Hybrid white shark optimizer with differential evolution for training multi-layer perceptron neural network

2.1 Overview of DE

DE, introduced by Rainer Storn and Kenneth Price in 1995, is a popular population-based optimization algorithm for solving multi-dimensional real-valued functions. DE operates using three main operators: mutation, crossover, and selection. The structure of DE is defined as follows:

Population initialization: DE begins by initializing a population of N individuals randomly within the search space. Each individual’s initial position X i ( 0 ) is generated within the lower and upper bounds of the search space, as shown in the following equation:

Mutation: DE creates a mutant vector V i ( t ) for each individual by adding the weighted difference between two randomly selected population vectors to a third vector, as shown in the following equation:

where F is the scaling factor controlling the mutation strength, and r 1 , r 2 , r 3 are distinct random indices from the population.

Crossover: The next step is to combine the mutant vector with the target vector to form a trial vector U i ( t ) . This is done through a crossover operation, as shown in the following equation:

where CR is the crossover probability, rand j is a random number between 0 and 1, and j rand is a randomly chosen index.

Selection: In the selection phase, the trial vector is compared to the target vector based on their fitness values. The vector with the better fitness value is selected for the next generation, as shown in the following equation:

Despite its robustness, DE has some limitations in terms of balancing exploration and exploitation. Excessive exploration caused by large mutation factors can slow down convergence, while too much exploitation, caused by smaller factors, can lead to premature convergence and getting stuck in local optima. The mutation step in equation (2) contributes significantly to exploration, but the overall diversity of the population may decline after several generations, making it prone to stagnation.

2.2 Overview of WSO

The WSO is a bio-inspired metaheuristic algorithm that simulates the hunting behavior of white sharks, particularly their ability to detect, pursue, and capture prey. The WSO algorithm operates by balancing exploration and exploitation through position and velocity updates, driven by both random and deterministic factors. Below, we outline the key steps and equations used in the WSO.

Population initialization: WSO starts by randomly initializing the positions of a population of white sharks ( w h i t e S h a r k s ) within the search space. The positions are initialized as in the following equation:

where X i ( 0 ) is the initial position of the i th white shark, and lb and ub are the lower and upper bounds of the search space, respectively.

Velocity initialization: The initial velocity of each white shark is set to zero, as shown below:

Fitness evaluation: The fitness of each white shark is computed using the objective function f o b j . The initial fitness of the population is calculated and stored as follows:

Velocity update: During each iteration, the velocity of each white shark is updated based on both the best-known global position ( g b e s t ) and the best position of a randomly chosen individual ( w b e s t ). This update is influenced by an inertia parameter μ and a weighting factor wr , as shown in the following equation:

where ν ( i ) is the index of a randomly selected white shark, and wr is a random weighting factor that ensures the stochastic nature of the velocity updates.

Position update: The position of each white shark is updated based on its velocity and the wavy motion of the shark. The position update is governed by the frequency f , which controls the wavy motion and is calculated based on the minimum and maximum frequencies ( f min and f max ), as shown in the following equation:

The new position of the white shark is then updated as follows (as shown in equation (10)):

The position update ensures that white sharks follow a wavy motion, characteristic of their natural hunting behavior.

Boundary check: After updating positions, WSO ensures that all individuals remain within the boundaries of the search space. This boundary correction is applied as the following equation:

Sensing mechanism and prey pursuit: WSO incorporates a prey-sensing mechanism where sharks adjust their positions based on their proximity to the global best position ( g b e s t ). If the shark senses prey, it uses a “sensing” parameter m v to either randomly relocate within the bounds or continue moving based on its velocity. The probability of pursuing prey is adjusted during iterations, and the sensing parameter m v is updated as shown below:

where a 0 and a 1 are constants controlling the decay of the sensing mechanism over time.

