3.1 Initialization

During initialization, the population distribution and parameter settings are defined. The population is initialized by setting the individual positions, X_i, where each individual represents a solution and the population consists of candidate solutions. The fitness (objective function value) for each individual is then calculated, and the individual with the best fitness becomes the initial global best solution . Each whale’s position is randomly initialized in the search space. Assuming the search space is j-dimensional, the initial position of the individual is given below:

(1)

where ub and lb are the upper and lower bounds of the decision variables; and Rand is a random value between 0 and 1.

3.2 Encircling prey

The WOA search space is the global solution space, with the prey’s position determined first. Since the location in the solution space is not known a priori, WOA assumes the current best solution is the target, and other whale individuals approach the current best solution. When 1, whales perform a shrinking encircling action, gradually approaching the target and focusing on local development near the best solution. This helps improve convergence and optimization accuracy but may lead to premature trapping in local optima. The specific position update formula is as follows:

(2)

(3)

where is the current global best solution; X is the current whale position; and A and C are coefficients used to control the shrinkage and exploration, influencing the attraction or repulsion of whales to the prey. These coefficients are calculated as follows:

(4)

(5)

(6)

where parameter a starts at 2 and linearly decreases to 0 with iterations; r is a random value between [0, 1]; T is the maximum number of iterations; and t is the current iteration.

3.3 Bubble-net attacking method

Spiral updating is a unique bubble-net hunting behavior of humpback whales, which gradually approach the prey by spiraling around it. It is a local exploration strategy. When 0.5, the spiral updating formula is as follows:

(7)

(8)

(9)

(10)

where represents the distance between the current individual position X and the leader ; b is a constant defining the shape of the spiral (usually set to 1); and the spiral coefficient l ranges between [–2, 1].

The exponential decay and periodic movement are used to simulate the whale’s path toward the prey, while maintaining some randomness to enhance the global search ability and avoid local convergence. As the whale approaches the prey, the spiral movement becomes tighter. However, in complex search spaces, WOA’s position updates rely heavily on randomness and are entirely based on the current global best solution. This approach can be effective in the early stages but tends to converge near local optima in the later stages, struggling to escape local optima and achieve a balance between global and local search strategies.

3.4 Search-for-prey

When whales have not yet located the prey, to ensure all whales fully explore the solution space, one whale is randomly selected to search for the prey. This random search enlarges the exploration range and prevents trapping in local optima. When 1, the update formulas are as follows:

(11)

(12)

where X_rand is a random individual from the current population; and A controls how far the individual moves away from the current best solution to ensure global search capability.

3.5 Pseudo-code of WOA

The pseudo-code of the WOA is shown in Algorithm 1.

Algorithm 1. WOA

3.6 Advantages and disadvantages of WOA

WOA randomly initializes the population position vectors and sets the maximum number of iterations and parameter a firstly. Then, the global best solution is determined by evaluating the fitness. The hunting behavior of whales is simulated by updating the target position through encircling the prey and using random search strategies to select prey. The population positions are updated based on the hunting results and continue until the maximum iterations or convergence condition is met, outputting the global best solution. This algorithm approximates the optimal solution using bionics principles, offering excellent search performance and convergence. It is simple, easy to implement, and applicable to various optimization problems. Compared to complex algorithms, WOA is more computationally efficient. However, in high-dimensional, multimodal problems, the population may focus too much on local development, reducing global search capability. The algorithm’s performance is sensitive to parameter settings, and improper adjustment of parameters can reduce search effectiveness. As iterations progress, the population diversity gradually diminishes, affecting the search capability. Additionally, increasing the number of iterations increases computational costs. To address these issues, this paper introduces an enhanced whale optimization algorithm with multiple strategies, GWOA, which uses Good Nodes Set initialization to generate a uniform distribution of population, incorporates Growth-based Encircling Prey strategy with a newly-designed inertia weight , incorporates a novel designed Adaptive Sine-Cosine strategy, Synergetic Search-for-Prey strategy and an improved Cauchy Mutation Strategy based on DE. Meanwhile, the updating method of parameter a was resigned to better balance exploration and exploitation. These enhancements comprehensively improve the global search capability, convergence speed, population diversity and robustness of the WOA.

