Enhanced Whale Optimization Algorithm with Novel Initialization Techniques for Global Optimization Problems in the IntelliWriter.io Tool

Enhanced Whale Optimization Algorithm with Novel Initialization Techniques for Global Optimization Problems in the IntelliWriter.io Tool Atif Ali Research Management Centre (RMC), Multimedia University, Cyberjaye 63100 Malaysia. [email protected] Salman Ghani Virk Riphah International University, Islamabad. [email protected] Ali Raza University of Gujrat, Pakistan [email protected] Taimoor Ali Khan UIIT PMAS Arid Agriculture University [email protected] Ali Rashid Mahmud Military College of Signals, National Univeristy of Science and Technology [email protected] Tariq Hanif Abbasian University islamababd Abstract: Nature inspires the Whale Optimization Algorithm (WOA), a well-liked metaheuristic for complex global optimization problems. WOA imitates the humpback whales' bubble-net hunting strategy using a populationbased approach with random initialization. However, a major disadvantage of WOA is its tendency to get stuck in local optima when used on complex problems. This work presents an improved version called the Improved Whale Optimization Algorithm (I-WOA), which is practically used in the intelliwriter.io (IW) tool by robx.ai research lab. This improved version improves the standard WOA by making it more capable of exploitation. In addition, we initialize using a quasi-random Torus sequence, which helps us get around convergence and diversity problems. For WOA, proper population initialization is crucial because it has a big impact on the convergence and diversity of the swarm. It has been demonstrated that quasi-random sequences work better than random distributions. We employed our suggested approach on a collection of common benchmark functions that are frequently utilized in studies. The experimental data unequivocally demonstrate the superior performance of our method. The simulations verify that the Improved Whale Optimization Algorithm performs better than the conventional WOA for function optimization. Keywords: Quasi-random sequence, swarm intelligence, intelliwriter.io, whale algorithm, exploration, and exploitation I. INTRODUCTION Optimization algorithms are crucial for resolving complicated real-world issues in various fields, such as finance, engineering, logistics, and artificial intelligence. Under specified constraints, these algorithms seek to maximize or minimize an objective function to arrive at the optimal solution. Because they effectively handle large, nonlinear, multi-modal optimization problems, natureinspired metaheuristic algorithms [1] have drawn much attention among the many optimization techniques available. Stochastic and deterministic algorithms are the two main categories into which optimization algorithms can be divided [2]. When starting from the same starting point, deterministic algorithms, which rely on gradients, generate the same solutions for every iteration. Whereas the final values tend to converge to the same optimal solutions within a given precision, stochastic algorithms do not use gradients and produce distinct solutions even when starting from the same points. Heuristics and metaheuristics are two more subcategories of population-based and stochastic algorithms. One well-liked nature-inspired metaheuristic that is often used to solve challenging optimization problems is swarm intelligence (SI)[3]. Furthermore, evolutionary algorithms and conventional neural networks are frequently used for data classification and optimization tasks. For numerous years, swarm intelligence (SI) techniques have been utilized to solve tough real-world optimization issues. The concept of SI was first proposed by Beni after observing how fish, birds, and insects work together to solve enormously difficult problems despite not being very intelligent. They can plan their nests or find the shortest path to a food source even though they lack much intelligence because they collaborate with each other and interact with the environment around them. This is what enables them to address challenges that would otherwise be impossible for any one individual among them. The Whale Algorithm and other optimization methods, such as Evolutionary