## 3 Results  
### Algorithm Performance  
The original AIS algorithms [38] is implemented using the Gbest approach. Comparing the dynamic hybrid approach with the Gbest alone ignores the other possible AIS BPSO approaches, namely Lbest and static hybrid. Thus the AIS Gbest algorithm in [38] is extended to implement Lbest and static hybrid approaches to provide a comprehensive comparison of the these algorithms with DH-BPSO. The Lbest simply replaces the global best position with the local best position in the original AIS algorithm in [38]. The static hybrid approach (AIS Hybrid) uses a combined Gbest and Lbest approach with the probability threshold \(p\) from Equation (1) fixed to 0.5. Thus, a broad range of AIS approaches were considered for the following analysis. Each of the four optimization algorithms were performed 500 times with a population of 10 particles and a maximum iteration of 200. The number of variables \(N\) was set to 100. The algorithms operate on four well-known test functions: Sphere, Griewank, Holder Table 2, and Eggholder functions [46]. The definitions and their global minima of the test functions are summarized in Table 1 below. The input vector \(\mathbf{x}\) to the Sphere and Griewank fucntions represents a particle's position as a list of raw bits \(x_{n}\). The Eggholder and Holder Table 2 functions take two real-valued parameters. To use these functions, each particles' position, expressed as a binary string of 100 bits, is converted to a real number using a sign-magnitude fixed precision \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Definition** & **Domain** & **Global Minimum** \\ \hline Sphere & \(f(\mathbf{x})=\sum\limits_{n=1}^{N}x_{n}^{2}\) & \(x_{n}\in\{0,\,1\}\) & \(f(0,0,\ldots,0)=0\) \\ Griewank & \(f(\mathbf{x})=1+\sum\limits_{n=1}^{N}\frac{x_{n}^{2}}{4000}-\prod\limits_{n=1}^ {N}\cos\left(\frac{x_{n}}{\sqrt{n}}\right)\) & \(x_{n}\in\{0,\,1\}\) & \(f(0,0,\ldots,0)=0\) \\ Holder Table 2 & \(f(x_{1},x_{2})=-\Big{|}\sin(x_{1})\cos(x_{2})\exp\left(\left|1-\frac{\sqrt{x_{ 1}^{2}+x_{2}^{2}}}{\pi}\right|\right)\Big{|}\) & \(x_{n}\in[-8,\,8]\) & \(f(\pm 8,\pm 6.48094)\) \\ Eggholder & \(f(x_{1},x_{2})=-(x_{2}+47)\sin(\sqrt{|x_{2}+\frac{x_{1}}{2}+47|})\) & \(x_{n}\in[-512,\,512]\) & \(f(512,404.2319)\) \\  & \(-x_{1}\sin\left(\sqrt{|x_{1}-(x_{2}+47|)}\right)\) & & \(=-959.6407\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mathematical definitions of functions used to test a set of BPSO algorithms. encoding scheme. First, each bit string is divided into two 50-bit strings. The leftmost bit of each string is the sign bit. For the Eggholder function, the next nine bits are used as the integer bits, leaving 40 precision bits. This naturally constrains the domain of \(x_{n}\) to \(\pm\)512. The encoding for the Holder Table 2 function uses three integer bits and 46 precision bits, naturally restricting the domain to \(\pm\)8. Restricting the domain to a power of 2 simplifies the enforcement of boundary conditions in binary algorithms such as AIS BPSO. The domain is also within the \(\pm\)10 domain of the Holder Table 2 function as defined in [46]. Figure 2 plots against iteration number each algorithm's average global best cost over 500 independent optimizations for the aforementioned test functions. From Figure 2,b, it is evident that Griewank and Sphere functions pose little difficulty in optimizing for any algorithm. The dynamic behavior of DH-BPSO is also clearly shown in that the DH-BPSO convergence curves mimic those of the AIS Lbest algorithm for a brief period at the beginning of the optimization. DH-BPSO quickly begins to accelerate, deviating away from the AIS Lbest curve and closely approaching the AIS Gbest and Hybrid curves some iterations later. Although the Griewank and Sphere function are multi-modal, they are bowl-shaped cost functions with a singular global minimum. Such functions are better optimized by Gbest algorithms which accelerate towards this global minimum once any one particle discovers it. This can be observed from Figure 2,b.