1. Introduction

The structure of this article is as follows: Section 2 is a literature review, reviewing and discussing past meta heuristic algorithms. The third and fourth chapters are the construction of the theoretical framework, with the third chapter explaining the original SO and the fourth chapter proposing three improvement strategies, including the fusion of RIME algorithm. Section 5, Section 6, and Section 7 are the experimental parts. Section 5 quantitatively calculates the population distribution of the ISO algorithm in space using three algorithms: Average Nearest Neighbor Distance, Sum of Squared Deviations (SSD), and Star Discrepancy, and uses cec2017 to determine the balance between exploration and development of ISO. Chapter 6 conducts two sets of experiments using the most classic and widely used 23 test functions and compares them with five other competitive algorithms. The first set of experiments compares the convergence performance of the other five competitive algorithms, and the second set of experiments discusses in detail the final solution results of these algorithms, including, Optimum Value,Standard Deviation,Average Value,Median Value,Minimum Value,And Wilcoxon rank sum was used to determine whether there were significant differences in these results. Section 7 applied the algorithm to a wider range of engineering scenarios, including UVA path planning and robot path planning, Wireless sensor network node deployment and Pressure vessel design, which spans across the mainstream applications of aerospace engineering, robotics technology, communication engineering, and mechanical engineering, is compared and analyzed with five competing algorithms. Section 8 summarizes the entire text and provides prospects for possible algorithm optimization and application scenarios in the future. The following figure clearly illustrates the structure of this article.

Figure 1. Enter Caption

Figure 1. Enter Caption

2. Literature Review

3. Snake Optimization Algorithm

3.7. Algorithm Flowchart

Figure 2. Algorithm Flowchart

Figure 2. Algorithm Flowchart

4. Proposed Method

4.3. Lens Imaging Reverse Learning

The population distributed in the function space can lead the group to find good resources due to its wide search range and flexible search methods[31]. Once some populations fall into local optima, the overall performance of the algorithm declines. Therefore, enhancing the search range of the finders is particularly important. General learning strategies have achieved good results in some optimization algorithms, but they have little impact on the algorithm’s performance. This is because general learning strategies can only perform reverse solving in local spaces, which, while enriching the population’s diversity, still have a narrow search range and lose the ability to explore the global space. [32]To improve the population’s search capability, an inverse learning strategy based on lens imaging is proposed and applied to the position update formulas of all individuals in the population to enhance the global search capability of ISO. $[lb,ub]$ is the search space for solutions, and y represents the function value. Suppose there is a system $P_i$ with a height of $h_i$, and its projection at point x is x; this system is reproduced as another system $P_i^{*}$ on the other side with a lens focal length that is halved midway, with a height of $h_i^{*}$ and a projection at point $x_i$ as $x_i^{*}$. According to the principle of lens imaging, we obtain[33]:

$${\displaystyle \frac{\frac{lb+ub}{2}-x_i}{x_i^{*}-\frac{lb+ub}{2}}}={\displaystyle \frac{h_i}{h_i^{*}}}$$

(25)

Let $k={\displaystyle \frac{h}{h^{\prime }}}$; then Equation (9) can be rewritten as:

$$x_i^{*}={\displaystyle \frac{lb+ub}{2}}+{\displaystyle \frac{lb+ub}{2k}}-{\displaystyle \frac{x_i}{k}}$$

(26)

Equation (26) is the transformation formula for the back focal length of the original lens on the halved side.

The scaling factor k is a freely selectable parameter that can be dynamically adjusted according to different problems. In this paper, the selection of the k value is based on the results of multiple experiments, which can maximize the solving ability of the ISO algorithm[33].

$$k={\left(1+{\left({\displaystyle \frac{t}{T}}\right)}^{0.5}\right)}^9$$

(27)

The pseudocode is as follows:

5. Population Initialization and Exploration-Exploitation Analysis

5.1. Population Initialization Analysis

The following figure shows the initialization state of the ISO population, and the table below presents the statistical results measured using three algorithms: Star Discrepancy, Average Nearest Neighbor Distance, and Sum of Squared Deviations (SSD).

Figure 3. The picture of the comparison between ISO with Sobol initialization and SO with random initialization.

Figure 3. The picture of the comparison between ISO with Sobol initialization and SO with random initialization.

Table 1. Precision Data

Table 1. Precision Data

Parameter	SO (Random Initialization)	ISO (Sobol Sequence)
Star Discrepancy	0.133966	0.047965
Average Nearest Neighbor Distance	0.069264	0.086739
Sum of Squared Deviations (SSD)	11.255603	9.990234

The three metrics—Star Discrepancy,Average Nearest Neighbor Distance and Sum of Squared Deviations (SSD) are crucial for evaluating the uniformity of population distributions. The specific calculations are as follows.

Star Discrepancy

Description: The star discrepancy measures the overall spatial uniformity of a point set within a unit cube. By calculating the maximum difference between the number of points within subrectangles and the expected number of points in a perfectly uniform distribution, the uniformity can be assessed. A smaller star discrepancy indicates a more uniform distribution, while a larger star discrepancy indicates less uniformity.[38]

Formula:

$$D^{*}=\underset{1\le i\le n}{max}\left|{\displaystyle \frac{1}{n}}\sum _{j=1}^n(p_j\le p_i)-\prod _{k=1}^d{x}_{ik}\right|$$

(28)

where n is the total number of points, d is the dimension, $p_j$ is a point in the set, $p_i$ is a reference point, $x_{ik}$ is the coordinate of point $p_i$ in the kdimension.

Average Nearest Neighbor Distance

Description: The average nearest neighbor distance measures the uniformity of point distributions in space. By calculating the distance from each point to its nearest neighbor and taking the average of these distances, one can determine the distribution pattern. A larger average nearest neighbor distance indicates a more uniform distribution, while a smaller distance indicates clustering.[39] Formula:

$$\mathrm{Average}\mathrm{Nearest}\mathrm{Neighbor}\mathrm{Distance}={\displaystyle \frac{1}{n}}\sum _{i=1}^n\underset{j\ne i}{min}d(p_i,p_j)$$

(29)

where n is the total number of points, $p_i$ and $p_j$ are points in the set, and $d(p_i,p_j)$ denotes the Euclidean distance between point $p_i$ and point $p_j$.

Sum of Squared Deviations (SSD)

Description: The sum of squared deviations evaluates the dispersion of a set of points relative to a central location. By calculating the squared distance from each point to the center and summing these values, one can quantify the spread of the point set. A smaller SSD indicates a more uniform distribution, while a larger SSD indicates greater dispersion.[40] Formula:

$$\mathrm{SSD}=\sum _{i=1}^n\sum _{j=1}^d{(p_{ij}-c_j)}^2$$

(30)

where n is the total number of points, d is the dimension, $p_{ij}$ is the coordinate of the i-th point in the j-th dimension, and $c_j$ is the coordinate of the central point in the j-th dimension.

The Sobol sequence initialization (ISO) demonstrates superior uniformity compared to random initialization (SO). Specifically, the ISO exhibits a higher Average Nearest Neighbor Distance (0.086739 compared to 0.069264). Moreover, the ISO shows a lower SSD (9.990234 compared to 11.255603) and a significantly lower Star Discrepancy (0.047965 compared to 0.133966). These results collectively affirm that ISO provides a more homogeneous initial population than SO, making it a preferable choice for applications requiring highly uniform distributions.

