In the experiment, we selected several optimization algorithms, namely WOA, SWOA, EWOA, PSO, SCA, FOA, FA, MFO, and BA, to compare their performance in different dimensions (10, 30, 50, 100). Additionally, we utilized the CEC2017 test function to evaluate the algorithms’ impact, allowing for a comparison of SEWOA and 19 other algorithms across various dimensions of the CEC2017 function(See Appendix 3 for details.).The specific experimental settings are as follows:
1.
Each chosen optimization algorithm was tested on problems with dimensions of 10, 30, 50, and 100.
2.
We evaluated the average performance of 30 independent runs for CEC2014 (See Appendix 2 for details.) and 23 base test functions (See Appendix 1 for details.), following a standardized test protocol.
3.
All experiments were conducted on a Windows 11 operating system using MATLAB 2022b software. The hardware configuration consisted of a 3200 MHz processor, 16G memory, and a GPU model of 3060, ensuring a fair comparison.
4.
Upon completion of each run, we recorded the AVG, STD, ARV, and RANK of the algorithm. From these data, optimized images and box plots were generated.
5.
In our comparative analysis, we not only compared similar whale optimization algorithms (WOA, SWOA, EWOA), but also conducted a comprehensive comparison with meta-heuristic optimization algorithms (PSO, SCA, FOA, FA, MFO, BA), the latest meta-heuristic optimization algorithms(COA, ZOA, SS, SSA) and the latest improved meta-heuristic strategies (CDLOBA48, CEBA, CIBA, RCBA, SABA, CWOAI, CWOAII62, ASCA_PSO, CGSCA, MSCA2). Parameter Settings and full names are shown in Table 1 below:
Table 1 Parameter settings and full names.
6.
Finally, SEWOA was applied to solve three engineering optimization problems: the three-bar truss design problem, the tension and compression spring design problem, and the pressure vessel design problem. This aimed to assess its performance in practical problem-solving scenarios.
Influence of nonlinear time-varying dynamic disturbance factor and improved spiral modeThe SEWOA algorithm enhances the initial population and the strategy for updating locations in the original WOA algorithm. To ascertain the effectiveness of the algorithm enhancement, we will compare the impacts of the two position improvements. Illustrated in Table 2, the enhanced initial population’s position is represented by “S,” while the improved search and foraging are denoted by “E.” In the table, “1” signifies that this position update strategy has been enhanced in WOA, whereas “0” indicates it has not been modified.Table 2 Representation of the two optimizations in WOA.The aforementioned four algorithms were subjected to a comparison based on 53 functions, and the prowess of the algorithms was substantiated through evaluation metrics such as average value (AVG), standard deviation (STD), average RANK (ARV), and final rank (RANK). As depicted in Table 3, a comprehensive comparison has been made between the three enhanced algorithms and the original WOA algorithm. The findings indicate that the SEWOA algorithm exhibits the lowest ARV and RANK, suggestive of its superior optimization strategy.Table 3 Comparison of different improvement strategies.As depicted in the illustration presented beneath Fig. 4, the fluctuations in the initial position are observed via F40, F51, and F53, and the curve is remarkably more pronounced. In comparison to the two unoptimized algorithms, the convergence rate of the algorithm initialized by the Soble sequence is expedited, thereby signifying that the initialization of the population position utilizing the Soble sequence contributes a certain enhancement to the convergence and accuracy of the algorithm. By scrutinizing F15 and F53, it becomes evident that the amelioration in search and foraging significantly assists in enhancing the population diversity during the latter stages of the algorithm. By considering F50 and F52, it is evident that the ramifications of the preceding several functions are manifest in the two, thereby suggesting that the combination of the two optimization methods has the capacity to more effectively optimize the algorithm’s performance.Figure 4Comparison of convergence curves of different improvement strategiesIn Fig. 5, the findings illustrate the optimization outcomes achieved through the implementation of diverse strategies to enhance WOA. Notably, SEWOA exhibits a consistent and reliable distribution of data in comparison to several alternative algorithms. By scrutinizing the upper limit, upper quartile, median, lower quartile, and lower limit indicators, it becomes apparent that SEWOA provides a more precise representation of the optimal value.Figure 5 Comparison of box plots for different improvement strategiesComparison with other meta-heuristic optimization algorithmsUpon reviewing the comparative analysis conducted in the preceding section, it is apparent that SEWOA demonstrates favorable efficacy when contrasted with the optimized WOA algorithm. In the forthcoming chapter, the identical parameter configurations employed earlier along with the consistent hardware environment will be utilized to juxtapose SEWOA against seven prevalent meta-heuristic algorithms: PSO, SCA, FOA, FA, MFO, BA. In the