Algorithms (EAs) like Gene Expression Programming (GEP), Genetic Algorithm (GA), Differential Evolution (DE), and Genetic Programming (GP), have a serious problem with premature convergence [4]. In heuristics, the ideas of intensification (exploitation) and diversification (exploration) are crucial. Exploitation is the capacity to search locally, while exploration is the capacity of any population-based algorithm to search globally. Swarm-based algorithms are highly dependent on preserving an equilibrium between exploration and exploitation. While excessive exploration and insufficient exploitation can impede the discovery of the ideal solution, excessive exploitation and insufficient exploration can cause premature convergence [5]. The standard Whale Optimization Algorithm (WOA) has a stronger local search ability (exploitation) than a global search ability (exploration). We have created an enhanced version, the Improved Whale Optimization Algorithm (IWOA), to improve its exploration capabilities. We used 2025 I-RIM Conference October 17-19, Rome, Italy ISBN: 9788894580570 10.5281/zenodo.17629632 39 nine well-known benchmark test functions to compare the proposed IWOA with the original WOA to verify its efficacy. According to the experimental findings, the suggested variant performs better overall and in terms of exploration than the original WOA on particular test functions. The paper is organized as follows: In Section 2, relevant literature is reviewed, and in Section 3, the operation of the original WOA Algorithm is explained. The methodology is described in Section 4, and the outcomes and conclusions of the suggested approach are covered in Section 5. Section 6 wraps up the work and makes some recommendations for future research. II. RELATED WORK This paper presents an opposition-based learning (OBL) that is unique in the improvement of initialization stage for evolutionary algorithms (EA) [6]. This proposed method makes the convergence faster by beginning closer to the global optimal solution hence considering both candidate solutions and their opposite. When applied to Differential Evolution (DE) algorithm, this technique was tested on 34 benchmark functions as used in intelliwriter.io where it was found out to have significantly sped up convergence rates as opposed when traditionally initializing with random values. It was shown that EAs become more robust and efficient through oppositional initializations especially while dealing with optimization problems having many dimensions. Another approach to improving how evolutionary algorithms initialize their populations is outlined in a paper written by the authors of [7]. In this method, clustering is used for identifying points of interest in a given search space whereas Cauchy deviates are employed to generate individuals around these areas. By diversifying initial populations more effectively and improving its quality through double strategy approach where by this way also enhances general performance speed up convergence rate additionally displays power of sophisticated initializing techniques over traditional random ones particularly with dimensions greater than three based on test results obtained from using several well-known test functions for optimization algorithms. This paper authors [8] suggest a new way to initialize PSO (Particle Swarm Optimization) algorithms for global optimization. Population diversity and convergence rate are improved through the WELL method by seeding populations with low-discrepancy sequences. WE-PSO (a proposed well-based PSO), standard PSO, Sobol-based PSO (SO-PSO), and Halton-based PSO (H-PSO) were all compared by the authors using fifteen widely used benchmark test problems. The WE-PSO was found to be more accurate and efficient than other methods. Additionally, when applied in training artificial neural networks (ANNs), its performance has been evaluated against traditional methods showing it to do better. It can be seen from these findings that complex initialization techniques may increase both robustness and efficiency of PSO algorithms. The update of the improvement spiral location model was proposed foundation of the authors’ improved WOA [9]. They suggested these modifications with an objective to tackle