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Fig. 1. The example of opposition-based optimization: (a) The contour plot of optimized function. (b) The optimized function.

Figure 1 The example of opposition-based optimization: (a) The contour plot of optimized function. (b) The optimized function.

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2.2. Opposition-based learning
2.2. Opposition-based learning
Fig. 2. The proposed OBSCA algorithm.
Fig. 2. The proposed OBSCA algorithm.
*Fixed-dimension multimodal (Liang, Suganthan, & Deb, 2005), these functions are a combination of the unimodal and multimodal functions with different shift, ro- tation, and biasing. The composite functions are similar to other real search spaces and they are used to evaluate the performance of the algorithms to balance between the exploitation and explo- ration. functions by character “F’ and its number such as F1, F2, F3,... , F29. The functions from F1 to F7 are unimodal optimization problems which have one extreme value point within a given search area. These functions are used to investigate the convergence speed and optimization precision of the algorithms, while the functions from F8 to F23 are multimodal optimization functions which have more than one local one extreme value points within a given search area. They are used to evaluate the abilities to jump out of lo- cal optimum and seeking the global excellent result. The rest of benchmark functions (F24 to F29) are composite functions intro- duced by CEC 2005 special session (Suganthan et al., May 2005), The number N of agents in the population is fixed to 30 cor- responding to the dimension 30 and the maximum number of iterations is 500. The experiments are implemented over ”Win- dows 7 (64bit)” that runs on "CPU Core2 Duo with 4GB ram”; and "Matlab 2014b” software is used. For statistical analysis, all the
*Fixed-dimension multimodal (Liang, Suganthan, & Deb, 2005), these functions are a combination of the unimodal and multimodal functions with different shift, ro- tation, and biasing. The composite functions are similar to other real search spaces and they are used to evaluate the performance of the algorithms to balance between the exploitation and explo- ration. functions by character “F’ and its number such as F1, F2, F3,... , F29. The functions from F1 to F7 are unimodal optimization problems which have one extreme value point within a given search area. These functions are used to investigate the convergence speed and optimization precision of the algorithms, while the functions from F8 to F23 are multimodal optimization functions which have more than one local one extreme value points within a given search area. They are used to evaluate the abilities to jump out of lo- cal optimum and seeking the global excellent result. The rest of benchmark functions (F24 to F29) are composite functions intro- duced by CEC 2005 special session (Suganthan et al., May 2005), The number N of agents in the population is fixed to 30 cor- responding to the dimension 30 and the maximum number of iterations is 500. The experiments are implemented over ”Win- dows 7 (64bit)” that runs on "CPU Core2 Duo with 4GB ram”; and "Matlab 2014b” software is used. For statistical analysis, all the
algorithms performed 30 independent runs over each benchmark problem.
algorithms performed 30 independent runs over each benchmark problem.
Comparison between SCA and OBSCA over 11 fitness functions. Table 3 Table 2 Table 4 The results of test the Influence of Dimensionality.
Comparison between SCA and OBSCA over 11 fitness functions. Table 3 Table 2 Table 4 The results of test the Influence of Dimensionality.
The average of unimodal and multimodal functions function values for all algorithms. Table 6
The average of unimodal and multimodal functions function values for all algorithms. Table 6
Standard deviation of unimodal and multimodal functions for all algorithms. Table 7
Standard deviation of unimodal and multimodal functions for all algorithms. Table 7
Fig. 5. Convergence curves for fitness function from F13 to F18.
Fig. 5. Convergence curves for fitness function from F13 to F18.
Fig. 7. Convergence curves for fitness function from F24 to F29.
Fig. 7. Convergence curves for fitness function from F24 to F29.
Fig. 8. The parameters of welded beam design problem.
Fig. 8. The parameters of welded beam design problem.
Fig. 9. The parameters of tension/compression spring problem. The optimization results of OBSCA and other algorithms for solving the welded beam design problem. 4.8.2. Tension/compression spring design The aim of this section is to analyze the performance of OBSCA to solve the tension/compression spring problem which aims to minimize the weight of tension/compression spring. In this prob- lem, there are three parameters namely, mean coil diameter (D), the number of active coils (N) and wire diameter (d). This problem is illustrated in Fig. 9 and it is defined as follows:
Fig. 9. The parameters of tension/compression spring problem. The optimization results of OBSCA and other algorithms for solving the welded beam design problem. 4.8.2. Tension/compression spring design The aim of this section is to analyze the performance of OBSCA to solve the tension/compression spring problem which aims to minimize the weight of tension/compression spring. In this prob- lem, there are three parameters namely, mean coil diameter (D), the number of active coils (N) and wire diameter (d). This problem is illustrated in Fig. 9 and it is defined as follows:
The optimization results of OBSCA and other literatures for pressure vessel design problem. Table 14 HPSO (He & Wang, 2007b), CPSO (He & Wang, 2007a), ACO (Kaveh & Talatahari, 2010), CSCA (Huang et al., 2007), ES (Mezura- Montes & Coello, 2008), GA (Coello & Montes, 2002) (Deb, 1997), Lagrangian Multiplier (Kannan & Kramer, 1994), Branch-bound (Sandgren, 1990), and GSA (Mirjalili et al., 2016), as shown in Table 14. From Table 14 we can conclude that the results of OB- SCA is better than all other algorithms.
The optimization results of OBSCA and other literatures for pressure vessel design problem. Table 14 HPSO (He & Wang, 2007b), CPSO (He & Wang, 2007a), ACO (Kaveh & Talatahari, 2010), CSCA (Huang et al., 2007), ES (Mezura- Montes & Coello, 2008), GA (Coello & Montes, 2002) (Deb, 1997), Lagrangian Multiplier (Kannan & Kramer, 1994), Branch-bound (Sandgren, 1990), and GSA (Mirjalili et al., 2016), as shown in Table 14. From Table 14 we can conclude that the results of OB- SCA is better than all other algorithms.
The optimization results of OBSCA and other literatures for tension/compression spring design problems. Table 13
The optimization results of OBSCA and other literatures for tension/compression spring design problems. Table 13
Fig. 10. The parameters of pressure vessel design problem. timal value of four variables (namely, the thickness of the head (T;,), thickness of the shell (Ts), the length of the cylindrical sec- tion without considering the head (L) and the inner radius (R)) that satisfied four constraints as (16):
Fig. 10. The parameters of pressure vessel design problem. timal value of four variables (namely, the thickness of the head (T;,), thickness of the shell (Ts), the length of the cylindrical sec- tion without considering the head (L) and the inner radius (R)) that satisfied four constraints as (16):
Related topics:
Evolutionary algorithmsSwarm IntelligenceOpposition-Based Learning
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