## 3 The Hybrid Evolutionary Optimization Method Coupling CA with GAs
### Communication Protocol
#### 3.4.2 Influence Function  
The core of the influence function is the manner and proportion in which each type of knowledge affects the population. Knowledge acts on each type of influence function to control the number of individuals affected by each type of influence function. Therefore, the proportion by which each type of knowledge affects the population is the relative role that each type of influence function has in the population. The proportion of the effect of the influence function is determined by the success rate of the knowledge effect, and is expressed as follows:  
\[P_{k}(T)= \begin{cases}1/N_{k}&\text{if }T=1\\ \alpha+\beta\times\frac{v_{k}(T-1)}{v(T-1)}&\text{if }(T\ mod\ K)=0\\ P_{k}(T-1)&\text{otherwise}\end{cases}  \tag{16}\]  
It satisfies the condition that \(\beta+\alpha N_{k}=1\), where \(N_{k}\) is the number of knowledge sources types, \(v_{k}(T-1)\) denotes the number of individuals influenced by knowledge \(k\) that are better than their parents in generation \(T-1\) and \(v(T-1)\) denotes the number of individuals influenced by all knowledge sources that are better than their parents in generation \(T-1\). The success rate of knowledge sources influenced in the previous generation determines the proportion of the effect of each knowledge source in the next generation. In order to allow each kind of knowledge source to always have the possibility of being used, we took \(\alpha=0.1\), \(\beta=0.7\), ensuring that the lower bound of \(P_{k}\) is 0.1 and the proportion of all knowledge sources in the first generation is the same, which is \(1/N_{k}\). Next, we introduced the proposed \(t\)-mutation operator into the influence function to develop a knowledge-guided \(t\)-mutation strategy. 1. Situational knowledge. Situational knowledge has a guiding role in the evolutionary process, and the effect of situational knowledge on the population space under the action of the \(t\)-mutation operator is noted as follows: \[x^{\prime}_{ij}= \begin{cases}x_{ij}+|(x_{ij}-E_{j})t(n)|&\text{if }x_{ij}<E_{j}\\ x_{ij}-|(x_{ij}-E_{j})t(n)|&\text{if }x_{ij}>E_{j}\\ x_{ij}+\gamma t(n)&\text{otherwise}\end{cases} \] (17) where \(x_{ij}\) is the \(j\)th dimensional design variable of the ith individual, \(x^{\prime}_{ij}\) is the \(j\)th dimensional design variable of the newly generated \(i\)th individual, \(\gamma\) is a constant and \(E_{j}\) is the \(j\)th dimensional design variable of the situational knowledge. 2. Normative knowledge. The normative knowledge guides the population to search in the dominant region, and the effect of the normative knowledge on the population space under the action of the \(t\)-mutation operator is noted as follows: \[x^{\prime}_{ij}= \begin{cases}x_{ij}+|(u_{j}-l_{j})t(n)|&\text{if }x_{ij}<l_{j}\\ x_{ij}-|(u_{j}-l_{j})t(n)|&\text{if }x_{ij}>u_{j}\\ x_{ij}+\mu(u_{j}-l_{j})t(n)&\text{otherwise}\end{cases} \] (18) where \(\mu\) is a constant, and \(u_{j}\) and \(l_{j}\) are the upper and lower bounds of the \(j\)th dimensional design variables preserved by the normative knowledge of the current generation belief space, respectively. 3. Historical knowledge. Historical knowledge is used to adjust the offset distance and direction when the optimization is trapped in a local optima, and the effect of historical knowledge on the population space under the action of the \(t\)-mutation operator is noted as follows: \[x^{\prime}_{ij}= \begin{cases}ex_{j}+d_{rj}t(n)&\text{for 45\% of the time}\\ ex_{j}+d_{sj}t(n)&\text{for 45\% of the time}\\ random(l^{\prime}_{j},u^{\prime}_{j})&\text{for 10\% of the time}\end{cases} \] (19) where \(ex_{j}\) is the jth dimensional design variable of the best individual ex stored in the historical knowledge and \(u^{\prime}_{j}\) and \(l^{\prime}_{j}\) are the upper and lower bounds, respectively.