# Search Patterns Based on Trajectories Extracted from the Response of Second-Order Systems

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## Abstract

**:**

## 1. Introduction

- A new physics-based optimization algorithm, namely SOA, is introduced. It uses search patterns obtained from the response of second-order systems.
- New search patterns are proposed as an alternative to those known in the literature.
- The statistical significance, convergence speed and exploitation-exploration ratio of SOA are evaluated against other popular metaheuristic algorithms.
- SOA outperforms other competitor algorithms on two sets of optimization problems.

## 2. Second-Order Systems

_{n}represent the damping ratio and ω

_{n}the natural frequency, respectively, while s symbolizes the Laplace domain.

#### 2.1. Underdamped Behavior ($0<\zeta <1)$

#### 2.2. Critically Damped Behavior ($\zeta =1)$

#### 2.3. Overdamped Behavior ($\zeta >1)$

## 3. Search Patterns in Metaheuristics

## 4. Proposed Search Patterns

## 5. Balance of Exploration and Exploitation

## 6. Proposed Metaheuristic Algorithm

#### 6.1. Initialization

#### 6.2. Trajectory Generation

#### 6.3. Reset of Bad Elements

#### 6.4. Avoid Premature Convergence Mechanism

## 7. Experimental Results

- ABC: Onlooker Bees = 50, acceleration coefficient = 1 [9].
- DE: crossover probability = 0.2, Betha = 1 [6].
- CMAES: Lambda = 50, father number = 25, sigma = 60, csigma = 0.32586, dsigma = 1.32586 [4].
- CSA: Flock = 50, awareness probability = 0.1, flight length = 2 [8].
- MFO: search agents = 50, “a” linearly decreases from 2 to 0 [18].
- SOA: the experimental results give the best algorithm performance with the next parameter set par1 = 0.7, par2 = 0.3 and par3 = 0.05.

**AB**) solution, the Median Best-so-far (

**MB**) solution and the Standard Deviation (

**SD**) of the best-so-far solutions. In the analysis, each optimization problem is solved using every algorithm 30 times. From this operation, 30 results are produced. From all these values, the mean value of all best-found solutions represents the Average Best-so-far (

**AB**) solution. Likewise, the median of all 30 results is computed to generate

**MB**and the standard deviation of the 30 data is estimated to obtain

**SD**of the best-so-far solutions. Indicators

**AB**and

**MB**correspond to the accuracy of the solutions, while

**SD**their dispersion, and thus, the robustness of the algorithm.

#### 7.1. Multimodal Functions

_{0}) was adopted that showed that there is no significant difference in the results. On the other hand, it is assumed as an alternative hypothesis (H

_{1}) that the result has a similar structure. For the Wilcoxon analysis, it is assumed a significance value of 0.05 considering 30 independent execution for each test function. Table 2 shows the p-values assuming the results of Table 2 (where n = 30) produced by the Wilcoxon study. For faster visualization, in the Table, we use the following symbols ▲ ▼, and ►. The symbol ▲ refers that the SOA algorithm produces significantly better solutions than its competitor. ▼ symbolizes that SOA obtains worse results than its counterpart. Finally, the symbol ► denotes that both compared methods produce similar solutions. A close inspection of Table 2 demonstrates that for functions ${f}_{1}$, ${f}_{4}$, ${f}_{5}$, ${f}_{6}$, ${f}_{8}$, ${f}_{9}$, ${f}_{10}$, ${f}_{11}$ and ${f}_{12}$ the proposed SOA scheme obtain better solutions than the other methods. On the other hand, for functions ${f}_{2}$, ${f}_{2}$ and ${f}_{7}$, it is clear that the groups SOA versus ABC, SOA versus CMAES, SOA versus DE and SOA versus MFO and EA-HC versus SCA present similar solutions.

#### 7.2. Unimodal Functions

**AB**,

**MB**and

**SD**obtained in the executions. According to Table 3, the SOA approach provides better performance than ABC, DE, DE, CMAES CSA and MFO for all functions. In general, this study demonstrates big differences in performance among the metaheuristic scheme, which is directly related to a better trade-off between exploration and exploitation produced by the trajectories of the SOA scheme. Considering the information from Table 3, Table 4 reports the results of the Wilcoxon analysis. An inspection of the $p$-values from Table 4, it is clear that the proposed SOA method presents a superior performance than each metaheuristic algorithm considered in the experimental study.

#### 7.3. Hybrid Functions

**AB**,

**MB**and

**SD**. From Table 5, it can be observed that the SOA method presents a superior performance than the other techniques in all functions. Table 6 reports the results of the Wilcoxon analysis assuming the index of the Average Best-so-far (

**AB**) values of Table 5. Since all elements present the symbol ▲, they validate that the proposed SOA method produces better results than the other methods. The remarkable performance of the proposed SOA scheme for hybrid functions is attributed to a better balance between exploration and exploitation of its operators provoked by the properties of the second system trajectories. This denotes that the SOA approach generates an appropriate number of promising search agents that allow an adequate exploration of the search space. On the other hand, a balanced number of candidate solutions is also produced that make it possible to improve the quality of the already-detected solutions, in terms of the objective function.

#### 7.4. Convergence Analysis

#### 7.5. Performance Evaluation with CEC 2017

## 8. Analysis and Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Name | Function | S | Dim | Minimum | |
---|---|---|---|---|---|

${f}_{1}\left(\mathrm{x}\right)$ | Levy | $si{n}^{2}\left(\pi {\omega}_{1}\right)+{\displaystyle {\displaystyle \sum}_{i=1}^{d-1}}{\left({\omega}_{i}-1\right)}^{2}\left[1+10si{n}^{2}\left(\pi {\omega}_{i}+1\right)+{\left({\omega}_{d}-1\right)}^{2}\left[1+si{n}^{2}\left(2\pi {\omega}_{d}\right)\right]\right]$ | ${\left[-10,10\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=0;$ ${x}^{*}=\left(1,\dots ,1\right)$ |

${f}_{2}\left(\mathrm{x}\right)$ | Mishra 1 | ${\left(1+{x}_{n}\right)}^{{x}_{n}};{x}_{n}=n-{{\displaystyle \sum}}_{i=1}^{n-1}{x}_{i}$ | ${\left[0,1\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=2;$ ${x}^{*}=\left(1,\dots ,1\right)$ |