the problem of losing variety in population of the algorithm at later stages. Furthermore, global search abilities were also enhanced while premature convergence prevented through introducing some random adjustive parameters, normal mutation operations together with opposite learning strategies into it. When tested on 23 benchmark functions including fixed-dimension, multimodal and unimodal functions, this modified method performed better than the standard WOA in terms of both speed of convergence as well accuracy. Based on these results it can be affirmed – IMWOA is stable and efficient approach towards resolution of hard optimization tasks. In this paper authors [10] propose a new method of multiuser detection learning in MIMO communication systems based on Improved Whale Optimized MLP Neural Network (IWMLP-NN).The IWMLP-NN improves signal detection in DS-CDMA systems for 3G/4G channels by combining a multilayer perceptron (MLP) neural network with optimized whale algorithm (IWOA).This approach enhances monitoring accuracy for arrival directions while reducing error rates through adapting weights dynamically when different dependent conditions occur (Kadarmideen & Wu, 2015).Compared with classic methods such as MUSIC algorithm or LMS/RMS algorithms, it has been observed that the bit error rate (BER) performance of this suggested technique is much better under different SNR values.The channel models included AWGN as well as Rayleigh fading ones.According to my experience so far this work shows how effective IWMLP-NN can be at improving MUD performance in complex communication environments. In this research [11] explain an improved form of the Particle Swarm Optimization algorithm (PSO) designed for optimal trajectory planning of robots in robotics with the focus on minimizing time and jerk. They propose an IWOA which is a combination of threshold mechanism as well as adaptive weight to balance between exploration ability of algorithm and exploitation capability thus enhancing its accuracy during global search by minimizing cost function based on time mean jerk. IWOA algorithm is used for trajectory planning of 6-axis welding industrial robot applying fifth order B-spline interpolation in joint space. The study found that compared to traditional methods, IWOA reduces considerably jerks and travel time leading to smoother more efficient movement of robots according to experimental results. This work demonstrates how these methods can be applied in complex manufacturing processes so as to improve stability & effectiveness among other things Authors [12] have introduced an improved version of Whale Optimization Algorithm (WOA) for feature selection. It does this by combining adaptive neighborhood and hybrid mutation strategies. This new algorithm, called HMNWOA, uses a hybrid mutation strategy of Cauchy and Gaussian 40 distributions together with an adaptive neighborhood radius which is aimed at enhancing the exploitation and exploration abilities of regular WOA. The performance of this method was evaluated using twelve standard datasets from UCI Repository and it was observed that it performed better in terms of selecting fewer features while maintaining high classification accuracy compared to other popular techniques for feature selection. According to the results obtained from the experiments, HMNWOA has succeeded in finding a compromise between exploration and exploitation thus ensuring strong search ability coupled with fast convergence. The Whale Optimization Algorithm (EWOA) tries to solve original WOA problems by this paper authors [13] who proposed it. The main problems of basic WOA are slow convergence and difficulty in escaping from local optima. Moreover, REWOA links some special strengthenings with random strategies for evolution in order to increase exploration effectiveness and exploitations efficiency within algorithmic search space. Additionally, proposed method stabilizes WOA via introducing adaptive weights, Gaussian perturbations and differential mutations. The REWOA is able to outperform WOA and other meta-heuristic algorithms not only on 23 benchmark functions but also on various Hammerstein model identification problems due to its quickness of reaching precise solutions during convergence