex algorithm is presented in Algorithm. ¹²

The statistical results of GANMA’s performance on 15 benchmark functions with dimensions (n) of 10, 20, and 30 are shown in Table 2. It also contains the final solutions’ best (Best), mean (Mean), and standard deviation (Std) across a 30-run period for each benchmark function. All benchmark functions for unimodal functions ( f ₁ - f ₄) have been solved in all three dimensions (10, 20, and 30). For the multimodal functions ( f ₅ – f ₉), the solutions for f ₅ and f ₉ occur in 10, 20, and 30 dimensions, whereas the solutions for f ₆ and f ₇ in 10 and 20 dimensions are almost optimal. The standard deviation range is 1 .62 E − 13 ∼ 7 .89 E + 00, 1 .45 E − 11 ∼ 1 .57 E + 01, and 1 .29 E − 11 ∼ 2 .95 E + 01, respectively, while the mean value’s variations range in the 10, 20, and 30 dimensions is 3 .18 E − 13 ∼ 1 .46 E − 01, 1 .11 E − 11 ∼ 2 .52 E − 01, and 4 .39 E − 11 ∼ 2 .84 E + 00.

Six shifted test functions have been chosen for this study to validate the performance of GANMA: three shifted multimodal test functions, denoted as f ₁₃ to f ₁₅; and three shifted unimodal test functions, denoted as f ₁₀ to f ₁₂, Sphere, Elliptic, and Rosenbrock. On functions f ₁₀, f ₁₁, f ₁₂ (in 10, 20, and 30), and f ₁₅ (in 10), GANMA achieved optimum solutions; on functions f ₁₃ (in 10) and f ₁₄ (in 10, 20, and 30), the solutions are nearly optimal. Even while GA is outperformed by the solutions of f ₁₃ and f ₁₅ (in 20 and 30) in GANMA, the solutions are still far from the optimal ones. Furthermore, the Std that GANMA found on five test functions is not too high, suggesting that GANMA’s performance on shifted test functions is steady.

Therefore, for all unimodal functions (in 10, 20, and 30 dimensions), GANMA can obtain the global optimum. GANMA can identify outcomes with negligible deviations from the global optimal value for multimodal functions. Except for f ₅ and f ₉ in dimensions 10, 20, and 30, the results of f ₆, and f ₇ in dimensions 10 and 20 are quite near to the optimal value. The outcomes produced by GANMA algorithms for shifted unimodal and multimodal functions are optimal or extremely near-optimal in all functions for all three dimensions, except f ₁₃ and f ₁₅ (in 20 and 30). The benefits of the GANMA algorithm include excellent robustness, high convergence accuracy, and steady performance in all scenarios, whether they involve multimodal functions, unimodal functions, or shifting unimodal and multimodal functions. This is shown in Table 2 under the various numbers of iterations for the corresponding dimensions, which are 300, 400, and 600 for the dimensions 10, 20, and 30, respectively.

To help further investigate the evolutionary behavior of various methods, the convergence curves of GANMA and GA for a few chosen benchmark functions are displayed in Figure 2, Figure 3, and Figure 4 for dimensions (n) = 10, 20, and 30, respectively. These graphs demonstrate the convergence behavior of methods that can help to analyze the evolutionary behavior of various algorithms. The y- and x-axes, respectively, represent the values of the fitness function and the number of iterations. The blue solid line shows the genetic algorithm (GA), while the suggested method GANMA is shown by the solid orange line.

Until the ideal solution is discovered, GA shows a decreasing trend for unimodal functions like f ₁, f ₂, f ₃, and f ₄. In contrast, GANMA presents a straight line for all three dimensions (n = 10, 20, and 30). Similar to this, for multimodal functions other than f ₅ and f ₇ (in 30), there is a greater similarity between the global optimum solution and the GANMA optimal solution in f ₅, f ₇ (in 10, and 20), and f ₆, f ₈ (in 10, 20, and 30). As a result, of these two algorithms, the lowest optimum solution and the fastest rate of convergence are found through GANMA. The curves for shifted functions, except f ₁₅ (in 20 and 30), demonstrate how well the proposed method was able to obtain the ideal solution for other functions like f ₁₀, f ₁₁, and f ₁₂ (in 10, 20, and 30). Out of these two methods, GANMA yields the lowest optimum solution and has the fastest convergence rate than GA’s in both the multimodal and shifted functions.

Examining the convergence curve and experimental results, it can be shown that GANMA generally performs exceptionally well on the 15 test functions, with a proportion of correct convergence to the global optimal solution that approaches 90%. When compared to GA and the NM algorithm, GANMA outperforms both in terms of exploration and exploitation. As a result, GANMA can meet the requirements for addressing diverse optimization issues, increasing its competitiveness.

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 (1) 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 (2)

Probability density and cumulative distribution plots for some different parameter values are given in Figure 1.

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 (1). 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 ) (3) 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 ] (4) LnL ( β , η ) = nlnβ − nβln η + ( β − 1 ) ∑ i = 1 n ln x i − η − β ∑ i = 1 n x i β (5)

The log-likelihood function is differentiated with respect to 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 (6) ∂ 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 (7)

By eliminating α from the above equations and simplifying the equations we get, η ̂ = ( 1 n ∑ i = 1 n x i β ) 1 β (8) 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 (9)

Eqn. (8) may be used to calculate the estimate η ̂ . However, because of Eqn. (9) 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. (5) is considered a fitness function for GA and NM methods.

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

6.1.1 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 5- 7 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 8- 11 are the absolute values of the biases and the MAE.

The MAE values for the shape parameter β are shown in Figure 8. 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 9. 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 10. 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 11. 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. ³¹

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. ³²

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. ³³ D = max | F o ( x ) − S N ( x ) | (16)

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 12 and 13. 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 12 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.