Analyzing of Usage Effect of the Distribution Functions for SMDO Algorithm via Benchmark Function with Matlab Toolbox

Öz

This paper presents solution comparisons of benchmark functions by using stochastic multi-parameters divergence (SMDO) method with using different distribution functions. Using benchmark functions is an important method in measuring the effectiveness of algorithms. Because benchmark functions are used by all algorithm producers while trying their algorithms and this provides a good tool for the others to compare their algorithms with similar procedures. Benchmark functions is used in this paper for the main purpose of analyzing randomization process. It is known that distribution functions take place a vital role in getting random numbers. These random numbers are used in stochastic methods through specifying step size. It is believed that a suitable random number generation can support the search processes of algorithms. In this study the effects of distribution functions on benchmark functions are analyzed. For this purpose, a program is developed with MATLAB. The comparisons via the help of this program is shown in tabular form. The results are analyzed from the viewpoint of whether developing the randomization process makes contribution to problem solving power of algorithms. In this study SMDO algorithm is analyzed with different distribution functions by using different benchmark functions. In addition, in the study, a useful friend-friendly Matlab toolbox is proposed in which SMDO algorithm can be tested over different benchmark functions according to different distribution functions. (https://www.mathworks.com/matlabcentral/fileexchange/75044-smdo-with-distribution-function-for-benchmarking)

Anahtar Kelimeler

• SMDO
• optimization
• stochastic
• benchmark functions

Analyzing of Usage Effect of the Distribution Functions for SMDO Algorithm via Benchmark Function with Matlab Toolbox

Abstract

This paper presents solution comparisons of benchmark functions by using stochastic multi-parameters
divergence (SMDO) method with different distribution functions. Using benchmark functions is an
important method in measuring the effectiveness of algorithms. Because benchmark functions are used by
all algorithm producers while trying their algorithms and this provides a good tool for the others to compare
their algorithms with similar procedures. Benchmark functions are used in this paper for the main purpose
of analyzing randomization process. It is known that distribution functions take place a vital role in getting
random numbers. These random numbers are used in stochastic methods through specifying step size. It
is believed that a suitable random number acquisition process can support the search processes of
algorithms. In this study the effects of distribution functions on benchmark functions are analyzed. For
this purpose, a program is developed with MATLAB. The comparisons via the help of this program is
shown in tabular form. The results are analyzed from the viewpoint of whether developing the
randomization process makes contribution to problem solving power of algorithms. In this study SMDO
algorithm is analyzed with different distribution functions by using different benchmark functions. In
addition, in the study, a useful friend-friendly Matlab toolbox is proposed in which SMDO algorithm can
be tested over different benchmark functions according to different distribution functions.
(https://www.mathworks.com/matlabcentral/fileexchange/75044-smdo-with-distribution-function-forbenchmarking)

Keywords

• SMDO
• optimization
• stochastic
• benchmark functions
• distribution functions

Kaynakça

1. [1] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms. 2005.
2. [2] R. Hemmecke, M. Köppe, J. Lee, and R. Weismantel, “Nonlinear integer programming,” in 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art, 2010.
3. [3] C. Zhang and R. Ordóñez, “Numerical optimization,” in Advances in Industrial Control, 2012.
4. [4] L. Bianchi, M. Dorigo, L. M. Gambardella, and W. J. Gutjahr, “A survey on metaheuristics for stochastic combinatorial optimization,” Nat. Comput., 2009, doi: 10.1007/s11047-008-9098-4.
5. [5] J. H. He, “Homotopy perturbation technique,” Comput. Methods Appl. Mech. Eng., 1999, doi: 10.1016/S0045-7825(99)00018-3.
6. [6] J. H. He, “Comparison of homotopy perturbation method and homotopy analysis method,” Appl. Math. Comput., 2004, doi: 10.1016/j.amc.2003.08.008.
7. [7] X. S. Yang, “Firefly algorithm, stochastic test functions and design optimization,” Int. J. Bio-Inspired Comput., 2010, doi: 10.1504/IJBIC.2010.032124.
8. [8] X. S. Yang, “Flower pollination algorithm for global optimization,” in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2012, doi: 10.1007/978-3-642-32894-7_27.

