5.1 Initialization

As shown in Fig 2 on the left, like most meta-heuristic algorithms, WOA uses pseodo-random number initialization for population initialization.

(1)

where is whale population initialized by pseodo-random number initialization; ub and lb are the upper limit and lower limit of the problem; Rand is a random number between 0 and 1.

5.2 Encircling prey

Humpback whales recognize the location of their prey and encircle them. Since the location of the optimal design in the search space is unknown, WOA assumes that the current optimal candidate solution is the target prey or a near-optimal solution. After defining the optimal search agent, other search agents will try to update their positions to the optimal search agent. This behavior is represented by Eqs 2 and 3.

(2)

(3)

where t is the current iteration; A and C are coefficient vectors; is the position of the current best solution; X is the position of the whale.

If the result of each iteration has a better solution, the fitness value of this position is smaller than the fitness value of this position, then the position vector at this time will be set as the new .

The formulae for vector A and C are given below:

(4)

(5)

(6)

where r is a vector of random numbers from 0 to 1; a decreases from 2 to 0 during the iteration, as shown in Fig 1; t is the current number of iterations; T is the maximum number of iterations.

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Fig 1. The update method of parameter a in the original WOA.

https://doi.org/10.1371/journal.pone.0320913.g001

5.3 Bubble-net attacking method

In addition to encircling their prey, whales also use the bubble net attack method to attack their prey. They trap their prey by blowing round bubbles in the water to create a water net. At around 15 meters underwater, the whales swim upwards in a spiral position and spit out bubbles of varying sizes to encircle their prey and bring them closer to a central area. The whale then swallows the prey in the bubble net in a near vertical position. Therefore, the bubble net attack method is divided into the following two strategies:

1. Shrinking encircling: This behaviour is achieved by reducing the value of A in Eq 4. The position is updated as shown in Eq 3.
2. Spiral updating position: The method first calculates the whale position X and the prey position in a straight line distance from the prey position as shown in Eq 7. And then creates a spiral equation to simulate the whale spiraling up to encircle the prey as shown in Eq 8.(7)(8)(9)(10)
   where b is a constant defining the shape of the logarithmic helix, usually set to 1; is a linearly varying parameter of [-2, -1]; t is the current number of iterations; T is the maximum number of iterations; Rand is a random number between 0 and 1; the spiral coefficient l takes the values [-2, 1].

WOA sets a 50% probability for each of the Shrinking encircling mechanism and spiral updating position mechanism of the whale updating its position as follows:

(11)

5.4 Search for prey

If the whale moves beyond the position where the prey exists, then the whale will abandon the previous moving direction and randomly search for other prey in other directions to avoid falling into a local optimum. The modeling of the whale searching for prey is as follows:

(12)

(13)

where is a random whale chosen from the current population; A and C are described in Eq 4 and Eq 6.

5.5 Advantages and disadvantages of WOA

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

Algorithm 1: WOA

Begin

  (Pseodo-random number initialization)

  Initialize the population using Pseodo-random number method;

  Initialize the parameters  ( T , N , p , etc. ) ;

  Calculate the fitness of each search agent;

  The best search agent is ;

    while t < T

     for each search agent

      Update a, A, C, l, and p;

      if p < 0 . 5       

if  | A | < 1

        (Search for prey)

       Update the position of the current search agent by      Eq 3;

       else

        (Encircling prey)

       Update the position of the current search agent by      Eq 13;

       end if

      else

        (Spiral updating)

       Update the position of the current search agent by      Eq 8;

      end if

     end for

     Check if any search agent goes beyond the search space and amend it;

     Calculate the fitness of each search agent;

     Update if there is a better solution;

     t = t + 1

    end while

  return

  End

From Algorithm 1, it can be seen that the late iteration of traditional WOA causes the WOA to have the disadvantages of weak global search ability, slow convergence speed, low accuracy of searching for the best and easy to fall into the local optimum due to the reduction of the diversity of the population. The development and exploration ability of traditional WOA is weak, and the parameters of WOA are difficult to balance the global exploration and local development of the whale algorithm. In addition, the traditional WOA does not take into account the possible differences in the guiding force of the prey to guide the whale for position updating. WOA has much room for improvement. Therefore, we proposed RWOA to address the above shortcomings of WOA.

6.1 Good nodes set

The traditional WOA uses Pseodo-random number initialization to generate the population, which is simple, direct and random, but the randomly generated population is not uniformly distributed in the whole solution space, it is denser somewhere and sparser somewhere, which can’t achieve the uniform distribution of the population in the whole search space, and it leads to the inefficiency of WOA in searching process due to the low quality of the population as shown in Fig 2 on the left. Therefore, RWOA uses Good Nodes Set Initialization to generate uniformly distributed populations.

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Fig 2. Pseodo-random number initialization vs good nodes set initialization.

https://doi.org/10.1371/journal.pone.0320913.g002

The concept of the Good Nodes Set was first proposed by the Chinese mathematician Loo-keng Hua. This method constructs a set of nodes that are uniformly distributed in the space, which provides a significant advantage over random initialization, especially in higher-dimensional spaces. The key intuition behind this approach is that the construction of the Good Nodes Set is independent of the number of dimensions, which ensures uniformity in any dimensional space. In contrast, random initialization often suffers from clustering of nodes in some regions and large gaps in others, especially as the dimensionality increases. The uniform distribution provided by the Good Nodes Set improves the search efficiency by avoiding such issues and facilitating better global exploration.