6. Benchmark Function Test

6.1. Convergence Behavior Analysis

In this section, we focus on analyzing the convergence behavior of different algorithms. For each set of results, the left figure presents the 3D image of the test function, vividly illustrating the complexity of the problem, while the right figure shows the convergence curves of different algorithms when solving various problems. All algorithms were configured with a population size of N=40 and a maximum of T=500 iterations.

he convergence curve results indicate that the ISO algorithm achieves rapid convergence and lower function values in most cases. Specifically, ISO performs similarly to other competitive algorithms on functions F8, F15, F16, F19, and F20, while achieving lower function results with fewer iterations on the remaining 18 test functions. Notably, ISO demonstrates superior performance on functions F1, F2, F3, F4, F7, and F10 compared to competitive algorithms. This underscores ISO’s superior convergence speed and its enhanced capability to explore global optimal solutions compared to SO and other competitive algorithms.

Figure 5. 23 Test Function

Figure 5. 23 Test Function

6.2. Detailed Analysis of the Benchmark Function

To comprehensively demonstrate the effectiveness of ISO, the maximum value, standard deviations, averages, medians, and minimum values obtained by six algorithms when solving 23 different functions were meticulously recorded. Boxplots were plotted based on the experimental results [47], and Wilcoxon rank-sum tests were conducted [48].In Wilcoxon rank-sum tests, a significance threshold of 5% was established. If the computed value falls below 5%, it denotes a statistically significant distinction between the two algorithms; if the computed value surpasses 5%, it signifies no statistically significant difference. NaN values indicate that the performance of the two algorithms is similar and cannot be compared.

Figure 6. Test Function F1-F6

Figure 6. Test Function F1-F6

Figure 7. Test Function F7-F12

Figure 7. Test Function F7-F12

Figure 8. Test function F13-F18

Figure 8. Test function F13-F18

Figure 9. Test Function F19-F23

Figure 9. Test Function F19-F23

Figure 10. Result of Boxplot (maximum value,Standard Deviation,Average Value,Median Value,Minimum Value)

Figure 10. Result of Boxplot (maximum value,Standard Deviation,Average Value,Median Value,Minimum Value)

Table 2. Precision Data

Table 2. Precision Data

	ISO	SO	RIME	WOA	GWO	SCA
F1 Maximum value	0	1.6156e-101	0.79632	6.2667e-90	2.9418e-32	0.018227
F1 Standard Deviation	0	3.0829e-96	0.82593	3.6085e-80	2.1056e-30	5.8823
F1 Average Value	0	2.1112e-96	1.6654	1.1703e-80	1.3072e-30	3.4484
F1 Median Value	0	5.977e-97	1.5769	6.7968e-86	3.3005e-31	0.38834
F1 Minimum Value	0	8.8973e-96	3.7043	1.1438e-79	5.9552e-30	15.6586
F2 Maximum value	0	1.5847e-46	0.53371	2.392e-57	2.0754e-19	4.9074e-05
F2 Standard Deviation	0	5.2612e-44	0.4858	1.0992e-52	6.9339e-19	0.0088126
F2 Average Value	0	3.0919e-44	0.90418	4.0029e-53	1.1986e-18	0.0077208
F2 Median Value	0	6.2431e-45	0.75306	2.2097e-55	9.5475e-19	0.0039093
F2 Minimum Value	0	1.7346e-43	2.0696	3.5024e-52	2.2958e-18	0.027479
F3 Maximum value	0	1.1664e-67	384.3923	19995.9104	1.1741e-09	1525.2676
F3 Standard Deviation	0	1.1251e-59	304.3303	8915.4857	6.7043e-07	4607.7335
F3 Average Value	0	5.4436e-60	854.9073	35898.743	4.7077e-07	10138.0775
F3 Median Value	0	4.0227e-63	827.1762	36953.5381	8.146e-08	11024.7254
F3 Minimum Value	0	3.1096e-59	1402.0287	48574.3146	1.5437e-06	16323.6006
F4 Maximum value	0	5.1604e-43	3.4574	3.6318	9.928e-09	9.027
F4 Standard Deviation	0	2.964e-41	1.6497	30.5907	1.2116e-07	6.7815
F4 Average Value	0	2.3189e-41	6.0868	41.9403	1.5419e-07	21.0788
F4 Median Value	0	7.0935e-42	5.6659	37.7864	1.2797e-07	22.0562
F4 Minimum Value	0	8.5465e-41	9.573	85.1509	3.9006e-07	29.6541
F5 Maximum value	24.7598	28.8231	52.2265	27.2556	26.179	82.5777
F5 Standard Deviation	0.59228	54.4177	458.7257	0.21881	0.67703	18885.0142
F5 Average Value	25.3956	54.7162	445.2294	27.67	27.1109	9078.5037
F5 Median Value	25.1402	28.9162	353.4091	27.7765	27.1375	2236.0169
F5 Minimum Value	26.4062	159.0777	1680.5587	27.866	27.9309	61772.1619
F6 Maximum value	0.00022812	0.12251	0.43999	0.028669	0.24964	5.1908
F6 Standard Deviation	0.0021647	0.23817	0.80692	0.058841	0.25713	11.0357
F6 Average Value	0.0021236	0.36652	1.5199	0.11855	0.54914	12.1774
F6 Median Value	0.0013819	0.29371	1.5159	0.11984	0.50366	7.2204
F6 Minimum Value	0.0072755	0.90824	3.017	0.22648	0.99279	35.4493
F7 Maximum value	5.9325e-06	9.8841e-05	0.0066106	7.2842e-05	0.00054522	0.013495
F7 Standard Deviation	1.8724e-05	0.00013572	0.013488	0.0037777	0.0013254	0.064203
F7 Average Value	3.4074e-05	0.00027541	0.030573	0.0027863	0.001424	0.066626
F7 Median Value	3.7737e-05	0.000272	0.030255	0.0014118	0.0010548	0.042689
F7 Minimum Value	6.2134e-05	0.00053759	0.047252	0.012901	0.0049689	0.21389
F8 Maximum value	-8941.9029	-11202.8618	-11212.2637	-12567.4537	-7241.0015	-3964.0364
F8 Standard Deviation	705.6836	419.4716	352.7549	1680.0918	470.9275	158.2354
F8 Average Value	-7979.4087	-10433.1947	-10551.4162	-10362.0449	-6462.9317	-3734.5521
