# A New Two-Stage Algorithm for Solving Optimization Problems

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. The New Two-Stage Optimization Algorithm

#### 3.1. Theory of the TSO Algorithm

#### 3.2. Mathematical Modeling of the TSO Algorithm

Algorithm 1 Pseudo-code of the TSO approach. | ||

Start the TSO algorithm. | ||

1. | Determine the range of decision variables, constraints and objective function of the problem. | |

2. | Create the initial population at random. | |

3. | Evaluate the objective function based on the initial population. | |

4. | For t = 1:T, with t being iteration number and T the maximum iteration: | |

5. | Update the good group. | |

6. | For i = 1:N, with N being the number of population members; | |

7. | For d = 1:m, with d being the contour and m the number of variables: | |

8. | Select the j’-th good member. | |

9. | Stage 1: Update ${{x}^{\prime}}_{i}^{d}$ based on (1). | |

10. | End for d = 1:m. | |

11. | Update $\overrightarrow{{X}_{i}}$ based on (2). | |

12. | For d = 1:m: | |

13. | Select the k’-th good member, with k ≠ j. | |

14. | Stage 2: Update ${{x}^{\prime}}_{i}^{d}$ based on (3). | |

15. | End for d = 1:m. | |

16. | Update $\overrightarrow{{X}_{i}}$ based on (4). | |

17. | End for i = 1:N. | |

18. | Save the best quasi-optimal solution. | |

19. | End for t = 1:T. | |

20. | Print the best quasi-optimal solution obtained by the TSO algorithm. | |

End the TSO algorithm. |

## 4. Simulation Study and Results

#### 4.1. Experimental Setup

#### 4.2. Evaluation for Unimodal Objective Functions

_{1}to F

_{7}are unimodal. The optimization results of the TSO algorithm and other mentioned algorithms for these objective functions are presented in Table 1. For all of these functions, the TSO algorithm performs better than the other eight algorithms. Note that the proposed algorithm provides exactly the global optimal solution for F

_{6}. In addition, for other functions, the TSO algorithm provides a solution very close to the global optimum, especially for F

_{1}and F

_{2}. These results show that the new proposed algorithm has a good efficiency in achieving a suitable quasi-optimal solution for this type of objective functions.

#### 4.3. Evaluation for High-Dimesional Multimodal Objective Functions

_{8}to F

_{13}are selected from high-dimension multimodal functions. Table 2 reports the results of optimizing these functions using the TSO algorithm and other algorithms. Note that the new algorithm performs better for all F

_{8}to F

_{13}. Especially for F

_{9}and F

_{11}, the TSO algorithm has achieved the global-optimal solution. An overview of the results in Table 2 shows that the proposed algorithm is able to solve this type of optimization problems more effectively compared to the other algorithms.

#### 4.4. Evaluation for Fixed-Dimesional Multimodal Objective Functions

_{14}to F

_{23}are used to evaluate the performance of the TSO algorithm and other algorithms for multimodal functions. The results are reported in Table 3. Notice that the new algorithm provides suitable quasi-optimal solutions for this type of functions. Although the MP algorithm also performs well, it is not competitive with the TSO algorithm for F

_{15}, F

_{17}, and F

_{20}. Thus, the new algorithm is more efficient than the other eight algorithms in optimizing this type of objective functions.

_{1}to F

_{7}of the unimodal type have no local optimum, and the global optimum solution for these functions is zero. Based on the plots of F

_{1}to F

_{7}, the TSO algorithm provides the best performance among the optimization algorithms. The GA algorithm is the worst optimizer for F

_{1}, F

_{2}, F

_{3}, and F

_{5}. The PSO algorithm is not a good optimizer for F

_{4}, F

_{6}, and F

_{7}. Note that the objective functions F

_{8}to F

_{13}are high-dimension multimodal type with local optimal solutions. Considering the plots drawn for these objective functions in Figure 2, it is clear that the TSO algorithm has good performance in solving these types of optimization problems. The distributions of quasi-optimal solutions in the TSO algorithm are very close to each other and therefore have very low SD. The objective functions F

_{14}to F

_{23}are fixed-dimension multimodal type with local optimal solutions. The superiority of the TSO algorithm in providing quasi-optimal solutions with low SD is evident in Figure 2 for F

_{14}, F

_{15}, F

_{20}, F

_{21}, F

_{22}, and F

_{23}. As reported in Table 3, the TSO algorithm and other eight algorithms provide similar performance in optimizing the objective functions F

_{16}, F

_{17}, F

_{18}, and F

_{19}. Thus, it is expected that the plots of these functions are similar and practically with no difference to each other.