speed-up phase demonstrated by experimental results presented hereinbefore this sentence according whereof there has been carried out test with respect to twenty three benchmark functions along different Hammerstein model identification problems. It is also shown how REWOA can be used for solving difficult optimization problems successfully. Whale Algorithm An algorithm called the Whale Optimization Algorithm (WOA) was developed by metaheuristic algorithm using nature as inspiration for tackling difficult global optimization problems [14]. This algorithm copies how humpback whales catch their prey by using bubble nets to encircle and trap them into spirals made up of characteristic bubbles. The process of optimization for WOA has two phases; exploration phase where the solution space is searched extensively and exploitation phase that exploits this information in order to find a better solution. For WOA to function, a population of potential solutions known as whales must first be initialized. These solutions use the most well-known locations in the search space to update their positions iteratively. The algorithm mimics how whales hunt by alternating between encircling and spiraling towards their prey. Using a dual approach, WOA can better find the global optimum solution and escape local optima by striking a balance between local and global search processes. For the WOA to work, you need to start by creating a group of solutions whales. These solutions improve their position in each iteration by using the well-known areas within the search space. The algorithm emulates the hunting behavior of whales which involves surrounding prey alternately with spiraling towards them. WOA has an advantage over other algorithms because it strikes both locally and globally at once hence being able to find a better solution towards global optima while avoiding being trapped at local ones. Algorithm 1: Standard WOA 1. Initialize the parameters: - Population size (N) - Maximum number of iterations (Max_iter) - Problem dimensions (D) 2. Generate initial population (X) randomly 3. Evaluate the fitness of the initial population 4. Identify the best solution (X_best) in the initial population 5. Set iteration counter (iter) to 0 6. Repeat until iter < Max_iter: a. For each whale (i) in the population: i. Update position using the following equations: - If rand < 0.5: - Update position using encircling prey mechanism: X_i = X_best - A * |C * X_best - X_i| - Else: - Update position using spiral updating position: X_i = X_best * e^(b * l) * cos(2 * π * l) + X_best ii. Apply boundary constraints to ensure positions are within the search space b. Evaluate the fitness of the updated population c. Update the best solution (X_best) if a better solution is found d. Increment iteration counter (iter) 7. Return the best solution (X_best) and its fitness 41 III. METHODOLOGY This study seeks to improve population initialization by using advanced quasirandom sequences instead of standard random initialization such as Torus, Sobol and Halton sequences. These sequences are known for their low discrepancy properties that enhance diversity and uniformity in the initial population. This is advanced in order to overcome the limitations of Standard WOA by providing a more comprehensive and efficient search process. Significant improvements have been shown in exploration exploitation abilities; convergence speed as well as precision were demonstrated when these quasi random numbers were used. Figure 1. Flow chart for standard WOA A. Random Number Generator To ensure all potential solutions are evenly distributed across the search space, a uniform distribution was used to initialize the population in this work [15]. Under the uniform distribution technique, each person is equally and randomly placed within the boundaries of the solution space. This leads to a broad range of starting points that are uniformly spaced, thereby increasing the diversity of the initial population for optimization. Uniformly distributed initial populations reduce the chances of premature convergence and promote exploration ability of algorithms by avoiding clustering and ensuring equal representation of every part of the exploration space. However, this provides