Fishing school mechanism: Another key feature of WSO is the “fishing school” mechanism, where white sharks follow the best global position while accounting for their distance from g b e s t . If a randomly generated number is smaller than the school size parameter s s , the shark updates its position according to the following rule (as shown in equation (13)):

where rand is a random number between 0 and 1, and X i − 1 ( t ) is the previous position of the neighboring shark.

Fitness update and global best update: After updating the positions, the fitness of each white shark is recalculated. If the new fitness of an individual is better than its previous fitness, the individual’s best position is updated. The global best position ( g b e s t ) is updated if the best individual’s fitness improves further, as shown in the following equation:

The White Shark Optimizer (WSO) presents several limitations that impact its performance in optimization problems. One significant drawback is its susceptibility to local optima trapping, particularly in high-dimensional and complex multimodal search spaces. While WSO incorporates mechanisms to balance exploration and exploitation, it may still converge prematurely to suboptimal solutions. Additionally, WSO’s performance is highly sensitive to certain key parameters, such as the frequency of the wavy motion ( f ), acceleration coefficient ( τ ), and movement forces ( p 1 , p 2 , m v , s s ). Improper tuning of these parameters can result in poor convergence behavior or inefficient searches.

Another limitation of WSO is the curse of dimensionality, as its effectiveness tends to degrade with increasing problem dimensions, making it less suitable for large-scale optimization tasks. Finally, although WSO attempts to balance exploration and exploitation dynamically, improper balance can lead to excessive wandering in the search space or premature convergence, limiting its ability to find optimal solutions efficiently.

2.3 Overview of FNNs and MLP

FNNs are a type of neural network where the connections between neurons flow in one direction: from the input layer, through any hidden layers, and finally to the output layer. In FNNs, information only moves forward, with no loops or feedback connections. One specific type of FNN is the MLP, which contains one or more hidden layers. MLPs are widely used for supervised learning tasks, including classification and regression, due to their ability to learn complex patterns in data.

The structure of an MLP consists of three main components: the input layer, hidden layer(s), and output layer. The input layer consists of n neurons, each corresponding to one of the n features in the input data. The hidden layer (or layers) contains h neurons, with each neuron in the hidden layer receiving inputs from all neurons in the previous layer. The output layer has m neurons, and the number of neurons in this layer depends on the type of problem being solved (e.g., one neuron for binary classification, multiple neurons for multi-class classification or regression).

Each connection between neurons has an associated weight, which is adjusted during training to minimize the error between the predicted outputs and the actual values. Additionally, each neuron has a bias term that helps the network fit the data better by shifting the activation function.

The computation in an MLP proceeds in several steps. First, each neuron in the hidden layer computes the weighted sum of its inputs from the input layer, adding a bias term. This can be expressed mathematically as follows:

where s j represents the weighted sum for the j th neuron in the hidden layer, where W i j is the weight connecting the i th input neuron to the j th hidden neuron, X i is the input value, and θ j is the bias for the j th hidden neuron.

After calculating the weighted sum, each hidden neuron applies a nonlinear activation function to introduce nonlinearity into the network, which allows the network to model complex relationships in the data. A commonly used activation function is the sigmoid function, which is defined as follows:

where S j represents the output of the j th hidden neuron after applying the sigmoid activation function to the weighted sum s j .

The outputs of the hidden layer neurons are then passed to the output layer. Each neuron in the output layer computes a weighted sum of the inputs from the hidden layer, similar to the process in the hidden layer:

where o k is the weighted sum for the k th output neuron, where w j k is the weight connecting the j th hidden neuron to the k th output neuron, and θ k is the bias for the k th output neuron.

Finally, the output layer applies an activation function to the weighted sum to produce the final output. For classification tasks, the sigmoid function is commonly used in the output layer as well:

where O k represents the final output of the k th output neuron after applying the sigmoid function to the weighted sum o k .

The key variables in the MLP structure are essential for understanding how the network processes inputs and generates outputs. The weights W i j and w j k determine how strongly each neuron in one layer is connected to the neurons in the next layer. The biases θ j and θ k allow each neuron to shift its activation function, improving the network’s ability to fit the data. The inputs X i represent the raw data fed into the network, while the outputs S j and O k are the intermediate and final results produced by the network, respectively.