F8 Median Value	-7675.3417	-10305.079	-10469.8854	-9965.6122	-6406.543	-3705.014
F8 Minimum Value	-7314.8199	-10014.2347	-10071.167	-8522.1804	-5790.5202	-3541.7477
F9 Maximum value	0	32.1315	35.0709	0	0	0.16486
F9 Standard Deviation	0	9.6948	16.224	0	1.895	38.8566
F9 Average Value	0	47.3503	58.4451	0	1.087	46.9693
F9 Median Value	0	47.4626	55.8667	0	1.7053e-13	52.2689
F9 Minimum Value	0	64.4491	88.9517	0	5.2149	99.3974
F10 Maximum value	4.4409e-16	4.4409e-16	1.5369	4.4409e-16	5.0182e-14	0.047233
F10 Standard Deviation	0	1.1235e-15	0.36392	2.4841e-15	7.1937e-15	10.1462
F10 Average Value	4.4409e-16	3.6415e-15	2.0332	5.4179e-15	5.7643e-14	10.2912
F10 Median Value	4.4409e-16	3.9968e-15	2.0954	5.7732e-15	5.7288e-14	10.3939
F10 Minimum Value	4.4409e-16	3.9968e-15	2.6224	7.5495e-15	6.7946e-14	20.352
F11 Maximum value	0	0	0.70668	0	0	0.32025
F11 Standard Deviation	0	0.19571	0.1102	0	0.0072452	0.27456
F11 Average Value	0	0.12263	0.87297	0	0.0044876	0.81096
F11 Median Value	0	0.026978	0.86309	0	0	0.87535
F11 Minimum Value	0	0.48677	1.0302	0	0.015867	1.2516
F12 Maximum value	8.7918e-06	0.64823	0.64372	0.005086	0.019088	0.64244
F12 Standard Deviation	0.00033743	2.0266	1.6162	0.011681	0.010848	85.404
F12 Average Value	0.00014159	2.4297	2.2166	0.017477	0.035523	37.3233
F12 Median Value	3.9058e-05	1.8708	1.9305	0.015435	0.038864	8.3931
F12 Minimum Value	0.0010994	7.3143	5.9443	0.044372	0.048986	278.8304
F13 Maximum value	0.10897	0.68249	0.10148	0.093222	0.11287	2.7887
F13 Standard Deviation	0.42285	0.72067	0.044075	0.15559	0.19083	3464.4011
F13 Average Value	0.86081	1.5917	0.15969	0.26413	0.51709	1600.5508
F13 Median Value	0.97321	1.3588	0.15913	0.20027	0.5127	24.2384
F13 Minimum Value	1.3428	2.6575	0.24852	0.48656	0.84402	10046.4838
F14 Maximum value	0.998	0.998	0.998	0.998	0.998	0.998
F14 Standard Deviation	1.813e-16	1.5457	6.6484e-12	3.0198	3.5827	0.95833
F14 Average Value	0.998	1.6899	0.998	2.2728	4.1415	1.5934
F14 Median Value	0.998	0.998	0.998	0.99815	2.9821	0.99823
F14 Minimum Value	0.998	5.9288	0.998	10.7632	10.7632	2.9821
F15 Maximum value	0.00030749	0.00032065	0.00040604	0.0003206	0.00030749	0.00055656
F15 Standard Deviation	0.0003688	0.0062831	0.0095202	0.00040904	0.0084458	0.00036924
F15 Average Value	0.00063972	0.0024876	0.006571	0.00078746	0.0043389	0.0010789
F15 Median Value	0.00054085	0.00050553	0.00067254	0.00073404	0.00030813	0.0010486
F15 Minimum Value	0.0012759	0.020363	0.020363	0.0015046	0.020363	0.0015135
F16 Maximum value	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316
F16 Standard Deviation	1.282e-16	1.9582e-16	1.2238e-07	9.7558e-11	2.1284e-08	3.4995e-05
F16 Average Value	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316
F16 Median Value	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316
F16 Minimum Value	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316	-1.0315
F17 Maximum value	0.39789	0.39789	0.39789	0.39789	0.39789	0.39796
F17 Standard Deviation	0	0	4.5491e-07	2.0914e-06	6.7395e-05	0.0012934
F17 Average Value	0.39789	0.39789	0.39789	0.39789	0.39791	0.39942
F17 Median Value	0.39789	0.39789	0.39789	0.39789	0.39789	0.39935
F17 Minimum Value	0.39789	0.39789	0.39789	0.39789	0.3981	0.4025
F18 Maximum value	3	3	3	3	3	3
F18 Standard Deviation	1.9244e-15	3.0553e-15	5.1877e-08	2.0542e-05	1.1567e-05	6.5547e-05
F18 Average Value	3	3	3	3	3	3.0001
F18 Median Value	3	3	3	3	3	3
F18 Minimum Value	3	3	3	3.0001	3	3.0002
F19 Maximum value	-3.8628	-3.8628	-3.8628	-3.8628	-3.8628	-3.8623
F19 Standard Deviation	8.6315e-16	7.4015e-16	5.2188e-07	0.0031652	0.0037075	0.0033693
F19 Average Value	-3.8628	-3.8628	-3.8628	-3.8594	-3.8604	-3.8563
F19 Median Value	-3.8628	-3.8628	-3.8628	-3.8597	-3.8626	-3.8546
F19 Minimum Value	-3.8628	-3.8628	-3.8628	-3.8539	-3.8549	-3.852
F20 Maximum value	-3.2031	-3.322	-3.322	-3.3218	-3.322	-3.1176
F20 Standard Deviation	0.041065	0.057431	0.057429	0.060089	0.1023	0.49659
F20 Average Value	-3.1637	-3.2863	-3.2863	-3.2927	-3.2182	-2.7968
F20 Median Value	-3.1675	-3.322	-3.322	-3.3208	-3.1985	-3.0029
F20 Minimum Value	-3.0977	-3.2031	-3.2031	-3.1688	-3.0272	-1.8081
F21 Maximum value	-10.1532	-10.1532	-10.1532	-10.1513	-10.1529	-7.4983
F21 Standard Deviation	2.1479	2.871	2.9233	2.6293	0.00097496	2.523
F21 Average Value	-9.4737	-8.4288	-6.8697	-8.1101	-10.1514	-2.9758
F21 Median Value	-10.1532	-10.1532	-5.1007	-10.1429	-10.1515	-2.6826
F21 Minimum Value	-3.3607	-2.6305	-2.6305	-5.0551	-10.1498	-0.49728
F22 Maximum value	-10.4029	-10.4029	-10.4029	-10.4025	-10.4023	-5.2831
F22 Standard Deviation	2.1119	3.0882	3.1212	2.7809	1.6676	0.86604
F22 Average Value	-9.735	-7.2831	-8.0118	-9.0997	-9.8738	-4.335
F22 Median Value	-10.4029	-7.2186	-10.4014	-10.3924	-10.4012	-4.6794
F22 Minimum Value	-3.7243	-2.7659	-3.7243	-2.7656	-5.1276	-2.7142
F23 Maximum value	-10.5364	-10.5364	-10.5363	-10.5355	-10.5361	-6.7449
F23 Standard Deviation	2.119	3.7908	2.6046	3.9107	0.00085842	1.6382
F23 Average Value	-9.8663	-6.4037	-8.9183	-6.8186	-10.5352	-3.7579
F23 Median Value	-10.5364	-5.3924	-10.5356	-7.0568	-10.5355	-3.477
F23 Minimum Value	-3.8354	-2.4217	-5.1281	-1.8594	-10.5337	-0.94217