${f}_{3}\left(\mathrm{x}\right)$ | Mishra 2 | ${\left(1+{x}_{n}\right)}^{{x}_{n}};{x}_{n}=n-{{\displaystyle \sum}}_{i=1}^{n-1}\frac{\left({x}_{i}+{x}_{i+1}\right)}{2}$ | ${\left[0,1\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=2;$ ${x}^{*}=\left(1,\dots ,1\right)$ |

${f}_{4}\left(\mathrm{x}\right)$ | Mishra 11 | $\left[\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left|{x}_{i}\right|-{\left({\displaystyle {\displaystyle \prod}_{i=1}^{n}}\left|{x}_{i}\right|\right)}^{\frac{1}{n}}\right]{}^{2}$ | ${\left[-10,10\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=0;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

${f}_{5}\left(\mathrm{x}\right)$ | Penalty 1 | $\frac{\pi}{30}\left\{\begin{array}{c}10{\mathrm{sin}}^{2}\left(\pi {y}_{1}\right)\\ +{{\displaystyle \sum}}_{i=1}^{n-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {y}_{i}+1\right)\right]\\ +{\left({y}_{n}-1\right)}^{2}\end{array}\right\}+{{\displaystyle \sum}}_{i=1}^{n}u\left({x}_{i},10,100,4\right);$ ${y}_{i}=1+\frac{{x}_{i}+1}{4};$ $u\left({x}_{i},a,k,m\right)=\left\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{m},{x}_{i}a\\ 0,-a\le {x}_{i}\le a\\ k{\left(-{x}_{i}-a\right)}^{m},{x}_{i}-a\end{array}\right.$ | ${\left[-50,50\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=0;$ ${x}^{*}=\left(-1,\dots ,-1\right)$ |

${f}_{6}\left(\mathrm{x}\right)$ | Perm1 | ${\displaystyle \sum}_{k=1}^{n}}{\left[{\displaystyle {\displaystyle \sum}_{i=1}^{n}}({i}^{k}+50)\left\{{\left({x}_{i}/i\right)}^{k}-1\right\}\right]}^{2$ | ${\left[-n,n\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=0;$ ${x}^{*}=\left(1,2,\dots ,n\right)$ |

${f}_{7}\left(\mathrm{x}\right)$ | Plateau | $30+{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left|{x}_{i}\right|$ | ${\left[-5.12,5.12\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=30;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

${f}_{8}\left(\mathrm{x}\right)$ | Step | ${\displaystyle \sum}_{i=1}^{n}}{\left(\u23a3{x}_{i}+0.5\u23a6\right)}^{2$ | ${\left[-100,100\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=0;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

${f}_{9}\left(\mathrm{x}\right)$ | Styblinski tang | $\frac{1}{2}{{\displaystyle \sum}}_{i=1}^{n}\left({x}_{i}^{4}-16{x}_{i}^{2}+5{x}_{i}\right)$ | ${\left[-5,5\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=-39.1659n;$ ${x}^{*}=\left(-2.90,\dots ,2.90\right)$ |

${f}_{10}\left(\mathrm{x}\right)$ | Trid | ${\displaystyle \sum}_{i=1}^{n}}{\left({x}_{i}-1\right)}^{2}-{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}{x}_{i}-1$ | ${\left[-{n}^{2},{n}^{2}\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=-n\left(n+4\right)\left(n-1\right)/6;$ ${x}^{*}=\left[i\left(n+1-i\right)\right]$ $\mathrm{for}i=1,\dots ,n$ |

${f}_{11}\left(\mathrm{x}\right)$ | Vincent | $-{{\displaystyle \sum}}_{i=1}^{n}\mathrm{sin}\left(10\mathrm{log}{x}_{i}\right)$ | ${\left[0.25,10\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=-n;$ ${x}^{*}=\left(7.70,\dots ,7.70\right)$ |

${f}_{12}\left(\mathrm{x}\right)$ | Zakharov | ${{\displaystyle \sum}}_{i=1}^{n}{x}_{i}^{2}+{\left({{\displaystyle \sum}}_{i=1}^{n}0.5i{x}_{i}\right)}^{2}+{\left({{\displaystyle \sum}}_{i=1}^{n}0.5i{x}_{i}\right)}^{4}$ | ${\left[-5,10\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=0;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

${f}_{13}\left(\mathrm{x}\right)$ | Rothyp | ${\displaystyle \sum}_{i=1}^{d}}{\displaystyle {\displaystyle \sum}_{j=1}^{i}}{x}_{j}^{2$ | ${\left[-65.536,65.536\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=0;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

${f}_{14}\left(\mathrm{x}\right)$ | Schwefel 2 | ${{\displaystyle \sum}}_{i=1}^{n}{\left({{\displaystyle \sum}}_{j=1}^{i}{x}_{i}\right)}^{2}$ | ${\left[-100,100\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=0;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

${f}_{15}\left(\mathrm{x}\right)$ | Sum2 | ${\displaystyle \sum}_{i=1}^{d}}{\displaystyle {\displaystyle \sum}_{j=1}^{i}}{x}_{j}^{2$ | ${\left[-10,10\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=0;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

${F}_{16}\left(\mathrm{x}\right)$ | Sum of different powers | ${\displaystyle \sum}_{i=1}^{d}}{\left|{x}_{i}\right|}^{i+1$ | ${\left[-1,1\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=0;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

${f}_{17}\left(\mathrm{x}\right)$ | Rastringin + Schwefel22 + Sphere | $10n+{{\displaystyle \sum}}_{i=1}^{n}\left[{x}_{i}{}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)\right]+\left({\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left|{x}_{i}\right|+{\displaystyle {\displaystyle \prod}_{i=1}^{n}}\left|{x}_{i}\right|\right)+\left({\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{2}\right)$ | ${\left[-100,100\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=0;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

${f}_{18}\left(\mathrm{x}\right)$ | Griewank + Rastringin + Rosenbrock | $\frac{1}{4000}{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}{}^{2}-{{\displaystyle \prod}}_{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1+10n+{{\displaystyle \sum}}_{i=1}^{n}\left[{x}_{i}{}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)\right]+{{\displaystyle \sum}}_{i=1}^{n-1}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]$ | ${\left[-100,100\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=n-1;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