Mathematically, the Good Nodes Set is defined as a sequence of nodes in a unit hypercube D-dimensional Euclidean space, where the nodes are spaced according to a specific non-random, deterministic pattern. The uniformity of the distribution can be understood through the fractional part of the product , ensuring that the nodes are spread evenly. This deterministic nature contrasts with the random nature of pseudo-random number initialization, where the nodes are arbitrarily distributed, leading to inefficiencies.

In this sense, Good Nodes Set Initialization enhances the population quality by providing a more structured and evenly distributed set of candidate solutions, improving both the exploration and exploitation abilities of the algorithm, as illustrated in Fig 2 on the right [37]. Good Nodes Set in D-dimensional space is described by Eq 14 as follows:

(14)

where {x} represents the fractional part of x; M is the number of points; r is a deviation parameter greater than zero; the constant C ( r , 𝜀 )  is associated only with r and 𝜀 is related to and is a constant greater than zero.

This set is called Good Nodes Set and each node p(k) in it is called a Good Node. Assume that the upper and lower bounds of the dimension of the search space are and , then the mapping formula for mapping the Good Nodes Set to the actual search space is:

(15)

6.2 Hybrid collaborative exploration (HCE) strategy

Red-billed Blue Magpie Optimization algorithm (RBMO) is a new meta-heuristic algorithm proposed by Shengwei Fu et al. in 2024 [38], which has a powerful search capability by simulating the hunting and food storage behaviour of red-billed blue magpies. Red-billed blue magpies use two strategies to search for food, and they usually act in small groups of 2-5 individuals or in groups of more than 10 individuals to improve the search efficiency.

Location updates were made using Eq 16 when red-billed blue magpies were searching for food in small groups:

(16)

where t represents the current number of iterations; X represents the location of the search agent; p represents the number of red-billed blue magpies in 2-5 cliques randomly selected from all search individuals; represents the randomly selected individual; represents the randomly selected search agent in the current iteration.

Location updates were made using Eq 17 when red-billed blue magpies were searching for food as a group:

(17)

where q represents the number of red-billed blue magpies in 10-n randomly selected cliques from all searched individuals and n is the population size.

In the search-for-prey phase, the WOA uses only the position of a randomly selected whale for searching, which may limit the diversity of search paths, especially in high-dimensional search spaces, and may lead to insufficient search capability. In addition, although WOA updates the position by randomly selecting a reference individual, this single randomness may not be sufficient to help the WOA explore regions far away from the current optimal solution, which reduces the global search capability. Therefore, Hybrid Collaborative Exploration mechanism in RWOA incorporates the idea of searching for food from RBMO algorithm by randomly selecting a small group of 2-5 whales or a group of 10-n whales in the whale population to search for food randomly, as shown in Fig 3. By introducing the multi-option positions of 2-5 or 10-n whales for reference, RWOA is able to combine the information of multiple individuals for position updating. This means that more directions and paths are considered in the RWOA position update, avoiding the limitations of a single position update and allowing it to jump and explore over a larger area while searching. This strategy of incorporating RBMO helps to enhance the global search capability of WOA, enabling the algorithm to better discover the global optimal solution. In addition, this strategy takes into account the position of the current best solution, effectively guiding the search of the whales towards better solutions. As the number of iterations increases, the reliance on the current best solution’s position gradually decreases. This means that, in the early stages, the position update is more influenced by the global best solution to guide the search, while as iterations progress, the reliance on the current best solution’s position diminishes, the search range expands, and greater emphasis is placed on diversity, thereby preventing premature convergence to local optima. This multi-option reference reduces the instability caused by a single random individual, improves the robustness of the algorithm, and makes WOA perform more stably in the face of various optimization problems.

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Fig 3. Hybrid collaborative exploration mechanism.

In the search for prey, whales adopt the cooperative strategy of the red-billed blue magpies, randomly selecting 2-5 whales or 10-n whales to search for prey.

https://doi.org/10.1371/journal.pone.0320913.g003

The formula for Hybrid Collaborative Exploration (HCE) strategy of RWOA is shown in Eq 18:

(18)

where A is the vector of coefficients; is the position of the current best solution; p and q represent the number of whales in 2-5 small groups or 10-n flocks randomly selected from all searched individuals respectively; denotes a random number from 0 to 1.

6.3 Spiral encircling prey strategy

As shown in Eq 3, the original WOA uses an encircling prey strategy, where the whale’s position is updated based on the distance to the prey. While effective in approaching the optimal solution, this strategy can lead to premature convergence and local optima, particularly in complex, multi-modal problems. The simplicity of the strategy limits exploration during local search and makes the algorithm sensitive to initial positions.

Spiral flight is a continuous and systematic search pattern that differs from traditional linear or local search methods, as shown in Fig 4. It allows the whales to generate more varied trajectories while encircling the prey. This spiral motion not only enhances the encircling capability towards the target but also helps avoid local convergence during the search process. The introduction of spiral motion improves the whales’ precision while encircling the prey, particularly when approaching the optimal solution. The nonlinear spiral trajectory makes the search trajectory more flexible, ensuring that the whale maintains ongoing exploration of the target while preventing quick convergence to a particular location, which may cause it to miss other potential optimal solutions. By incorporating the spiral strategy, the whales’ movement direction and distance become more diverse, enabling exploration of the solution space around the prey in multiple directions, thus enhancing the diversity of the search process. Fig 5 is a schematic diagram of the Spiral Encircling Prey mechanism.