#### 4.5. Statistical Testing

## 5. Discussion

## 6. Conclusions and Future Works

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Objective Function | Variables’ Interval |
---|---|

${F}_{1}\left(x\right)={\displaystyle \sum}_{i=1}^{m}{x}_{i}^{2}$ | ${\left[-100,100\right]}^{m}$ |

${F}_{2}\left(x\right)={\displaystyle \sum}_{i=1}^{m}\left|{x}_{i}\right|+{\displaystyle \prod}_{i=1}^{m}\left|{x}_{i}\right|$ | ${\left[-10,10\right]}^{m}$ |

${F}_{3}\left(x\right)={\displaystyle \sum}_{i=1}^{m}{\left({\displaystyle \sum}_{j=1}^{i}{x}_{i}\right)}^{2}$ | ${\left[-100,100\right]}^{m}$ |

${F}_{4}\left(x\right)=\mathrm{max}\left\{\left|{x}_{i}\right|,1\le i\le m\right\}$ | ${\left[-100,100\right]}^{m}$ |

${F}_{5}\left(x\right)={\displaystyle \sum}_{i=1}^{m-1}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2})\right]$ | ${\left[-30,30\right]}^{m}$ |

${F}_{6}\left(x\right)={\displaystyle \sum}_{i=1}^{m}{\left(\left[{x}_{i}+0.5\right]\right)}^{2}$ | ${\left[-100,100\right]}^{m}$ |

${F}_{7}\left(x\right)={\displaystyle \sum}_{i=1}^{m}i{x}_{i}^{4}+\mathrm{rand}\left(0,1\right)$ | ${\left[-1.28,1.28\right]}^{m}$ |

Objective Function | Variables’ Interval |
---|---|

${F}_{8}\left(x\right)={\displaystyle \sum}_{i=1}^{m}-{x}_{i}\mathrm{sin}\left(\sqrt{\left|{x}_{i}\right|}\right)$ | ${\left[-500,500\right]}^{m}$ |

${F}_{9}\left(x\right)={\displaystyle \sum}_{i=1}^{m}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]$ | ${\left[-5.12,5.12\right]}^{m}$ |

${F}_{10}\left(x\right)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{m}{\displaystyle \sum}_{i=1}^{m}{x}_{i}^{2}}\right)-\mathrm{exp}\left(\frac{1}{m}{\displaystyle \sum}_{i=1}^{m}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+e$ | ${\left[-32,32\right]}^{m}$ |

${F}_{11}\left(x\right)=\frac{1}{4000}{\displaystyle \sum}_{i=1}^{m}{x}_{i}^{2}-{\displaystyle \prod}_{i=1}^{m}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | ${\left[-600,600\right]}^{m}$ |

${F}_{12}\left(x\right)=\frac{\pi}{m}\left\{10\mathrm{sin}\left(\pi {y}_{1}\right)+{\displaystyle \sum}_{i=1}^{m}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {y}_{i+1}\right)\right]+{\left({y}_{n}-1\right)}^{2}\right\}+{\displaystyle \sum}_{i=1}^{m}u\left({x}_{i},10,100,4\right),$ where $u\left({x}_{i},a,i,n\right)=\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{n}{x}_{i}-a,\\ 0-a{x}_{i}a,\\ k{\left(-{x}_{i}-a\right)}^{n}{x}_{i}-a\end{array}$ | ${\left[-50,50\right]}^{m}$ |

${F}_{13}\left(x\right)=0.1\left\{{\mathrm{sin}}^{2}\left(3\pi {x}_{1}\right)+{\displaystyle \sum}_{i=1}^{m}{\left({x}_{i}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(3\pi {x}_{i}+1\right)\right]+{\left({x}_{n}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(2\pi {x}_{m}\right)\right]\right\}+{\displaystyle \sum}_{i=1}^{m}u\left({x}_{i},5,100,4\right)$ | ${\left[-50,50\right]}^{m}$ |