The experimental results from the boxplots illustrate the numerical values obtained by different algorithms when solving the 23 test functions multiple times. They specifically display the maximum and minimum values, the median, the first and third quartiles, and the interquartile range of the solution results. From the plots, it can be observed that, except for functions F8, F15, and F20, the ISO algorithm yields the smallest function values and exhibits minimal errors across multiple solving attempts, demonstrating excellent convergence performance. However, its solving capability for functions F8, F15, and F20 is slightly inferior, indicating that there is still room for improvement in these areas.

The Wilcoxon rank-sum test results show that there are no NaN values in the table, and most p-values are less than 0.05, indicating significant differences between ISO and other competing algorithms, including SO. In summary, based on the analyses in Section 6, ISO demonstrates superior overall performance on benchmark functions compared to competitive algorithms. This highlights the efficacy of the strategies employed in ISO, including Sobol sequence initialization, integration of the RIME algorithm, and lens imaging reverse learning.

Table 3. Rank-Sum Test Results

Table 3. Rank-Sum Test Results

	SO	RIME	WOA	GWO	SCA
F1	6.3864e-05	6.3864e-05	6.3864e-05	6.3864e-05	6.3864e-05
F2	6.3864e-05	6.3864e-05	6.3864e-05	6.3864e-05	6.3864e-05
F3	6.3864e-05	6.3864e-05	6.3864e-05	6.3864e-05	6.3864e-05
F4	6.3864e-05	6.3864e-05	6.3864e-05	6.3864e-05	6.3864e-05
F5	1.8267e-04	1.8267e-04	1.8267e-04	3.2984e-04	1.8267e-04
F6	1.8267e-04	1.8267e-04	1.8267e-04	1.8267e-04	1.8267e-04
F7	1.8267e-04	1.8267e-04	1.8267e-04	1.8267e-04	1.8267e-04
F8	1.8267e-04	1.8267e-04	0.0010	1.8267e-04	1.8267e-04
F9	6.3864e-05	6.3864e-05	1	2.1655e-04	6.3864e-05
F10	9.6605e-05	6.3864e-05	1.8923e-04	5.7206e-05	6.3864e-05
F11	0.0149	6.3864e-05	1	0.0779	6.3864e-05
F12	1.8267e-04	1.8267e-04	1.8267e-04	1.8267e-04	1.8267e-04
F13	0.0312	0.0140	0.0113	0.0211	1.8267e-04
F14	0.0384	1.3093e-04	1.3093e-04	1.3093e-04	1.3093e-04
F15	0.8501	0.1620	0.3847	0.2730	0.0073
F16	0.0891	1.2855e-04	1.2855e-04	1.2855e-04	1.2855e-04
F17	1	6.3864e-05	6.3864e-05	6.3864e-05	6.3864e-05
F18	0.0060	1.7661e-04	1.7661e-04	1.7661e-04	1.7661e-04
F19	0.1851	1.0997e-04	1.0997e-04	1.0997e-04	1.0997e-04
F20	9.0134e-04	0.0022	0.0028	0.2404	3.2138e-04
F21	0.1770	0.0055	0.0044	0.0167	7.2031e-04
F22	0.0323	0.0029	0.0023	0.0029	0.0013
F23	0.0124	0.0027	7.2031e-04	0.0027	5.4476e-04

7. Engineering Application

7.1. UAV Path Planning

For decades, Unmanned Aerial Vehicles (UAVs) have been recognized as one of the most challenging and promising technologies in the field of aviation. They are widely applied in agriculture[49,50], power line inspection[51,52], aerial photography[53,54], and national defense[55,56]. Path planning has gradually become a critical technology for UAV, garnering extensive research from scholars.[57] UVA flying in the sky are regarded as particles of negligible size.The primary objective of UAV path planning is to design a flight path towards the target that minimizes the overall cost while meeting the performance requirements of the UAV.[58]In this problem, the maximum number of iterations is $T=100$, and the population size is $N=15$. In UAV path planning, the primary considerations are threat cost and fuel cost, as presented in Eq. (32).

$$Ft=k_1\times F_{tc}+k_2\times F_{fc}$$

(34)

Here, $Ft$ indicates the total cost, $F_{tc}$ represents the threat cost calculated using Eq. (33), $F_{fc}$ denotes the fuel cost calculated by Eq. (34), and $k_i(i=1,2)$ are weight coefficients. The constraints on these coefficients are given in Eq. (35).

$$F_{tc}=\sum _{j=2}^{\dim -1}\sqrt{{\left({\displaystyle \frac{x_{end}}{j+1}}\right)}^2+{(path\left(j\right)-path(j-1))}^2}$$

(35)

where $x_{end}$ represents the abscissa value of the flight endpoint, and $path\left(j\right)$ represents the jth subpath.

$$F_{fc}=\sum _{j=1}^6{F}_{tc}\left(j\right)\sum _{b=1}^6{\left({\displaystyle \frac{1}{\mathrm{dis}(C_b,P_b,(j-0.1)·dx,(0.9·path(j-1)+0.1·path\left(j\right)))}}\right)}^4$$

(36)

where $C_b$ is the coordinate of the bth threat center, $P_b$ is the threat level of the bth threat center, and dis is the distance function between two points.

$$\left\{\begin{array}{c}{k}_i\ge 0\hfill \\ \sum _{i=1}^k{k}_i=1\hfill \end{array}\right.$$

(37)

Figure 11. Visual Comparison of UAV Path Planning

Figure 11. Visual Comparison of UAV Path Planning

Figure 12. Convergence Curves of Optimization Algorithms

Figure 12. Convergence Curves of Optimization Algorithms

Table 4. Precision Data

Table 4. Precision Data

ISO	SO	RIME	GWO	SCA	WOA
70.892	73.207	73.151	72.152	74.522	73.133

The ISO path is notably smoother than competing algorithms, and the algorithm converges faster. The revamped approach enhances its effectiveness in addressing 2D unmanned aerial vehicle path planning problems.

7.2. Robot Path Planning

Mobile robots play an important role in a wide range of fields such as the automation industry [59], military security [60], agricultural operations [61], mining exploration [62], and warehouse management [63]. One of their core tasks is path planning, which aims to comprehensively consider the geometric dimensions, physical limitations, and time efficiency of the robot, and plan a collision-free and optimized travel route in complex working environments [64]. Unlike motion planning, which delves into dynamic characteristics, path planning focuses more on finding a path that minimizes travel time and accurately reflects environmental features from a kinematic perspective.

To achieve this goal, robots need to construct or receive spatial models of their operating environment. These models can be explicit, actively created and stored by the robots [65], or implicit, indirectly utilized through algorithms. These spatial models can exist in various forms, including discretized representations such as grid maps, where space is uniformly divided into small blocks, or non-uniform discretization forms such as topological maps, which focus more on representing critical paths and the connection relationships between nodes. Additionally, there are continuous mapping methods that accurately describe the environment by defining a continuous coordinate sequence of polygon boundaries and paths around the boundaries. Although this method is more efficient in memory usage, in the field of robot path planning, discretized maps are favored for their ability to naturally transform into graph structures. The graph structure not only intuitively reflects environmental characteristics, but also benefits from the rich search and optimization algorithm resources in the field of graph theory, making the calculation process both efficient and concise. Therefore, although continuous mapping has its unique advantages, in practical operation, discretized maps are still the mainstream choice for robot path planning [66].

In this experiment, a robot walking on the ground is regarded as a particle of negligible size. Primary considerations encompass the intricate obstacle space and the optimization of the fitness function. The upper limit of the iteration count is set to T = 100 and the population size to N = 15.To simultaneously assess the feasibility and convergence performance of the algorithm, this experiment simulated a complex spatial environment for the robot.The detailed description is as follows: The matrix G provided undergoes a sequence of operations to produce a new matrix G. These operations are outlined as follows: The initial matrix G, with dimensions m × n, is extended by transposing it and vertically concatenating it with itself:

Matrix Extension: The initial matrix G, sized $m\times n$, is extended by transposing and vertically concatenating it with itself:

$$G^{\prime }=\left[\begin{array}{c}{G}^T\\ G^T\end{array}\right]$$

(38)

where $G^T$ denotes the transpose of G. Further Extension The resultant matrix $G^{\prime }$, now sized $2m\times n$, is further extended by transposing and vertically concatenating it with itself again:

$$G^{\prime \prime }=\left[\begin{array}{c}{G}^{\prime T}\\ G^{\prime T}\end{array}\right]$$

(39)

resulting in a matrix $G^{\prime \prime }$ of size $4m\times 2n$. Matrix Flipping The extended matrix $G^{\prime \prime }$ undergoes a flipping operation where the upper half rows are swapped with the corresponding lower half rows:

$$\forall i\in \left[1,{\displaystyle \frac{\mathrm{num}}{2}}\right],\forall j\in [1,\mathrm{num}]:\left(\begin{array}{cc}{G}^{\prime \prime }(i,j)\hfill & \leftrightarrow G^{\prime \prime }(\mathrm{num}+1-i,j)\hfill \end{array}\right)$$

(40)

where num is the total number of rows in $G^{\prime \prime }$. Through these operations, the initial matrix G is transformed into the matrix $G^{\prime \prime }$, simulating the complex and challenging obstacle space We define the fitness function $f_{\mathrm{fitness}}(x,G)$ that evaluates each path by checking if the path passes through an obstacle and by calculating the total length of the path. Starting Point End Point Definition:

- The start point S is fixed to the upper left corner of the map $(1,1)$.

