# XOR Binary Gravitational Search Algorithm with Repository: Industry 4.0 Applications
## 5 Simulation Results
### Benchmark Optimization Problem
**Sphere function [21]:** a differentiable, separable, unimodal function without local minima  
\[f_{1}(X)=\sum_{i=1}^{N}x_{i}^{2} \tag{59}\]  
**Multiplication of squares:** a differentiable, non-separable, unimodal function without local minima  
\[f_{2}(X)=\prod_{i=1}^{N}ix_{i}^{2} \tag{60}\]  
**Griewark function [22]:** a differentiable, non-separable, multimodal function with local minimas  
\[f_{3}\left(X\right)=\frac{1}{4000}\sum_{i=1}^{N}x_{i}^{2}-\prod_{i=1}^{N}cos \left(\frac{x_{i}}{\sqrt{i}}\right) \tag{61}\]  
**Schwefel function [23]:** a differentiable, separable, multimodal function with local minima  
\[f_{4}(X)=-\sum_{i=1}^{N}x_{i}sin(\sqrt{\left|x_{i}\right|}) \tag{62}\]  
**Rastrigin Function [22]:** a differentiable, separable, multimodal function with local minima  
\[f_{5}(X)=10.N+\sum_{i=1}^{N}\left(x_{i}^{2}-10 cos\left(2\pi x_{i}\right)\right) \tag{63}\]  
**Styblinski-Tang [24]:** a differentiable, separable, multimodal function with local minima  
\[f_{6}(X)=\frac{1}{N}\sum_{i=1}^{N}\left(x_{i}^{4}-16x_{i}^{2}+5x_{i}\right) \tag{64}\]  
**Michalewicz function [22]:** a differentiable, separable, multimodal function with local minima  
\[f_{7}(X)=-\sum_{i=1}^{N}sin(x_{i})\Big{(}sin(ix_{i}^{2}/\pi)\Big{)}^{20} \tag{65}\]  
**Ackley function [25]:** a differentiable, non-separable, multimodal function with local minima  
\[f_{8}\left(X\right)=-ae^{-b\sqrt{\frac{1}{4\sum_{i=1}^{N}x_{i}^{2}}}}-e^{\frac {1}{d}\sum_{i=1}^{N}\cos(cx_{i})}+\mathbf{a}+e^{1} \tag{66}\]  
**Rotated hyperellipsoid function [22]:** a differentiable, scalable, unimodal function without local minima  
\[f_{9}\left(X\right)=\sum_{i=1}^{N}\ \sum_{j=1}^{i}x_{j}^{2} \tag{67}\]  
**Alpine 1 function [22]:** a non-differentiable, separable, multimodal function with local minima  
\[f_{10}\left(X\right)=\sum_{i=1}^{N}\left|x_{i}\sin\left(x_{i}\right)\ +0.1x_{i}\right| \tag{68}\]  
**Cosine mixture function [22]:** a differentiable, separable, multimodal function with local minima  
\[f_{2}(X)=\prod_{i=1}^{N}ix_{i}^{2} \tag{60}\]0The results of comparison between the proposed algorithm and five other existing binary optimization algorithms are presented in \[f_{2}(X)=\prod_{i=1}^{N}ix_{i}^{2} \tag{60}\]1 Optimization process in each case is repeated for 30 times and the mean values are reported. It can be seen that the proposed optimization algorithm performs well in optimizing uni-modal as well as multi modal optimization problems. In 28 cases out of 44 cases, the proposed method outperforms other investigated optimization algorithms. It is further observed that when the proposed method is not the best optimization algorithm, it is still very close to the desired results. For instance, in the case of optimization function number 1, a uni-modal optimization problem, the proposed version of BGSA outperforms other algorithms in all different number of dimensions except for the case of \(n=1\) where it is the second best optimization algorithm after BDA. Optimization function number 11, \(f_{11}(x)\), is a multi-modal optimization problem with several local minimums for which the proposed XOR BGSA outperforms the other five investigated optimization algorithms in all four dimensions. It can further be observed from the Table 2 that BDA ranks the second after the proposed optimization algorithm.