${f}_{19}\left(\mathrm{x}\right)$ | Ackley + Penalty2 + Rosenbrock + Schwefel2 | $(-20exp(-0.2\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{2}})-\mathit{exp}\left(\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+exp)+\left(0.1\right\{\mathrm{sin}(3\pi {x}_{i})+{\displaystyle {\displaystyle \sum}_{i=1}^{n}}({x}_{i}-1){}^{2}[1+sin{}^{2}(3\pi {x}_{i}+1)]+[({x}_{n}-1){}^{2}[1+sin{}^{2}(2\pi {x}_{n})]]\}+{{\displaystyle \sum}}_{i=1}^{n}u({x}_{i},5.100,4))+\left({\displaystyle {\displaystyle \sum}_{i=1}^{n-1}}\right[100({x}_{i+1}-{x}_{i}^{2}){}^{2}+{\left({x}_{i}-1\right)}^{2}])+\left({\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left|{x}_{i}\right|+{\displaystyle {\displaystyle \prod}_{i=1}^{n}}\left|{x}_{i}\right|\right)$ | ${\left[-100,100\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=\left(1.1n\right)-1;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

${f}_{20}\left(\mathrm{x}\right)$ | Ackley + Griewnk + Rastringin + Rosenbrock + Schwefel22 | $-20{e}^{-0.2\sqrt{\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}^{2}}}-{e}^{\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}\mathrm{cos}\left(2\pi {x}_{i}\right)}+20+e+\frac{1}{4000}{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}{}^{2}-{{\displaystyle \prod}}_{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1+10n+{{\displaystyle \sum}}_{i=1}^{n}\left[{x}_{i}{}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)\right]+{{\displaystyle \sum}}_{i=1}^{n-1}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]+{{\displaystyle \sum}}_{i=1}^{n}\left|{x}_{i}\right|+{{\displaystyle \prod}}_{i=1}^{n}\left|{x}_{i}\right|$ | ${\left[-100,100\right]}^{n}$ | 30 | $f\left({x}^{*}\right)=n-1;$ ${x}^{*}=\left(0,\dots ,0\right)$ |

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**Figure 2.**Temporal responses of second-order system considering its different behaviors: Underdamped $(0<\zeta <1)$, critically damped $\left(\zeta =1\right)$, and overdamped $(\zeta >1)$.

**Figure 4.**Some examples of trajectories produced by using different values for ζ. (

**a**) ${x}_{i,1}^{k}\leftarrow \zeta =0$ and ${x}_{i,2}^{k}\leftarrow \zeta =1$, (

**b**) ${x}_{i,1}^{k}\leftarrow \zeta =0.1$ and ${x}_{i,2}^{k}\leftarrow \zeta =0.5$, (

**c**) ${x}_{i,1}^{k}\leftarrow \zeta =1$ and ${x}_{i,2}^{k}\leftarrow \zeta =1.67$ and (

**d**) ${x}_{i,1}^{k}\leftarrow \zeta =0.5$ and ${x}_{i,2}^{k}\leftarrow \zeta =1$.

**Figure 5.**Flowchart of the proposed metaheuristic method based on the response of second-order systems.

ABC | DE | CMAES | CSA | PSO | MFO | SOA | ||
---|---|---|---|---|---|---|---|---|

AB | 8.9132622 | 0.7932535 | 2.8976 × 10^{−19} | 55.918504 | 0.2012803 | 27.983271 | 0.1119774 | |

${\mathit{f}}_{\mathbf{1}}\left(x\right)$ | MD | 8.4392750 | 0.7993166 | 2.4779 × 10^{−19} | 57.038263 | 4.1459 × 10^{−23} | 25.365199 | 1.0714 × 10^{−10} |

SD | 2.6748059 | 0.1378538 | 1.5343 × 10^{−19} | 6.2032578 | 1.1022968 | 12.283254 | 0.2272439 | |

AB | 2 | 2 | 2 | 1,897,783.3 | 27.4 | 2 | 2 | |

${\mathit{f}}_{\mathbf{2}}\left(x\right)$ | MD | 2 | 2 | 2 | 35,691.155 | 2 | 2 | 2 |

SD | 9.9512 × 10^{−12} | 0 | 0 | 9,636,632.1 | 113.43495 | 0 | 0 | |

AB | 2 | 2 | 2 | 3,620,834.4 | 34.723128 | 2 | 2 | |

${\mathit{f}}_{\mathbf{3}}\left(x\right)$ | MD | 2 | 2 | 2 | 676,981.46 | 9 | 2 | 2 |

SD | 2.3308 × 10^{−11} | 0 | 0 | 7,265,352.7 | 113.36672 | 0 | 0 | |

AB | 0.1371551 | 0.002 | 1.7942 × 10^{−6} | 0.0862919 | 2.2285 × 10^{−8} | 5.5194 × 10^{−10} | 1.164 × 10^{−11} | |

${\mathit{f}}_{\mathbf{4}}\left(x\right)$ | MD | 0.1349076 | 0.01 | 0 | 0.0892256 | 0 | 0 | 7.7118 × 10^{−12} |

SD | 0.0399861 | 0.123 | 5.4833 × 10^{−6} | 0.0213332 | 1.2206 × 10^{−7} | 3.0231 × 10^{−9} | 1.0995 × 10^{−11} | |

AB | 13,781,291 | 1,331,987.7 | 22,307.195 | 44,274,761 | 82.539625 | 85.756149 | 71.964984 | |

${\mathit{f}}_{\mathbf{5}}\left(x\right)$ | MD | 13,876,263 | 1,365,502.1 | 72.377516 | 46,153,728 | 81.698488 | 85.665615 | 72.362277 |

SD | 3,237,147.7 | 306,385.34 | 50,923.637 | 10,118,180 | 7.1916726 | 3.2857088 | 0.9929591 | |

AB | 1.152 × 10^{85} | 5.850 × 10^{81} | 1.812 × 10^{83} | 6.429 × 10^{83} | 1.397 × 10^{81} | 3.0883 × 10^{81} | 1.0051 × 10^{81} | |

${\mathit{f}}_{\mathbf{6}}\left(x\right)$ | MD | 4.622 × 10^{84} | 3.405 × 10^{81} | 7.928 × 10^{82} | 2.939 × 10^{83} | 5.977 × 10^{80} | 7.9607 × 10^{80} | 4.901 × 10^{80} |

SD | 1.685 × 10^{85} | 7.33 × 10^{81} | 3.047 × 10^{83} | 7.551 × 10^{83} | 2.197 × 10^{81} | 5.6651 × 10^{81} | 1.9457 × 10^{81} | |

AB | 30.033333 | 30 | 30 | 58.766666 | 30 | 33.633333 | 30 | |

${\mathit{f}}_{\mathbf{7}}\left(x\right)$ | MD | 30 | 30 | 30 | 59 | 30 | 30 | 30 |