Objective Function | Variables’ Interval |
---|---|

${F}_{14}\left(x\right)={\left(\frac{1}{500}+{\displaystyle \sum}_{j=1}^{25}\frac{1}{j+{\sum}_{i=1}^{2}{\left({x}_{i}-{a}_{ij}\right)}^{6}}\right)}^{-1}$ | ${\left[-65.53,65.53\right]}^{2}$ |

${F}_{15}\left(x\right)={\displaystyle \sum}_{i=1}^{11}{\left[{a}_{i}-\frac{{x}_{1}\left({b}_{i}^{2}+{b}_{i}{x}_{2}\right)}{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}\right]}^{2}$ | ${\left[-5,5\right]}^{4}$ |

${F}_{16}\left(x\right)=4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}$ | ${\left[-5,5\right]}^{2}$ |

${F}_{17}\left(x\right)={\left({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}-6\right)}^{2}+10\left(1-\frac{1}{8\pi}\right)\mathrm{cos}{x}_{1}+10$ | [−5, 10]$\times $[0, 15] |

${F}_{18}\left(x\right)=[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\left(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}\right)]\times [30+{\left(2{x}_{1}-3{x}_{2}\right)}^{2}\times \left(18-32{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2}\right)]$ | ${\left[-5,5\right]}^{2}$ |

${F}_{19}\left(x\right)=-{\displaystyle \sum}_{i=1}^{4}{c}_{i}\mathrm{exp}(-{\displaystyle \sum}_{j=1}^{3}{a}_{ij}{\left({x}_{j}-{P}_{ij}\right)}^{2})$ | ${\left[0,1\right]}^{3}$ |

${F}_{20}\left(x\right)=-{\displaystyle \sum}_{i=1}^{4}{c}_{i}\mathrm{exp}(-{\displaystyle \sum}_{j=1}^{6}{a}_{ij}{\left({x}_{j}-{P}_{ij}\right)}^{2})$ | ${\left[0,1\right]}^{6}$ |

${F}_{21}\left(x\right)=-{\displaystyle \sum}_{i=1}^{5}{[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+6{c}_{i}]}^{-1}$ | ${\left[0,10\right]}^{4}$ |

${F}_{22}\left(x\right)=-{\displaystyle \sum}_{i=1}^{7}{[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+6{c}_{i}]}^{-1}$ | ${\left[0,10\right]}^{4}$ |

${F}_{23}\left(x\right)=-{\displaystyle \sum}_{i=1}^{10}{[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+6{c}_{i}]}^{-1}$ | ${\left[0,10\right]}^{4}$ |