4.1 Good nodes set initialization

The core idea behind Good Nodes Set Initialization is to select a set of high-quality initial nodes (i.e., “good nodes”) to improve global search efficiency and optimization solution quality. The basic principle is to select several excellent nodes as the initial solutions or starting nodes for the population at the beginning of the algorithm. These nodes are usually determined through heuristic methods, statistics, or other techniques, rather than being generated purely randomly. This provides the optimization algorithm with a set of starting nodes that are closer to the global optimal solution. The traditional WOA algorithm uses pseudo-random number generation to generate the population. This method is simple and exhibits strong randomness, but it also has the problem of uneven population distribution. Randomly generated populations may be overly dense in some areas and sparse in others, resulting in uneven coverage of the search space, which negatively affects the efficiency and performance of the WOA algorithm during the search process. As shown in the Fig 1, GWOA uses Good Nodes Set initialization to generate a uniformly distributed population, improving the quality of the population.

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Fig 1. Populations generated by the initialization of the good nodes set.

https://doi.org/10.1371/journal.pone.0322494.g001

Let D_n be an n-dimensional Euclidean space’s unit cube, where there exists a point set:

(13)

where C(r, ) is a constant that depends on r and ( > 0). Then; P_M is called the Good Points Set.

The value of r is calculated as follows:

(14)

where p is the smallest prime number satisfying (p-3)/2 . The Good Nodes Set is mapped to the actual search space using the following mapping formula:

(15)

where ub_j and lb_j represent the upper and lower bounds of the dimension.

By providing high-quality initial nodes, Good Nodes Set initialization can accelerate the convergence speed of the algorithm, reduce unnecessary computations, and improve the quality of the solutions. It effectively avoids local optima issues caused by random initialization, especially in complex multi-modal optimization problems. Furthermore, it ensures uniform coverage of the search space, enhancing global search ability and preventing bias, thus improving the optimization algorithm’s performance and efficiency. This advantage is not only evident in two-dimensional spaces but also shows in high-dimensional spaces, as the construction of the Good Nodes Set itself is independent of the dimension.

4.2 Growth-based encircling prey

In the original prey encirclement strategy, position updates are based on a fixed linear relationship, making the search process overly dependent on the best solution. The search range remains unchanged throughout the process, which may cause the algorithm to over-explore in the early stages or fall into local optima during the exploitation phase. Shi et al. were the first to introduce the concept of inertia weight into the PSO algorithm, which led to significant performance improvement [34]. Inspired by inertia weights, this paper proposes an inertia weight update mechanism based on the Sigmoid function. The inertia weight is calculated as follows:

(16)

where is an inertia weight that decreases from 0.9 to 0; t represents the current iteration; and T is the maximum number of iterations.

The following are the common calculation formulas for :

(17)

(18)

(19)

As shown in the Fig 2, the major advantage of the proposed inertia weight lies in its smooth decrease. In the early stages, a larger gives the search agents strong exploration ability, allowing them to search the solution space widely. As iterations proceed, the reduction in reduces the dependence on the optimal solution for position updates, shifting from global exploration to local refinement. This mechanism balances exploration and exploitation at different stages, preventing premature convergence to a local optimum. Near the global optimal solution, it prevents large steps from causing excessive deviations from the optimal solution. It also makes position updates more flexible and avoids over-reliance on leaders.

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Fig 2. Comparison of Common and the Proposed .

https://doi.org/10.1371/journal.pone.0322494.g002

The Growth-based Encircling Prey strategy is defined as follows:

(20)

where is the proposed inertia weight; is the current global best solution; A is a coefficient vector; and represents the distance between the current individual position X and the leader , as shown in Eq 7.

4.3 Adaptive sine-cosine strategy

The Sine-Cosine Algorithm (SCA) is a population-based global optimization algorithm proposed by Seyedali Mirjalili [35]. It is inspired by the oscillatory behavior of the sine and cosine functions. The core idea of the SCA is to simulate the oscillatory behavior of the sine and cosine functions to find the global optimum. The algorithm is simple, easy to implement, and requires few parameters, making it suitable for various optimization problems. The optimization process of SCA is based on two main update mechanisms: the convergence property of the sine function and the oscillatory property of the cosine function. This allows SCA to explore the search space effectively and avoid local optima. To enhance the global search ability of WOA, we have introduced an adaptive sine-cosine strategy inspired by SCA. The adaptive sine-cosine strategy incorporates the oscillatory behavior of the sine function and the spiral updating behavior of the cosine function, enabling the algorithm to adaptively explore in the early stages and avoid local optima while balancing exploration and exploitation. The adaptive sine-cosine strategy is modeled as follows:

(21)

(22)

(23)

(24)

(25)

where r is a random number between 0 and 1; l is a random number that controls the whale’s movement pattern when searching for prey, the formula is shown in Eq 10; is the distance between the current individual X and the global best solution , the formula is shown in Eq 7; b is a spiral factor that adjusts the speed of the exponential function; r₁ is a dynamic factor that controls the range of the sine strategy; t represents the current iteration; T is the maximum number of iterations; r₂ is a random angle between 0 and 2; r₃ is a random factor between 0 and 2; and r₄ controls the periodic behavior of the update position.

The adaptive sine-cosine strategy introduces sine and cosine functions to randomly adjust the search direction and uses random factor r₃ to control the amplitude of the update. This provides more search dimensions and dynamic position updates, preventing the algorithm from sticking to a single hunting pattern that could cause it to get stuck in local optima. In the original WOA algorithm, the global exploration ability is strong in the early stages of a search, but as iterations progress, the search range shrinks, potentially slowing down the search speed in later stages. To address this, the Adaptive Sine-Cosine strategy introduces adaptive shrinking control parameters r₄ and r₁, allowing broad exploration in the early stages and gradual narrowing toward the target, reducing oscillations and deviations, which helps accelerate convergence. Additionally, the strategy uses the random factor r to control the selection between the sine and cosine hunting methods, allowing whale agents to perform both global and local searches. This enables dynamic control of the search phase increases the flexibility of the hunting behavior. Furthermore, the introduction of random angles and diversified range r₄ enhances diversity and jumping ability, improving the chances of escaping local optima. The non-linear form of the sine-cosine strategy effectively explores the complex fitness function’s search space, improving the algorithm’s adaptability to high-dimensional problems. This strategy enhances global search capability, avoids local optima, and helps the algorithm find the optimal solution faster and more efficiently. It is especially beneficial for complex, high-dimensional optimization problems, where it can find the global optimum in fewer iterations while exhibiting stronger robustness and adaptability.

4.4 Synergetic search-for-prey

In the original Search-for-prey strategy, a random whale is selected each time for the search. This may cause unnecessary fluctuations in the solution space, particularly in the middle stages of the algorithm, potentially leading to an unstable search process or even jumping out of the global optimum region. The strategy is overly reliant on the randomly chosen individual, which can cause the algorithm to get stuck in local optima, especially in complex solution spaces. To reduce the excessive randomness of this strategy, we propose a new Synergetic Search-for-Prey strategy. In the Synergetic Search-for-Prey strategy, position updates are made by referencing both the global optimal solution and the average position of the current whale agents, focusing the search near the current optimal solution. This enhances local search ability and helps explore the current region more effectively and avoids unnecessary long-distance jumps. The Synergetic Search-for-Prey strategy also avoids over-reliance on a single individual, improving the diversity of the search. By introducing both the global optimal solution and the average position of all whale agents, and introducing a random disturbance term. The position update formula better combines the overall information of the whale population. This enables the algorithm to escape local optima, enhancing its global search ability. The Synergetic Search-for-Prey strategy is modeled as follows:

(26)

(27)

where X_m represents the average position of the whale agents, calculated in Eq 27; r is a random number uniformly distributed in the interval [0, 1]; X is the position of the current whale agent; is the position of the current global optimal solution.

4.5 Improved cauchy mutation strategy based on differential evolution

The standard position update strategy in WOA relies on biomimetic principles like prey encirclement and spiral updates. These strategies generally focus on searching near the current optimal solution, particularly in the later stages when whale agents converge on a specific area, reducing the exploration of the search space. To enhance search diversity, a mutation strategy can be introduced after position updates to perturb the current solution and help whale agents escape local optima, allowing for broader exploration. This is particularly useful when solving complex problems with multiple local optima, as the mutation strategy helps agents avoid premature convergence.

Therefore, this paper proposes an improved Cauchy Mutation Strategy based on Differential Evolution. This strategy combines the mutation mechanism of the Cauchy distribution with the differential evolution approach, enhancing the global search capability of the algorithm. It can effectively avoid stagnation near local optima and improve the exploration and convergence properties in complex optimization problems.