SD | 0.18257419 | 0 | 0 | 1.77498583 | 0 | 5.4550409 | 0 | |

AB | 9.2 | 8.0666666 | 1.0666666 | 19,543.266 | 0.0333333 | 2000.0333 | 0 | |

${\mathit{f}}_{\mathbf{8}}\left(x\right)$ | MD | 9 | 8 | 0 | 19,797 | 0 | 0 | 0 |

SD | 2.5784250 | 1.9464084 | 3.1941037 | 2077.7459 | 0.1825741 | 4842.3277 | 0 | |

AB | −745.05202 | −1125.4815 | −1127.8626 | −725.09353 | −1071.7869 | −1031.2617 | −1146.3478 | |

${\mathit{f}}_{\mathbf{9}}\left(x\right)$ | MD | −743.4462 | −1174.9722 | −1132.5748 | −719.10777 | −1068.9596 | −1033.6178 | −1145.2467 |

SD | 25.137593 | 78.706 | 25.809999 | 25.815652 | 34.055847 | 34.244188 | 10.928362 | |

AB | 110,282.54 | 665,278.86 | −4930 | 1,170,939.0 | 45,556.260 | 222,833.73 | −501.79356 | |

${\mathit{f}}_{\mathbf{10}}\left(x\right)$ | MD | 96,461.061 | 673,449.49 | −4930 | 1,126,234.2 | 5076.8152 | 71,582.051 | −332.82466 |

SD | 44,933.417 | 129,147.27 | 3.7318 × 10^{−9} | 159,175.64 | 75,990.880 | 305,159.37 | 663.92006 | |

AB | −18.26109 | −26.056561 | −29.6576 | −16.504756 | −28.367666 | −28.863589 | −30 | |

${\mathit{f}}_{\mathbf{11}}\left(x\right)$ | MD | −18.131984 | −26.092349 | −29.9286 | −16.183447 | −28.14029 | −29.070145 | −30 |

SD | 1.6366873 | 0.5428906 | 0.466 | 1.16917142 | 1.5408536 | 1.2837415 | 0 | |

AB | 1502.3129 | 369.60375 | 786.36819 | 519.17242 | 196.95838 | 261.52332 | 11.905761 | |

${\mathit{f}}_{\mathbf{12}}\left(x\right)$ | MD | 1457.6865 | 368.82916 | 778.72369 | 465.93238 | 213.00730 | 252.76747 | 0.3841574 |

SD | 420.70611 | 35.389136 | 215.93893 | 228.69860 | 86.542862 | 106.52353 | 29.544101 |

SOA | SOA | SOA | SOA | SOA | SOA | |
---|---|---|---|---|---|---|

Function | vs. | vs. | vs. | vs. | vs. | vs. |

ABC | CMAES | CSA | DE | MFO | PSO | |

${f}_{1}\left(\mathrm{x}\right)$ | 7.13 × 10^{−9}▲ | 2.61 × 10^{−8}▲ | 2.40 × 10^{−11}▲ | 9.77 × 10^{−7}▲ | 3.97 × 10^{−11}▲ | 6.87 × 10^{−8}▲ |

${f}_{2}\left(\mathrm{x}\right)$ | 1.21 × 10^{−12}► | 1► | 1.21 × 10^{−12}▲ | 1► | 1► | 4.13 × 10^{−9}▲ |

${f}_{3}\left(\mathrm{x}\right)$ | 1.21 × 10^{−12}► | 1► | 1.21 × 10^{−12}▲ | 1► | 1► | 8.33 × 10^{−7}▲ |

${f}_{4}\left(\mathrm{x}\right)$ | 6.48 × 10^{−12}▲ | 2.43 × 10^{−8}▲ | 6.48 × 10^{−12}▲ | 1.10 × 10^{−7}▲ | 1.18 × 10^{−7}▲ | 5.79 × 10^{−8}▲ |

${f}_{5}\left(\mathrm{x}\right)$ | 3.02 × 10^{−11}▲ | 2.18 × 10^{−6}▲ | 3.02 × 10^{−11}▲ | 3.02 × 10^{−11}▲ | 8.30 × 10^{−1}▲ | 9.94 × 10^{−8}▲ |

${f}_{6}\left(\mathrm{x}\right)$ | 3.69 × 10^{−11}▲ | 1.29 × 10^{−9}▲ | 2.87 × 10^{−10}▲ | 6.73 × 10^{−8}▲ | 1.75 × 10^{−5}▲ | 9.52 × 10^{−4}▲ |

${f}_{7}\left(\mathrm{x}\right)$ | 3.34 × 10^{−1}► | 1► | 1.57 × 10^{−12}▲ | 3.34 × 10^{−1}► | 2.23 × 10^{−5}▲ | 3.34 × 10^{−7}▲ |

${f}_{8}\left(\mathrm{x}\right)$ | 3.96 × 10^{−6}▲ | 1.10 × 10^{−7}▲ | 7.87 × 10^{−12}▲ | 3.28 × 10^{−6}▲ | 2.45 × 10^{−1}▲ | 5.58 × 10^{−7}▲ |

${f}_{9}\left(\mathrm{x}\right)$ | 2.97 × 10^{−11}▲ | 5.75 × 10^{−8}▲ | 2.97 × 10^{−11}▲ | 7.72 × 10^{−8}▲ | 4.20 × 10^{−4}▲ | 1.83 × 10^{−5}▲ |

${f}_{10}\left(\mathrm{x}\right)$ | 3.02 × 10^{−11}▲ | 2.85 × 10^{−11}▲ | 3.02 × 10^{−11}▲ | 3.02 × 10^{−11}▲ | 2.57 × 10^{−7}▲ | 1.86 × 10^{−3}▲ |

${f}_{11}\left(\mathrm{x}\right)$ | 2.80 × 10^{−11}▲ | 1.12 × 10^{−07}► | 2.80 × 10^{−11}▲ | 3.00 × 10^{−11}▲ | 8.88 × 10^{−1}► | 8.86 × 10^{−6}▲ |

${f}_{12}\left(\mathrm{x}\right)$ | 3.02 × 10^{−11}▲ | 3.02 × 10^{−11}▲ | 3.02 × 10^{−11}▲ | 3.02 × 10^{−11}▲ | 1.78 × 10^{−10}▲ | 9.76 × 10^{−10}▲ |

