# Multi-Criteria Decision Making in the PMEDM Process by Using MARCOS, TOPSIS, and MAIRCA Methods
## 5 Multi-Criteria Decision Making
### Results and Remarks  
Table 6 presents the ranking results of options when applying three methods, MARCOS, TOPSIS, and MAIRCA. Moreover, Figure 2 shows a chart used for comparing the results of MCDM by using different methods. The vertical axis represents the values of the quantities when ranking the alternatives by different methods. Specifically, these methods include \(f(K_{i})\) (when using the MARCOS method), \(\text{R}_{\text{i}}\) (the TOPSIS method), and \(\text{Q}_{\text{i}}\) (the MAIRCA method). From the results, the following observations are proposed:  
1. The ranking order of alternatives is different when using the three methods, MARCOS, TOPSIS, and MAIRCA. 2. All the three above-mentioned methods lead to the same result, i.e., A13 is the best option, which indicates that determining the best alternative does not depend on the decision-making method used. This observation is also consistent with the results when applying these MCDM methods to the turning process [6]. 3. The MAIRCA and TOPSIS methods have up to 14/18 alternatives rated the same (except for options A5, A7, A8, and A18), which proves that these two methods have given quite similar results and can be used interchangeably. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Trial**} & \multicolumn{3}{c}{\(\text{k}_{\text{ij}}\)} & \multicolumn{3}{c}{\(\text{l}_{\text{ij}}\)} & \multirow{2}{*}{\(\text{D}_{\text{i}}\text{}^{+}\)} & \multirow{2}{*}{\(\text{D}_{\text{i}}\text{}^{-}\)} & \multirow{2}{*}{\(\text{R}_{\text{i}}\)} & \multirow{2}{*}{**Rank**} \\ \cline{2-2} \cline{6-9}  & **Ra** & & & & & & & \\ \hline A1 & 0.1248 & 0.0492 & 0.0710 & 0.0212 & 0.1851 & 0.1971 & 0.5157 & 11 \\ A2 & 0.1790 & 0.0904 & 0.1019 & 0.0390 & 0.1725 & 0.1672 & 0.4922 & 12 \\ A3 & 0.4710 & 0.2617 & 0.2681 & 0.1127 & 0.2296 & 0.0916 & 0.2851 & 17 \\ A4 & 0.1265 & 0.0538 & 0.0720 & 0.0232 & 0.1832 & 0.1961 & 0.5171 & 10 \\ A5 & 0.3008 & 0.4756 & 0.1712 & 0.2049 & 0.1130 & 0.2077 & 0.6477 & 5 \\ A6 & 0.4237 & 0.0547 & 0.2412 & 0.0236 & 0.2582 & 0.0271 & 0.0948 & 18 \\ A7 & 0.1202 & 0.0941 & 0.0684 & 0.0405 & 0.1656 & 0.2006 & 0.5478 & 8 \\ A8 & 0.1694 & 0.2570 & 0.0964 & 0.1107 & 0.1025 & 0.1936 & 0.6540 & 4 \\ A9 & 0.2790 & 0.2968 & 0.1588 & 0.1279 & 0.1272 & 0.1527 & 0.5456 & 9 \\ A10 & 0.2181 & 0.4689 & 0.1242 & 0.2020 & 0.0660 & 0.2311 & 0.7779 & 3 \\ A11 & 0.2960 & 0.1308 & 0.1685 & 0.0563 & 0.1857 & 0.1057 & 0.3626 & 16 \\ A12 & 0.2276 & 0.0961 & 0.1296 & 0.0414 & 0.1792 & 0.1400 & 0.4386 & 15 \\ A13 & 0.1754 & 0.4777 & 0.0999 & 0.2058 & 0.0416 & 0.2498 & 0.8573 & 1 \\ A14 & 0.2522 & 0.1567 & 0.1436 & 0.0675 & 0.1625 & 0.1329 & 0.4498 & 14 \\ A15 & 0.1779 & 0.0840 & 0.1012 & 0.0362 & 0.1750 & 0.1675 & 0.4891 & 13 \\ A16 & 0.1024 & 0.1405 & 0.0583 & 0.0605 & 0.1453 & 0.2135 & 0.5951 & 6 \\ A17 & 0.2180 & 0.4780 & 0.1241 & 0.2059 & 0.0658 & 0.2343 & 0.7807 & 2 \\ A18 & 0.1805 & 0.1725 & 0.1027 & 0.0743 & 0.1388 & 0.1737 & 0.5559 & 7 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculated results and ranking of alternatives by TOPSIS method. * To achieve \"minimum\" Ra and \"maximum\" MRS simultaneously, the proposed optimum process parameters include the following values: C\({}_{\rm p}\) = 0.5 (g/l), T\({}_{\rm on}\) = 8 (\(\mu\)s), T\({}_{\rm Toff}\) = 12 (\(\mu\)s), IP = 15 (A), and SV = 5 (V). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Trial**} & \multicolumn{3}{c}{**Rank**} \\ \cline{2-4}  & **MARCOS** & **TOPSIS** & **MAIRCA** \\ \hline A1 & 8 & 11 & 11 \\ A2 & 12 & 12 & 12 \\ A3 & 15 & 17 & 17 \\ A4 & 9 & 10 & 10 \\ A5 & 5 & 5 & 4 \\ A6 & 18 & 18 & 18 \\ A7 & 7 & 8 & 7 \\ A8 & 6 & 4 & 5 \\ A9 & 11 & 9 & 9 \\ A10 & 4 & 3 & 3 \\ A11 & 17 & 16 & 16 \\ A12 & 16 & 15 & 15 \\ A13 & 1 & 1 & 1 \\ A14 & 14 & 14 & 14 \\ A15 & 13 & 13 & 13 \\ A16 & 3 & 6 & 6 \\ A17 & 2 & 2 & 2 \\ A18 & 10 & 7 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Ranking of alternatives by different methods of multi-criteria decision making.