First, a new intermediate solution is generated using the differential evolution strategy:

(28)

where X_i represents the position of the current ith agent; is the position of the current best individual; X₁, X₂, X₃ are positions of three randomly selected different individuals from the population; and F is a factor controlling the scaling of the differential vector, calculated as below:

(29)

where Rand is a random number between 0 and 1.

Next, Cauchy Mutation is applied to the intermediate solution :

(30)

where cauchy(0,1,1,dim) represents the perturbation generated by the Cauchy distribution.

Finally, boundary checks and adjustments are made to avoid population degradation in WOA, it as shown in Eq 31:

(31)

If the fitness of the mutated solution is better than the original solution X_i, then will replace X_i.

4.6 The pseudo-code of GWOA

The pseudo-code of the GWOA is shown in Algorithm 2.

4.7 Computational complexity analysis

4.7.1 Time complexity analysis.

The time complexity mainly depends on the computational effort required during each iteration. We need to analyze the key steps involved in each iteration.

Time Complexity of WOA:

• The initialization of WOA requires generating a random matrix of size SearchAgentsno dim, and the time complexity is O(SearchAgentsno dim);
• The objective function calculation involves running the outer loop for Maxiter iterations. The inner loop operates on SearchAgentsno search agents. The operations include checking boundary conditions and calculating the objective function value fitness = fobj, which has a time complexity of O(dim) because the objective function is computed for each position dimension. The leader update compares the objective function values of each search agent, which has a time complexity of O(1). Therefore, the overall objective function calculation time complexity is O(Maxiter SearchAgentsno dim);
• In each iteration, the positions of all search agents are updated based on certain mathematical formulas. Each search agent needs to traverse dim dimensions, so the time complexity for updating each agent’s position is O(dim). Each position update involves multiple formulas (e.g., updating A, C, D), most of which have a time complexity of O(1). Therefore, the time complexity for position updates is O(SearchAgentsno dim).

Algorithm 2. GWOA

Thus, the overall time complexity of WOA is O(Maxiter SearchAgentsno dim).

GWOA introduces some improvements compared to WOA, but its time complexity is similar to WOA.

Time Complexity of GWOA:

• The initialization process is the same as in WOA, with a time complexity of O(SearchAgentsno dim);
• The objective function calculation follows the same procedure as WOA. The outer loop runs Maxiter times, and the inner loop iterates over SearchAgentsno search agents. The time complexity for calculating the objective function and updating the leader is O(dim) for each agent;
• GWOA uses more complex strategies, including inertia weight updates, cosine-sine strategies, and multi-differential Cauchy mutation strategies. These increase the computational effort, and each update involves operations over dim dimensions, which has a time complexity of O(dim).

Therefore, the time complexity of GWOA is O(Maxiter SearchAgentsno dim) + O(Maxiter SearchAgentsno dim). The total time complexity is still O(Maxiter SearchAgentsno dim), but due to additional operations (such as Cauchy mutation and cosine-sine strategies), the actual computational effort is slightly higher.

5.1 Ablation study

In this section, we remove five improvement strategies from GWOA: GWOA without the Good Nodes Set initialization is named GWOA1; GWOA with the Growth-based Prey Encirclement Strategy replaced by the original WOA prey encirclement mechanism is named GWOA2; GWOA with the Synergetic Search-for-Prey strategy replaced by the original WOA prey search strategy is named GWOA3; GWOA with the Adaptive Sine-Cosine (ASC) strategy replaced by the original WOA spiral updating is named GWOA4; and GWOA without the Differential Evolution-based Enhanced Cauchy Mutation strategy is named GWOA5. The uniform settings include the maximum iteration T=500 and population size N=30. Each algorithm runs 30 times on 23 benchmark functions for performance analysis. The iteration curves are shown as Fig 3.