ABC | DE | CMAES | CSA | PSO | MFO | SOA | ||
---|---|---|---|---|---|---|---|---|

AB | 25.648891 | 23.234006 | 0.0382940 | 117,864.48 | 2433.8148 | 16,893.538 | 4.416 × 10^{−16} | |

${f}_{13}\left(\mathrm{x}\right)$ | MD | 26.054364 | 22.218747 | 1.503 × 10^{−23} | 120,885.37 | 5.9145 × 10^{−10} | 10,737.418 | 4.1862 × 10^{−16} |

SD | 8.3305663 | 5.5158397 | 0.1428856 | 14,742.855 | 4322.0341 | 19,501.276 | 2.4373 × 10^{−16} | |

AB | 0.0136600 | 0.0145526 | 1.2398 × 10^{−5} | 51.476735 | 4.0467 × 10^{−13} | 7.8643202 | 1.5 × 10^{−20} | |

${f}_{14}\left(\mathrm{x}\right)$ | MD | 0.0144199 | 0.0144921 | 1.3237 × 10^{−20} | 51.798031 | 7.5065 × 10^{−14} | 2.8398 × 10^{−7} | 1.3059 × 10^{−20} |

SD | 0.0041468 | 0.0037530 | 3.908 × 10^{−5} | 5.9378289 | 7.8227 × 10^{−13} | 14.024260 | 9.6811 × 10^{−21} | |

AB | 0.5442777 | 0.569612 | 19.789197 | 2466.2017 | 20 | 403.33365 | 1.2053 × 10^{−18} | |

${f}_{15}\left(\mathrm{x}\right)$ | MD | 0.4952416 | 0.5733913 | 0.2257190 | 2478.7246 | 8.9126 × 10^{−12} | 200.00000 | 1.025 × 10^{−18} |

SD | 0.1986349 | 0.1376955 | 39.959487 | 344.56578 | 66.436383 | 520.92974 | 6.4946 × 10^{−19} | |

AB | 0.0006647 | 1.8659 × 10^{−10} | 6.9743 × 10^{−10} | 0.0068342 | 5.6588 × 10^{−24} | 9.5555 × 10^{−19} | 0 | |

${f}_{16}\left(\mathrm{x}\right)$ | MD | 0.0004306 | 1.2946 × 10^{−10} | 7.1112 × 10^{−10} | 0.0066858 | 3.6687 × 10^{−29} | 2.8339 × 10^{−22} | 0 |

SD | 0.0006187 | 1.8814 × 10^{−10} | 4.3714 × 10^{−10} | 0.0032037 | 3.0973 × 10^{−23} | 3.3608 × 10^{−18} | 0 |

SOA | SOA | SOA | SOA | SOA | SOA | |
---|---|---|---|---|---|---|

Function | vs. | vs. | vs. | vs. | vs. | vs. |

ABC | CMAES | CSA | DE | MFO | PSO | |

${f}_{13}\left(x\right)$ | 2.80 × 10^{−11}▲ | 3.86 × 10^{−1}▲ | 2.80 × 10^{−11}▲ | 2.80 × 10^{−11}▲ | 1.75 × 10^{−9}▲ | 1.22 × 10^{−4}▲ |

${f}_{14}\left(x\right)$ | 5.51 × 10^{−9}▲ | 2.11 × 10^{−1}▲ | 2.72 × 10^{−11}▲ | 3.22 × 10^{−9}▲ | 1.07 × 10^{−4}▲ | 1.74 × 10^{−2}▼ |

${f}_{15}\left(x\right)$ | 1.58 × 10^{−1}▲ | 3.02 × 10^{−11}▲ | 3.02 × 10^{−11}▲ | 1.81 × 10^{−1}▲ | 5.26 × 10^{−4}▲ | 3.11 × 10^{−1}▲ |

${f}_{16}\left(x\right)$ | 1.21 × 10^{−12}▲ | 1.21 × 10^{−12}▲ | 1.21 × 10^{−12}▲ | 1.21 × 10^{−12}▲ | 1.21 × 10^{−12}▲ | 1.21 × 10^{−12}▲ |

ABC | DE | CMAES | CSA | PSO | MFO | SOA | ||
---|---|---|---|---|---|---|---|---|

AB | 396.75458 | 7.7178366 | 3.1526 × 10^{−9} | 20,330.245 | 334.63082 | 23,758.790 | 0.8147792 | |

${f}_{17}\left(\mathrm{x}\right)$ | MD | 210.31173 | 7.7772734 | 2.7959 × 10^{−9} | 19,814.613 | 6.9181 × 10^{−7} | 20,077.849 | 4.3564 × 10^{−11} |

SD | 497.02847 | 1.3190384 | 1.1535 × 10^{−9} | 2074.4742 | 1832.8485 | 18,166.603 | 2.5336865 | |

AB | 212.40266 | 75.917033 | 105.99474 | 731.38151 | 65.728942 | 161.36081 | 30.785661 | |

${f}_{18}\left(\mathrm{x}\right)$ | MD | 212.09269 | 75.575979 | 31.783896 | 741.08639 | 65.904649 | 116.86683 | 28.998449 |

SD | 25.805145 | 10.604761 | 84.809720 | 68.759154 | 14.502755 | 107.40820 | 3.5251022 | |

AB | 221,724.73 | 1264.0862 | 57.570417 | 70,494,770 | 87.951413 | 80.887783 | 31.999808 | |

${f}_{19}\left(\mathrm{x}\right)$ | MD | 200,128.19 | 1278.1579 | 32.661016 | 65,110,859 | 84.267959 | 78.040599 | 31.999808 |

SD | 114,341.27 | 268.82145 | 54.602147 | 23,890,494 | 25.256576 | 23.169503 | 6.4582 × 10^{−10} | |

AB | 319.57592 | 49.741282 | 97.542580 | 867.75158 | 122.92508 | 802.21444 | 30.307556 | |

${f}_{20}\left(\mathrm{x}\right)$ | MD | 299.50312 | 49.708103 | 29.002196 | 879.05543 | 65.254961 | 685.34608 | 29 |

SD | 64.709010 | 4.4720540 | 93.393879 | 103.61875 | 127.74698 | 500.43174 | 4.0252793 |

SOA | SOA | SOA | SOA | SOA | SOA | |
---|---|---|---|---|---|---|

Function | vs. | vs. | vs. | vs. | vs. | vs. |

ABC | CMAES | CSA | DE | MFO | PSO | |

${f}_{17}\left(x\right)$ | 4.35 × 10^{−11}▲ | 6.63 × 10^{−5}▲ | 2.92 × 10^{−11}▲ | 8.16 × 10^{−8}▲ | 5.40 × 10^{−10}▲ | 6.55 × 10^{−2}▲ |