## References

- Zelinka, I.; Snasael, V.; Abraham, A. Handbook of Optimization: From Classical to Modern Approach; Springer: New York, NY, USA, 2012. [Google Scholar]
- Beck, A.; Stoica, P.; Li, J. Exact and approximate solutions of source localization problems. IEEE Trans. Signal Process.
**2008**, 56, 1770–1778. [Google Scholar] [CrossRef] - Beheshti, Z.; Shamsuddin, S.M.H. A review of population-based meta-heuristic algorithms. Int. J. Adv. Soft Comput. Appl.
**2013**, 5, 1–35. [Google Scholar] - Palacios, C.A.; Reyes-Suarez, J.A.; Bearzotti, L.A.; Leiva, V.; Marchant, C. Knowledge discovery for higher education student retention based on data mining: Machine learning algorithms and case study in Chile. Entropy
**2021**, 23, 485. [Google Scholar] [CrossRef] - Kawamoto, K.; Hirota, K. Random scanning algorithm for tracking curves in binary image sequences. Int. J. Intell. Comput. Med. Sci. Image Process.
**2008**, 2, 101–110. [Google Scholar] [CrossRef] - Dhiman, G. SSC: A hybrid nature-inspired meta-heuristic optimization algorithm for engineering applications. Knowl.-Based Syst.
**2021**, 222, 106926. [Google Scholar] [CrossRef] - Dehghani, M.; Mardaneh, M.; Malik, O.P.; Guerrero, J.M.; Sotelo, C.; Sotelo, D.; Nazari-Heris, M.; Al-Haddad, K.; Ramirez-Mendoza, R.A. Genetic algorithm for energy commitment in a power system supplied by multiple energy carriers. Sustainability
**2020**, 12, 10053. [Google Scholar] [CrossRef] - Das, S.; Suganthan, P.N. Differential evolution: A survey of the state-of-the-art. IEEE Trans. Evol. Comput.
**2011**, 15, 4–31. [Google Scholar] [CrossRef] - Holland, J.H. Genetic algorithms. Sci. Am.
**1992**, 267, 66–73. [Google Scholar] [CrossRef] - Hofmeyr, S.A.; Forrest, S. Architecture for an artificial immune system. Evol. Comput.
**2000**, 8, 443–473. [Google Scholar] [CrossRef] [PubMed] - Singh, P.R.; Abd Elaziz, M.; Xiong, S. Ludo game-based metaheuristics for global and engineering optimization. Appl. Soft Comput.
**2019**, 84, 105723. [Google Scholar] [CrossRef] - Ji, Z.; Xiao, W. Improving decision-making efficiency of image game based on deep Q-learning. Soft Comput.
**2020**, 24, 8313–8322. [Google Scholar] [CrossRef] - Derrac, J.; Garcia, S.; Molina, D.; Herrera, F. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol. Comput.
**2011**, 1, 3–18. [Google Scholar] [CrossRef] - Ramirez-Figueroa, J.A.; Martin-Barreiro, C.; Nieto, A.B.; Leiva, V.; Galindo, M.P. A new principal component analysis by particle swarm optimization with an environmental application for data science. Stoch. Environ. Res. Risk Assess.
**2021**. [Google Scholar] [CrossRef] - Cheng, Z.; Song, H.; Wang, J.; Zhang, H.; Chang, T.; Zhang, M. Hybrid firefly algorithm with grouping attraction for constrained optimization problem. Knowl. Based Syst.
**2021**, 220, 106937. [Google Scholar] [CrossRef] - Rao, R.V.; Savsani, V.J.; Vakharia, D. Teaching–learning-based optimization: A novel method for constrained mechanical design optimization problems. Comput. Aided Des.
**2011**, 43, 303–315. [Google Scholar] [CrossRef] - Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef][Green Version] - Faramarzi, A.; Heidarinejad, M.; Mirjalili, S.; Gandomi, A.H. Marine predators algorithm: A nature-inspired metaheuristic. Expert Syst. Appl.
**2020**, 152, 113377. [Google Scholar] [CrossRef] - Kaur, S.; Awasthi, L.K.; Sangal, A.; Dhiman, G. Tunicate swarm algorithm: A new bio-inspired based metaheuristic paradigm for global optimization. Eng. Appl. Artif. Intell.
**2020**, 90, 103541. [Google Scholar] [CrossRef] - Mirjalili, S.; Lewis, A. The whale optimization algorithm. Adv. Eng. Softw.
**2016**, 95, 51–67. [Google Scholar] [CrossRef] - Rashedi, E.; Nezamabadi-Pour, H.; Saryazdi, S. GSA: A gravitational search algorithm. Inf. Sci.
**2009**, 179, 2232–2248. [Google Scholar] [CrossRef] - Montazeri, Z.; Niknam, T. Optimal utilization of electrical energy from power plants based on final energy consumption using gravitational search algorithm. Electr. Eng. Electromech.
**2018**, 4, 70–73. [Google Scholar] [CrossRef][Green Version] - Dehghani, M.; Montazeri, Z.; Dehghani, A.; Malik, O.P.; Morales-Menendez, R.; Dhiman, G.; Nouri, N.; Ehsanifar, A.; Guerrero, J.M.; Ramirez-Mendoza, R.A. Binary spring search algorithm for solving various optimization problems. Appl. Sci.
**2021**, 11, 1286. [Google Scholar] [CrossRef] - Hashim, F.A.; Houssein, E.H.; Mabrouk, M.S.; Al-Atabany, W.; Mirjalili, S. Henry gas solubility optimization: A novel physics-based algorithm. Future Gener. Comput. Syst.
**2019**, 101, 646–667. [Google Scholar] [CrossRef] - Daniel, W.W. Applied Nonparametric Statistics; PWS-Kent Publisher: Boston, MA, USA, 1990. [Google Scholar]
- Korosec, P.; Eftimov, T. Insights into exploration and exploitation power of optimization algorithm using DSCTool. Mathematics
**2020**, 8, 1474. [Google Scholar] [CrossRef] - Givi, H.; Dehghani, M.; Montazeri, Z.; Morales-Menendez, R.; Ramirez-Mendoza, R.A.; Nouri, N. GBUO: “The good, the bad, and the ugly” optimizer. Appl. Sci.
**2021**, 11, 2042. [Google Scholar] [CrossRef] - Kang, H.; Bei, F.; Shen, Y.; Sun, X.; Chen, Q. A diversity model based on dimension entropy and its application to swarm intelligence algorithm. Entropy
**2021**, 23, 397. [Google Scholar] [CrossRef] [PubMed] - Abualigah, L.; Diabat, A.; Mirjalili, S.; Abd Elaziz, M.; Gandomi, A.H. The arithmetic optimization algorithm. Comput. Methods Appl. Mech. Eng.
**2021**, 376, 113609. [Google Scholar] [CrossRef]