only a basic method against which more sophisticated initialization strategies such as Torus, Sobol or Halton sequences can be compared for their effect on the performance of optimization algorithms. The initial population was distributed uniformly in a twodimensional space as illustrated in Figure 2, showing how points were spread evenly throughout the area. Figure 2. Random Number Generator B. Sobol 42 Optimization algorithms use a Sobol [16] sequence to start off populations of points. This sequence helps to increase initial diversity in the population as it is known for creating uniformly distributed points in multi-dimensional space. It boosts search capability by ensuring that there is enough exploration within any given area since all parts of the search space are covered equally often. The research made use of the Sobol sequence for the initialization of the Improved Whale Optimization Algorithm (IWOA) population. In turn, this led to faster convergence rates and higher precision in performance measures. It is employed so as to exhibit its effectiveness in dealing with strong search dynamics along with complex function optimization tests. When compared against completely random initialization methods, Sobol sequences can be seen as capable of bridging the gap between clusters which might appear if local optima were reached prematurely. The global optimum is more likely to be found when a wider region is searched therefore offering thorough coverage across such space through employing Sobol sequences which ensures systematicity throughout such coverage while increasing chances of finding the global optimum. The performance of the Sobol sequence in handling robust search dynamics and complex function optimization tests was demonstrated in this study.. Figure 3. Sobol Uniform Distribution C. Holton Sequence The Halton sequence [17] is well-known for its ability to produce points that are evenly distributed across several dimensions, which greatly enhances initial population diversity. This kind of distribution is crucial as it ensures exhaustive coverage over the entire search space thereby improving algorithmic exploration unlike regular random initialization methods that may lead to non-uniformity or clustering represented by some areas having more points than others because when you use Halton sequences there isn’t any region with higher point density than another. Such equity bars premature attraction towards local optima making global optima easier to find. In the Improved Whale Optimization Algorithm (IWOA), researchers employed Halton sequences during population initialization stage. It was noted that application of these sequences yielded better measures of performance like faster convergence rates and higher solution accuracies thereby showing how effective they can be in sustaining multitude whiles solving highly intricate problems optimally. Figure 4. Holton Uniform Distribution D. Torus Sequence To begin with, optimization algorithms use the Torus sequence, a highly developed quasi-random sequence [18], to establish populations. In contrast to conventional random initialization methods that can result in clustering and uneven search space coverage, the Torus sequence guarantees a more uniform and low-discrepancy distribution of points. This approach increases diversity within an initial population thereby enhancing exploration capabilities of an algorithm. The Torus sequence can be systematically applied throughout the entire search domain which prevents premature convergence while promoting global optima identification. By employing the Torus sequence for initializing populations in Improved Whale Optimization Algorithm (IWOA) convergence rate and accuracy were improved significantly. This shows how powerful this technique is when it comes to maintaining general effectiveness during optimization procedure through preserving variety among individuals forming a population. Figure 5. Torus Uniform Distribution E. Improved Whale Algorithm The Improved Whale Optimization Algorithm (EWOA) is designed to overcome the deficiencies of the original method and improve its performance. IWOA utilizes advanced quasi-random sequences like Torus, Sobol, Halton etc., when initializing populations which