- The end point E is the lower right corner of the map $(m,n)$, where G is a matrix of size $m\times n$. Path definition: A path $\mathbf{route}$ consists of a start point S, a midpoint x and an end point E:

$$\mathbf{route}=[S,x,E]$$

(41)

where x is a series of coordinate points representing the intermediate path points from the start point to the end point.

Obstacle Checking: Each point in the path $\mathbf{route}$ is checked and the number of obstacles the path passes through is calculated $n_B$:

$$n_B=\sum _{j=2}^{\dim -1}(G(\mathbf{route}\left(j\right),j)=1)$$

(42)

Path length calculation: If the path does not pass through any obstacle, i.e. $n_B=0$, a smooth path $\mathbf{path}$ is generated and the total length of the path is calculated:

$$f_{\mathrm{fitness}}(x,G)=\sum _{i=1}^{\left|\mathbf{path}\right|-1}\sqrt{{(\mathbf{path}(i+1,1)-\mathbf{path}(i,1))}^2+{(\mathbf{path}(i+1,2)-\mathbf{path}(i,2))}^2}$$

(43)

Collision penalty: If the path passes through an obstacle, i.e. $n_B>0$, the adaptation value is the map area multiplied by the number of collisions:

$$f_{\mathrm{fitness}}(x,G)=m·n·n_B$$

(44)

The fitness function: Combining the above, the final form of the fitness function is:

$$f_{\mathrm{fitness}}(x,G)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{m·n·n_B}}\sum _{i=1}^{\left|\mathbf{path}\right|-1}\sqrt{{(\mathbf{path}(i+1,1)-\mathbf{path}(i,1))}^2+{(\mathbf{path}(i+1,2)-\mathbf{path}(i,2))}^2}\hfill & \mathrm{if}{n}_B>0\hfill \\ 0\hfill & \mathrm{if}{n}_B=0\hfill \end{array}\right.$$

(45)

Figure 13. Visual Comparison of robot path planning in complex scenarios

Figure 13. Visual Comparison of robot path planning in complex scenarios

Table 5. Precision Data

Table 5. Precision Data

Experiment	ISO	SO	RIME	WOA	SCA	GWO
1	Success (68.4081)	Failure (71.0511)	Failure (53.4309)	Failure (57.3173)	Failure (83.4763)	Failure (53.6302)
2	Success (70.0802)	Failure (85.3329)	Failure (54.1934)	Failure (53.826)	Success (87.9804)	Failure (50.4595)
3	Success (72.376)	Failure (79.9242)	Failure (56.6274)	Failure (49.7342)	Failure (80.1427)	Failure (59.6362)
4	Success (71.117)	Failure (71.9905)	Failure (51.5289)	Failure (51.3301)	Success (80.7291)	Failure (51.0745)
5	Success (71.799)	Failure (79.826)	Failure (50.7607)	Failure (50.8605)	Failure (76.9859)	Failure (52.1692)
6	Success (69.3769)	Success (81.929)	Failure (50.8476)	Failure (50.207)	Success (91.7876)	Failure (64.3891)

Through data analysis, it can be demonstrated that ISO surpasses SO and four other competing algorithms when facing complex optimization spaces, boasting a 100% success rate in search, showcasing superior adaptability and robustness. Furthermore, in successfully completing tasks, Compared to SO, ISO reduces the robot’s path length by 15.4%, and compared to SCA, ISO reduces the robot’s path length by 19.1% on average.

7.3. Wireless Sensor Network Node Deployment

With the development of distributed environments such as grid computing[67,68,69], peer-to-peer networks[70,71,72] and cloud computing[73,74,75] ,wireless sensor networks (WSNs) have become more popular than ever. WSNs represent an emerging computing and networking paradigm that can be defined as networks composed of miniature, small, inexpensive, and highly intelligent devices known as sensor nodes.[76] In Wireless sensor network node deployment,We deploy n wireless sensor nodes in a two-dimensional planar region A[77]. Each node can obtain its position and has an identical sensing radius R. The sensing radius R denotes the maximum distance a sensor node can detect, meaning all points within a circular area centered at node $s_i$ with radius R are covered by this node. Our objective is to optimize the distribution of nodes to maximize the proportion of the area that is covered. For this problem, set the maximum iteration count T = 500 and the population size N = 200.

Let $S=\{s_1,s_2,\dots ,s_n\}$ represent all WSN nodes in region A, where $s_i=(x_i,y_i)$ denotes the coordinates of node $s_i$ in A. For any grid point $p_j$ ( $j=1,2,\dots ,L\times L$ ), with coordinates $(x_j,y_j)$, the distance between node $s_i$ and grid point $p_j$ is given by:

$$d_{ij}=\sqrt{{(x_i-x_j)}^2+{(y_i-y_j)}^2}$$

(46)

The coverage of grid point $p_j$ by node $s_i$ is described by:

$$C_{ij}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{d}_{ij}\le R\hfill \\ 0\hfill & \mathrm{if}{d}_{ij}>R\hfill \end{array}\right.$$

(47)

Where R is the sensing radius of the sensor, varying for different types of sensors (4, 6, 8).

Define an $L\times L$ matrix M where each element indicates whether the corresponding grid point is covered. Initially, all elements of M are 0. For each node $s_i$, compute all grid points it covers and set the corresponding elements in M to 1. The coverage rate $C_s$ is defined as the ratio of covered grid points to the total number of grid points:

$$C_s={\displaystyle \frac{{\sum }_{i=1}^{L^2}M\left(i\right)}{L^2}}$$

(48)

Our optimization objective is to maximize the coverage rate $C_s$. Since $C_s$ represents the proportion of the covered area, the optimization problem can be formulated as:

$$max{C}_s$$

(49)

We conducted tests using three different sensing radii, validating the algorithm’s performance and robustness in optimizing wireless network coverage under varying radii:

Figure 14. The Coverage Effectiveness and Convergence Curves of Sensor Wireless Network Coverage.

Figure 14. The Coverage Effectiveness and Convergence Curves of Sensor Wireless Network Coverage.

Table 6. Precision Data

Table 6. Precision Data

ISO	SO	RIME	GWO	SCA	WOA
0.8571	0.71137	0.83802	0.78817	0.67697	0.80669

Qualitative analysis of six maps of wireless network coverage shows that ISO , compared to SO and four other competing algorithms, consistently achieves more uniform coverage. Additionally, quantitative evidence from convergence curve graphs confirms that ISO exhibits stronger solving capabilities compared to competing algorithms, resulting in the attainment of the maximum network coverage area.

7.3.1. Pressure Vessel Design

Pressure vessels are widely used where it is required to hold the fluids at high pressure as well as temperature.[78] They are majorly used in processing plants, nuclear power plants, and oil refining industries. The Pressure Vessel Design(PVD) problem features a structure as shown in Fig. 18. Our design philosophy is to maximize cost-effectiveness while fully meeting practical application needs, by finely adjusting four key parameters to achieve this goal.[79] These four crucial optimization dimensions are: vascular thickness ($T_s$), head thickness ($T_h$), inner radius (R), and head length (L). In order to quantify and optimize the comprehensive impact of these parameters, we have constructed an accurate mathematical model, as shown in Equation (21). This model not only reflects the complex relationship between design parameters and performance, but also provides us with a scientific basis for optimizing design, ensuring effective control and reduction of overall costs while meeting usage requirements.

$$\begin{array}{cc}\hfill & \mathrm{Consider}\overrightarrow{x}=[x_1,x_2,x_3,x_4]=[T_s,T_h,R,L],\hfill \\ \hfill & \mathrm{Minimize}f\left(\overrightarrow{x}\right)=0.6224x_1{x}_3{x}_4+1.7781x_2{x}_3^2+3.1661x_1^2{x}_4+19.84x_2^2{x}_3,\hfill \\ \hfill & \mathrm{Subject}\mathrm{to}\left\{\begin{array}{c}{g}_1\left(\overrightarrow{x}\right)=-x_1+0.0193x_3\le 0,\hfill \\ g_2\left(\overrightarrow{x}\right)=-x_2+0.00954x_3\le 0,\hfill \\ g_3\left(\overrightarrow{x}\right)=-\pi x_3^2{x}_4-{\displaystyle \frac{4}{3}}\pi x_3^3+1296000\le 0,\hfill \\ g_4\left(\overrightarrow{x}\right)=x_4-240\le 0,\hfill \end{array}\right.\hfill \end{array}$$

(50)

$$\mathrm{Parameter}\mathrm{range}0\le x_1,x_2\le 99,10\le x_3,x_4\le 200.$$

(51)

Figure 15. Pressure Vessel Design

Figure 15. Pressure Vessel Design

Table 7. Optimal values for Variables and Their Corresponding Optimal Value and Ranking for Different Algorithms.