As illustrated by equations (15)–(18), the relationship between the inputs and outputs in an MLP is determined by the network’s weights and biases. These parameters are adjusted during the training process, which seeks to find the optimal set of weights and biases that minimize the error between the predicted outputs and the actual values. In Section 3, the hybrid WSODE will be applied to train the MLP by optimizing the weights and biases.

3.1 WSODE mathematical model

In this section, we introduce the WSODE algorithm, which is a hybrid approach combining the strengths of DE and the WSO. The goal of WSODE is to leverage DE for exploration and WSO for exploitation, aiming to enhance optimization performance. The procedure for WSODE can be broken down into two main phases: the DE phase and the WSO phase.

Population initialization: Initialize the population randomly within the search space (as shown in equation (19)).

where X i ( 0 ) is the initial position of the i th individual, N is the population size, and lb and ub are the lower and upper bounds of the search space, respectively.

DE phase: Run DE for a predefined number of iterations to explore the search space and find a good initial solution.

Mutation: Create a mutant vector by adding the weighted difference between two population vectors to a third vector (as shown in equation (20)).

where F is the mutation factor, r 1 , r 2 , r 3 are distinct random indices, and t is the current iteration .

Crossover: Create a trial vector by combining elements from the target vector and the mutant vector (as shown in equation (21)).

where CR is the crossover probability, rand j is a random number between 0 and 1, and j rand is a randomly chosen index.

Selection: Select the better vector between the trial vector and the target vector (as shown in equation (22)).

WSO phase: Use the best solutions from DE as the initial population for WSO to refine the solutions and find the global optimum. Velocity update: Update the velocity of each individual based on the best positions found so far (as shown in equation (23)).

where μ is an inertia parameter, wr is a weighting factor, and wbest ν ( i ) ( t − 1 ) is a randomly chosen individual from the best solutions.

Position update: Update the position of each individual based on its velocity (as shown in equation (24)).

where f is a factor controlling the wavy motion.

Boundary check and correction: Ensure that individuals remain within the search space boundaries (as shown in equation (25)).

Best solution update: Update the best positions found so far (as shown in equation (26)).

Tracking the best solution:

where ccurve ( t ) records the best objective function value found up to iteration t (as shown in equation (27)).

The hybrid WSODE algorithm as shown in Algorithm 1, combines the DE and WSO techniques to enhance optimization performance. In the initialization phase, the population is randomly initialized within the search space, covering a wide range of potential solutions (as shown in equation (19)). During the DE phase, a mutant vector is created by adding the weighted difference between two population vectors to a third vector (as shown in equation (20)). A trial vector is then formed by combining elements from the target vector and the mutant vector (as shown in equation (21)). The better vector between the trial vector and the target vector is selected based on their fitness values (as shown in equation (22)). This process is repeated for a predefined number of iterations to explore the search space.

In the WSO phase, as shown in Figure 1, the velocity of each individual is updated based on the best positions found so far (as shown in equation (23)). The position of each individual is then updated based on its velocity (as shown in equation (24)). A boundary check and correction ensure that individuals remain within the search space boundaries (as shown in equation (25)). The best positions found so far are updated based on the fitness values (as shown in equation (26)). This phase continues for the remaining number of iterations to refine the solutions and find the global optimum. Throughout the iterations, the algorithm records the best objective function value found up to each iteration (as shown in equation (27)).

Figure 1

WSODE Flowchart. Source: Created by the authors.

3.2 Exploration, exploitation, and local optima avoidance features of WSODE

WSODE adeptly balances exploration and exploitation through its integration of DE and the WSO. Each phase of the algorithm is tailored to enhance either exploration or exploitation, ensuring a comprehensive search of the solution space and effective refinement of potential solutions.