greatly increases diversity and evenness among initial populations. This modified initialization technique ensures more homogenous distribution over the entire search space hence preventing premature convergence while boosting chances for locating global optima. Furthermore, IWOA employs controlled parameters for enhancing exploitation capability that results 43 into faster convergence during solution search thus making it more robust and efficient. Meanwhile, IWOA increased exploration potential guarantees thorough coverage within the whole solution space while the enhanced exploitation mechanism allows for fine-tuning towards an optimal solution. Extensive simulations and benchmark studies have demonstrated that IWOA outperforms standard WOA in terms of convergence speed, solution accuracy and overall optimization effectiveness. Algorithm 2: The Improved WOA 1. Initialize the parameters: - Population size (N) - Maximum number of iterations (Max_iter) - Problem dimensions (D) - Quasi-random sequence method (e.g., Torus, Sobol, Halton) 2. Generate the initial population (X) using the chosen quasi-random sequence method 3. Evaluate the fitness of the initial population 4. Identify the best solution (X_best) in the initial population 5. Set iteration counter (iter) to 0 6. Repeat until iter < Max_iter: a. For each whale (i) in the population: i. Update position using the following equations: - If rand < 0.5: - Update position using encircling prey mechanism: X_i = X_best - A * |C * X_best - X_i| - Else: - Update position using spiral updating position: X_i = X_best * e^(b * l) * cos(2 * π * l) + X_best ii. Apply boundary constraints to ensure positions are within the search space b. Evaluate the fitness of the updated population c. Update the best solution (X_best) if a better solution is found d. Increment iteration counter (iter) 7. Return the best solution (X_best) and its fitness IV. RESULTS AND DISCUSSION The Improved Whale Optimization Algorithm (IWOA), was tested in the context of optimization on a 10th Gen Intel Core i7 at 2.9 GHz machine. The proposed IWOA is evaluated over ten well-known benchmark test functions with respect to the WOA algorithm. We selected these functions aimed to consider a variety performance metrics including exploration, exploitation and convergence speed. The test function definitions and their minimum values can be found in Table I, while the experimental results showing the effectiveness of our proposed methods are illustrated in Table II. A. Parameter Setting We choose a population of 40 for initialization and consider 1000, 2000, 3000, 4000, and 5000 iterations. Also, for each iteration the dimensions are set to 10, 20, 30, 40, and 50. B. Analysis The different initialization strategies employed by the Improved Whale Optimization Algorithm (IWOA) are thoroughly examined in this section. In particular, we assess how well Sobol, Halton, and Torus sequences perform compared to the conventional random initialization technique used by the original WOA. This analysis aims to comprehend how these quasi-random sequences affect the algorithm's optimization efficacy, exploration and exploitation capabilities, and convergence speed. We methodically assess each initialization strategy using a set of benchmark test functions. Our goal in conducting this comparative study is to demonstrate how they can improve convergence accuracy and prevent premature convergence, which can improve the standard WOA. As indicated in Table II, the outcomes of these comparisons offer insightful information about how well these initialization techniques perform when optimizing complex functions. TABLE I. Definitions and Properties of Benchmark Test Functions Sr. No Fn Name Equation Range Optimal Value F1 Sphere min𝑓(𝑡) = ∑𝑡𝑖2 𝐷 𝑖=1 [-100, 100] 0 44 F2 Schwefel's 2.22 min𝑓(𝑡) = ∑|𝑡𝑖| 𝐷 𝑖=1 + ∏|𝑡𝑖| 𝐷 𝑖=1 [-10, 10] 0 F3 Schwefel's 1.2 min𝑓(𝑡) = ∑(∑𝑡𝑗 𝑖 𝑗=1 )2 𝐷 𝑖=1 [-100, 100] 0 F4 Schwefel's 2.21 min𝑓(𝑡) =𝑚𝑎𝑥𝑖{|𝑡𝑖|,1≤𝑖≤𝐷 } [-100, 100] 0 F5 Generalized Rosen brock's min𝑓(𝑡) = ∑[100 (𝑡𝑖+1− 𝑡𝑖2)2+ (𝑡𝑖− 1)2] 𝐷−1 𝑖=1 [-30, 30] 0 F6 Step Function min𝑓(𝑡) = ∑(⌊𝑡𝑖+0.5⌋)2 𝐷 𝑖=1 [-100, 100] 0 F7 Quartic Function min𝑓(𝑡) = ∑𝑖.𝑡𝑖4+𝑟𝑎𝑛𝑑𝑜𝑚[0,1] 𝐷 𝑖=1 [-1.28, 1.28] 0 F8 Rotated hyper ellipsoid min𝑓(𝑡) = ∑∑𝑡𝑗2 𝑖 𝑗=1 𝐷 𝑖=1 [-65.536, 65.536] 0 F9 Moved axis min𝑓(𝑡) = ∑5𝑖.