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In order to evaluate the performance of SPBO for optimizing the benchmark functions, ten of the CEC-2005 benchmark functions are considered. The ten different CEC-2005 benchmark functions among twenty-five are presented in Table 1.1. To compare the performance of SPBO with other optimization methods, the results obtained by using SPBO are compared to those obtained by using the different optimization methods namely PSO, TLBO, CS, and SOS. Twenty-five individual runs are performed for each of the functions and for each of the algorithms.

The performance of the algorithms selected is evaluated on the basis of the optimal result obtained and on the basis of convergence mobility. For the purposes of analysis, the algorithms are found to converge when the gap between the optimal function result and the result obtained crosses below 1×10^-5. The results obtained below 1×10^-5 are considered as equal to zero. The parameters of PSO, TLBO, CS and SOS are considered according to the dimension of the benchmark function. But SPBO does not have any parameter and the size of the population needs not vary according to the dimension of the benchmark functions. With the increase of dimension of the functions, the size of the population of the proposed SPBO needs not to be increased. That’s why the population size of SPBO for all the considered benchmark functions is considered to be constant. It is considered as 20 for the proposed SPBO. In order to have a fair comparison of the performance of all the algorithms, the analysis is done based on the number of fitness function evaluations (NFFE) taken to converge.

Figure 1.1 Flowchart of the SPBO algorithm.

Table 1.1 CEC 2005 benchmark function [67].

Problem	Type of the function	Name of the functions	F(x*)	Initial range	Bounds	Dimension (D)
F1	Unimodal	Shifted Sphere Function	-450	[-100,100]D	[-100,100] D	30
F2	Unimodal	Shifted Schwefel’s Problem 1.2	-450	[-100,100] D	[-100,100] D	30
F3	Unimodal	Shifted Rotated High Conditioned Elliptic Function	-450	[-100,100] D	[-100,100] D	30
F4	Unimodal	Shifted Schwefel’s Problem 1.2 with Noise in Fitness	-450	[-100,100] D	[-100,100] D	30
F5	Unimodal	Schwefel’s Problems 2.6 with Global Optimum on Bounds	-310	[-100,100] D	[-100, 100] D	30
F6	Basic multimodal	Shifted Rosenbrock’s Function	390	[-100, 100] D	[-100, 100] D	30
F7	Basic multimodal	Shifted Rotated Griewank’s Function without Bounds	-180	[0, 600] D	[0, 600] D	30
F8	Basic multimodal	Shifted Rotated Ackley’s Function with Global Optimum on Bounds	-140	[-32, 32] D	[-32, 32] D	30
F9	Basic multimodal	Shifted Rastrigin’s Function	-330	[-5, 5] D	[-5, 5] D	30
F10	Basic multimodal	Shifted Rotated Rastrigin’s Function	-330	[-5, 5] D	[-5, 5] D	30