**Figure 2.**Plots of the objective function average with y-axis in logarithm scale for the indicated algorithm and function.

Genetic | PSO | GS | TLBO | GWO | WO | TS | MP | TSO | ||
---|---|---|---|---|---|---|---|---|---|---|

F_{1} | AV | 13.2405 | 1.7740 × 10^{−5} | 2.0255 × 10^{−17} | 8.3373 × 10^{−60} | 1.09 × 10^{−58} | 2.1741 × 10^{−9} | 7.71 × 10^{−38} | 3.2715 × 10^{−21} | 1.2 × 10^{−163} |

SD | 4.7664 × 10^{−15} | 6.4396 × 10^{−21} | 1.1369 × 10^{−32} | 4.9436 × 10^{−76} | 5.1413 × 10^{−74} | 7.3985 × 10^{−25} | 7.00 × 10^{−21} | 4.6153 × 10^{−21} | 2.65 × 10^{−180} | |

F_{2} | AV | 2.4794 | 0.3411 | 2.3702 × 10^{−8} | 7.1704 × 10^{−35} | 1.2952 × 10^{−34} | 0.5462 | 8.48 × 10^{−39} | 1.57 × 10^{−12} | 2.29 × 10^{−86} |

SD | 2.2342 × 10^{−15} | 7.4476 × 10^{−17} | 5.1789 × 10^{−24} | 6.6936 × 10^{−50} | 1.9127 × 10^{−50} | 1.7377 × 10^{−16} | 5.92 × 10^{−41} | 1.42 × 10^{−12} | 1.05 × 10^{−99} | |

F_{3} | AV | 1536.896 | 589.492 | 279.3439 | 2.7531 × 10^{−15} | 7.4091 × 10^{−15} | 1.7634 × 10^{−8} | 1.15 × 10^{−21} | 0.0864 | 5.83 × 10^{−70} |

SD | 6.6095 × 10^{−13} | 7.1179 × 10^{−13} | 1.2075 × 10^{−13} | 2.6459 × 10^{−31} | 5.6446 × 10^{−30} | 1.0357 × 10^{−23} | 6.70 × 10^{−21} | 0.1444 | 4.06 × 10^{−77} | |

F_{4} | AV | 2.0942 | 3.9634 | 3.2547 × 10^{−9} | 9.4199 × 10^{−15} | 1.2599 × 10^{−14} | 2.9009 × 10^{−5} | 1.33 × 10^{−23} | 2.6 × 10^{−8} | 1.91 × 10^{−70} |

SD | 2.2342 × 10^{−15} | 1.9860 × 10^{−16} | 2.0346 × 10^{−24} | 2.1167 × 10^{−30} | 1.0583 × 10^{−29} | 1.2121 × 10^{−20} | 1.15 × 10^{−22} | 9.25 × 10^{−9} | 4.56 × 10^{−83} | |