𝑡𝑖2 𝐷 𝑖=1 [-5.12, 5.12] 0 F10 ChungReynolds min𝑓(𝑡) = (∑𝑡𝑖2 𝐷 𝑖=1 )2 [-100, 100] 0 TABLE II. COMPARISON OF STANDARD WOA, TORUS, SOBOL, AND HALTON INITIALIZATION TECHNIQUES Functions Iter*Dim WOW WOA - S WOA-H WOA-T F1 1000*10 2000*20 3000*30 4000*40 5000*50 1.46E-102 3.96E-120 9.63E-135 3.01E-145 1.47E-159 6.99E-102 2.74E-122 6.80E-136 4.24E-147 1.44E-160 2.21E-102 7.92E-120 2.67E-134 5.18E-146 3.86E-159 2.16E-103 1.47E-123 5.47E-136 2.43E-148 3.69E-160 F2 1000*10 2000*20 3000*30 4000*40 5000*50 1.70E-61 1.08E-72 9.31E-83 4.46E-91 8.21E-99 9.71E-62 1.53E-73 5.67E-83 2.40E-92 6.16E-99 1.45E-60 7.10E-74 1.54E-81 1.06E-89 5.64E-97 7.80E-62 4.10E-74 3.06E-85 1.62E-93 1.30E-100 F3 1000*10 2000*20 3000*30 4000*40 5000*50 1.01E+02 3.21E+03 1.64E+04 4.25E+04 7.26E+04 2.02E+02 6.21E+02 4.64E+04 2.25E+03 4.26E+04 7.01E+02 3.21E+03 7.64E+03 2.25E+04 1.26E+04 2.62E+01 3.21E+02 1.64E+03 4.25E+03 7.26E+03 F4 1000*10 2000*20 3000*30 4000*40 5000*50 2.65E+00 6.85E-01 5.70E-02 1.35E-02 4.83E-01 1.67E+00 1.33E-01 2.70E-02 1.16E-02 2.43E-01 2.21E+00 1.43E-01 4.50E-02 4.20E-02 3.89E-01 3.50E+00 5.41E-01 4.21E-02 2.31E-02 5.31E-01 F5 1000*10 2000*20 3000*30 4000*40 5000*50 1.16E+01 2.84E+01 2.55E+01 3.56E+01 4.58E+01 2.12E+00 1.58E+01 2.44E+01 3.43E+01 4.63E+01 1.13E+01 2.52E+01 2.66E+01 3.57E+01 4.32E+01 1.77E+00 1.62E+01 2.62E+01 3.58E+01 4.60E+01 F6 1000*10 2000*20 3000*30 4000*40 6.29E-04 8.71E-04 5.97E-04 3.00E-04 5.04E-04 4.78E-04 4.17E-04 2.17E-04 5.04E-04 8.92E-04 4.76E-04 5.96E-04 3.07E-04 2.23E-04 3.14E-04 1.81E-04 45 5000*50 4.47E-04 4.22E-04 5.32E-04 3.95E-04 F7 1000*10 2000*20 3000*30 4000*40 5000*50 2.25E-49 8.21E-87 1.89E-129 0.00E+00 0.00E+00 3.56E-53 9.67E-91 2.41E-131 0.00E+00 0.00E+00 5.25E-52 8.57E-89 1.21E-132 0.00E+00 0.00E+00 3.12E-53 7.33E-93 1.02E-135 0.00E+00 0.00E+00 F8 1000*10 2000*20 3000*30 4000*40 5000*50 9.07E-52 3.77E-94 5.87E-131 0.00E+00 0.00E+00 4.21E-52 3.93E-95 5.87E-128 0.00E+00 0.00E+00 3.07E-54 3.77E-93 5.87E-137 0.00E+00 0.00E+00 2.19E-53 3.77E-95 5.87E-138 0.00E+00 0.00E+00 F9 1000*10 2000*20 3000*30 4000*40 5000*50 1.35E-199 8.27E-239 3.71E-263 4.12E-294 0.00E+00 2.23E-203 9.10E-245 6.81E-269 9.11E-296 0.00E+00 2.21E-147 4.64E-240 2.71E-271 3.17E-297 0.00E+00 2.44E-208 5.95E-245 2.66E-272 7.00E-299 0.00E+00 F10 1000*10 2000*20 3000*30 4000*40 5000*50 7.23E-106 2.44E-122 2.38E-137 3.37E-149 9.25E-161 1.53E-107 1.05E-122 1.52E-138 1.25E-149 2.10E-161 1.16E-106 2.09E-120 2.00E-136 1.09E-149 1.50E-161 7.02E-107 8.63E-122 1.02E-138 1.16E-149 2.11E-163 The Torus initialization technique performs better than the Sobol, Halton, and standard WOA initialization methods in all dimensions and iterations, as shown by the F1, F2, F3, F6, F7, and F9 results. However, as can be seen from the results for F4, the Sobol initialization technique outperforms the other initialization techniques. For F5, the Sobol initialization method works extremely well for 30 and 40 dimensions, while the Torus initialization method performs better for 10 and 20 dimensions. The Torus initialization method performs better once more at 50 dimensions. Regarding F8, the Halton initialization method performs admirably in 10 dimensions, but the Torus initialization method performs better in every other dimension. According to the results, the Torus initialization technique works better at 30, 40, and 50 dimensions for F10, while the Sobol initialization technique performs well at 10 and 20 dimensions. Figure 1 Performance Comparison of Quasi-Random Initialization Techniques F1. Figure 2 Performance Comparison of Quasi-Random Initialization Techniques F2. 46 Figure 3 Performance Comparison of Quasi-Random Initialization Techniques F3. Figure 4 Performance Comparison of Quasi-Random Initialization Techniques F4. Figure 5 Performance Comparison of Quasi-Random Initialization Techniques F5. Figure 6 Performance Comparison of Quasi-Random Initialization Techniques F6. Figure 7 Performance Comparison of Quasi-Random Initialization Techniques F7. Figure 8 Performance Comparison of Quasi-Random Initialization Techniques F8. Figure 9 Performance Comparison of Quasi-Random Initialization Techniques F9. Figure 10 Performance Comparison of Quasi-Random Initialization Techniques F10. 47