${f}_{18}\left(x\right)$ | 1.16 × 10^{−7}▲ | 7.28 × 10^{−4}▲ | 3.02 × 10^{−11}▲ | 6.63 × 10^{−7}▲ | 2.01 × 10^{−1}▲ | 5.20 × 10^{−6}▲ |

${f}_{19}\left(x\right)$ | 3.02 × 10^{−11}▲ | 3.02 × 10^{−11}▲ | 3.02 × 10^{−11}▲ | 3.02 × 10^{−11}▲ | 1.17 × 10^{−4}▲ | 4.94 × 10^{−5}▲ |

${f}_{20}\left(x\right)$ | 4.91 × 10^{−11}▲ | 2.71 × 10^{−5}▲ | 2.98 × 10^{−11}▲ | 6.73 × 10^{−5}▲ | 5.43 × 10^{−11}▲ | 8.11 × 10^{−5}▲ |

ABC | CMAES | CSA | DE | MFO | PSO | SOA | ||
---|---|---|---|---|---|---|---|---|

AB | 31,543,508.93 | 16,816,680,165 | 62,350,211,795 | 92,576,614.71 | 9,869,100,268 | 5,754,474,631 | 389,770,985.4 | |

${\mathit{F}}_{\mathbf{1}}\left(\mathit{x}\right)$ | MD | 31,272,497.07 | 10,317,001,637 | 62,180,271,044 | 90,540,094.66 | 8,120,831,786 | 5,716,932,117 | 348,397,609.3 |

SD | 11,711,445.55 | 18,287,769,833 | 6,701,490,748 | 19,876,885.3 | 6,276,129,768 | 3,760,493,838 | 175,590,731.3 | |

AB | 4.9441 × 10^{32} | 1.9584 × 10^{42} | 9.584 × 10^{43} | 4.6409 × 10^{32} | 1.32094 × 10^{37} | 1.8354 × 10^{43} | 2.4753 × 10^{19} | |

${\mathit{F}}_{\mathbf{2}}\left(\mathit{x}\right)$ | MD | 8.92 × 10^{31} | 7.2683 × 10^{41} | 5.1378 × 10^{42} | 9.165 × 10^{31} | 5.8224 × 10^{31} | 1.5174 × 10^{31} | 4.8076 × 10^{17} |

SD | 8.1141 × 10^{32} | 3.0961 × 10^{42} | 2.7242 × 10^{44} | 9.8904 × 10^{32} | 5.35797 × 10^{37} | 1.0053 × 10^{44} | 1.2579 × 10^{20} | |

AB | 143,585.796 | 206,258.738 | 105,227.778 | 184,983.575 | 141,570.4456 | 79,667.257 | 50,153.7766 | |

${\mathit{F}}_{\mathbf{3}}\left(\mathit{x}\right)$ | MD | 143,614.585 | 201,904.141 | 104,066.094 | 187,839.624 | 132,229.1115 | 69,068.2299 | 47,032.1654 |

SD | 17,888.2525 | 52,000.2984 | 14,953.9048 | 27,184.1578 | 57,378.84773 | 39,077.8042 | 20,560.6136 | |

AB | 558.75833 | 3855.6622 | 16,385.5764 | 558.09385 | 1026.441012 | 937.069787 | 547.952065 | |

${\mathit{F}}_{\mathbf{4}}\left(\mathit{x}\right)$ | MD | 562.142677 | 3566.57161 | 16,520.3929 | 556.117066 | 856.5465617 | 890.326594 | 544.342148 |

SD | 20.2914617 | 1429.47447 | 3040.62335 | 23.4459834 | 629.655489 | 361.015317 | 20.9881356 | |

AB | 730.381143 | 825.655628 | 951.170114 | 721.540918 | 692.7089629 | 629.566432 | 631.994737 | |

${\mathit{F}}_{\mathbf{5}}\left(\mathit{x}\right)$ | MD | 732.333214 | 847.872393 | 951.759754 | 720.602523 | 694.7441044 | 634.793518 | 630.746916 |

SD | 12.7345432 | 66.2497783 | 22.7361791 | 9.29337642 | 42.18671231 | 29.110803 | 26.6185453 | |

AB | 603.83984 | 669.750895 | 691.817816 | 604.901193 | 632.8602211 | 612.05324 | 609.329809 | |

${\mathit{F}}_{\mathbf{6}}\left(\mathit{x}\right)$ | MD | 603.786524 | 668.724864 | 691.322797 | 604.969262 | 632.4713596 | 611.116445 | 608.748467 |

SD | 0.60630828 | 9.3375723 | 6.33324596 | 0.550883 | 8.875276889 | 5.83130625 | 2.23326029 | |

AB | 977.074304 | 889.858004 | 1868.59237 | 983.896545 | 1085.414574 | 850.625181 | 933.44086 | |

${\mathit{F}}_{\mathbf{7}}\left(\mathit{x}\right)$ | MD | 977.307172 | 899.416046 | 1847.81046 | 988.619865 | 1067.152488 | 837.191947 | 939.402368 |

SD | 13.674818 | 39.0420544 | 125.687304 | 17.0217457 | 139.9764841 | 43.9666771 | 25.6723087 | |

AB | 1033.74747 | 1047.6713 | 1181.82055 | 1023.57155 | 991.3574401 | 914.911462 | 920.212894 | |

${\mathit{F}}_{\mathbf{8}}\left(\mathit{x}\right)$ | MD | 1035.62163 | 1026.79246 | 1181.41759 | 1023.90313 | 992.7702619 | 916.328125 | 921.260403 |

SD | 13.5776161 | 86.6169315 | 29.0041369 | 12.1336359 | 43.34646427 | 24.9852693 | 21.5473475 | |

AB | 1926.96476 | 900 | 15,096.7081 | 6434.26285 | 6487.6152 | 2436.07714 | 3251.16746 | |

${\mathit{F}}_{\mathbf{9}}\left(\mathit{x}\right)$ | MD | 1839.55339 | 900 | 15,358.9047 | 6327.72188 | 5976.801998 | 2313.97194 | 2702.63581 |