The exploration capabilities of WSODE are primarily driven by the DE component. DE is renowned for its ability to traverse the search space extensively, preventing the algorithm from getting trapped in local optima. This phase involves the mutation operation, where DE generates mutant vectors by adding the weighted differences between randomly selected population vectors to another vector (equation (20)). The crossover operation combines elements from the target vector and the mutant vector to produce a trial vector (equation (21)). This helps maintain diversity in the population and explores different combinations of solutions. Finally, DE ensures that only the best solutions are carried forward by selecting the better vector between the trial and target vectors (equation (22)). This selection process guarantees that the population evolves towards better solutions.

The exploitation capabilities of WSODE are primarily harnessed through the WSO component, which fine-tunes the solutions obtained from the DE phase. In the WSO phase, the velocity of each individual is updated based on the best positions found so far (equation (23)). The position of each individual is then updated based on its velocity (equation (24)). To ensure that individuals remain within the feasible search space, a boundary check and correction mechanism are employed (equation (25)). The best positions are continually updated based on the fitness values of the solutions, ensuring convergence towards the global optimum (equation (26)). Local optima avoidance is a crucial feature of WSODE. The DE phase introduces diversity through its mutation operation (equation (20)), which consistently generates new solutions from different areas of the search space, reducing the likelihood of the population getting stuck in suboptimal regions. Additionally, the crossover operation (equation (21)) further enhances this diversity by combining different vectors, ensuring the algorithm explores a wide range of solutions.

In the WSO phase, local optima avoidance is enhanced by the use of the velocity updates (equation (31)), which drive individuals toward both global and local best solutions while maintaining a degree of randomness through the weighting factor wr . The wavy motion (equation (9)) also helps introduce nonlinearity into the position updates, allowing the individuals to avoid settling into local optima. Moreover, the adaptive prey-sensing mechanism (equation (12)) enables the sharks to dynamically adjust their positions based on their proximity to the global best, ensuring that they continue exploring rather than prematurely converging. The “fishing school” mechanism (equation (13)) further enhances the algorithm’s ability to escape local optima by allowing individuals to adjust their positions relative to neighboring solutions, fostering collaboration within the population and encouraging exploration in different directions.

3.3 Solving single objective optimization problems using WSODE

WSODE is adept at solving single objective optimization problems, where the goal is to find the best solution that minimizes or maximizes a given objective function. The formulation of the objective function and the step-by-step process of solving such problems using WSODE are outlined below.

A single objective optimization problem can be mathematically formulated as:

where f ( X ) is the objective function to be optimized, and X represents the vector of decision variables within the search space defined by the lower and upper bounds lb and ub , respectively.

The algorithm begins by initializing the population randomly within the search space, ensuring a diverse set of initial solutions as described by equation (19). In the DE phase, mutant vectors are generated using equation (20), and trial vectors are created by combining elements from the target and mutant vectors as per equation (21). The better vectors are then selected based on their fitness values, ensuring only the best solutions are carried forward, as formalized by equation (22).

In the WSO phase, the velocities of individuals are updated based on the best positions found so far (equation (23)), and their positions are updated accordingly (equation (24)). A boundary check and correction ensure solutions remain within search space boundaries (equation (25)), and the best solutions are updated based on their fitness values (equation (26)).

The convergence of the optimization process is tracked by recording the best objective function value found up to each iteration using equation (27). The algorithm concludes by returning the best solution found and the convergence curve, indicating the progression of the optimization process.

3.4 Computational complexity analysis of WSODE

The computational complexity of WSODE is analyzed by examining both the DE and WSO phases. Each phase contributes to the overall complexity, which depends on the population size N , the dimensionality of the problem d i m , and the maximum number of iterations itemax .

DE phase complexity: In the DE phase, the algorithm starts by initializing a population of N individuals, each with d i m dimensions. The main operations during each iteration include mutation, crossover, and selection:

Mutation involves selecting three random vectors from the population and applying a mutation strategy. This operation takes constant time per individual, i.e., O ( 1 ) , since it only requires arithmetic operations.

