## 2 Materials and Methods
### Principles of IDE Algorithm and Multi-Objective Optimization
#### 2.4.2 Improved Self-Adaptive Differential Evolutionary Algorithm  
In this section, the proposed improved IDE has been introduced and the adaption mechanism has been described. Adaptive Regulation Mechanisms for Scale Factor and Crossover Rate  
Selecting \(F\) and \(CR\) depends highly on the specified issue that was applied, and that is almost difficult. Overall, small values of \(F\) and \(CR\) lead to a faster convergence speed, whereas large values of \(F\) and \(CR\) contribute to rise the population diversity and can prevent premature convergence. In order to obtain a balance from the speed of convergence to premature convergence, having an adaptive regulation mechanism for \(F\) and \(CR\) will be essential. So, to develop the global searching ability with escaping the local optimal and preventing the premature convergence, an improved approach and technique is proposed to develop the total efficiency of the IMODE algorithm in this research. In the proposed algorithm, a heuristic searching optimization approach according to the basics of genetics and immune adaptive (IA) method is proposed. During the process of population search, IA maintains the potential solutions populations. As a result, in order to reach the parameters of the optimal scale factor, the principles of immune approach are used. The strategies of parameter F for updating are adjusted as the following steps:  
1. Computing values of affinity: IA utilizes the affinity as a differentiator between the qualities of solutions; besides, antibodies with a higher affinity are more likely to be opted, and at the higher probability, are more likely to be surviving for the next generation. Hence, we describe the affinity in Equation (23). \[\mathrm{affinity}_{\mathrm{i,t}}=\mathrm{r}\times(1-\mathrm{r})^{\mathrm{i-1}} \tag{23}\]where, the coefficient \(r\in[0,\,0.5]\); \(i\) also by arranging them based on their performance values, shows the numerical order of every antibody in the population. In order to calculate the values of affinity for the parameters of scale factor, we define Equation (24) as follows:  
\[\mathrm{A}_{(\mathrm{F}_{\mathrm{i}})}=\left\{ \begin{array}{c}\left|\frac{ \mathrm{f}\left(\mathrm{x}_{\mathrm{i}}^{+}-\mathrm{f}\left(\mathrm{x}_{ \mathrm{i}}^{+}\right)\right)}{\mathrm{f}\left(\mathrm{x}_{\mathrm{i}}^{+} \right)}\right|,\,f\left(\mathrm{x}_{\mathrm{i}}^{+}\right)<f(\mathrm{x}_{ \mathrm{i}}^{\mathrm{t}})\\ 0,\text{ otherwise}\end{array} \right. \tag{24}\]  
Based on their affinity, the parameters of scale factor should be arranged between the best and the worst, where the best twenty percent mutation parameters \(MF^{T+1}\) (new) have been stored in the memory based on MF. 1. Updating the MF memory based on MF: \[\mathrm{MF}_{\mathrm{j}}^{T+1}=\left\{ \begin{array}{c}\mathrm{MF}_{ \mathrm{j}+\alpha\mathrm{N}^{\prime}}^{T+1},\,\mathrm{j}\leq\mathrm{N}- \alpha\mathrm{N}\\ \mathrm{MF}_{\mathrm{j}+\alpha\mathrm{N}-\mathrm{N}^{\prime}}^{T+1}(\mathrm{ new}),\text{ otherwise}\end{array} \right.\] (25)
2. Generating the scale factor parameters for the next generation. Now, the scale factor parameters in the next generation should be updated as follows: \[\mathrm{F}_{\mathrm{j}}^{T+1}=\left\{ \begin{array}{c}\mathrm{MF}_{\mathrm{j} }^{T+1},\,\mathrm{P}_{\mathrm{i}}\leq\mathrm{P}_{\mathrm{F}}\\ \mathrm{\tilde{F}}_{\mathrm{j}}^{T+1},\text{ otherwise}\end{array} \right.\] (26)  
where the initial \(F_{j}^{T+1}\) has been generated by a normal distribution \(N(0.5,1)\) through \([0,1]^{2}\), \(P_{i}\) represents a random number in the distance within [0, 1], and \(P_{F}\) represents the regulating probability. Considering these items and based on the reference [47], the crossover rate (\(CR\)) values have been updated as:  
\[\mathrm{CR}_{\mathrm{i}}^{t+1}=\left\{ \begin{array}{cc}\mathrm{rand}_{3}, \,\,\,\mathrm{if}&\mathrm{rand}_{4}<\tau\\ \mathrm{CR}_{\mathrm{i}}^{t},&\mathrm{otherwise}\end{array} \right. \tag{27}\]  
where, \(\tau=0.9\) represents the adjusting probability in our proposed method.