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Fig 3. Iteration curves in ablation study.

https://doi.org/10.1371/journal.pone.0322494.g003

From the figure, it is observed that the initialization of the Good Nodes Set generates a uniformly distributed whale population. This uniform distribution helps algorithm explore the solution space more effectively and quickly, overcoming the problem of poor optimization efficiency that may arise from improper initial population selection in WOA. As a result, the convergence speed and accuracy are improved on functions F5, F15, and F21-F23. The Growth-based Encircling Prey strategy enhances the dynamic balance between exploration and exploitation in the algorithm, overcoming the issues of local optima and slow convergence speed that may occur with fixed parameter settings in WOA. In particular, it focuses more on local exploitation in the later stages, which is advantageous when handling functions F1-F6, F9-F11. The Synergetic Search-for-Prey strategy increases dependence on the optimal solution and introduces randomness to provide more search directions for the whale individuals. This improvement reduces the insufficient exploration problem that may arise in WOA for complex problems. The random perturbation further enhances the global search ability of the algorithm, demonstrating better performance on functions F1-F4, F7, F10-F11. The Adaptive Sine-Cosine (ASC) strategy considers the distance between the whale’s current position and the current best solution, thereby strengthening the exploration ability in the early stages. This strategy effectively reduces the risk of WOA getting trapped in local optima in high-dimensional and complex optimization problems, improving the global search capability. It enables the algorithm to converge quickly to the optimal solution for functions like F1-F6, F12. From F1-F6, F21-F22, the improved Cauchy Mutation strategy based on DE introduces a new perturbation mechanism, effectively helping the whale individuals escape from the current local optimal area and avoid early convergence to suboptimal solutions. This strategy helps the algorithm maintain population diversity during the search process, preventing the solution aggregation phenomenon that may occur in WOA in some cases.

Through these improvement strategies, GWOA shows significant improvements in global search ability, convergence speed, and solution accuracy compared to the traditional WOA, making it more effective in solving complex multi-constrained optimization problems.

5.2 Qualitative analysis experiment

In the qualitative analysis, we record the individual search history of GWOA on benchmark functions, the exploration-exploitation ratio during the iteration process, and population diversity. This allows us to comprehensively evaluate the improvement effects of GWOA. The improvement points of GWOA over WOA can be summarized in the following Fig 4. The experimental setup includes a maximum iteration count of T=500, function dimension Dim=30 and a population size of N=30. The results of the experiment are shown in the Fig 5, Fig 6, Fig 7 below:

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Fig 4. GWOA vs WOA flowchart.

https://doi.org/10.1371/journal.pone.0322494.g004

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Fig 5. Results of the GWOA qualitative analysis (F1-F8).

https://doi.org/10.1371/journal.pone.0322494.g005

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Fig 6. Results of the GWOA qualitative analysis (F9-F18).

https://doi.org/10.1371/journal.pone.0322494.g006

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Fig 7. Results of the GWOA qualitative analysis (F17-f23).

https://doi.org/10.1371/journal.pone.0322494.g007

From the population search history in the figure, it can be observed that GWOA, benefiting from the Good Points Set strategy, explores most of the area in unimodal functions F1-F7 and quickly converges to the global optimum. In multimodal and compositional functions F8-F23, the trajectory covers multiple potential optimal areas before converging and focusing on the neighborhood of the optimal solution. This demonstrates a good balance between global exploration and local exploitation in GWOA. In functions F20-F23, local optima were found, indicating that GWOA, when facing complex function problems, may increase computation time and converge to a local optimum.

Further analysis of the changes in the exploration-exploitation ratio during the iteration process shows that a higher exploration ratio helps escape local optima in the early stages, while a higher exploitation ratio aids in accelerating convergence in the later stages. The Growth-based Encircling Prey strategy introduces an adaptive parameter adjustment mechanism to dynamically adjust key parameters in the algorithm. As a result, in functions F5-F6, F8, F11-14, and F17-23, GWOA’s exploration-exploitation ratio exhibits a favorable state. This strategy enhances the algorithm’s search capability at different stages, with initial exploration focusing on discovery and later-stage exploitation driving convergence. Its dynamic balancing capability reflects the strong adaptability of the algorithm. In functions F1-F4, F7, F9-F10, and F15-F16, the Synergetic Search-for-Prey strategy and the Adaptive Sine-Cosine (ASC) strategy contribute to fast convergence in the early stages, demonstrating GWOA’s ability to quickly find the optimal solution. GWOA incorporates the improved Cauchy mutation strategy based on Differential Evolution to enhance search diversity. In complex functions such as F14-F23, GWOA maintains higher population diversity, preserving global search ability and avoiding premature convergence. However, in functions F1-F13, rapid convergence leads to a sharp decline in population diversity.

In summary, the performance of GWOA reflects its ability to balance exploration and exploitation, maintain diversity, and achieve convergence precision. GWOA converges quickly and stably in unimodal, functions, indicating strong exploitation capability. In multimodal and compositional functions, GWOA maintains diversity and eventually converges, demonstrating enhanced global search ability. However, in some functions, an imbalance between exploration and exploitation may lead to premature convergence.