Table 7. Optimal values for Variables and Their Corresponding Optimal Value and Ranking for Different Algorithms.

Algorithm	h	l	t	b	Optimal value	Ranking
ISO	0.1991	3.3339	9.1857	0.1991	1.6712	1
SO	0.19372	3.437	9.192	0.19883	1.6757	2
RIME	0.4176	2.3298	4.9704	0.68011	3.1046	6
SCA	0.19728	3.8021	9.181	0.19971	1.7338	4
GWO	0.1942	3.4412	9.1891	0.19897	1.6776	3
WOA	0.125	8.3518	8.3518	0.24085	2.3073	5

From the results in Table 17, it is evident that ISO outperform all other competitors, achieving a minimum cost of 1.6712.

8. Conclusion and Outlook

References

1. J. Tang, G. Liu, and Q. T. Pan, "A Review on Representative Swarm Intelligence Algorithms for Solving Optimization Problems: Applications and Trends," IEEE/CAA J. Autom. Sinica, vol. 8, no. 10, pp. 1627-1643, Oct. 2021. [CrossRef]
2. Breiman, L., Cutler, A. A deterministic algorithm for global optimization. Mathematical Programming 58, 179–199 (1993). [CrossRef]
3. Tansel Dokeroglu, Ender Sevinc, Tayfun Kucukyilmaz, Ahmet Cosar,A survey on new generation metaheuristic algorithms,Computers & Industrial Engineering,Volume 137,2019,106040,ISSN 0360-8352. [CrossRef]
4. P. Agrawal, H. F. Abutarboush, T. Ganesh and A. W. Mohamed, "Metaheuristic Algorithms on Feature Selection: A Survey of One Decade of Research (2009-2019)," in IEEE Access, vol. 9, pp. 26766-26791, 2021. [CrossRef]
5. Adam, S. P., Alexandropoulos, S. A. N., Pardalos, P. M., & Vrahatis, M. N. (2019). No free lunch theorem: A review. Approximation and optimization: Algorithms, complexity and applications, 57-82. [CrossRef]
6. Fatma A. Hashim, Abdelazim G. Hussien,Snake Optimizer: A novel meta-heuristic optimization algorithm,KnowledgeBasedSystems,Volume 242,2022,108320. [CrossRef]
7. Peng, L., Yuan, Z., Dai, G. et al. A Multi-strategy Improved Snake Optimizer Assisted with Population Crowding Analysis for Engineering Design Problems. J Bionic Eng 21, 1567–1591 (2024). [CrossRef]
8. Tsakiridis, N. L., Theocharis, J. B., & Zalidis, G. C. (2016). DECO3R: A Differential Evolution-based algorithm for generating compact Fuzzy Rule-based Classification Systems. Knowledge-Based Systems, 105, 160-174. [CrossRef]
9. Su, H., Zhao, D., Heidari, A. A., Liu, L., Zhang, X., Mafarja, M., & Chen, H. (2023). RIME: A physics-based optimization. Neurocomputing, 532, 183-214. [CrossRef]
10. Fattahi, E., Bidar, M., & Kanan, H. R. (2018). Focus group: an optimization algorithm inspired by human behavior. International Journal of Computational Intelligence and Applications, 17(01), 1850002. [CrossRef]
11. Chakraborty A, Kar A K. Swarm intelligence: A review of algorithms[J]. Nature-inspired computing and optimization: Theory and applications, 2017: 475-494. [CrossRef]
12. Mirjalili S, Mirjalili S. Genetic algorithm[J]. Evolutionary algorithms and neural networks: Theory and applications, 2019: 43-55. [CrossRef]
13. Espejo P G, Ventura S, Herrera F. A survey on the application of genetic programming to classification[J]. IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), 2009, 40(2): 121-144. [CrossRef]
14. Maheri A, Jalili S, Hosseinzadeh Y, et al. A comprehensive survey on cultural algorithms[J]. Swarm and Evolutionary Computation, 2021, 62: 100846. [CrossRef]
15. Bertsimas, D., & Tsitsiklis, J. (1993). Simulated annealing. Statistical science, 8(1), 10-15. [CrossRef]
16. Kennedy, J., & Eberhart, R. (1995, November). Particle swarm optimization. In Proceedings of ICNN’95-international conference on neural networks (Vol. 4, pp. 1942-1948). ieee.
17. Han, K. H., & Kim, J. H. (2002). Quantum-inspired evolutionary algorithm for a class of combinatorial optimization. IEEE transactions on evolutionary computation, 6(6), 580-593. [CrossRef]
18. Xu, L. D. (1994). Case based reasoning. IEEE potentials, 13(5), 10-13. [CrossRef]
19. Qiang, W., & Zhongli, Z. (2011, August). Reinforcement learning model, algorithms and its application. In 2011 International Conference on Mechatronic Science, Electric Engineering and Computer (MEC) (pp. 1143-1146). IEEE.
20. Dorigo, M., & Stützle, T. (2019). Ant colony optimization: overview and recent advances (pp. 311-351). Springer International Publishing.
21. Abualigah, L., Shehab, M., Alshinwan, M., & Alabool, H. (2020). Salp swarm algorithm: a comprehensive survey. Neural Computing and Applications, 32(15), 11195-11215. [CrossRef]
22. Hashim, F. A., & Hussien, A. G. (2022). Snake Optimizer: A novel meta-heuristic optimization algorithm. Knowledge-Based Systems, 242, 108320. [CrossRef]
23. Yan, C., & Razmjooy, N. (2023). Optimal lung cancer detection based on CNN optimized and improved Snake optimization algorithm. Biomedical Signal Processing and Control, 86, 105319. [CrossRef]
24. Zheng, W., Pang, S., Liu, N., Chai, Q., & Xu, L. (2023). A compact snake optimization algorithm in the application of WKNN fingerprint localization. Sensors, 23(14), 6282. [CrossRef]
25. Hu, G., Yang, R., Abbas, M., & Wei, G. (2023). BEESO: Multi-strategy boosted snake-inspired optimizer for engineering applications. Journal of Bionic Engineering, 20(4), 1791-1827. [CrossRef]
26. Osuna-Enciso, V., Cuevas, E., & Castañeda, B. M. (2022). A diversity metric for population-based metaheuristic algorithms. Information Sciences, 586, 192-208. [CrossRef]
27. Agushaka, J. O., & Ezugwu, A. E. (2022). Initialisation approaches for population-based metaheuristic algorithms: a comprehensive review. Applied Sciences, 12(2), 896. [CrossRef]
28. Joe, S., & Kuo, F. Y. (2008). Constructing Sobol sequences with better two-dimensional projections. SIAM Journal on Scientific Computing, 30(5), 2635-2654. [CrossRef]
29. Su, H., Zhao, D., Heidari, A. A., Liu, L., Zhang, X., Mafarja, M., & Chen, H. (2023). RIME: A physics-based optimization. Neurocomputing, 532, 183-214. [CrossRef]
30. Zhong, R., Yu, J., Zhang, C., & Munetomo, M. (2024). SRIME: a strengthened RIME with Latin hypercube sampling and embedded distance-based selection for engineering optimization problems. Neural Computing and Applications, 36(12), 6721-6740. [CrossRef]
31. Osuna-Enciso, V., Cuevas, E., & Castañeda, B. M. (2022). A diversity metric for population-based metaheuristic algorithms. Information Sciences, 586, 192-208. [CrossRef]
32. Learning, L. (2021). Image Formation by Lenses. Fundamentals of Heat, Light & Sound.
33. Ouyang, C., Zhu, D., & Qiu, Y. (2021). Lens learning sparrow search algorithm. Mathematical Problems in Engineering, 2021(1), 9935090. [CrossRef]