F_{5} | AV | 310.4273 | 50.26245 | 36.10695 | 146.4564 | 36.8607 | 41.7767 | 28.8615 | 46.049 | 28.4397 |

SD | 2.0972 × 10^{−13} | 1.5888 × 10^{−14} | 3.0982 × 10^{−14} | 1.9065 × 10^{−14} | 2.6514 × 10^{−14} | 2.5421 × 10^{−24} | 4.76 × 10^{−3} | 0.4219 | 1.83 × 10^{−15} | |

F_{6} | AV | 14.55 | 20.25 | 0 | 0.4435 | 0.6423 | 1.6085 × 10^{−9} | 7.10 × 10^{−21} | 0.398 | 0 |

SD | 3.1776 × 10^{−15} | 1.2564 | 0 | 4.2203 × 10^{−16} | 6.2063 × 10^{−17} | 4.6240 × 10^{−25} | 1.12 × 10^{−25} | 0.1914 | 0 | |

F_{7} | AV | 5.6799 × 10^{−3} | 0.1134 | 0.0206 | 0.0017 | 0.0008 | 0.0205 | 3.72 × 10^{−4} | 0.0018 | 2.75 × 10^{−5} |

SD | 7.7579 × 10^{−19} | 4.3444 × 10^{−17} | 2.7152 × 10^{−18} | 3.87896 × 10^{−19} | 7.2730 × 10^{−20} | 1.5515 × 10^{−18} | 5.09 × 10^{−5} | 0.001 | 8.49 × 10^{−20} |

**Table 2.**Results of applying the indicted algorithm on the listed high-dimension multimodal objective function.

Genetic | PSO | GS | TLBO | GWO | WO | TS | MP | TSO | ||
---|---|---|---|---|---|---|---|---|---|---|

F_{8} | AV | −8184.4142 | −6908.6558 | −2849.0724 | −7408.6107 | −5885.1172 | −1663.9782 | −5740.3388 | −3594.16321 | −12536.9 |

SD | 833.2165 | 625.6248 | 264.3516 | 513.5784 | 467.5138 | 716.3492 | 41.5 | 811.3265 | 1.30 × 10^{−11} | |

F_{9} | AV | 62.4114 | 57.0613 | 16.2675 | 10.2485 | 8.5265 × 10^{−15} | 4.2011 | 5.70 × 10^{−3} | 140.1238 | 0 |

SD | 2.5421 × 10^{−14} | 6.3552 × 10^{−15} | 3.1776 × 10^{−15} | 5.5608 × 10^{−15} | 5.6446 × 10^{−30} | 4.3692 × 10^{−15} | 1.46 × 10^{−3} | 26.3124 | 0 | |

F_{10} | AV | 3.2218 | 2.1546 | 3.5673 × 10^{−9} | 0.2757 | 1.7053 × 10^{−14} | 0.3293 | 9.80 × 10^{−14} | 9.6987 × 10^{−12} | 4.44 × 10^{−15} |

SD | 5.1636 × 10^{−15} | 7.9441 × 10^{−16} | 3.6992 × 10^{−25} | 2.5641 × 10^{−15} | 2.7517 × 10^{−29} | 1.9860 × 10^{−16} | 4.51 × 10^{−12} | 6.1325 × 10^{−12} | 7.06 × 10^{−31} | |

F_{11} | AV | 1.2302 | 0.0462 | 3.7375 | 0.6082 | 0.0037 | 0.1189 | 1.00 × 10^{−7} | 0 | 0 |

SD | 8.4406 × 10^{−16} | 3.1031 × 10^{−18} | 2.7804 × 10^{−15} | 1.9860 × 10^{−16} | 1.2606 × 10^{−18} | 8.9991 × 10^{−17} | 7.46 × 10^{−7} | 0 | 0 | |

F_{12} | AV | 0.047 | 0.4806 | 0.0362 | 0.0203 | 0.0372 | 1.7414 | 0.0368 | 0.0851 | 7.42 × 10^{−4} |

SD | 4.6547 × 10^{−18} | 1.8619 × 10^{−16} | 6.2063 × 10^{−18} | 7.7579 × 10^{−19} | 4.3444 × 10^{−17} | 8.1347 × 10^{−12} | 1.5461 × 10^{−2} | 0.0052 | 1.75 × 10^{−18} | |