5.3 Comparison of different metaheuristic algorithms

To further verify the superiority of GWOA, we selected Dung Beetle Optimization Algorithm (DBO), Grey Wolf Optimizer (GWO), Harris Hawk Optimization Algorithm (HHO), Sine Cosine Algorithm (SCA), and Whale Optimization Algorithm (WOA), An enhanced whale optimization Algorithm (eWOA) [37], Modified Whale Optimization Algorithm (MWOA) [38], a multi-strategy WOA (MSWOA) [39], Attraction Repulsion Optimization Algorithm (AROA) [40], an improved version of the sand cat swarm optimization algorithm (ISCSO) [41] as comparisons. The detailed information of the algorithm is shown in Table A1 and the detailed parameter settings of the algorithm are shown in Table A2. The algorithms are tested on selected benchmark functions with the following unified settings: maximum iteration T=500, population size N=30 and function dimension Dim=30. Each algorithm runs 30 times on 23 classical benchmark functions to record the average fitness (Ave), standard deviation (Std), p-values of Wilcoxon rank-sum test, and Friedman values for performance analysis.

5.3.1 Parametric analysis.

As shown in Fig 8 and Table 4. GWOA performs significantly better than other algorithms on F1-F7, with both the Avg and Std being 0. The Avg and Std of other algorithms on these functions are notably higher than those of GWOA. This indicates that GWOA is able to find the optimal solution and performs very stably. The convergence speed and stability of GWOA allow it to quickly find the global optimum in unimodal function problems. GWOA also shows superior performance on F9, F11-F13, and F15, suggesting that it performs well on these multimodal functions, benefiting from the Synergetic Search-for-Prey strategy and its global search ability. It is better able to escape local minima and find the global optimum. The smoothness of its curve demonstrates GWOA’s robustness and strong adaptability and consistency for different types of optimization problems. GWOA also performs excellently on F8 and F14, thanks to the application of the Good Points Set initialization. In the early stages, GWOA can quickly narrow the search space, making the solution space more regular, allowing it to find the global optimum quickly and stably. On F10 and F16-F19, the GWOA curve is similar to other algorithms because most algorithms can find very close solutions, showing minimal performance differences. On F20-F23, GWOA performs well but the results are similar to those of some other algorithms. This is because these algorithms converge to near-optimal solutions, with no significant differences. However, GWOA has a smaller Std, indicating stronger stability. By integrating various improvement strategies, GWOA possesses powerful global search ability and high stability, performing outstandingly on both unimodal and multimodal functions. In complex compositional functions, the results of GWOA are similar to those of other excellent algorithms like MSWOA and eWOA, as these algorithms can also quickly find the global optimum or near-optimal solution in these cases.

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Fig 8. Iteration curves for comparison of different algorithms.

https://doi.org/10.1371/journal.pone.0322494.g008

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Table 4. Ave and Std of different algorithms.

https://doi.org/10.1371/journal.pone.0322494.t004

However, in performance evaluation of optimization algorithms, Avg and Std are often used to measure convergence and stability, but they may not directly reflect the algorithm’s superiority. Relying solely on these two metrics has some limitations in comparing different algorithms. So non-parametric statistical methods, such as Wilcoxon rank-sum test and Friedman test, are often used for more in-depth analysis and more reliable performance validation.

5.3.2 Non-parametric Wilcoxon rank-sum test and non-parametric Friedman test.

The Friedman test is a non-parametric statistical method used to compare differences between three or more related samples. It serves as a non-parametric alternative to analysis of variance (ANOVA) and is applicable in situations where multiple experiments are conducted on the same dataset using different algorithms. This test can assess statistical differences between algorithms and identify whether significant differences exist. The performance of multiple algorithms across different datasets or test environments is ranked. The ranks are then summed, and the significance of the inter-group differences is determined using a chi-squared distribution. The Friedman test effectively reduces bias between samples, thereby enabling a fairer comparison of algorithms. As shown in the Table 5, there is a clear difference between GWOA and other excellent algorithms. GWOA’s average Friedman value is 2.3021, ranking first.