9. [9] M. Ghaemi and M. R. Feizi-Derakhshi, “Forest optimization algorithm,” Expert Syst. Appl., 2014, doi: 10.1016/j.eswa.2014.05.009.
10. [10] F. Glover, “Tabu Search—Part I,” ORSA J. Comput., 1989, doi: 10.1287/ijoc.1.3.190.
11. [11] F. Glover, “Tabu Search—Part II,” ORSA J. Comput., 1990, doi: 10.1287/ijoc.2.1.4.
12. [12] X. S. Yang, “A new metaheuristic Bat-inspired Algorithm,” in Studies in Computational Intelligence, 2010, doi: 10.1007/978-3-642-12538-6_6.
13. [13] X. S. Yang, “Harmony search as a metaheuristic algorithm,” Studies in Computational Intelligence. 2009, doi: 10.1007/978-3-642-00185-7_1.
14. [14] M. Jain, V. Singh, and A. Rani, “A novel nature-inspired algorithm for optimization: Squirrel search algorithm,” Swarm Evol. Comput., 2019, doi: 10.1016/j.swevo.2018.02.013.
15. [15] M. Karimzadeh and F. Keynia, “Woodpecker Mating Algorithm ( WMA ): a nature-inspired algorithm for solving optimization problems,” vol. 11, no. 1, pp. 137–157, 2020.
16. [16] “Probability density function - MATLAB pdf.” [Online]. Available: https://www.mathworks.com/help/stats/prob.normaldistribution.pdf.html. [Accessed: 11-Mar-2020].
17. [17] “Normal Distribution - MATLAB & Simulink.” [Online]. Available: https://www.mathworks.com/help/stats/normal-distribution.html. [Accessed: 11-Mar-2020].
18. [18] “Beta Distribution - MATLAB & Simulink.” [Online]. Available: https://www.mathworks.com/help/stats/beta-distribution.html. [Accessed: 11-Mar-2020].
19. [19] “Binomial Distribution - MATLAB & Simulink.” [Online]. Available: https://www.mathworks.com/help/stats/binomial-distribution.html. [Accessed: 11-Mar-2020].
20. [20] “Extreme Value Distribution - MATLAB & Simulink.” [Online]. Available: https://www.mathworks.com/help/stats/extreme-value-distribution.html. [Accessed: 11-Mar-2020].
21. [21] B. B. Alagoz, A. Ates, and C. Yeroglu, “Auto-tuning of PID controller according to fractional-order reference model approximation for DC rotor control,” Mechatronics, vol. 23, no. 7, 2013, doi: 10.1016/j.mechatronics.2013.05.001.
22. [22] C. Yeroǧlu and A. Ateş, “A stochastic multi-parameters divergence method for online auto-tuning of fractional order PID controllers,” J. Franklin Inst., vol. 351, no. 5, 2014, doi: 10.1016/j.jfranklin.2013.12.006.
23. [23] A. Ates, C. Yeroglu, B. B. Alagoz, and B. Senol, “Tuning of fractional order PID with master-slave stochastic multi-parameter divergence optimization method,” in 2014 International Conference on Fractional Differentiation and Its Applications, ICFDA 2014, 2014, doi: 10.1109/ICFDA.2014.6967388.
24. [24] A. Ateş and C. Yeroglu, “SMDO Algoritması ile İki Serbestlik Dereceli FOPID Kontrol Çevrimi Tasarımı Two Degrees of Freedom FOPID Control Loop Design via SMDO Algorithm,” pp. 6–11.
25. [25] A. Ates and C. YEROĞLU, “Online Tuning of Two Degrees of Freedom Fractional Order Control Loops,” Balk. J. Electr. Comput. Eng., vol. 4, no. 1, pp. 5–11, 2016, doi: 10.17694/bajece.52491.
26. [26] A. Ates et al., “Fractional Order Model Identification of Receptor-Ligand Complexes Formation by Equivalent Electrical Circuit Modeling,” 2019 Int. Conf. Artif. Intell. Data Process. Symp. IDAP 2019, 2019, doi: 10.1109/IDAP.2019.8875913.
27. [27] A. Ates, B. B. Alagoz, Y. Q. Chen, C. Yeroglu, and S. H. HosseinNia, “Optimal Fractional Order PID Controller Design for Fractional Order Systems by Stochastic Multi Parameter Divergence Optimization Method with Different Random Distribution Functions,” pp. 9–14, 2020, doi: 10.1109/iccma46720.2019.8988599.
28. [28] “Unconstained.” [Online]. Available: http://www-optima.amp.i.kyoto-u.ac.jp/member/student/hedar/Hedar_files/TestGO_files/Page364.htm. [Accessed: 16-Apr-2020].