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Fig 4. Simulation of spiral flight.

https://doi.org/10.1371/journal.pone.0320913.g004

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Fig 5. Spiral encircling prey mechanism.

The whales encircle the prey in a spiral pattern.

https://doi.org/10.1371/journal.pone.0320913.g005

(19)

(20)

(21)

where r is a random number from 0 to 1; Z and L are spiral coefficients.

6.4 Enhanced spiral updating strategy

While the original spiral updating strategy of WOA aids in local search, the lack of randomness and perturbation mechanisms may cause whale individuals to converge around the leader (current best solution) in a short time, limiting the ability to escape local optima. To address this, this paper proposed an Enhanced Spiral Updating strategy by introducing Levy flight into the Spiral Updating strategy.

Levy flight is a random process whose distribution characteristics lie between Brownian motion and more extreme jumping behaviors. The concept originated from the work of the mathematician Paul Levy in the 1920s. Inspired by foraging behavior in nature and jumping phenomena in complex systems, Levy flight combines short-range exploration with long-distance jumps, resembling the foraging paths of predators such as sharks, birds, and insects. Fig 6 illustrates a simulation of Levy flight. It is a random walk model based on the Levy distribution, which has a higher probability of producing longer step sizes, enabling whales to explore vast distances through jumps. In global search scenarios, Levy flight allows for large jumps, which helps avoid getting trapped in local optima, while still maintaining some local search capability through shorter steps. By incorporating unequal probabilities for both long and short jumps, Levy flight enhances the diversity of the search process, making it better suited to adapt to more complex solution space structures. The step size L(s) of Levy flight is calculated as follows:

(22)

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Fig 6. Simulation of Levy flight.

https://doi.org/10.1371/journal.pone.0320913.g006

where u and ν are normally distributed; β=1.5.

(23)

(24)

The calculation of is given by:

(25)

By introducing Levy flight, the long step sizes inherent in the Levy flight mechanism help individuals leap freely across the entire search space. This significantly boosts the jumping capability of the Spiral Updating strategy, thereby increasing the chance of escaping local optima and enhancing the global search ability. The introduction of Levy flight allows whales to not only follow the spiral trajectory for local search but also utilize larger jumps for global exploration, which helps prevent the algorithm from converging prematurely to local optima. This strategy enhances the diversity and flexibility of whale movements, particularly in cases where the search space is large and the solution space is complex, effectively avoiding premature convergence. Enhanced Spiral Updating strategy was modeled as follows:

(26)

where represents the position of the current best solution; L(s) is the step size of the Levy flight; b is a constant that defines the shape of a logarithmic spiral, usually set to 1; the value of spiral coefficient l is  [ − 2 , 1 ] , and l is calculated in Eq 10; X is the whale’s position; is the parameter of the linear change of  [ − 2 , − 1 ] , calculated in Eq 9; Rand is a random number between 0 and 1.

6.5 Enhanced cauchy mutation based on differential evolution

The standard position update strategies for WOA mainly rely on bionic principles, such as prey encirclement and spiral update. These strategies are usually more focused on searching in the vicinity of the current optimal solution, especially in the later stages of the iteration, the WOA may gradually converge to a particular region, leading to a decrease in the exploration ability of the search space. By introducing a mutation strategy after the position update, a new perturbation can be introduced into the solution space, generating a certain degree of mutation that allows individual whales to jump out of the current local optimal region and explore a wider solution space. In addition, for algorithms like WOA that rely on optimal solutions in the population for search, the variation strategy can help individuals jump out of the locked region so that the solution of WOA does not fall into a local optimum, especially when dealing with complex problems with multiple local optimal solutions. By incorporating a variation strategy into the WOA, the possible limitations of the WOA in the later convergence phase can be compensated for, ensuring that the WOA continues to explore rather than converging to a sub-optimal solution early on. We incorporate a novel Cauchy mutation strategy based on Differential Evolution into RWOA.

First, we generate an intermediate solution by Differential Evolution.

(27)

where represents the current position of the individual; represents the position of the current best individual; , and represent the positions of three different individuals randomly selected from the population respectively; F is the factor controlling the scaling of the difference vector, which is calculated as in Eq 27.

(28)

where Rand denotes a random number between 0 and 1.

Subsequently, execute Cauchy mutation on the generated intermediate solutions :

(29)

where δ denotes the perturbation term sampled from the Cauchy distribution:

(30)

Then, after applying the mutation operation to the intermediate solution, boundary checking and adjustment are required, otherwise the population degradation of WOA may occur. The boundary checking is performed as follows:

(31)

where ub and lb represent the upper and lower bounds of the problem respectively.

Finally, if the mutated solution has a better fitness than the original solution , then the solution is updated with .

6.6 Redesign of parameter a

Although integrating these strategies improves WOA’s performance, the traditional linear update of parameter a from 2 to 0 no longer meets RWOA’s needs. The linear method results in a smooth behavior change, especially at later stages, where the minimal reduction slows convergence. To address this, a Sigmoid-based parameter a update was proposed to balance global exploration and local exploitation, as shown in Eq 32. Fig 7 compares the proposed Sigmoid method with the original linear one.