F_{13} | AV | 1.2085 | 0.5084 | 0.002 | 0.3293 | 0.5763 | 0.3456 | 2.9575 | 0.4901 | 1.08 × 10^{−4} |

SD | 3.2272 × 10^{−16} | 4.9650 × 10^{−17} | 4.2617 × 10^{−14} | 2.1101 × 10^{−16} | 2.4825 × 10^{−16} | 3.25391 × 10^{−12} | 1.5682 × 10^{−12} | 0.1932 | 3.41 × 10^{−17} |

**Table 3.**Results of applying the indicated algorithm on the listed fixed-dimension multimodal objective function.

Genetic | PSO | GS | TLBO | GWO | WO | TS | MP | TSO | ||
---|---|---|---|---|---|---|---|---|---|---|

F_{14} | AV | 0.9986 | 2.1735 | 3.5913 | 2.2721 | 3.7408 | 0.998 | 1.9923 | 0.998 | 0.998 |

SD | 1.5640 × 10^{−15} | 7.9441 × 10^{−16} | 7.9441 × 10^{−16} | 1.9860 × 10^{−16} | 6.4545 × 10^{−15} | 9.4336 × 10^{−16} | 2.6548 × 10^{−7} | 4.2735 × 10^{−16} | 8.69 × 10^{−16} | |

F_{15} | AV | 5.3952 × 10^{−2} | 0.0535 | 0.0024 | 0.0033 | 0.0063 | 0.0049 | 0.0004 | 0.003 | 0.0003 |

SD | 7.0791 × 10^{−18} | 3.8789 × 10^{−19} | 2.9092 × 10^{−19} | 1.2218 × 10^{−17} | 1.1636 × 10^{−18} | 3.4910 × 10^{−18} | 9.0125 × 10^{−4} | 4.0951 × 10^{−15} | 1.82 × 10^{−19} | |

F_{16} | AV | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 |

SD | 7.9441 × 10^{−16} | 3.4755 × 10^{−16} | 5.9580 × 10^{−16} | 1.4398 × 10^{−15} | 3.9720 × 10^{−16} | 9.9301 × 10^{−16} | 2.6514 × 10^{−16} | 4.4652 × 10^{−16} | 8.65 × 10^{−17} | |

F_{17} | AV | 0.4369 | 0.7854 | 0.3978 | 0.3978 | 0.3978 | 0.4047 | 0.3991 | 0.3979 | 0.3978 |

SD | 4.9650 × 10^{−17} | 4.9650 × 10^{−17} | 9.9301 × 10^{−17} | 7.4476 × 10^{−17} | 8.6888 × 10^{−17} | 2.4825 × 10^{−17} | 2.1596 × 10^{−16} | 9.1235 × 10^{−15} | 9.93 × 10^{−17} | |

F_{18} | AV | 4.3592 | 3 | 3 | 3.0009 | 3 | 3 | 3 | 3 | 3 |

SD | 5.9580 × 10^{−16} | 3.6741 × 10^{−15} | 6.9511 × 10^{−16} | 1.5888 × 10^{−15} | 2.0853 × 10^{−15} | 5.6984 × 10^{−15} | 2.6528 × 10^{−15} | 1.9584 × 10^{−15} | 4.97 × 10^{−16} | |

F_{19} | AV | −3.85434 | −3.8627 | −3.8627 | −3.8609 | −3.8621 | −3.8627 | −3.8066 | −3.8627 | −3.8627 |

SD | 9.9301 × 10^{−17} | 8.9371 × 10^{−15} | 8.3413 × 10^{−15} | 7.3483 × 10^{−15} | 2.4825 × 10^{−15} | 3.1916 × 10^{−15} | 2.6357 × 10^{−15} | 4.2428 × 10^{−15} | 6.95 × 10^{−16} | |

