## 5 A Real Case Study
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline  & **a1** & **a2** & **a3** & **b1** & **b2** & **b3** & **b4** & **c1** & **c2** & **c3** & **d1** \\ \hline c(X1 \(\geq\) X2) & 0.00 & 0.00 & 1.00 & 0.00 & 1.00 & 0.62 & 0.43 & 0.24 & 0.00 & 0.46 & 0.83 \\ c(X1 \(\geq\) X3) & 0.05 & 0.00 & 1.00 & 0.36 & 0.00 & 0.92 & 0.74 & 0.20 & 0.00 & 0.50 & 0.61 \\ c(X1 \(\geq\) X4) & 0.00 & 0.00 & 1.00 & 0.00 & 0.93 & 0.68 & 0.74 & 0.19 & 0.00 & 0.41 & 0.45 \\ c(X2 \(\geq\) X1) & 0.26 & 0.85 & 0.00 & 0.70 &  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**d(Xi \(\geq\) Xj)** & **a1** & **a2** & **a3** & **b1** & **b2** & **b3** & **b4** & **c1** & **c2** & **c3** & **d1** \\ \hline d(X1 \(\geq\) X2) & 0.00 & 0.00 & 1.00 & 0.00 & 0.91 & 0.47 & 0.34 & 0.21 & 0.00 & 0.36 & 0.61 \\ d(X1 \(\geq\) X3) & 0.08 & 0.00 & 1.00 & 0.30 & 0.02 & 0.68 & 0.56 & 0.19 & 0.00 & 0.39 & 0.46 \\ d(X1 \(\geq\) X4) & 0.00 & 0.00 & 0.88 & 0.00 & 0.68 & 0.51 & 0.56 & 0.17 & 0.00 & 0.33 & 0.35 \\ d(X2 \(\geq\) X1) & 0.22 & 0.63 & 0.00 & 0.52 & 0.00 & 0.00 & 0.00 & 0.33 & 0.00 & 0.00 \\ d(X2 \(\geq\) X2) & 0.28 & 0.11 & 0.12 & 0.80 & 0.00 & 0.23 & 0.24 & 0.00 & 0.17 & 0.05 & 0.00 \\ d(X2 \(\geq\) X3) & 0.00 & 0.00 & 0.00 & 0.37 & 0.00 & 0.06 & 0.24 & 0.00 & 0.00 & 0.00 & 0.00 \\ d(X3 \(\geq\) X1) & 0.00 & 0.54 & 0.00 & 0.00 & 0.02 & 0.00 & 0.00 & 0.00 & 0.18 & 0.00 & 0.00 \\ d(X3 \(\geq\) X2) & 0.00 & 0.00 & 0.00 & 0.00 & 0.91 & 0.00 & 0.00 & 0.05 & 0.00 & 0.00 & 0.17 \\ d(X3 \(\geq\) X4) & 0.00 & 0.00 & 0.00 & 0.00 & 0.68 & 0.00 & 0.02 & 0.01 & 0.00 & 0.00 & 0.00 \\ d(X4 \(\geq\) X1) & 0.35 & 0.72 & 0.00 & 0.18 & 0.00 & 0.00 & 0.00 & 0.00 & 0.68 & 0.00 & 0.00 \\ d(X4 \(\geq\) X2) & 0.15 & 0.12 & 0.20 & 0.00 & 0.25 & 0.00 & 0.00 & 0.06 & 0.37 & 0.06 & 0.29 \\ d(X4 \(\geq\) X3) & 0.41 & 0.21 & 0.30 & 0.45 & 0.00 & 0.19 & 0.02 & 0.03 & 0.52 & 0.08 & 0.13 \\ \hline
**d(Xi \(\geq\) Xj)** & **d2** & **d3** & **d4** & **e1** & **e2** & **e3** & **f1** & **f2** & **f3** & **f4** \\ \hline d(X1 \(\geq\) X2) & 0.67 & 0.78 & 0.00 & 0.55 & 0.00 & 0.13 & 0.00 & 0.06 & 0.00 & 0.00 & \\ d(X1 \(\geq\) X3) & 0.89 & 0.96 & 0.00 & 0.58 & 0.00 & 0.36 & 0.06 & 0.29 & 0.00 & 0.00 & \\ d(X1 \(\geq\) X4) & 0.30 & 0.43 & 0.00 & 0.14 & 0.00 & 0.41 & 0.00 & 0.00 & 0.00 & 0.00 & \\ d(X2 \(\geq\) X1) & 0.00 & 0.00 & 0.35 & 0.00 & 0.27 & 0.00 & 0.23 & 0.00 & 0.52 & 0.19 \\ d(X2 \(\geq\) X2) & 0.24 & 0.20 & 0.00 & 0.05 & 0.00 & 0.25 & 0.26 & 0.25 & 0.00 & 0.00 \\ d(X2 \(\geq\) X3) & 0.00 & 0.00 & 0.21 & 0.00 & 0.00 & 0.31 & 0.09 & 0.00 & 0.41 & 0.00 \\ d(X3 \(\geq\) X1) & 0.00 The ranking order using ELECTRI III is X1 \(>\) X2 \(>\) X3 \(>\) X4. In order to check the validity of the results, TOPSIS and VIKOR methods were used to reorder the alternative SMPV plans as shown in Table 2.3 The result obtained by TOPSIS method is X1 \(>\) X4 \(>\) X3 \(>\) X2, while the result achieved by VIKOR method is X1 \(>\) X4 \(>\) X3 \(>\) X2. from these three rankings, alternative plan X1 is the optimal selection, no matter what method was used. That is to say, the alternative X1 is much better than the remaining alternatives, and there is no doubt in choosing X1 first. However, the rankings for X2, X3, and X4 are different between these three methods. There is the veto threshold, which indicates when the value of alternative Xi is lower than the value of alternative Xj, and the lower value exceeds the veto threshold, and it is not recognized that Xi is preferred to the Xi in general. However, it is the other term in TOPSIS method and VIKOR method, and some really bad performance in a certain criterion can be tolerated and remedied by other good performances in other criteria. In that case, an alternative with some fatal defect in a certain criterion of an alternative may be neglected, which leads to an unsatisfactory selection. When an alternative is vetoed better than the other alternative in ELECTRE III, it can still come out in front in the TOPSIS and VIKOR methods. That why the X2, X3, X4 ranked differently in TOPSIS and VIKOR methods.