## 2 Materials and Methods
### Principles of IDE Algorithm and Multi-Objective Optimization
#### 2.4.1 Differential Evolutionary Algorithm  
DE algorithm, today, acts one of the highly efficient alternatives to optimize complicated multi-objective issues. DE algorithm dives a fast rate of convergence, robustness with a global optimization capability. To generate the vectors of trial parameter, the principle operation of this algorithm consists of the mutation, crossover, and selection operations. The principal operating process of the basic DE has been provided in reference [46]. Of all the operational above-mentioned items, mutation operation can act as the core of DE algorithm. Several and various mutations mechanisms have been presented in the literature review and the frequently utilized five different techniques which are listed in the following equations:  
DE/rand/1:  
\[\mathrm{v}_{\mathrm{i}}^{\mathrm{t}}=\mathrm{x}_{\mathrm{r}_{1}}^{\mathrm{t}}+ \mathrm{F}\times\left(\mathrm{x}_{\mathrm{r}_{2}}^{\mathrm{t}}-\mathrm{x}_{ \mathrm{r}_{3}}^{\mathrm{t}}\right) \tag{14}\]  
DE/rand/2:  
\[\mathrm{v}_{\mathrm{i}}^{\mathrm{t}}=\mathrm{x}_{\mathrm{r}_{1}}^{\mathrm{t}}+ \mathrm{F}\times\left[\left(\mathrm{x}_{\mathrm{r}_{2}}^{\mathrm{t}}-\mathrm{x }_{\mathrm{r}_{3}}^{\mathrm{t}}\right)+\left(\mathrm{x}_{\mathrm{r}_{4}}^{ \mathrm{t}}-\mathrm{x}_{\mathrm{r}_{5}}^{\mathrm{t}}\right)\right] \tag{15}\]  
DE/best/1:  
\[\mathrm{v}_{\mathrm{i}}^{\mathrm{t}}=\mathrm{x}_{\mathrm{best}}^{\mathrm{t}}+ \mathrm{F}\times\left(\mathrm{x}_{\mathrm{r}_{1}}^{\mathrm{t}}-\mathrm{x}_{ \mathrm{r}_{2}}^{\mathrm{t}}\right) \tag{16}\]  
DE/best/2:  
\[\mathrm{v}_{\mathrm{i}}^{\mathrm{t}}=\mathrm{x}_{\mathrm{best}}^{\mathrm{t}}+ \mathrm{F}\times\left[\left(\mathrm{x}_{\mathrm{r}_{1}}^{\mathrm{t}}-\mathrm{x }_{\mathrm{r}_{2}}^{\mathrm{t}}\right)+\left(\mathrm{x}_{\mathrm{r}_{3}}^{ \mathrm{t}}-\mathrm{x}_{\mathrm{r}_{4}}^{\mathrm{t}}\right)\right] \tag{17}\]  
DE/current to best/1:  
\[\mathrm{v}_{\mathrm{i}}^{\mathrm{t}}=\mathrm{x}_{\mathrm{best}}^{\mathrm{t}}+ \mathrm{F}\times\left[\left(\mathrm{x}_{\mathrm{best}}^{\mathrm{t}}-\mathrm{x }_{\mathrm{i}}^{\mathrm{t}}\right)+\left(\mathrm{x}_{\mathrm{r}_{1}}^{\mathrm{t }}-\mathrm{x}_{\mathrm{r}_{2}}^{\mathrm{t}}\right)\right] \tag{18}\]  
where, \(t=1,2,3,\ldots\) represents the generation (time), indices \(r_{i}(i=1,\;2,\ldots,5)\) give the mutually exclusive integers which generated randomly through \(\left[1\,,\mathit{NP}\right]\), that differ from index \(i\); \(x_{\mathit{best}}^{\mathrm{t}}\) represents the best vector that denotes the best amount of fitness of the population in generation \(t\); also \(F\) gives the scale factor. The crossover operation also consists of the binomial crossover approach and the index cross approach. The most commonly utilized bi-nominal crossover operation has been used in this research work. Hence, we should define the following equation as:  
\[\mathrm{u}_{\mathrm{i,\ j}}^{\mathrm{t}}=\left\{ \begin{array}{ccc}\mathrm{v}_{ \mathrm{i,\ j}}^{\mathrm{t}}&\mathrm{if}&(\mathrm{rand}\leq\mathrm{CR})\ \ or\ \ \ \mathrm{j=sn,}\\ &\mathrm{x}_{\mathrm{i,\ j}}^{\mathrm{t}},&\mathrm{otherwise}\end{array} \right. \tag{19}\]  
where \(sn\in[1,n]\) represents a random integer; \(u_{i,\ j}^{\mathrm{t}}\) gives the \(jth\) part of the \(ith\) new trail vector, also \(CR\) denotes the rate of crossover through the range \([0,1]\). In addition, if the experimental vector performs better than the target vector, in the next generation, the experimental vector will be replaced by the greedy selection vector in the selection operation with the target vector. \[\mathrm{x}_{\mathrm{i}}^{\mathrm{t+1}}=\left\{ \begin{array}{ccc}\mathrm{u}_ {\mathrm{i}}^{\mathrm{t}}&\mathrm{if}&(\mathrm{f}(\mathrm{u}_{\mathrm{i}}^{ \mathrm{t}})\ \leq\ \mathrm{f}(\mathrm{x}_{\mathrm{i}}^{\mathrm{t}}))\\ \mathrm{x}_{\mathrm{i}}^{\mathrm{t}},&\mathrm{otherwise}\end{array} \right. \tag{20}\]  
In the proposed strategy, the degree of solution limitation violation \(x\) on the \(jth\) limit can be obtained using Equation (21). \[\mathrm{G}_{\mathrm{j}}(\mathrm{x})=\left\{