34. Osuna-Enciso, V., Cuevas, E., & Castañeda, B. M. (2022). A diversity metric for population-based metaheuristic algorithms. Information Sciences, 586, 192-208. [CrossRef]
35. Cuevas, E., Luque, A., Morales Castañeda, B., & Rivera, B. (2024). A Measure of Diversity for Metaheuristic Algorithms Employing Population-Based Approaches. In Metaheuristic Algorithms: New Methods, Evaluation, and Performance Analysis (pp. 49-72). Cham: Springer Nature Switzerland.
36. Cuevas, E., Diaz, P., Camarena, O., Cuevas, E., Diaz, P., & Camarena, O. (2021). Experimental analysis between exploration and exploitation. Metaheuristic Computation: A Performance Perspective, 249-269. [CrossRef]
37. Yang, X. S., Deb, S., & Fong, S. (2014). Metaheuristic algorithms: optimal balance of intensification and diversification. Applied Mathematics & Information Sciences, 8(3), 977. [CrossRef]
38. Clément, F., Doerr, C., & Paquete, L. (2022). Star discrepancy subset selection: Problem formulation and efficient approaches for low dimensions. Journal of Complexity, 70, 101645. [CrossRef]
39. Thiémard, E. (2001). An algorithm to compute bounds for the star discrepancy. journal of complexity, 17(4), 850-880. [CrossRef]
40. Bansal, P. P., & Ardell, A. J. (1972). Average nearest-neighbor distances between uniformly distributed finite particles. Metallography, 5(2), 97-111. [CrossRef]
41. Kobayashi, K., & Salam, M. U. (2000). Comparing simulated and measured values using mean squared deviation and its components. Agronomy Journal, 92(2), 345-352. [CrossRef]
42. Salleh, M. N. M., Hussain, K., Cheng, S., Shi, Y., Muhammad, A., Ullah, G., & Naseem, R. (2018). Exploration and exploitation measurement in swarm-based metaheuristic algorithms: An empirical analysis. In Recent Advances on Soft Computing and Data Mining: Proceedings of the Third International Conference on Soft Computing and Data Mining (SCDM 2018), Johor, Malaysia, February 06-07, 2018 (pp. 24-32). Springer International Publishing.
43. Mirjalili, S. (2016). SCA: a sine cosine algorithm for solving optimization problems. Knowledge-based systems, 96, 120-133. [CrossRef]
44. Mirjalili, S., & Lewis, A. (2016). The whale optimization algorithm. Advances in engineering software, 95, 51-67. [CrossRef]
45. Mirjalili, S., Mirjalili, S. M., & Lewis, A. (2014). Grey wolf optimizer. Advances in engineering software, 69, 46-61. [CrossRef]
46. Garcà a-Martà nez, C., Gutiérrez, P. D., Molina, D., Lozano, M., & Herrera, F. (2017). Since CEC 2005 competition on real-parameter optimisation: a decade of research, progress and comparative analysis’s weakness. Soft Computing, 21, 5573-5583. [CrossRef]
47. Sun, Y., & Genton, M. G. (2011). Functional boxplots. Journal of Computational and Graphical Statistics, 20(2), 316-334. [CrossRef]
48. Wilcoxon, F., Katti, S., & Wilcox, R. A. (1970). Critical values and probability levels for the Wilcoxon rank sum test and the Wilcoxon signed rank test. Selected tables in mathematical statistics, 1, 171-259.
49. Tsouros, D. C., Bibi, S., & Sarigiannidis, P. G. (2019). A review on UAV-based applications for precision agriculture. Information, 10(11), 349.(6), 4233-4284. [CrossRef]
50. Delavarpour, N., Koparan, C., Nowatzki, J., Bajwa, S., & Sun, X. (2021). A technical study on UAV characteristics for precision agriculture applications and associated practical challenges. Remote Sensing, 13(6), 1204. [CrossRef]
51. Yang, L., Fan, J., Liu, Y., Li, E., Peng, J., & Liang, Z. (2020). A review on state-of-the-art power line inspection techniques. IEEE Transactions on Instrumentation and Measurement, 69(12), 9350-9365. [CrossRef]
52. Zhang, Y., Yuan, X., Li, W., & Chen, S. (2017). Automatic power line inspection using UAV images. Remote Sensing, 9(8), 824. [CrossRef]
53. Li, X., & Yang, L. (2012, August). Design and implementation of UAV intelligent aerial photography system. In 2012 4th International Conference on Intelligent Human-Machine Systems and Cybernetics (Vol. 2, pp. 200-203). IEEE.
54. ZHANG, J. (2021). Review of the light-weighted and small UAV system for aerial photography and remote sensing. National Remote Sensing Bulletin, 25(3), 708-724. [CrossRef]
55. Lyu, C., & Zhan, R. (2022). Global analysis of active defense technologies for unmanned aerial vehicle. IEEE Aerospace and Electronic Systems Magazine, 37(1), 6-31. [CrossRef]
56. .
57. Roberge, V., Tarbouchi, M., & Labonté, G. (2012). Comparison of parallel genetic algorithm and particle swarm optimization for real-time UAV path planning. IEEE Transactions on industrial informatics, 9(1), 132-141. [CrossRef]
58. Puente-Castro, A., Rivero, D., Pazos, A., & Fernandez-Blanco, E. (2022). A review of artificial intelligence applied to path planning in UAV swarms. Neural Computing and Applications, 34(1), 153-170. [CrossRef]
59. Mac, T. T., Copot, C., Tran, D. T., & De Keyser, R. (2016). Heuristic approaches in robot path planning: A survey. Robotics and Autonomous Systems, 86, 13-28. [CrossRef]
60. Zhang, H., Wang, Y., Zheng, J., & Yu, J. (2018). Path planning of industrial robot based on improved RRT algorithm in complex environments. IEEE Access, 6, 53296-53306. [CrossRef]
61. Kunchev, V., Jain, L., Ivancevic, V., & Finn, A. (2006). Path planning and obstacle avoidance for autonomous mobile robots: A review. In Knowledge-Based Intelligent Information and Engineering Systems: 10th International Conference, KES 2006, Bournemouth, UK, October 9-11, 2006. Proceedings, Part II 10 (pp. 537-544). Springer Berlin Heidelberg.
62. Chakraborty, S., Elangovan, D., Govindarajan, P. L., ELnaggar, M. F., Alrashed, M. M., & Kamel, S. (2022). A comprehensive review of path planning for agricultural ground robots. Sustainability, 14(15), 9156. [CrossRef]
63. Weimin, Z. H. A. N. G., Yue, Z. H. A. N. G., & Hui, Z. H. A. N. G. (2022). Path planning of coal mine rescue robot based on improved A* algorithm. Coal Geology & Exploration, 50(12), 20. [CrossRef]
64. Vivaldini, K. C., Galdames, J. P., Bueno, T. S., Araújo, R. C., Sobral, R. M., Becker, M., & Caurin, G. A. (2010, March). Robotic forklifts for intelligent warehouses: Routing, path planning, and auto-localization. In 2010 IEEE International Conference on Industrial Technology (pp. 1463-1468). IEEE.