SD | 333.208904 | 0 | 1637.81695 | 887.500363 | 2250.718589 | 920.338184 | 1462.45439 | |

AB | 8559.09425 | 8026.51416 | 8695.92279 | 7235.23129 | 5260.902021 | 5136.84827 | 4524.56624 | |

${\mathit{F}}_{\mathbf{10}}\left(\mathit{x}\right)$ | MD | 8598.36183 | 7995.60926 | 8710.13364 | 7246.56367 | 5291.679788 | 4941.91541 | 4512.99611 |

SD | 327.683849 | 246.346496 | 312.712801 | 234.084094 | 711.3287215 | 845.232077 | 357.978315 | |

AB | 1594.97778 | 19,382.0122 | 7888.11046 | 1813.68615 | 4011.754904 | 1465.88225 | 1248.42954 | |

${\mathit{F}}_{\mathbf{11}}\left(\mathit{x}\right)$ | MD | 1594.04404 | 18,978.0433 | 7566.7015 | 1774.91924 | 2427.207284 | 1465.34072 | 1247.83322 |

SD | 90.044155 | 9762.53738 | 2030.84932 | 246.845029 | 3525.449844 | 125.920596 | 32.5337336 | |

AB | 22,035,530.6 | 4,281,754,760 | 8,661,838,227 | 92,723,516.6 | 91,292,958.57 | 354,557,569 | 4,401,381.17 | |

${\mathit{F}}_{\mathbf{12}}\left(\mathit{x}\right)$ | MD | 20,991,516.3 | 4,358,279,185 | 8,610,836,607 | 93,208,101.7 | 23,663,273.6 | 246,467,712 | 3,700,514.39 |

SD | 7,163,344.91 | 1,489,535,737 | 2,020,262,530 | 18,696,916.2 | 134,545,916.4 | 411,606,206 | 3,405,855.44 | |

AB | 19,266.9698 | 3,652,661,382 | 6,874,450,894 | 3,979,238.98 | 38,881,437.05 | 111,741,897 | 26,817.0052 | |

${\mathit{F}}_{\mathbf{13}}\left(\mathit{x}\right)$ | MD | 18,527.1646 | 3,905,472,944 | 7,021,073,065 | 3,726,551.68 | 186,879.0885 | 4,517,059.75 | 16,966.594 |

SD | 9421.82494 | 1,271,069,390 | 2,248,014,249 | 1659,861.04 | 193,180,811.6 | 371,472,002 | 23,502.6252 | |

AB | 131,154.772 | 6,816,811.85 | 2,093,007.76 | 270,517.596 | 369,042.9999 | 333,813.274 | 38,907.1977 | |

${\mathit{F}}_{\mathbf{14}}\left(\mathit{x}\right)$ | MD | 109,800.311 | 5,506,707.5 | 1,692,106.9 | 248,357.925 | 137,669.8479 | 99,365.0474 | 24,004.7416 |

SD | 74,612.9136 | 4,656,160.18 | 1,375,028.96 | 113,787.245 | 640,038.5333 | 1,180,870.46 | 40,344.6637 | |

AB | 8915.09332 | 519,343,539 | 512,525,748 | 522,995.483 | 58,693.31574 | 86,213.8881 | 7888.58976 | |

${\mathit{F}}_{\mathbf{15}}\left(\mathit{x}\right)$ | MD | 5428.6522 | 428,173,625 | 475,300,661 | 514,399.613 | 35,343.82469 | 63,456.3421 | 3430.78723 |

SD | 12,023.8419 | 350,472,927 | 266,935,668 | 282,908.145 | 74,805.66654 | 61,784.9118 | 8715.94813 | |

AB | 3371.67432 | 4820.88686 | 5247.63056 | 2937.15125 | 3160.316909 | 2823.54207 | 2647.38414 | |

${\mathit{F}}_{\mathbf{16}}\left(\mathit{x}\right)$ | MD | 3388.42168 | 4838.74796 | 5230.02992 | 2987.83654 | 3135.128472 | 2823.76408 | 2599.69633 |

SD | 194.740886 | 282.787939 | 336.291377 | 168.503372 | 330.2217186 | 407.164335 | 272.998599 | |

AB | 2395.86777 | 3482.11031 | 3378.54815 | 2177.58318 | 2441.465302 | 2294.79893 | 2174.93599 | |

${\mathit{F}}_{\mathbf{17}}\left(\mathit{x}\right)$ | MD | 2389.4562 | 3468.22585 | 3372.19373 | 2187.84011 | 2454.470182 | 2268.3003 | 2159.79812 |

SD | 102.889656 | 285.667211 | 324.223747 | 85.3307491 | 250.2293907 | 294.2882 | 194.053384 | |

AB | 5,046,671.6 | 37,362,365 | 23,998,373.2 | 2,228,087.66 | 4,549,050.542 | 1,341,517.26 | 818,677.628 | |

${\mathit{F}}_{\mathbf{18}}\left(\mathit{x}\right)$ | MD | 4,728,588.97 | 33,258,799.8 | 22,232,367.5 | 2,033,807.45 | 1,182,780.284 | 682,252.246 | 300,231.736 |

SD | 2,468,179.06 | 22,078,270.5 | 13,062,667.1 | 914,951.248 | 9,866,226.402 | 1,857,506.17 | 1,622,268.27 | |

AB | 16,681.7816 | 620,387,919 | 616,153,747 | 551,809.083 | 16,702,987.12 | 13,973,268.1 | 4686.90195 | |

${\mathit{F}}_{\mathbf{19}}\left(\mathit{x}\right)$ | MD | 7134.24743 | 509,584,907 | 600,010,030 | 481,336.814 | 143,655.9903 | 541,084.259 | 3334.90276 |

SD | 24,639.1966 | 466,495,950 | 315,926,138 | 399,361.441 | 46,487,264.39 | 44,969,899.6 | 3974.19763 | |

AB | 2776.66582 | 2804.5126 | 2945.57725 | 2439.77572 | 2697.417352 | 2452.70485 | 2439.26615 | |

${\mathit{F}}_{\mathbf{20}}\left(\mathit{x}\right)$ | MD | 2764.33673 | 2834.35184 | 2955.48441 | 2445.1877 | 2627.891978 | 2488.70557 | 2449.42599 |