ensuing Table 4, the recorded data shall encompass the mean value, standard deviation, and additional evaluation metrics.Table 4 Comparison of different meta-heuristic algorithms.The table displays the ARV value of SEWOA, which is 2.69811320754717, positioning it at the forefront with a notable lead. Upon comparing it with seven other metaheuristic algorithms, it becomes apparent that SEWOA outperforms the remaining seven algorithms in terms of search effectiveness. The same experimental results further validate the exceptional performance of SEWOA across various aspects, such as unimodal, multimodal, and fixed dimension.To present a more visually compelling demonstration of SEWOA’s superiority over other algorithms, convergence curves of SEWOA and the original swarm optimization algorithms are depicted in Fig. 6. The convergence graph portrays SEWOA’s inclination towards earlier convergence when contrasted with other algorithms, showcasing its pronounced advantages in terms of optimization precision and convergence speed.Figure 6 Comparison of convergence curves of different meta-heuristic algorithms.Comparison with the latest meta-heuristic optimization algorithmsThe SEWOA algorithm performs well in AVG on most test functions. For example, in F2, F3, F7 and other functions, the score of SEWOA algorithm is significantly higher than other algorithms, which indicates that SEWOA algorithm has a strong ability to find the optimal solution, showing its high efficiency in complex problems.As shown in Table 5. The STD of SEWOA algorithm is small, especially in F1, F3, F6 and other functions, and the standard deviation is significantly lower than that of other algorithms. The standard deviation measures the volatility of the algorithm’s results, and a smaller standard deviation means that the SEWOA algorithm’s performance is more consistent and stable across different test functions. This stability is crucial for practical applications, as it guarantees the reliability of the algorithm in different environments.Not only does the SEWOA algorithm achieve a high average score in most test functions, but its smaller standard deviation indicates superior stability and robustness. In optimization problems, the stability and consistency of the algorithm are the key factors, because it affects the reliability of the algorithm in practical applications. Therefore, SEWOA algorithm is considered to be the best performing algorithm in this test and has high practical application potential.Table 5 Comparison of the latest optimization algorithms.As shown in Fig. 7. The SEWOA algorithm performs significantly better on these benchmark functions than many traditional optimization algorithms. Specifically, on F2 and F6 functions, the SEWOA algorithm can approach the global optimal solution faster and maintain a low optimal value, which indicates that it has a strong advantage in terms of global search ability. For F8, F9, F10, and F16 functions, the SEWOA algorithm also performs better than other algorithms, which shows its stability and efficiency when dealing with complex functions. Especially for the high-dimensional complex function such as F16, SEWOA algorithm can still effectively avoid the trap of local optimal solution and find the solution closer to the global optimal.Figure 7Comparison of convergence curves of different latest meta-heuristic algorithms.Comparison of algorithms with different dimensionsIn order to delve deeper into the impact of dimensions on the optimization performance of SEWOA, this study conducts experiments on SEWOA using the IEEE CEC2014 function, with dimensions set at 10, 30, 50, and 100, respectively. The evaluation times are set at 1,000,000, 300,000, 500,000, and 1,000,000, concurrently, while 30 independent repeated experiments are conducted. The aforementioned experiments include WOA, SWOA, EWOA, as well as the original optimization algorithms PSO, SCA, FOA, FA, MFO, and BA for comparison.The experimental results, as shown in Table 6, reveal that when the dimension is 10, the optimal solution obtained by SEWOA surpasses that of the other 10 algorithms, exhibiting the smallest ARV value and securing the top rank. As the dimension expands to 30, 50, and 100, the optimal solution acquired by SEWOA remains the best, featuring the smallest ARV value and maintaining its top-ranking position. These experiments substantiate that the SEWOA algorithm demonstrates superior optimization capacity across varying dimensions.Table 6 Comparison of algorithms for functions of different dimensions.CEC2017 test function for algorithm impact comparisonThe CEC2017 has updated and optimized the benchmark functions to better reflect the characteristics and challenges of real-world problems. This includes more reasonable constraint settings, a greater number of local and global optimal solutions, and more complex topological structures, all aimed at more accurately evaluating the performance of algorithms. For further testing the performance of SEWOA, the CEC2017 test functions were selected.In this section, the CEC2017 test functions were applied, and two improved strategies were incorporated into WOA, SWOA, and EWOA. Additionally, meta-heuristic algorithms such as PSO、SCA、FOA、FA、MFO、BA, as