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Fig 7. Comparison between the update methods of the original parameter a and the proposed a (β=10, β=15, β=20 and β=25).

https://doi.org/10.1371/journal.pone.0320913.g007

The Sigmoid update reduces a slowly at first, rapidly in the middle, and slowly again at later stages. This simulates complex variation, enabling distinct convergence behaviors: slow early reduction supports broad exploration, rapid mid-stage decrease enhances convergence speed, and slower late-stage reduction balances exploration and exploitation for fine local searches. This strategy improves adaptability and final solution accuracy, making the algorithm more effective and reliable for complex problems.

(32)

where β is the scaling factor of the Sigmoid function and value of β was tested in the parameter sensitivity analysis experiment; t is the current iteration number; T is the maximum number of iterations.

The pseudo-code of the complete RWOA is provided in Algorithm 2.

Algorithm 2: RWOA

Begin

  (Good Nodes Set Initialization)

  Initialize the population using Good Nodes Set method;

  Initialize the parameters  ( T , N , p , etc. ) ;

  Calculate the fitness of each search agent;

  The best search agent is ;

    while t < T

     for each search agent

      Update a, A, C, l, and p;

      if p < 0 . 5

       if  | A | < 1

        (Spiral Encircling Prey)

       Update the position of the current search agent by Eq 19;

       else

        (Hybrid Collaborative Exploration)

       Update the position of the current search agent by Eq 18;

       end if

      else

        (Enhanced Spiral Updating)

       Update the position of the current search agent by Eq 26;

      end if

     end for

        (Enhanced Cauchy Mutation based on Differential Evolution)

     for each search agent

      Generate an intermediate solution by DE through Eq 27;

      Execute Cauchy mutation to generate an another intermediate

    solution

      by Eq 29;

       if is within the boundary

        = ;

       else

        if exceeds the upper boundary ub

        set it to the upper boundary ub;

        else if exceeds the lower boundary lb

        set it to the lower boundary lb;

        end if

       end if

     end for

     Calculate the fitness of each search agent;

     Update if there is a better solution;

     t = t + 1

    end while

  return

  End

6.7 Time complexity analysis

Assume that the time complexity of Pseodo-random number initialization in WOA is O(ND). During each iteration, and the total time complexity of position updates is O(ND). Therefore, the total time complexity per iteration is O(ND). If the algorithm iterates T times, the total time complexity of WOA is calculated as:

Total Time Complexity 1 = Initialization + T * (the total time complexity per iteration) = O(ND) + T * O(ND) = O ( T ∗ ND ) 

Assume that the time complexity of Good Nodes Set Initialization in RWOA is O(ND). During each iteration, and the total time complexity of position updates is O(ND), the time complexity of Enhanced Cauchy Mutation based on Differential Evolution is O(ND), Therefore, the total time complexity per iteration is O(ND). If the algorithm iterates T times, the total time complexity of RWOA is calculated as:

Total Time Complexity 2 = Initialization + T * (the total time complexity per iteration) =O(ND) + T * O(ND) = O ( T ∗ ND ) 

In summary, the time complexity of RWOA and WOA are the same, both are O ( T ∗ ND ) .

7.1 Ablation study

In this ablation study, we excluded each of the six improvement strategies from the RWOA:

• RWOA1: RWOA without Good Nodes Set Initialization;
• RWOA2: RWOA without Hybrid Collaborative Exploration strategy;
• RWOA3: RWOA without Spiral Encircling Prey strategy;
• RWOA4: RWOA without Enhanced Spiral Updating strategy;
• RWOA5: RWOA without Enhanced Cauchy Mutation based on Differential Evolution;
• RWOA6: RWOA with the original update method of parameter a;

The number of iterations T=500 and the number of populations N=30 were set uniformly. Each algorithm was run 30 times individually on 23 classical Benchmark Functions for performance analysis. The Friedman values of the algorithms were recorded. And the iteration curves were shown in Fig 8.

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Fig 8. Iteration curves of the algorithms in ablation study.

https://doi.org/10.1371/journal.pone.0320913.g008

As shown in Fig 8, the Good Nodes Set Initialization generated whale individuals that were more uniformly distributed in the solution space. This high-quality population provided a significant advantage when addressing problems such as F5-F6, F12-F15, and F21-F23. The Hybrid Collaborative Exploration strategy integrated RBMO’s cluster hunting strategy, allowing RWOA to update positions by incorporating the positional information of multiple individual whales. This avoided the limitations associated with single-position updates, enabling RWOA to explore a larger search range. Consequently, the global search capability of WOA was enhanced, strengthening its ability to solve complex functions. The Spiral Encircling Prey Strategy, incorporating a nonlinear step size through Spiral flight, increased the randomness and introduced nonlinear fluctuations in position updates. This addition brought both periodicity and unpredictability to the algorithm, allowing it to consistently avoid local optima and prevent premature convergence, particularly when tackling complex problems such as F5-F6 and F15-F17. The Enhanced Spiral Updating strategy, which introduced Levy flight, allowed whale individuals to effectively escape local optima while spiraling upward. The incorporation of step size of Levy flight enhanced the algorithm’s global exploration ability and significantly improved its convergence speed. This strategy is particularly advantageous when dealing with functions such as F1-F4 and F9-F11. The Enhanced Cauchy Mutation based on Differential Evolution integrated the concepts of Differential Evolution and Cauchy mutation, helping the algorithm to generate superior solutions after position updates. The new update method for parameter a based on the Sigmoid function, effectively balanced exploration and exploitation. This novel convergence behavior improved the algorithm’s adaptability and optimization precision. As shown in Table 3, the average Friedman value of RWOA is 2.9217, ranking first. This indicates that RWOA was the optimal choice.