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Table 5. Results of Wilcoxon rank-sum test and Friedman test for different algorithms.

https://doi.org/10.1371/journal.pone.0322494.t005

The Wilcoxon rank-sum test is a non-parametric statistical method designed to assess whether there are significant differences in the distributions of two independent samples. The core idea is to merge the two datasets, rank them, and calculate the statistic based on the ranks to infer distribution differences. Unlike traditional methods that rely on mean and variance, the Wilcoxon test is not affected by outliers, providing more robust results, particularly when handling non-normally distributed data. In the comparison of optimization algorithms, the Wilcoxon test is an effective tool for determining whether there is a real difference in the performance of two algorithms. If the p-value of the test is less than the preset significance level (typically 0.05), the performance difference between the two algorithms can be considered statistically significant, rather than due to random fluctuations. As shown in the Table 5 below, “+” represents a significant difference, “=” indicates equal results, and “–” denotes no obvious difference. GWOA shows significant differences compared to most algorithms, while there are more ties when compared to the excellent variant MWOA, as both ultimately converge to the optimal solution.

5.4 Scalability Comparison Experiment on Different Metaheuristic Algorithms

In the benchmark functions, F1-F13 are expandable functions. To explore the performance of GWOA in different dimensions, the experiments extend F1-F13 to 50 and 100 dimensions for further analysis. The algorithm’s parameter settings remain unchanged, with a maximum number of iterations set to T=500 and a population size of N=30. Then, the algorithm is run 30 times on each function, and the p-values of the Wilcoxon rank-sum test and the Friedman value are recorded. The experimental results are shown in Table 6:

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Table 6. Wilcoxon Rank-sum Test and Friedman Test Results for Different Algorithms in 50 and 100 Dimensions.

https://doi.org/10.1371/journal.pone.0322494.t006

The results show that GWOA maintains good performance in high-dimensional environments, consistently ranking first in the Friedman Rank. In the Wilcoxon rank-sum test, it exhibits significant differences compared to other comparative algorithms. This is sufficient to demonstrate that GWOA has a strong competitive advantage over other optimization algorithms.

5.5 Overall effectiveness of GWOA

To further validate the performance of GWOA, this study uses a useful metric called Overall Efficiency (OE) [42] to summarize the performance results of GWOA and other algorithms. In the Table 7 below, w denotes win, t denotes tie, and l denotes loss. The OE of each algorithm is calculated using the following formula Eq 32:

(32)

where N is the total number of tests; L is the total number of failed tests for each algorithm.

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Table 7. Overall effectiveness of GWOA and other algorithms.

https://doi.org/10.1371/journal.pone.0322494.t007

Table 7 summarizes the performance results of GWOA and other algorithms. The results show that although the performance of GWOA decreases at high dimensions, its overall efficiency is 74.46%, making it the most effective algorithm. GWOA demonstrates excellent performance on classical benchmark functions and shows significant differences from the selected comparison algorithms. This quantitatively validate GWOA’s effectiveness

6.1 Pressure vessel design

The Pressure Vessel design problem is a classic engineering optimization problem, where the goal is to minimize the manufacturing cost of a pressure vessel. x₁, x₂, x₃ and x₄ are the design parameters representing the thickness of the vessel shell, thickness of the vessel head, inner diameter of the vessel, and length of the vessel (excluding the head) respectively. The structure of a pressure vessel as shown in the Fig 9. And the Pressure Vessel design problem is modeled as follows.

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Fig 9. The structure of a pressure vessel.

https://doi.org/10.1371/journal.pone.0322494.g009

Variable:

Objective function:

(33)

Subject to:

(34)

(35)

(36)

(37)

Variable range:

Where:

(38)

The objective function is subject to individual constraints, g1 is the constraint on the ratio between the thickness and the radius; g2 represents the linear relationship between the inner diameter and the radius; g3 corresponds to the volume requirement for the container design; g4 is the length requirement for the container design.

In practice, the selection of suitable materials is often constrained by suppliers, cost, and physical properties such as corrosion resistance and thermal stability. Additionally, the manufacturing process of pressure vessels is complex and may involve processes such as welding and casting, which present challenges for design accuracy and cost.

6.2 Tension/compression spring design

In the Tension/Compression Spring design problem, the goal is to find the optimal spring parameter combination that meets performance requirements, typically minimizing the spring’s volume or weight. The Tension/Compression Spring design as shown in the Fig 11. x₁ is the spring wire diameter ranging from 0.05 to 2.0, controlling the spring’s strength and stiffness. x₂ is the spring outer diameter, ranging from 0.25 to 1.3, determining the spring’s spatial size. x₃ is the number of active coils, ranging from 2.0 to 15.0, influencing the spring’s deformation ability. The Tension/Compression Spring design is modeled as follows.