F_{20} | AV | −2.8239 | −3.2619 | −3.0396 | −3.2014 | −3.2523 | −3.2424 | −3.3206 | −3.3211 | −3.3219 |

SD | 3.97205 × 10^{−16} | 2.9790 × 10^{−16} | 2.1846 × 10^{−14} | 1.7874 × 10^{−15} | 2.1846 × 10^{−15} | 7.9441 × 10^{−16} | 5.6918 × 10^{−15} | 1.1421 × 10^{−11} | 1.89 × 10^{−15} | |

F_{21} | AV | −4.3040 | −5.3891 | −5.1486 | −9.1746 | −9.6452 | −7.4016 | −5.5021 | −10.1532 | −10.1532 |

SD | 1.5888 × 10^{−15} | 1.4895 × 10^{−15} | 2.9790 × 10^{−16} | 8.5399 × 10^{−15} | 6.5538 × 10^{−15} | 2.3819 × 10^{−11} | 5.4615 × 10^{−13} | 2.5361 × 10^{−11} | 5.96 × 10^{−16} | |

F_{22} | AV | −5.1174 | −7.6323 | −9.0239 | −10.0389 | −10.4025 | −8.8165 | −5.0625 | −10.4029 | −10.4029 |

SD | 1.2909 × 10^{−15} | 1.5888 × 10^{−15} | 1.6484 × 10^{−12} | 1.5292 × 10^{−14} | 1.9860 × 10^{−15} | 6.7524 × 10^{−15} | 8.4637 × 10^{−14} | 2.8154 × 10^{−11} | 1.79 × 10^{−15} | |

F_{23} | AV | −6.5621 | −6.1648 | −8.9045 | −9.2905 | −10.1302 | −10.0003 | −10.3613 | −10.5364 | −10.5364 |

SD | 3.8727 × 10^{−15} | 2.7804 × 10^{−15} | 7.1497 × 10^{−14} | 1.1916 × 10^{−15} | 4.5678 × 10^{−15} | 9.1357 × 10^{−15} | 7.6492 × 10^{−12} | 3.9861 × 10^{−11} | 9.33 × 10^{−16} |

**Table 4.**Results of the Friedman rank test for evaluating the indicated algorithm and type of objective function.

Function | TSO | MP | TS | WO | GWO | TLBO | GS | PSO | Genetic | ||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | Unimodal (F _{1}–F_{7}) | Friedman value | 7 | 37 | 16 | 42 | 27 | 28 | 37 | 56 | 57 |

Friedman rank | 1 | 5 | 2 | 6 | 3 | 4 | 5 | 7 | 8 | ||

2 | High-dimension multimodal (F _{8}–F_{13}) | Friedman value | 6 | 33 | 27 | 38 | 24 | 25 | 32 | 37 | 40 |

Friedman rank | 1 | 6 | 4 | 8 | 2 | 3 | 5 | 7 | 9 | ||

3 | Fixed-dimension multimodal (F _{14}–F_{23}) | Friedman value | 10 | 15 | 33 | 33 | 31 | 35 | 38 | 45 | 55 |

Friedman rank | 1 | 2 | 4 | 4 | 3 | 5 | 6 | 7 | 8 | ||

4 | All 23 functions | Friedman value | 23 | 85 | 76 | 113 | 82 | 88 | 107 | 138 | 152 |

Friedman rank | 1 | 4 | 2 | 7 | 3 | 5 | 6 | 8 | 9 |

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**MDPI and ACS Style**

Doumari, S.A.; Givi, H.; Dehghani, M.; Montazeri, Z.; Leiva, V.; Guerrero, J.M.
A New Two-Stage Algorithm for Solving Optimization Problems. *Entropy* **2021**, *23*, 491.
https://doi.org/10.3390/e23040491

**AMA Style**

Doumari SA, Givi H, Dehghani M, Montazeri Z, Leiva V, Guerrero JM.
A New Two-Stage Algorithm for Solving Optimization Problems. *Entropy*. 2021; 23(4):491.
https://doi.org/10.3390/e23040491

**Chicago/Turabian Style**

Doumari, Sajjad Amiri, Hadi Givi, Mohammad Dehghani, Zeinab Montazeri, Victor Leiva, and Josep M. Guerrero.
2021. "A New Two-Stage Algorithm for Solving Optimization Problems" *Entropy* 23, no. 4: 491.
https://doi.org/10.3390/e23040491