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Table 3. Friedman values of the algorithms in the ablation study.

https://doi.org/10.1371/journal.pone.0320913.t003

7.2 Parameter sensitivity analysis experiment

The purpose of conducting the parameter sensitivity analysis experiment is to observe the changes in algorithm performance by adjusting the value of the scaling factor β, and thereby select the most suitable parameter value. In the code provided, β controls the update rate of parameter a, which determines the shape of the Sigmoid function. This parameter affects the rate at which a decreases from 2 to 0, directly controlling the balance between exploration and exploitation in the iterative process of the algorithm. The larger the value of β, the more abrupt the change of a from 2 to 0. Conversely, the smaller the value of β, the smoother the change of a. The main significance of this experiment is to optimize the performance of the algorithm, enhance its adaptability, and ensure that it performs well across different optimization problems. By adjusting β, the algorithm’s balance between global search in the early stages and local search in the later stages can be better controlled, thereby affecting the final optimization outcome. In this experiment, β values of 10, 15, 20, and 25 were selected for testing. The number of iterations, T, was set to 500, and the population size, N, was set to 30. Each algorithm was run 30 times individually on 23 classical benchmark functions for performance analysis. The Friedman values of the algorithms were recorded. As shown in Table 4, the RWOA with β = 25 achieved the smallest average Friedman value and ranked first. Therefore, β = 25 was chosen for this study.

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Table 4. Friedman values of the RWOAs.

https://doi.org/10.1371/journal.pone.0320913.t004

7.3 Qualitative analysis experiment

In the qualitative analysis experiment, we set the number of iterations to T = 500 and the population size to N = 30, and ran RWOA independently on 23 benchmark functions in Table 2 to analyze the search history, exploration-exploitation ratio, and population diversity of RWOA. In addition, we provided the landscape of the benchmark functions and the iteration curves for reference. The results of the qualitative analysis were shown in Fig 9, Fig 10 and Fig 11, which includes:

• Landscapes of benchmark functions;
• Search history of the whale population;
• Exploration-exploitation ratio;
• Population diversity curves;
• Iteration curves.

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Fig 9. Results of qualitative analysis experiment (F1-F8).

https://doi.org/10.1371/journal.pone.0320913.g009

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Fig 10. Results of qualitative analysis experiment (F1-F8).

https://doi.org/10.1371/journal.pone.0320913.g010

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Fig 11. Results of qualitative analysis experiment (F1-F8).

https://doi.org/10.1371/journal.pone.0320913.g011

The search history represents the positions and distribution of the whale individuals. In the search history graph of the whale population, the red circles indicate the global optimum position, while the blue circles represent the search history of the whale individuals. Notably, RWOA effectively explored the entire search space. In single-modal functions such as F1-F4, RWOA exhibited fast convergence, with whale individuals finding the optimal solution within a limited number of iterations, leading to a concentrated distribution of individuals in the solution space. In the case of complex multi-modal functions like F8 and F17-F23, where many local optima exist, RWOA first performed rapid global exploration and then refined its search in promising regions. As a result, the whale individuals traversed a large portion of the solution space, and the search history was mainly concentrated around the optimal solution. In terms of balancing exploration and exploitation, RWOA performed excellently, effectively managing this trade-off. When dealing with functions like F1-F4 and F9-F1, RWOA showed a rapid increase in the exploitation ratio during the early iterations, demonstrating its strong exploration capability. In the case of functions like F14-F23, RWOA exhibited a higher exploitation ratio in the early iterations, with a slow decline in the exploitation ratio. This highlights its robust global exploration ability and local exploitation capability. For functions such as F5-F8 and F14-F23, the diversity curve of RWOA’s population consistently fluctuated and maintains high values. This indicates that RWOA can maintain high population diversity when handling complex multi-modal functions, effectively preventing premature convergence caused by population clustering in certain areas.

7.4 Comparative experiment of different algorithms

To further validate the superiority of RWOA, we selected Zebra Optimization Algorithm (ZOA) [40], Rime Optimization Algorithm (RIME) [41], Improved Sand Cat Swarm Optimization Algorithm (ISCSO) [42], Grey Wolf Optimizer (GWO) [8], Harris Hawks Optimization (HHO) [4], Attraction-Repulsion Optimization Algorithm (AROA) [43], MWOA [44], MSWOA [45], and Whale Optimization Algorithm (WOA) [3] for comparison, and tested them on the benchmark functions listed in Table 2. The parameter settings for each algorithm are shown in Table 5. The number of iterations was uniformly set to T=500, and the population size to N=30. Each algorithm was independently run 30 times on 23 benchmark functions, and the average fitness (Ave), standard deviation (Std), p-values of the Wilcoxon rank-sum test, and Friedman values were recorded for performance analysis. The experimental results are shown in Fig 12, Tables 7 and 8.

