# Parametric Investigation of Die-Sinking EDM of Ti6 Al4V Using the Hybrid Taguchi-Rams-RATMI Method
## 3 RAMS-RATMI Method  
[PERSON] et al. [36] developed the RAMS-RATMI method in 2022 to address complex engineering issues. First, options are divided into advantageous and non-benefit categories using the RAMS approach, which is an extension of the ranking alternatives by perimeter similarity (RAPS) method. By comparing the perimeter of the triangle formed by the advantageous components with the ideal alternative, the optimal alternative is narrowed down. By comparing the triangle's median to the best option, RAMS rates the alternatives [37]. [PERSON] et al. [38] introduced the RATMI method, which is an expansion of the RAMS and multiple criteria ranking by alternative trace (MCRAT) techniques. The majority index is a novel metric for rating alternatives that is created by combining the alternative trace and the median similarity of the alternatives. Step 1: A relevant decision matrix is created with'm' no. of experimental runs and 'n' no. of responses:  
\[M= \begin{bmatrix}x_{11}&x_{12}&\dots&x_{1n}\\ x_{21}&x_{22}&\dots&x_{2n}\\ \dots&\dots&\dots&\dots\\ x_{m1}&x_{m2}&\dots&x_{mn}\end{bmatrix}  \tag{1}\]  
The value that was obtained of the 'j-th' response during the 'i-th' experimental trial is denoted by 'x\({}_{\text{ij}}\)'. Step 2: Due to the multidimensionality of the problem data, making decisions is difficult. The multidimensional decision space must be converted into a nondimensional space in order to prevent issues. It is best to maximize certain characteristics and minimize  
Figure 3: Surface roughness measurement. others. The answer values in the original choice matrix could have varied scales and dimensions, thus two equations must be used to convert them into dimensionless elements:  
\[r_{ij}=f(x)=\left\{ \begin{array}{ll}\frac{x_{ij}}{max_{ij}},&i=1,2,\ldots,m\ ( for\ maximization)\\ \frac{min_{ij}}{x_{ij}},&i=1,2,\ldots,m\ (for\ minimization)\end{array} \right. \tag{2}\]  
Step 3: The weighted normalized decision matrix is created by constructing a decision matrix based on the normalized decision matrix and its corresponding criteria weights. \[U=\left[u_{ij}\right]_{m\times m}=w_{j}r_{ij};\ \left\{ \begin{array}{ll}i=1,2, \ldots,m\\ j=1,2,\ldots,n\end{array} \right. \tag{3}\]  
where '\(w_{i}\)' is the weight of criteria \(j\)', which can be ascertained using the AHP approach or by consulting a panel of experts. Each weight must add up to a total of one. Step 4: The optimal alternative can be determined by analyzing each component of the solution:  
\[q_{ij}=max\big{(}u_{ij}\big{)};\ \left\{ \begin{array}{ll}i=1,2,\ldots,m\\ j=1,2,\ldots,n\end{array} \right. \tag{4}\]  
The optimal alternative is represented by the following set:  
\[Q=\left\{q_{1},q_{2},\ldots,q_{n}\right\} \tag{5}\]  
Step 5: The optimal alternative can be decomposed into two sets or two components:  
\[Q=Q^{max}\cup Q^{min} \tag{6}\]  
\[Q=\left\{q_{1},q_{2},\ldots,q_{k}\right\}\cup\left\{q_{1},q_{2},\ldots,q_{h} \right\};k+h=j \tag{7}\]  
where 'k' denotes the total number of criteria that must be maximized and 'h' the total number of criteria that mustbe minimized. Step 6: The experimental trials are further divided into two sets or components, following the same process as the previous step:  
\[U_{i}=U_{i}^{max}\cup U_{i}^{min};i=1,2,\ldots,m \tag{8}\]  
\[U_{i}=\left\{U_{i1},U_{i2},\ldots,U_{ik}\right\}\cup\left\{U_{i1},U_{i2}, \ldots,U_{ih}\right\};i=1,2,\ldots,m \tag{9}\]  
Step 7: The formula may be used to determine each component of the ideal alternative's magnitude:  
\[Q_{k}=\sqrt{Q_{1}^{2}+Q_{2}^{2}+\ldots+Q_{k}^{2}} \tag{10}\]  
\[Q_{h}=\sqrt{Q_{1}^{2}+Q_{2}^{2}+\ldots+Q_{h}^{2}} \tag{11}\]  
Similarly, it may be calculated as follows for every experimental trial:  
\[U_{ik}=\sqrt{u_{i1}^{2}+u_{i2}^{2}+\ldots+u_{ik}^{2}};i=1,2,\ldots,m \tag{12}\]  
\[U_{ih}=\sqrt{u_{i1}^{2}+u_{i2}^{2}+\ldots+u_{ih}^{2}} \tag{13}\]  
Step 8: MCRAT (ranking by alternatives trace). Here, matrix 'F' with the best possible alternative components is constructed:  
\[F=\left[ \begin{matrix}Q_{k}&0\\ 0&Q_{h}\end{matrix}