well as the currently improved meta-heuristic algorithms CDLOBA, CEBA, CIBA, RCBA, SABA, CWOAI, CWOAII, ASCA_PSO, CGSCA, and MSCA2 were included. The tests were conducted using the CEC2017 test functions with varying dimensionalities of 10, 30, 50, and 100. For each dimensional test, 50 search agents were independently executed 50 times. The average best solution was calculated and ranked accordingly.Table 7 shows the impact of SEWOA and the twenty algorithms on different dimensions of the CEC2017 functions. When the dimensions are 10, 30, 50, and 100, the Average Relative Value (ARV) of SEWOA is 2.6, 2.6, 2.4, and 2.5, respectively. Amongst the twenty algorithms compared, SEWOA ranks first in terms of ARV. This demonstrates that SEWOA performs well even in more complex scenarios represented by the CEC2017 test functions.Table 7 Comparison of algorithms on CEC2017 test functions.Engineering design experimental problemsThree bar truss design problems The three-bar truss design problem is another structural optimization problem in the field of civil engineering, in which the following operation is performed for two bar lengths in order to minimize the weight subject to stress, deflection and buckling constraints:$$\min\;f(x)=(2\sqrt2x_1+x_2)l$$The constraints are:\(\begin{gathered} {g_1}(x)=\frac{{\sqrt 2 {x_1}+{x_2}}}{{\sqrt 2 {x_1}^{2}+2{x_1}{x_2}}}P – \upsigma \leqslant 0 \hfill \\ {g_2}(x)=\frac{{{x_2}}}{{\sqrt 2 {x_1}^{2}+2{x_1}{x_2}}}P – \upsigma \leqslant 0 \hfill \\ {g_3}(x)=\frac{1}{{\sqrt 2 {x_2}+{x_1}}}P – \upsigma \leqslant 0 \hfill \\ \end{gathered}\)\(l=100 cm,P={{2{\text{k}}N} \mathord{\left/ {\vphantom {{2{\text{k}}N} {c{m^2},\upsigma }}} \right. \kern-0pt} {c{m^2},\upsigma }}={{2{\text{k}}N} \mathord{\left/ {\vphantom {{2{\text{k}}N} {c{m^2}}}} \right. \kern-0pt} {c{m^2}}}\)It can be seen from Table 8 that compared with the other three algorithms, under the condition of ensuring the basic requirements, SEWOA has the lowest consumption cost, only requiring 263.9037, which fully shows that the performance of SEWOA is better than other algorithms.Table 8 Comparison results of three-bar truss design problems.Tension spring design problemThe goal of the pull-and-compress spring design problem is to minimize the weight of the pull-and-compress spring while satisfying the constraints of minimum deflection, vibration frequency, and shear stress, The problem consists of three continuous decision variables, namely, the diameter of the spring coil, the diameter of the spring coil, and the number of coils, which are denoted by \(x_1,x_2,x_3\), respectively. The mathematical modeling formula is as follows:$$\hbox{min} f(x)=({x_3}+2){x_2}{x_1}^{2}$$The constraints are:\(\begin{gathered} {g_1}(x)=1 – \frac{{x_{2}^{3}{x_3}}}{{71785x_{1}^{4}}} \leqslant 0 \hfill \\ {g_2}(x)=\frac{{4x_{2}^{2} – {x_1}{x_2}}}{{12566\left( {{x_2}x_{1}^{3} – x_{1}^{4}} \right)}}+\frac{1}{{5108x_{1}^{2}}} – 1 \leqslant 0 \hfill \\ {g_3}(x)=1 – \frac{{140.45{x_1}}}{{x_{2}^{2}{x_3}}} \leqslant 0 \hfill \\ {g_4}(x)=\frac{{{x_1}+{x_2}}}{{1.5}} – 1 \leqslant 0 \hfill \\ \end{gathered}\)\(0.05 \leqslant {x_1} \leqslant 2,0.25 \leqslant {x_2} \leqslant 1.3,2 \leqslant {x_3} \leqslant 150\)It can be seen from Table 9 that SEWOA has achieved good results in solving the special problem of the tension and compression spring design problem, obtaining the lowest cost of 0.012667, which also verifies that SEWOA has a good effect in solving such problems.Table 9 Comparison results of tension and compression spring design problems.Pressure vessel design issuesThe objective of the pressure vessel design problem is to minimize the pressure vessel fabrication cost. Both ends of the pressure vessel are capped with a lid, and the cap at one end of the head is hemispheric. \(L,R,T_s,T_h\) is the four optimization variables of the pressure vessel design problem, \(L\) is the section length of the cylinder part without considering the head, \(R\) is the inner wall diameter of the cylinder part, and \(T_s,T_h\) represents the wall thickness of the cylinder part and the wall thickness of the head, respectively. The objective function and four optimization constraints of the problem are expressed as follows.\(x=\left[ {{x_1},{x_2},{x_3},{x_4}} \right]=\left[ {{T_s},{T_h},R,L} \right]\)\(\hbox{min} f(x)=0.6224{x_1}{x_3}{x_4}+1.7781{x_2}x_{3}^{2}+3.1661x_{1}^{2}{x_4}+19.84x_{1}^{2}{x_3}\)The constraints are:\(\begin{gathered} {g_1}\left( x \right)= – {x_1}+0.0193{x_3} \leqslant 0 \hfill \\ {g_2}\left( x \right)= – {x_2}+0.0954{x_3} \leqslant 0 \hfill \\ {g_3}\left( x \right)= – \pi x_{3}^{2} – {{4\pi x_{3}^{3}} \mathord{\left/ {\vphantom {{4\pi x_{3}^{3}} 3}} \right. \kern-0pt} 3}+1,296,000 \leqslant 0 \hfill \\ {g_4}\left( x \right)={x_4} – 240 \leqslant 0 \hfill \\ \end{gathered}\)\(0 \leqslant {x_1} \leqslant 100,0 \leqslant {x_2} \leqslant 100,10 \leqslant {x_3} \leqslant 100,10 \leqslant {x_4} \leqslant 100\)It can be seen from Table 10 that SEWOA has better cost control ability than other algorithms in solving the pressure vessel design problem, and solves this engineering problem at the lowest cost, which fully proves that SEWOA is better than other algorithms.Table 10 Comparison results of pressure vessel design problems.