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Table 5. Parameter settings for algorithms.

https://doi.org/10.1371/journal.pone.0320913.t005

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Table 6. Details of the metaheuristic algorithms.

https://doi.org/10.1371/journal.pone.0320913.t006

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Table 7. Comparative results of each algorithm in comparative experiment.

https://doi.org/10.1371/journal.pone.0320913.t007

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Fig 12. Iteration curves of different algorithms in comparison experiment.

https://doi.org/10.1371/journal.pone.0320913.g012

As can be seen from Fig 12, Tables 7 and 8, the average fitness (Ave) and standard deviation (Std) of RWOA are superior to other algorithms in F1-F13 and F15-F23. This proves that RWOA had better robustness, adaptability and stability than other algorithms when dealing with uni-modal problems or most complex multi-modal problems. In F1 and F3, both MWOA and RWOA found the optimal solution within a limited number of iterations. In F9-F11, the performance of both RWOA and ZOA, ISCSO, HHO, MWOA is the best, which also proves that RWOA had the ability to find the optimal solution to complex problems without losing to other algorithms. However, in F14, although RWOA had been optimized to the optimal solution for many times, its overall average fitness and standard deviation are slightly worse than RIME, indicating that RWOA was less stable than RIME when solving problems like F14, confirming the famous ’No free lunch (NFL) theorem’: RWOA is not perfect, it performs well in the vast majority of benchmark functions, but may perform slightly worse than individual algorithms on a few functions.

However, in the performance evaluation of optimization algorithm, the convergence and stability of the algorithm are usually measured by average fitness and standard deviation, which can not directly reflect the performance of the algorithm. Comparing the performance of different algorithms by average fitness and standard deviations alone has certain limitations, so non-parametric tests are often introduced: Wilcoxon rank sum tests and Friedman tests.

The Wilcoxon rank-sum test is a non-parametric test that is used to statistically test the difference between the medians of two independent samples. In the comparison of optimization algorithms, when we wish to compare whether there is a significant difference between the effects of two algorithms, Wilcoxon rank-sum test can help us determine whether the difference is statistically significant. If the p-value of Wilcoxon rank-sum test is less than a set of significance levels (typically 0.05), it can be assumed that the performance difference between the two algorithms is significant and not just due to random error. As can be seen from Table 8, RWOA had significant differences with RIME, GWO and MWOA in all benchmark functions.

The Friedman test is a non-parametric analysis of variance used to compare the performance of multiple algorithms on different test problems. It can identify if there are statistically significant differences between algorithms. Friedman test can effectively eliminate the bias between samples by comparing the performance of multiple algorithms in multiple data sets or test environments, providing a fairer comparison and avoiding the multiple comparison problem caused by comparing multiple algorithms individually. As shown in Table 8, the average Friedman value of RWOA is 1.6833, ranking first among the selected SOTA algorithms.

Table 9 summarizes all performance results of RWOA and other algorithms by a useful metric named overall effectiveness (OE). In Table 9, w indicates win, t indicates tie and l indicates loss. The OE of each algorithm is computed by Eq 33 [46].

(33)

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

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Table 8. Result of non-parametric tests of different algorithms.

https://doi.org/10.1371/journal.pone.0320913.t008

RWOA with overall effectiveness of 100.00% was the most effective algorithm. And RWOA was competitive with other SOTA algorithms on the benchmark functions. The results revealed the ability of RWOA to handle optimization problems. In summary, after comparing average fitness and standard deviation, p-values of Wilcoxon rank-sum test and Friedman values of Friedman test, RWOA performed best, which proved the superiority of RWOA.

8.1 Three-bar truss

The three-bar truss is a simple structural system consisting of three members, as shown in Fig 13. It is commonly used to support concentrated loads and is widely applied in engineering fields such as bridges, buildings, and aerospace. The three-bar truss design problem is a classic structural optimization problem, often used to study the mechanical behavior of simple structures under external loading conditions. In the three-bar truss design problem, the objective is to optimize the cross-sectional areas of the truss members to minimize material usage while ensuring that the structure meets the required mechanical performance.

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Fig 13. The structure of a three-bar truss.

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This optimization problem involves a nonlinear objective function, three nonlinear inequality constraints, and two continuous decision variables and . The objective function for the three-bar truss design problem can be described as follows:

Variable:

Minimize:

(34)

Subject to:

(35)

(36)

(37)

Where:

Variable range:

The experimental results are shown in Fig 14 and Table 10. From Table 10, it can be observed that RWOA significantly outperformed other algorithms in terms of stability, with the best optimization accuracy among all algorithms. This indicates that RWOA had a significant advantage when handling such optimization problems.

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Fig 14. Iteration curves of the algorithms in three-bar truss design.

