## 3 The Hybrid Evolutionary Optimization Method Coupling CA with GAs
### Proposed \(t\)-Mutation Operator  
Evolutionary algorithms require good exploration capabilities in the early stages and good exploitation capabilities in the later stages of evolution. The t distribution contains the degree of freedom parameter n, which approaches the standard Gaussian distribution infinitely when \(n\rightarrow+\infty\) and the t distribution is the standard Cauchy distribution when \(n=1\). That is, the standard Gaussian distribution and the standard Cauchy distribution are two boundary special cases of the t distribution. The probability density functions of the standard Gaussian distribution and the standard Cauchy distribution are shown in Figure 5. Obviously, the application of the Cauchy operator can produce a larger mutation step, which is conducive to the algorithm to guide individuals to jump out of the local optimal solution and ensure the exploration ability of the algorithm, and Gaussian distribution shows a better exploitation ability.  
Figure 4: Structure of historical knowledge.  
Figure 5: Probability density functions of the Cauchy and Gaussian distributions.  
Population diversity is considered as the primary reason for premature convergence, which determines the search capability of the algorithm. In evolutionary algorithms, population diversity decreases over time as evolution proceeds. Therefore, population diversity can be used to determine the stage of evolution; we can thus use the population diversity to construct the degree of freedom n. By changing the degree of freedom parameter n, the mutation scale changes adaptively with evolution to balance the exploitation and exploration capabilities of the algorithm. In this paper, we introduce population variance and population entropy to determine population diversity. The expression of population variance \(D_{T}\) in the \(T\)th generation is as follows:  
\[D_{T}=\frac{1}{N\times l}\Bigg{(}\sum_{i=1}^{N}\sum_{j=1}^{l}x_{i}^{j}-\overline {x^{j}}\Bigg{)} \tag{10}\]  
where \(x_{i}^{j}\) is the \(j\)th gene value of the \(i\)th individual, \(N\) is the number of populations and \(l\) is the individual coding length. The expression of \(\overline{x^{j}}\) is as follows:  
\[\overline{x^{j}}=\frac{1}{N}\Bigg{(}\sum_{i=1}^{N}x_{i}^{j}\Bigg{)} \tag{11}\]  
The solution space \(A\) of the optimization problem is divided equally into \(L\) small spaces, and the number of individuals belonging to the \(i\)th space \(A_{i}\) in generation \(T\) is \(|A_{i}|\). The expression of population entropy \(S_{T}\) in the \(T\)th generation is as follows:  
\[S_{T}=-\sum_{i=1}^{L}p_{i}log(p_{i}) \tag{12}\]  
where  
\[p_{i}=\frac{|A_{i}|}{N} \tag{13}\]  
From the definitions of population variance and population entropy, it is clear that population variance reflects the degree of dispersion of individuals in the population and that population entropy reflects the number of individual types in the population. Therefore, the \(t\)-mutation operator \(t(n)\) can be constructed based on the population variance and population entropy. The degree of freedom parameter \(n\) is expressed as follows:  
\[n=\bigg{[}1-ln\bigg{(}\frac{D_{T}+S_{T}}{D_{max}+S_{max}}\bigg{)}\bigg{]} \tag{14}\]  
where \([\ ]\) is the least integer function, and \(D_{max}\) and \(S_{max}\) are the maximum values of population variance and population entropy, respectively. Obviously, the degree of freedom parameter n of the \(t\)-mutation operator is 1 in the first generation and increases gradually as evolution proceeds, then the degree of freedom parameter \(n\) converges to positive infinity in the late evolutionary stage, and the \(t\) distribution becomes a standard Gaussian distribution. The \(t\)-mutation operator can ensure the exploration capability of the algorithm in the early evolutionary stage and the exploitation capability of the algorithm in the late evolutionary stage.