Crossover is performed over the d i m dimensions for each individual. The algorithm iterates over all the dimensions, combining the target and mutant vectors, as shown in equation (28). This step has a complexity of O ( d i m ) .

Selection compares the trial and target vectors based on their fitness values, and the better vector is retained for the next generation. The selection step is performed for each individual, and the fitness is evaluated in constant time, i.e., O ( 1 ) , as shown in equation (29).

The DE phase runs for D E _ i t e r m a x = itemax 2 iterations. Therefore, the total complexity of the DE phase is O ( N × dim × itemax 2 ) , as shown in equation (30).

WSO phase complexity: After the DE phase, WSO takes over and refines the solutions further. The main operations in the WSO phase include velocity updates, position updates, and fitness evaluations.

Velocity update is performed for each individual by calculating the difference between the current position and the best-known solutions, scaled by a random weighting factor. This operation is performed over all d i m dimensions and has a complexity of O ( d i m ) , as shown in in the following equation:

Position update involves updating each individual’s position based on the velocity and wavy motion. This step also runs over d i m dimensions and has a complexity of O ( d i m ) , as shown in the following equation:

Fitness evaluation is carried out after updating the positions. The fitness of each individual is evaluated in constant time, i.e., O ( 1 ) , assuming the objective function evaluation is constant.

The WSO phase runs for itemax iterations, so the total complexity of the WSO phase is O ( N × dim × itemax ) , as shown in in the following equation:

Overall complexity: The total computational complexity of WSODE is the sum of the complexities of both the DE and WSO phases. This is represented as:

Simplifying the expression results in the overall complexity of WSODE

where N is the population size, dim is the dimensionality of the problem, and itemax is the total number of iterations.

4.1 IEEE congress on evolutionary computation CEC2022 benchmark description

The assessment of WSODE efficacy leveraged a comprehensive array of benchmark functions from the CEC2022 competition. These functions are crafted to probe the capabilities and flexibility of evolutionary computation algorithms in diverse optimization environments. The suite includes various types of functions: unimodal, multimodal, hybrid, and composition. Unimodal functions, exemplified by the Shifted and Full Rotated Zakharov function (F1), evaluate the fundamental search capabilities and convergence attributes of the algorithms. In contrast, multimodal functions, such as the Shifted and Full Rotated Levy function (F5), present multiple local optima, thus testing the algorithms’ global search proficiency. Hybrid functions, with Hybrid function 3 (F8) as an instance, integrate aspects of different problem domains to reflect more intricate and practical optimization challenges. Furthermore, composition functions, notably Composition function 4 (F12), combine various problem landscapes into a singular test scenario, assessing the adaptability and resilience of the algorithms.

4.2 IEEE congress on evolutionary computation CEC2021 benchmark description

The CEC2021 benchmark functions are a set of standardized test functions designed to evaluate and compare the performance of optimization algorithms. These functions encompass various types of optimization challenges, including unimodal, multimodal, hybrid, and composition functions, each presenting unique complexities and characteristics. Unimodal functions have a single global optimum, while multimodal functions contain multiple local optima, making them challenging for optimization algorithms to navigate. Hybrid functions combine features from different types of functions, and composition functions blend multiple sub-functions to create highly complex landscapes. The primary goal of these benchmarks is to provide a rigorous and diverse set of problems that can thoroughly test the robustness, efficiency, and accuracy of optimization techniques in a controlled and consistent manner.