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8.2 Tension/compression spring

The extension/compression spring, as shown in Fig 15, plays a crucial role in modern industry, with widespread applications in fields such as automotive, home appliances, and electronics. Its design optimization not only helps improve product performance and extend service life, but also reduces costs and enhances manufacturing efficiency. Through reasonable design optimization, the spring can achieve optimal performance in dynamic working environments and meet various stringent requirements. The optimization objective of the design problem for the extension/compression spring is the minimization of its mass. The problem needs to be solved under constraints such as shear force, deflection, fluctuation frequency, and outer diameter. There are three design variables in this problem: coil diameter d, mean coil diameter D, and number of coils N. There are also four constraints, to . The mathematical model of the problem is as follows,

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Fig 15. The structure of a tension/compression spring.

https://doi.org/10.1371/journal.pone.0320913.g015

Variable:

Minimize:

(38)

Subject to:

(39)

(40)

(41)

(42)

Where:

Variable range:

The experimental results are presented in Fig 16, and Table 10. As shown in Table 10, the stability of RWOA in the tension/compression spring design problem significantly surpassed other algorithms, and it achieved the highest optimization accuracy among all algorithms. This indicates that RWOA had a significant advantage in handling such optimization problems.

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Fig 16. Iteration curves of the algorithms in tension/compression spring design.

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8.3 Speed reducer

A speed reducer is a mechanical transmission device and one of the key components of a gearbox, shown in Fig 17. It is primarily used to reduce the rotational speed of an electric motor or other power sources while increasing the output torque. The reducer achieves this speed reduction through gears, worm gears, or other transmission mechanisms. It is typically applied in situations where there is a need to decrease the rotational speed, increase torque, or adjust the direction of motion.

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Fig 17. The structure of a speed reducer.

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In the optimization design of a reducer, the goal is to minimize the weight of the reducer. This problem involves seven variables, which are as follows: the width of the gear teeth , the gear module , the number of teeth on the small gear , the length of the first shaft between the bearings , the length of the second shaft between the bearings , the diameter of the first shaft , and the diameter of the second shaft . Furthermore, this problem also involves eleven constraints, to . The mathematical formulation of the problem is as follows,

Variable:

Minimize:

(43)

Subject to:

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

Where:

Variable range:

The experimental results are presented in Fig 18, and Table 10. As shown in Table 10, the stability of RWOA in the Speed Reducer design problem significantly surpassed other algorithms, and it achieved the highest optimization accuracy among all algorithms. This indicates that RWOA had a significant advantage in handling such optimization problems.

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Fig 18. Iteration curves of the algorithms in speed reducer design.

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8.4 Cantilever beam

A cantilever beam is a common structural form, fixed at one end and free at the other, as shown in Fig 19. The cantilever beam design problem is a classic engineering structural optimization problem, with the objective of minimizing material usage or beam weight while satisfying constraints on strength, stability, and other factors. This optimization problem is widely used in civil engineering, mechanical design, and aerospace fields.

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Fig 19. The structure of a cantilever beam.

https://doi.org/10.1371/journal.pone.0320913.g019

The cantilever beam consists of five hollow square cross-section units. As shown in Fig 19, each unit is defined by one variable, and the thickness is constant. Therefore, the design problem includes five structural parameters, which correspond to five decision variables, denoted as , , , , . The objective function for the cantilever beam design problem can be expressed as:

Variable:

Minimize:

(55)

Subject to:

(56)

Where:

Variable range:

The experimental results are presented in Fig 20 and Table 10. As shown in Table 10, the stability of RWOA in the Cantilever Beam design problem significantly surpassed other algorithms, and it achieved the highest optimization accuracy among all algorithms. This indicates that RWOA had a significant advantage in handling such optimization problems.

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Fig 20. Iteration curves of the algorithms in cantilever beam design.

https://doi.org/10.1371/journal.pone.0320913.g020

8.5 Pressure vessel

A pressure vessel is a common mechanical structure used in fields such as chemical engineering, aerospace, and medical applications. The pressure vessel design problem is a classic structural optimization problem, where the goal is to minimize the manufacturing costs of the pressure vessel, including pairing, forming, and welding processes. The design of the pressure vessel is shown in Fig 21, with caps sealing both ends of the vessel. The cap at one end is hemispherical. and represent the wall thickness of the cylindrical section and the head, respectively, while is the inner diameter of the cylindrical section, and is the length of the cylindrical section, excluding the head. Thus, , , , and are the four optimization variables of the pressure vessel design problem. The objective function and four optimization constraints are represented as follows:

Variable:

Minimize:

(57)

Subject to:

(58)

(59)

(60)

(61)

Where:

Variable range:

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Fig 21. The structure of a pressure vessel.

https://doi.org/10.1371/journal.pone.0320913.g021

The experimental results are presented in Fig 22 and Table 10. As shown in Table 10, the stability of RWOA in the Pressure Vessel design problem significantly surpassed other algorithms, and it achieved the highest optimization accuracy among all algorithms. This indicates that RWOA had a significant advantage in handling such optimization problems.

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Fig 22. Iteration curves of the algorithms in pressure vessel design.

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8.6 I-beam

An I-beam, named for its cross-sectional shape resembling the letter I, is a type of steel with high strength and low self-weight. It is widely used in various engineering structures. Its superior mechanical properties make it applicable in multiple fields, particularly in structures subjected to bending moments and axial forces. The objective of I-beam design optimization is to select the geometric parameters of the I-beam (such as width, height, thickness, etc.) in a way that maximizes its performance. This typically involves maximizing its load-bearing capacity, minimizing material usage, controlling structural deformations, and reducing costs. Optimizing I-beam design in engineering can enhance the safety, economy, and efficiency of structures. As shown in Fig 23, the I-beam design optimization problem involves four variables (, , and ) and two constraints ( and ). , , and represent the web height, flange width, web thickness, and flange thickness of the I-beam, respectively. The objective function for the I-beam design problem can be described as:

Variable:

Maximize:

(62)

Subject to:

(63)

(64)

Where:

Variable range:

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Fig 23. The structure of an I-beam.

https://doi.org/10.1371/journal.pone.0320913.g023

The experimental results are shown in Fig 24 and Table 10. As shown in Table 10, RWOA significantly outperformed the other algorithms in terms of both optimization accuracy and stability for the I-beam design problem. This demonstrates that RWOA had superior solving capabilities when handling this type of problem. This indicates that RWOA had a significant advantage in handling such optimization problems.