SD | 94.7983596 | 199.062724 | 106.815418 | 88.0385339 | 220.4185834 | 179.132846 | 147.240337 | |

AB | 2523.44927 | 2641.22467 | 2739.23609 | 2514.75813 | 2506.000485 | 2434.7453 | 2421.58999 | |

${\mathit{F}}_{\mathbf{21}}\left(\mathit{x}\right)$ | MD | 2524.34506 | 2643.05906 | 2741.53087 | 2518.37232 | 2499.78219 | 2433.21201 | 2426.23287 |

SD | 15.4046594 | 36.2479731 | 41.9080485 | 10.9762507 | 40.05305083 | 23.6970887 | 21.141776 | |

AB | 4836.58643 | 9577.04104 | 9095.47501 | 6970.19197 | 6645.22701 | 5532.20097 | 4165.49182 | |

${\mathit{F}}_{\mathbf{22}}\left(\mathit{x}\right)$ | MD | 4242.35722 | 9932.08496 | 9075.49459 | 6750.16911 | 6848.788542 | 6523.54829 | 3767.22155 |

SD | 2051.83573 | 1437.66121 | 763.435864 | 1310.37051 | 1508.145086 | 1932.03672 | 1801.85242 | |

AB | 2872.69194 | 3047.39462 | 3398.54677 | 2848.15024 | 2821.343894 | 2904.06627 | 2785.54406 | |

${\mathit{F}}_{\mathbf{23}}\left(\mathit{x}\right)$ | MD | 2872.47313 | 3045.83859 | 3425.04431 | 2848.07645 | 2814.326953 | 2917.30159 | 2781.95288 |

SD | 10.9306383 | 35.8015482 | 80.91846 | 9.83856624 | 34.39809682 | 57.5467798 | 22.306126 | |

AB | 3029.76315 | 3180.2796 | 3622.26109 | 3048.38906 | 2970.576725 | 3097.14113 | 2998.32425 | |

${\mathit{F}}_{\mathbf{24}}\left(\mathit{x}\right)$ | MD | 3029.47271 | 3188.3269 | 3620.25868 | 3050.32895 | 2969.979831 | 3072.58458 | 2996.90489 |

SD | 12.2834187 | 30.8119037 | 99.4471973 | 14.170605 | 27.40350786 | 61.1357559 | 35.0296809 | |

AB | 2918.77003 | 3333.22102 | 6228.94202 | 2976.74179 | 3234.960327 | 3006.6561 | 2931.4471 | |

${\mathit{F}}_{\mathbf{25}}\left(\mathit{x}\right)$ | MD | 2919.85456 | 2980.52118 | 6253.4295 | 2977.53996 | 3142.80023 | 2992.45175 | 2927.85824 |

SD | 9.81487821 | 759.162584 | 864.95632 | 16.7265818 | 310.0654881 | 102.116534 | 17.6935953 | |

AB | 5896.23764 | 8340.37382 | 11,171.1937 | 5643.82962 | 5776.420284 | 5702.64017 | 4464.46857 | |

${\mathit{F}}_{\mathbf{26}}\left(\mathit{x}\right)$ | MD | 5889.23731 | 8393.26188 | 11,286.1812 | 5659.37067 | 5796.165191 | 5707.34766 | 4933.20537 |

SD | 115.549909 | 452.740525 | 720.862035 | 138.81176 | 469.5232252 | 888.194496 | 945.460049 | |

AB | 3235.30385 | 3408.04414 | 4035.72773 | 3228.08324 | 3245.912414 | 3361.79871 | 3225.24773 | |

${\mathit{F}}_{\mathbf{27}}\left(\mathit{x}\right)$ | MD | 3235.68049 | 3407.30733 | 4021.85789 | 3228.32551 | 3242.855656 | 3349.63975 | 3224.0298 |

SD | 6.57578664 | 33.3663071 | 190.024017 | 3.10051948 | 23.16796867 | 69.4288619 | 11.3386465 | |

AB | 3328.57792 | 6580.85938 | 7194.47112 | 3389.18088 | 4365.338523 | 3623.74347 | 3292.31393 | |

${\mathit{F}}_{\mathbf{28}}\left(\mathit{x}\right)$ | MD | 3326.19 | 6795.88776 | 7205.84449 | 3392.76022 | 4053.737772 | 3542.19939 | 3289.42285 |

SD | 17.9291186 | 558.703896 | 801.137793 | 29.1593474 | 920.6971447 | 251.617478 | 36.390848 | |

AB | 4459.90622 | 5654.19067 | 6442.47253 | 4169.15715 | 4120.329591 | 4068.18331 | 3734.31699 | |

${\mathit{F}}_{\mathbf{29}}\left(\mathit{x}\right)$ | MD | 4476.48304 | 5655.83405 | 6379.40471 | 4183.03085 | 4153.333265 | 4030.03404 | 3709.79866 |

SD | 160.899424 | 250.599138 | 525.490575 | 134.81878 | 272.9470092 | 334.451062 | 167.593361 | |

AB | 401,261.311 | 629,251,110 | 736,794,634 | 317,868.587 | 876,056.4301 | 2,818,096.31 | 27,288.2215 | |

${\mathit{F}}_{\mathbf{30}}\left(\mathit{x}\right)$ | MD | 343,273.6 | 484,558,438 | 795,847,326 | 289,961.292 | 213,206.5063 | 1,323,187.36 | 23,028.0874 |

SD | 260,605.287 | 448,734,951 | 231,471,227 | 159,645.774 | 1,156,943.744 | 3,490,925.37 | 12,690.9933 |

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## Share and Cite

**MDPI and ACS Style**

Cuevas, E.; Becerra, H.; Escobar, H.; Luque-Chang, A.; Pérez, M.; Eid, H.F.; Jiménez, M.
Search Patterns Based on Trajectories Extracted from the Response of Second-Order Systems. *Appl. Sci.* **2021**, *11*, 3430.
https://doi.org/10.3390/app11083430

**AMA Style**

Cuevas E, Becerra H, Escobar H, Luque-Chang A, Pérez M, Eid HF, Jiménez M.
Search Patterns Based on Trajectories Extracted from the Response of Second-Order Systems. *Applied Sciences*. 2021; 11(8):3430.
https://doi.org/10.3390/app11083430

**Chicago/Turabian Style**

Cuevas, Erik, Héctor Becerra, Héctor Escobar, Alberto Luque-Chang, Marco Pérez, Heba F. Eid, and Mario Jiménez.
2021. "Search Patterns Based on Trajectories Extracted from the Response of Second-Order Systems" *Applied Sciences* 11, no. 8: 3430.
https://doi.org/10.3390/app11083430