# Multi-Criteria Decision Making in the PMEDM Process by Using MARCOS, TOPSIS, and MAIRCA Methods
## 2 Methods of MCDM
### MAIRCA Method  
To apply the MAIRCA method, it is necessary to perform the following steps [17]:  
Step 1: Forming the initial matrix in the same way as in the MARCOS method.  
Step 2: Calculating the preferences according to alternative selection \(P_{A_{j}}\). It is assumed that there is no preference for alternatives. Then, the priority for the criteria remains the same and it can be found as follows:  
\[P_{A_{j}}=\frac{1}{m},\ j=1,\ 2,\ \ldots,\ n \tag{19}\]  
Step 3: Determining the elements \(t_{pij}\) of the theoretical rating matrix by:  
\[t_{pij}=P_{A_{j}}\cdot w_{j},\ i=1,\ 2,\ \ldots,\ m;j=1,\ 2,\ \ldots \tag{20}\]  
in which \(w_{j}\) is the weight of the \(j\)th criterion.  
Step 4: Determining the elements of real rating matrix \(t_{rij}\) by:  
\[t_{rij}=t_{pij}\cdot\left(\frac{x_{ij}-x_{i}^{-}}{x_{i}^{+}-x_{i}^{-}}\right)\ \text{if the criterion $j$ is bigger is better.} \tag{21}\]  
\[t_{rij}=t_{pij}\cdot\left(\frac{x_{ij}-x_{i}^{+}}{x_{i}^{-}-x_{i}^{+}}\right) \ \text{if the criterion $j$ is smaller is better.} \tag{22}\]  
Step 5: Calculating the total gap matrix \(g_{ij}\) by using the following equation:  
\[g_{ij}=t_{p_{ij}}-t_{r_{ij}} \tag{23}\]Step 6: Determining the final values of the criterion functions (\(Qi\)) using the alternatives. These values are found by the sum of the gaps (\(g_{ij}\)) according to alternatives or by the following formula:  
\[Q_{i}=\sum_{i=1}^{m}g_{ij} \tag{24}\]  
**3. Calculating the Weights of Criteria**  
In this work, the MEREC method was employed to find the weights of the criteria through the following steps [26]:  
Step 1: Forming the initial matrix as in the MARCOS method.  
Step 2: Determining the normalized matrix elements by:  
\[h_{ij}=\frac{minx_{ij}}{x_{ij}}\text{ if the criterion }j\text{ is bigger is better.} \tag{25}\]  
\[h_{ij}=\frac{x_{ij}}{maxx_{ij}}\text{ if the criterion }j\text{ is smaller is better.} \tag{26}\]  
Step 3: Finding the performance of the alternatives \(S_{i}\) according to the following equation:  
\[S_{i}=ln\left[1+\left(\frac{1}{n}\sum_{j}\left|ln\left(h_{ij}\right)\right| \right)\right] \tag{27}\]  
Step 4: Calculating the performance of \(i\)th alternative \(S^{\prime}_{ij}\) concerning the removal of the \(j\)th criterion by using the following equation:  
\[S^{\prime}_{ij}=Ln\left[1+\left(\frac{1}{n}\sum_{k,\ k\
eq j}\left|ln\left(h _{ij}\right)\right|\right)\right] \tag{28}\]  
Step 5: Calculating the removal effect of the \(j\)th criterion \(E_{j}\) by:  
\[E_{j}=\sum_{i}\left|S^{\prime}_{ij}-S_{i}\right| \tag{29}\]  
Step 6: Determining the weight for the criteria by using the following equation:  
\[w_{j}=\frac{E_{j}}{\sum_{k}E_{k}} \tag{30}\]  
**4. Experimental Setup**  
To solve the proposed MCDM problem, an experiment was performed. The input process parameters of this experiment are given in Table 1. In addition, the Taguchi method with L18 (\(2^{1}+3^{4}\)) design was chosen for the experiment. Figure 1 depicts the experimental setup using a Sodic A30 EDM machine (Japan), graphite electrodes (TOKAI Carbon Co., LTD, Tokyo, Japan), workpieces with 90 CrSi tool steel (China), 100 nm SiC powder (China), and Total Diel MS 7000 dielectric solution (France). After conducting experiments, the surface roughness (Ra) was measured, and the material-removal speed (MRS) was calculated. The experimental plan, the measurement values of workpieces SR, and the calculated values of the material-removal speed (MRS) (the average result of three measurements) are given in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{5}{c}{**Input Parameters**} & \multicolumn{2}{c}{**Output Results**} \\ \cline{2-7}  & \(\mathbf{C_{p}}\) & \(\mathbf{S_{p}}\) & \(\mathbf{T_{on}}\) & \(\mathbf{T_{off}}\) & \(\mathbf{SV}\) & \(\mathbf{Ra(\mu m)}\) & \(\mathbf{MRS(g/h)}\) \\ \hline  
\hline \end{tabular}
\end{table}
Table 2: Experimental plan and output results.  
Figure 1: Experimental setup.