4.3 IEEE congress on evolutionary computation CEC2017 benchmark description

The CEC2017 benchmark suite, launched at the IEEE congress on evolutionary computation in 2017, provides a sophisticated array of test functions specifically designed to evaluate and enhance the capabilities of optimization algorithms. This suite is a significant update from previous iterations, incorporating new challenges that accurately reflect the evolving complexities found in real-world optimization scenarios. The suite is systematically organized into different categories including unimodal, multimodal, hybrid, and composition test functions, each tailored to assess distinct aspects of algorithmic performance. Unimodal functions within the suite test the algorithms’ ability to refine solutions in relatively simple environments, focusing on the depth of exploitation required to achieve optimal results. Multimodal functions, by contrast, challenge algorithms on their exploratory capabilities, essential for identifying global optima in landscapes populated with numerous local optima. Hybrid functions examine the versatility of algorithms in handling a mix of these environments, while composition functions are designed to test the algorithms’ proficiency in managing a combination of several complex scenarios concurrently. The design of the CEC2017 functions aims to rigorously evaluate not just the accuracy and speed of algorithms in reaching optimal solutions, but also their robustness, scalability, and adaptability in response to dynamic and noisy environments. This comprehensive testing is crucial for advancing metaheuristic algorithms and other evolutionary computation techniques, ensuring they are sufficiently robust and versatile for practical applications. The CEC2017 benchmark suite thus stands as a vital resource for the optimization community, offering a structured and challenging environment for the continuous evaluation and refinement of algorithms. It plays a pivotal role in driving the innovation and development of advanced optimization methods that are capable of addressing the complex and dynamic challenges present in various sectors.

5.1 Setting parameters for benchmark testing

The setting of parameters is essential for the uniform evaluation of optimization algorithms using the benchmark functions established by the CEC competitions in 2022, 2021, and 2017. These established parameters ensure a standardized environment that facilitates comparative analysis of different evolutionary algorithms. Table 1 provides a summary of these settings across all benchmarks.

The chosen population size of 30 and dimensionality of 10 strike an effective balance between computational manageability and the complexity needed for significant testing. A limit of 1,000 function evaluations ensures adequate iterations for the algorithms to demonstrate their potential for convergence, without imposing undue computational demands. The specified search range of [ − 100 , 100 ] D provides a wide and uniform testing space across all dimensions.

The benchmark functions frequently incorporate rotation and shifting to enhance the complexity, better mimicking real-world optimization scenarios. Excluding noise focuses the results on the algorithms’ ability to adeptly navigate complex environments, rather than dealing with random variations. This structured setting enables an equitable comparison among different algorithms, showcasing their strengths and weaknesses within a broad spectrum of benchmark tests (Table 2).

In conducting a detailed statistical evaluation of various optimization algorithms, we utilized key statistical metrics including the mean, standard deviation (STD). The mean is crucial as it represents the central tendency, summarizing the average results achieved by the algorithms across multiple trials and providing an overview of their general performance levels. The STD is employed to measure the extent of variation or dispersion from the mean, which sheds light on the reliability and uniformity of the results from different tests. These measures are essential for determining the stability and predictability of the algorithms’ effectiveness.

5.2 Discussion of WSODE results on IEEE congress on evolutionary computation CEC 2022 benchmarks

The WSODE algorithm demonstrates outperforming performance across the CEC2022 benchmark suite, designed to evaluate optimization algorithms on complex, high-dimensional, multi-modal, and deceptive landscapes, which present significant challenges for conventional metaheuristics. As shown in Table ??, WSODE consistently outperforms or performs comparably to state-of-the-art algorithms such as WSO, GWO, WOA, MFO, FOX, SHIO, DBO, OHO, SCA, FVIM, and SHO. The results showcase WSODE’s capability to balance exploration and exploitation effectively, which is essential for avoiding premature convergence and achieving high-quality solutions across diverse problem landscapes.

In Function F1, which assesses unimodal performance where algorithms can effectively test their convergence capabilities, WSODE achieves the best performance with a mean value of 3.00 × 1 0 2 , ranking first overall. This result indicates WSODE’s superior convergence speed and precision, as it consistently finds the global optimum more reliably than other algorithms. The close ranking in Function F2, where WSODE ranks second with a mean value of 4.06 × 1 0 2 , reflects its robustness and stability in handling various optimization scenarios, outclassing other popular algorithms like GWO, WOA, and MFO. This result indicates WSODE’s adaptability to challenges posed by slightly more complex unimodal landscapes, demonstrating its reliability in diverse optimization tasks.