Variable:

Objective function:

(39)

Subject to:

(40)

(41)

(42)

(43)

Variable range:

Where:

(44)

where g1 represents the design requirements for the stress and dimensions of the spring material; g2 is the optimized design of the spring’s material strength and dimensions, ensuring the spring’s safety and stability; g3 refers to the spring’s elastic modulus and response, ensuring the spring is neither too stiff nor too soft under working conditions; g4 pertains to the spring’s dimensional ratio, ensuring a reasonable overall size ratio of the spring.

Spring design typically requires very precise manufacturing processes, as even slight deviations can lead to performance instability. Springs may face fatigue failure issues over time, especially under high load and frequent operation conditions.

6.3 Piston lever design

The Piston Lever design problem revolves around optimizing the geometric parameters of the piston lever to meet design objectives and constraints while improving performance. The structure of a piston lever is shown in the Fig 13. The key variables typically involved are x₁ for the horizontal distance of the lever, x₂ for the vertical distance, x₃ for the piston lever diameter, and x₄ for the length of the piston lever. The goal is to minimize the material volume, thus reducing the piston lever’s weight and cost. The Piston Lever design problem can be described as:

Variable:

Objective function:

(45)

Subject to:

(46)

(47)

(48)

(49)

Variable range:

where

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

Where g1 refers to the torque balance design, ensuring that the piston system generates sufficient torque to balance the external load; g2 is the maximum torque limitation design, which represents the maximum torque generated by the piston rod during operation; g3 pertains to the dimensional design requirements, ensuring the structural safety and stability of the system; g4 is the geometric design of the piston rod, ensuring its mechanical performance during operation.

The manufacturing precision of gears significantly impacts the overall system efficiency and noise levels, thus requiring high-precision processing techniques. The efficiency of the gear transmission system is often affected by factors such as friction, lubrication, and the quality of the gear surface, which necessitates the consideration of lubrication methods and material selection during the design process.

6.4 Speed reducer design

The Speed Reducer design problem aims to optimize the structural parameters of the speed reducer to minimize the system’s weight. The structure of a speed reducer is shown in Fig 15. The design involves 7 key variables: the gear diameter x₁, the center distance x₂, gear-related parameters x₃, the diameter of the gear shaft x₄, the diameter of another gear shaft x₅, the thickness of the gear shaft x₆, the thickness of another gear shaft x₇. The Speed Reducer design problem is defined below:

Variable:

Objective function:

(59)

Subject to:

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

Variable range:

Where:

(71)

where g1 refers to the power transmission efficiency design; g2 is the strength of the transmission system; g3 and g4 are the strength of the bearings and components; g5 and g6 pertain to the strength and stiffness of the shafts; g7 is the load capacity design; g8 refers to the gear size ratio design; g9 is the geometric ratio design of the gears; g10 and g11 are the relative size designs of the shafts and gears.

6.5 Results and analysis

As the experimental results shown in Fig 10, Fig 12, Fig 14, Fig 16 and Table 8, GWOA outperforms other algorithms in engineering design problems by virtue of its excellent stability and optimization accuracy. It’s able to converge to the optimal solution quickly and accurately.

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Fig 10. Iteration curves in pressure vessel design.

https://doi.org/10.1371/journal.pone.0322494.g010

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Fig 11. The structure of a tension/compression spring.

https://doi.org/10.1371/journal.pone.0322494.g011

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Fig 12. Iteration curves in tension/compression spring design.

https://doi.org/10.1371/journal.pone.0322494.g012

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Fig 13. The structure of a piston lever.

https://doi.org/10.1371/journal.pone.0322494.g013

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Fig 14. Iteration curves in piston lever design.

https://doi.org/10.1371/journal.pone.0322494.g014

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Fig 15. The structure of a speed reducer.

https://doi.org/10.1371/journal.pone.0322494.g015

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Fig 16. Iteration curves in speed reducer design.

https://doi.org/10.1371/journal.pone.0322494.g016

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Table 8. Results of the algorithms in solving engineering design optimization problems.

https://doi.org/10.1371/journal.pone.0322494.t008