65. Raja, P., & Pugazhenthi, S. (2012). Optimal path planning of mobile robots: A review. International journal of physical sciences, 7(9), 1314-1320. [CrossRef]
66. Souissi, O., Benatitallah, R., Duvivier, D., Artiba, A., Belanger, N., & Feyzeau, P. (2013, October). Path planning: A 2013 survey. In Proceedings of 2013 international conference on industrial engineering and systems management (IESM) (pp. 1-8). IEEE.
67. Lim, H. B., Teo, Y. M., Mukherjee, P., Lam, V. T., Wong, W. F., & See, S. (2005, November). Sensor grid: Integration of wireless sensor networks and the grid. In The IEEE Conference on Local Computer Networks 30th Anniversary (LCN’05) l (pp. 91-99). IEEE.
68. Sanchez-Ibanez, J. R., Pérez-del-Pulgar, C. J., & Garcà a-Cerezo, A. (2021). Path planning for autonomous mobile robots: A review. Sensors, 21(23), 7898. [CrossRef]
69. Farman, H., Javed, H., Ahmad, J., Jan, B., & Zeeshan, M. (2016). Grid‐based hybrid network deployment approach for energy efficient wireless sensor networks. Journal of Sensors, 2016(1), 2326917. [CrossRef]
70. Lim, H. B., Teo, Y. M., Mukherjee, P., Lam, V. T., Wong, W. F., & See, S. (2005, November). Sensor grid: Integration of wireless sensor networks and the grid. In The IEEE Conference on Local Computer Networks 30th Anniversary (LCN’05) l (pp. 91-99). IEEE.
71. Carbajo, R. S., & Mc Goldrick, C. (2017). Decentralised peer-to-peer data dissemination in wireless sensor networks. Pervasive and Mobile Computing, 40, 242-266. [CrossRef]
72. Krčo, S., Cleary, D., & Parker, D. (2005). Enabling ubiquitous sensor networking over mobile networks through peer-to-peer overlay networking. Computer communications, 28(13), 1586-1601. [CrossRef]
73. Park, H., & Hutchinson, S. (2014, May). Robust optimal deployment in mobile sensor networks with peer-to-peer communication. In 2014 IEEE International Conference on Robotics and Automation (ICRA) (pp. 2144-2149). IEEE.
74. Dash, S. K., Mohapatra, S., & Pattnaik, P. K. (2010). A survey on applications of wireless sensor network using cloud computing. International Journal of Computer Science & Emerging Technologies, 1(4), 50-55.
75. Ahmed, K., & Gregory, M. (2011, December). Integrating wireless sensor networks with cloud computing. In 2011 seventh international conference on mobile ad-hoc and sensor networks (pp. 364-366). IEEE.
76. Elamin, W. M., Endan, J. B., Yosuf, Y. A., Shamsudin, R., & Ahmedov, A. (2015). High pressure processing technology and equipment evolution: a review. Journal of Engineering Science & Technology Review, 8(5). [CrossRef]
77. Ballesteros, A., Ahlstrand, R., Bruynooghe, C., Von Estorff, U., & Debarberis, L. (2012). The role of pressure vessel embrittlement in the long term operation of nuclear power plants. Nuclear engineering and design, 243, 63-68. [CrossRef]
78. Chen, X., Fan, Z., Chen, Y., Zhang, X., Cui, J., Zheng, J., & Shou, B. (2018, July). Development of lightweight design and manufacture of heavy-duty pressure vessels in China. In Pressure Vessels and Piping Conference (Vol. 51593, p. V01BT01A023). American Society of Mechanical Engineers.
79. Khanduja, N., & Bhushan, B. (2021). Recent advances and application of metaheuristic algorithms: A survey (2014–2020). Metaheuristic and evolutionary computation: algorithms and applications, 207-228. [CrossRef]
80. Yaghoubzadeh-Bavandpour, A., Bozorg-Haddad, O., Zolghadr-Asli, B., & Gandomi, A. H. (2022). Improving approaches for meta-heuristic algorithms: A brief overview. Computational Intelligence for Water and Environmental Sciences, 35-61. [CrossRef]
81. Talbi, E. G. (2021). Machine learning into metaheuristics: A survey and taxonomy. ACM Computing Surveys (CSUR), 54(6), 1-32. [CrossRef]
82. Chiroma, H., Gital, A. Y. U., Rana, N., Abdulhamid, S. I. M., Muhammad, A. N., Umar, A. Y., & Abubakar, A. I. (2020). Nature inspired meta-heuristic algorithms for deep learning: recent progress and novel perspective. In Advances in Computer Vision: Proceedings of the 2019 Computer Vision Conference (CVC), Volume 1 1 (pp. 59-70). Springer International Publishing.
83. da Costa Oliveira, A. L., Britto, A., & Gusmão, R. (2023). Machine learning enhancing metaheuristics: a systematic review. Soft Computing, 27(21), 15971-15998. [CrossRef]
84. Hassanzadeh, M. R., & Keynia, F. (2021). An overview of the concepts, classifications, and methods of population initialization in metaheuristic algorithms. Journal of Advances in Computer Engineering and Technology, 7(1), 35-54.
85. Han, Y., Chen, W., Heidari, A. A., Chen, H., & Zhang, X. (2024). Balancing Exploration–Exploitation of Multi-verse Optimizer for Parameter Extraction on Photovoltaic Models. Journal of Bionic Engineering, 21(2), 1022-1054. [CrossRef]
86. Sharma, P., & Raju, S. (2024). Metaheuristic optimization algorithms: A comprehensive overview and classification of benchmark test functions. Soft Computing, 28(4), 3123-3186. [CrossRef]
87. Hussain, K., Salleh, M. N. M., Cheng, S., & Naseem, R. (2017). Common benchmark functions for metaheuristic evaluation: A review. JOIV: International Journal on Informatics Visualization, 1(4-2), 218-223. [CrossRef]
88. Yang, X. S. (2010). Engineering optimization: an introduction with metaheuristic applications. John Wiley & Sons.
89. Bozorg-Haddad, O., Solgi, M., & Loáiciga, H. A. (2017). Meta-heuristic and evolutionary algorithms for engineering optimization. John Wiley & Sons.
90. Hidary, J. D., & Hidary, J. D. (2019). Quantum computing: an applied approach (Vol. 1). Cham: Springer.
91. Fauseweh, B. (2024). Quantum many-body simulations on digital quantum computers: State-of-the-art and future challenges. Nature Communications, 15(1), 2123. [CrossRef]
92. Dahi, Z. A., & Alba, E. (2022). Metaheuristics on quantum computers: Inspiration, simulation and real execution. Future Generation Computer Systems, 130, 164-180. [CrossRef]
93. Gharehchopogh, F. S. (2023). Quantum-inspired metaheuristic algorithms: comprehensive survey and classification. Artificial Intelligence Review, 56(6), 5479-5543. [CrossRef]
94. Rylander, B., Soule, T., Foster, J. A., & Alves-Foss, J. (2000, July). Quantum Genetic Algorithms. In GECCO (p. 373).
95. Ross, O. H. M. (2019). A review of quantum-inspired metaheuristics: Going from classical computers to real quantum computers. Ieee Access, 8, 814-838. [CrossRef]
96. Nielsen, M6. A., & Chuang, I. L. (2010). Quantum computation and quantum information. Cambridge university press. [CrossRef]