For Function F3, which includes deceptive elements to challenge the search strategy, WSODE ranks first with a mean value of 6.00 × 1 0 2 and exhibits a notably low STD, underscoring its ability to avoid deception traps and maintain consistency in identifying optimal solutions. In Functions F4 and F5, characterized by multi-modality and complex local optima configurations, WSODE maintains top-tier performance, securing second place in both functions. This performance, with a mean value of 8.16 × 1 0 2 for Function F4 and a leading value of 9.00 × 1 0 2 in Function F5, highlights WSODE’s resilience and adaptability to the intricate landscape of local optima.

In the highly rugged landscape of Function F6, WSODE secures first place with a mean value of 1.80 × 1 0 3 , surpassing competitors like WOA, FOX, and SHO. This outcome underscores WSODE’s capability to efficiently explore the search space and identify high-quality solutions in highly complex, multi-modal scenarios. Furthermore, WSODE consistently ranks first or second in Functions F7, F8, and F9, where the landscape complexity increases with the addition of narrow, deep basins and deceptive valleys. Specifically, WSODE ranks first in Function F7 with a mean value of 2.01 × 1 0 3 , displaying its robustness and consistency in complex, nonlinear landscapes.

For the complex multi-modal scenarios in Functions F10 and F11, WSODE achieves first place with mean values of 2.51 × 1 0 3 and 2.60 × 1 0 3 , respectively. These results validate WSODE’s strength in maintaining a strong balance between exploration of diverse areas and exploitation of promising solutions, ensuring that it does not converge prematurely on suboptimal points. Finally, WSODE ranks first in Function F12 with a mean value of 2.86 × 1 0 3 , further cementing its effectiveness in tackling high-dimensional, complex optimization problems with deceptive and rugged landscapes (Table 3).

The Wilcoxon signed-rank test results for the CEC 2022 benchmarks (Table 4) show that the WSODE optimizer demonstrates robust performance across most comparisons. Although WSODE faces challenges from DE, with only 1 win, 5 losses, and 6 ties, it performs better against GWO, achieving 8 wins, 1 loss, and 3 ties. WSODE exhibits dominance over WOA and BOA, winning all 12 functions in both cases without any losses or ties. Similarly, WSODE outperforms MFO with 6 wins, 2 losses, and 4 ties, and shows a strong advantage over SHIO with 11 wins and 1 tie. Against COA, WSODE secures 11 wins and suffers only 1 loss. In flawless comparisons, WSODE wins all functions against OHO (12 wins), SCA (11 wins, 1 tie), and GJO (11 wins, 1 tie). Against SHO, WSODE achieves 10 wins, 1 loss, and 1 tie.

5.3 Discussion of the WSODE results on IEEE congress on evolutionary computation CEC 2021

The WSODE algorithm demonstrates significant performance improvements across the CEC2021 benchmark suite, which includes diverse optimization landscapes with varying properties, designed to test algorithms on different aspects such as multi-modality, ruggedness, separability, and deceptive traps. Table 5 presents a detailed comparison of WSODE’s performance against other leading optimizers, including WSO, CMAES, particle swarm optimization (PSO), MVO, DO, MFO, SHIO, SDE, BAT, and FOX, across ten test functions (C1 to C10). These results highlight WSODE’s adaptability and superior ability to navigate complex optimization landscapes.

Function C1 is a high-dimensional unimodal function, aimed at evaluating an algorithm’s convergence speed and precision in simpler landscapes without local minima. WSODE achieves a mean value of 3.45 × 1 0 − 33 with a STD of 1.04 × 1 0 − 27 , ranking third overall. This performance demonstrates WSODE’s strong convergence capability, as it reaches near-global optima, though WSO and MFO achieve slightly better results on this function.