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Fig 24. Iteration curves of the algorithms in I-beam design.

https://doi.org/10.1371/journal.pone.0320913.g024

8.7 Piston lever

In a factory, piston levers are critical components in various important pieces of equipment, such as pumps and compressors, where their performance directly impacts the reliability and efficiency of the entire system. Fig 25 is the structure of a piston lever. The piston lever design problem focuses on determining the optimal piston dimensions and material selection to ensure maximum performance in transmission systems. This design influences the efficiency, stability, and durability of mechanical systems. The optimization task seeks to balance dimensions, weight, material choice, and manufacturing cost in order to find the optimal piston size and material that maximize system efficiency and cost-effectiveness. Therefore, the piston lever design problem is of paramount importance in factories, and optimizing these key components is essential to ensuring the optimal performance of the system. The optimization involves four variables: piston length , piston diameter , material property , and transmission rod length , which affect the system’s mechanical properties, dynamic performance, and cost. The piston lever design problem can be described as:

Variable:

Minimize:

(65)

Subject to:

(66)

(67)

(68)

(69)

Variable range:

Where:

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Fig 25. The structure of a piston lever.

https://doi.org/10.1371/journal.pone.0320913.g025

The experimental results are shown in Fig 26 and Table 10. As shown in Table 10, RWOA significantly outperformed the other algorithms in terms of both optimization accuracy and stability for the piston lever design problem. This demonstrates that RWOA had superior solving capabilities when handling this type of problem. This indicates that RWOA had a significant advantage in handling such optimization problems.

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Fig 26. Iteration curves of the algorithms in piston lever design.

https://doi.org/10.1371/journal.pone.0320913.g026

8.8 Reactor network

Fig 27 is the structure of a reactor network. The reactor network design problem aims to optimize the configuration of chemical reactors in chemical plants, ordering to achieve a more efficient chemical reaction process. This involves selecting reactor types, configuring their arrangements, and allocating fluid flow rates, with the objective of maximizing product concentration. Optimization is achieved by adjusting the configuration and operating conditions of reactors to improve reaction efficiency, reduce energy consumption, or enhance product quality. The problem includes four variables representing the concentrations at different reaction stages: for the reactant concentration in the first reactor, for the product concentration in the first reactor, for the reactant concentration in the second reactor, and for the final product concentration. The constraints through are defined as follows: represents the balance of reactants and products in the first reactor, enforces mass conservation between the first and second reactors, maintains equilibrium for the reactant concentration between the reactors, and ensures mass conservation between intermediate and final products. The problem of reactor network design is modeled below.

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Fig 27. The structure of a reactor network.

https://doi.org/10.1371/journal.pone.0320913.g027

Variable:

Minimize:

(70)

Subject to:

(71)

(72)

(73)

(74)

(75)

Where:

Variable range:

The experimental results are shown in Fig 28 and Table 10. As shown in Table 10, RWOA significantly outperformed the other algorithms in terms of both optimization accuracy and stability for the Reactor Network design problem. This demonstrates that RWOA had superior solving capabilities when handling this type of problem. This indicates that RWOA had a significant advantage in handling such optimization problems.

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Fig 28. Iteration curves of the algorithms in reactor network design.

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8.9 Gas transmission system

The gas transmission system, as shown in Fig 29, is a crucial component of the modern energy supply chain, widely used in various industries, urban natural gas supply, and multinational energy transportation. Since the transportation of natural gas relies on Gas Transmission Compressors and pipeline networks, the design optimization of these devices is essential to ensuring energy transmission efficiency and reducing energy waste. The objective of the Gas Transmission Compressor optimization problem is to design and optimize the parameters of the natural gas transmission compressor, so that the compressor can deliver optimal performance under different working conditions, reduce energy consumption, extend service life, and minimize costs. The Gas Transmission Compressor optimization problem involves four design variables and one constraint. The meanings of the variables to are: indicates the length between compressor stations; indicates the compression ratio denoting inlet pressure to the compressor; indicates the pipe inside diameter; indicates the gas speed on the output side. The mathematical modeling of the Gas Transmission Compressor optimization problem is as follows:

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Fig 29. The structure of a gas transmission system.

https://doi.org/10.1371/journal.pone.0320913.g029

Variable:

Minimize:

(76)

Subject to:

(77)

Where:

Variable range:

The experimental results are shown in Fig 30 and Table 10. As shown in Table 10, RWOA significantly outperformed the other algorithms in terms of both optimization accuracy and stability for the Gas Transmission System design problem. This demonstrates that RWOA had superior solving capabilities when handling this type of problem. This indicates that RWOA had a significant advantage in handling such optimization problems.

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Fig 30. Iteration curves of the algorithms in gas transmission system design.

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