## 4 Design Example: The High-Speed Press
### Solution of the HMADM Problem Based on the Proposed HTOPSIS  
According to the normalization approaches for heterogeneous attributes described in Section 3.1, attribute \(a_{1}\) should be normalized with Equation (2), attributes \(a_{2}\), \(a_{3}\) and \(a_{4}\) should be normalized with Equations (5)-(7), attribute \(a_{5}\) should be normalized with Equation (8). After the normalized decision matrix is obtained as listed in Table 4, the weighted normalized decision matrix can be calculated as listed in Table 5. With the PISs and NISs of the five attributes determined as listed in Table 6 according to the corresponding expressions given in Tables 1 and 2, the relative closeness coefficients of all the alternative configurations can be calculated, based on the results of which the alternative configurations can be ranked according to their \(CC_{i}^{*}\) values, see Table 7. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Alternative** & \(a_{1}\) **(mm)** & \(a_{2}\) **(MPa)** & \(a_{3}\) **(Hz)** & \(a_{4}\) **(Hz)** & \(a_{5}\) \\ \hline
1 & 0.134 & [58.22,60.25] & [1.92,5.05] & [38.12,43.43] & E: (0.8,1.0,1.0) \\
2 & 0.145 & [61.21,61.88] & [9.25,13.52] & [51.63,55.05] & VG: (0.6,0.8,0.9) \\
3 & 0.140 & [59.92,60.48] & [3.11,7.82] & [45.86,51.38] & G: (0.3,0.5,7.7) \\
4 & 0.132 & [61.62,62.12] & [18.53,21.98] & [55.86,60.12] & VG: (0.6,0.8,0.9) \\
5 & 0.138 & [58.75,60.38] & [1.28,6.23] & [39.25,46.09] & G: (0.3,0.5,0.7) \\
6 & 0.136 & [61.92,63.25] & [15.32,20.34] & [59.87,61.35] & E: (0.8,1.0,1.0) \\
7 & 0.142 & [59.23,60.12] & [10.11,14.42] & [48.03,53.19] & G: (0.3,0.5,0.7) \\
8 & 0.131 & [57.25,59.34] & [3.22,8.19] & [38.05,41.4] & F: (0.1,0.3,0.4) \\
9 & 0.128 & [56.32,58.67] & [6.38,10.93] & [40.27,45.39] & VG: (0.6,0.8,0.9) \\ \hline \end{tabular}
\end{table} 
Table 3: The attribute data for nine alternative structural configurations. As can be seen from Table 7, the 4 th design has the maximum relative closeness coefficient of 0.7072, therefore it is chosen as the optimal configuration for the high-speed press. The 9 th design is the worst, since it has the minimum relative closeness coefficient of 0.3428. Figure 4 is the high-speed press developed based on the 4 th configuration, all the performance indices of which satisfy customer requirements pretty well. The developed high-speed press has been successfully put into production, demonstrating the feasibility and validity of the proposed HTOPSIS in the structural optimization of high-speed press. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Alternative** & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{a_{3}}\) & \(\mathbf{a_{4}}\) & \(\mathbf{a_{5}}\) \\ \hline \hline
1 & 0.0132 & [\(-\)0.0081,0.0904] & [\(-\)0.0105,0.0189] & [0.0434,0.0784] & (0.0755,0.0944,0.0944) \\
2 & 0.0154 & [0.0068,0.0612] & [0.0266,0.0618] & [0.0108,0.0333] & (0.0567,0.0755,0.0850) \\
3 & 0.0144 & [\(-\)0.0156,0.0351] & [\(-\)0.0045,0.0329] & [\(-\)0.0091,0.0273] & (0.0283,0.0472,0.0661) \\
4 & 0.0128 & [0.0202,0.0690] & [0.0737,0.1047] & [0.0387,0.0668] & (0.0567,0.0755,0.0850) \\
5 & 0.014 & [\(-\)0.0124,0.0732] & [\(-\)0.026,0.0248] & [0.0258,0.0710] & (0.0283,0.0472,0.0661) \\
6 & 0.0136 & [0.0299,0.1057] & [0.0574,0.0964] & [0.0652,0.0749] & (0.0755,0.0944,0.0944) \\
7 & 0.0148 & [\(-\)0.0039,0.0576] & [0.0310,0.0664] & [\(-\)0.0130,0.0211] & (0.0283,0.0472,0.0661) \\
8 & 0.0126 & [0.0215,0.1220] & [\(-\)0.0040,0.0348] & [0.0387,0.0789] & (0.0094,0.0283,0.0378) \\
9 & 0.012 & [0.0433,0.1523] & [0.0121,0.0487] & [0.0304,0.0642] & (0.0567,0.0755,0.0850) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Ideal solutions corresponding to the five attributes.