## 2 Method and Materials
### Process Modeling and Optimization
#### 2.3.2 Methodology of DE Optimization  
Differential evolution (DE) is a population-based technique that is widely utilized in engineering applications to resolve various optimization challenges [35]. Real-world optimization problems have two fundamental steps: the model construction of the process and solving the optimization model [36]. Optimization problems are always associated with either maximizing or minimizing the effect of the output response factor with input variables. The function can be represented by \(f(x)\) Maximize/Minimize  
The DE algorithm consists of a population of size \(N\), and the population matrix can be shown with Equation (2),  
\[x_{n,\ i}^{g}=\left[x_{n,\ 1}^{g},\ x_{n,\ 2}^{g},\ x_{n,\ 3}^{g},\ \ldots,\ x_{n,\ D,\ }^{g}\right] \tag{2}\]  
where \(g\) is the generation, \(n=1,2,\ 3,\ldots,N\) and \(D\) is the number of variables. The initial data of input variables with response factors are to be gathered from the experimentation. The lower and upper limit bounds of solution search space can be used to produce the initial candidate solutions during the initialization phase. The \(t^{\text{th}}\) chromosome offspring is produced as Equation (3):  
\[ \begin{split} y_{i,j}=x_{i,j}+K\Big{(}x_{best,j}-x_{i,j}\Big{)}+F \big{(}x_{r1,j}-x_{r2,j}\big{)}\\ for\ i=1,2,\ \ldots N\ and\ j=1,\ 2,\ \ldots D\end{split}  \tag{3}\]  
\(K\): crossover scale coefficient  
\(F\): mutationscale coefficient  
\(r_{1}\) and \(r_{2}\): random integers in the range [1:N] such that \(r_{1}\
eq r_{2}\), \(r_{1}\), \(r_{2}\
eq i\) (distinct and not equal to \(i\))  
The selection procedure allows DE to maintain population size in each generation while determining the survival of a target (parent) or a trial (offspring) solution in the subsequent iteration of the search process. Following the formation of the new population in the next generation, repetitive processes of mutation, crossover, and selection are carried out constantly until the termination criteria are satisfied. The selection process of DE is mathematically described as;  
\[x_{i}^{g+1}=\left\{ \begin{array}{l}y_{i}^{g}\ if\ Fit\ \left(y_{i}^{g}\right) \leq Fit\ \left(x_{i}^{g}\right)\\ x_{i}^{g}\ otherwise\end{array} \right. \tag{4}\]  
For, \(i=1,2,\ 3\ldots\ N\)  
Every child will compete against its corresponding parent, and the surviving will all make up the population of parents for the following generation. The fitness value of an individual solution can be determined if the most recent trial vector \(y_{i}^{g}\) creates an improved fitness value, then the present target vector \(x_{i}^{g}\) will be substituted by \(y_{i}^{g}\) in the subsequent iteration. The FE simulations and DE optimization technique has been employed to avail the optimal combinations of control factors, such as billet size ratio (z), reduction ratio (\(r\)), punch angle (\(a\)), and land height (\(h\)) to yield for lower values of the response factors, effective stress, strain, and punch force, while maximizing strain rate. The objective or fitness function (Fit) of the DE optimization technique is to minimize the Fit subjected to \begin{table}
\begin{tabular}{c c} \hline \hline
**Particulars** & **MSE at the End of 2500 Epochs** \\ \hline AISI 1010 model & 0.00804 \\ AISI 1018 model & 0.00679 \\ AISI 1045 model & 0.00847 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean square error (MSE) for different AISI grades steel. inequality constraints \(160\leq a\leq 170\), \(2\leq h\leq 4\) and therefore determined as shown with Equation (5).