In Function C2, a unimodal function with added separability, WSODE ranks fourth with a mean value of 6.77 × 1 0 2 and a STD of 2.33 × 1 0 2 . The separable nature of C2 allows algorithms to optimize variables independently, and WSODE’s competitive ranking highlights its proficiency in tackling such challenges, despite the close competition with WSO and CMAES.

Function C3 presents a multi-modal, deceptive landscape, introducing numerous local optima to test an algorithm’s exploration efficiency. WSODE ranks fourth with a mean value of 3.06 × 1 0 1 and a STD of 2.37, showing its capability to maintain a strong balance between exploration and exploitation, even though WSO and MFO exhibit slightly better results in this context. In Function C4, which adds ruggedness to the multi-modal landscape, WSODE ranks third with a mean value of 1.86 and a STD of 2.34 × 1 0 − 1 , underscoring its ability to handle rugged, complex search spaces where robust exploration is necessary.

Function C5 introduces nonseparability and a high degree of variable interaction, challenging algorithms to explore global optima without decomposing the function. WSODE ranks second with a mean value of 1.46 and a STD of 3.54, outperformed only by WSO. This result reflects WSODE’s effective handling of nonseparable problems, outperforming traditional algorithms like CMAES, PSO, and MFO, which tend to struggle with such intricacies.

For Function C6, a deceptive and highly rugged multi-modal function, WSODE ranks first with a mean value of 3.32 × 1 0 − 1 and a STD of 2.96 × 1 0 − 1 . This result emphasizes WSODE’s superior exploration abilities, as it effectively navigates the deceptive traps to locate high-quality solutions. Similarly, for Function C7, which features complex landscapes with narrow, deep basins, WSODE also ranks first with a mean value of 3.36 × 1 0 − 1 and a STD of 2.81 × 1 0 − 1 . This performance highlights WSODE’s robustness in locating optima in tightly constrained areas of the search space.

Function C8, a high-dimensional, multimodal landscape with numerous local optima, poses a challenge for exploitation-dominant algorithms. WSODE achieves the best performance with a mean value of 4.07 × 1 0 − 16 and a STD of 5.90 × 1 0 − 16 , outperforming all other algorithms. This result underscores WSODE’s excellent balance of exploration and exploitation, as it avoids becoming trapped in local optima and successfully converges to global solutions.

In Function C9, which introduces deceptive features to test resilience against premature convergence, WSODE ranks second with a mean value of 1.60 × 1 0 − 14 and a STD of 8.16 × 1 0 − 15 . Although CMAES performs slightly better, WSODE’s close ranking highlights its robust adaptability to deceptive optimization challenges, maintaining high-quality results consistently across functions.

Finally, Function C10, a nonseparable, high-dimensional problem with deceptive traps and ruggedness, represents one of the most challenging functions in the CEC2021 suite. WSODE ranks first with a mean value of 4.90 × 1 0 1 and a STD of 2.11 × 1 0 − 1 , demonstrating its strong optimization capabilities in handling complex, rugged, and deceptive landscapes.

As shown in Table 6, the Wilcoxon signed-rank test results demonstrate that the WSODE optimizer consistently achieves significant performance advantages over other optimizers on the CEC 2021 benchmark set. WSODE secures a majority of significant wins across comparisons, achieving flawless or near-flawless results against several algorithms, including WOA, BOA, and OHO, with no losses. While WSODE demonstrates robustness across all comparisons, it encounters minimal challenges from GWO and MFO, which manage to achieve a single loss and several ties, respectively.

5.4 WSODE comparison results on IEEE congress on evolutionary computation 2017

The performance of WSODE was evaluated against several prominent optimizers, including WSO, CMAES, COA, RSA, BBO, AVOA, SDE, SCA, WOA, DO, MFO, SHIO, and AOA, on the benchmark functions provided by the IEEE CEC 2017 competition (F1–F15). As detailed in Table 7, WSODE exhibits superior performance across a majority of the tested functions, achieving the lowest mean values in numerous cases, which underscores its high optimization efficacy.