## 3 HTOPSIS for Solving the HMADM Model with Deviation and Fixation Attributes
### Ranking of the Alternative Structural Configurations  
Once PISs and NISs for the heterogeneous attributes in the HMADM model have been determined from Tables 1 and 2, the distances of the \(i\)th alternative configuration deviating from PIS and NIS can be calculated as  
\[d_{i}^{*}=\sqrt{\sum\limits_{j=1}^{n}\left(d_{ij}^{*}\right)^{2}}\ (i=1,2,3, \cdots,m), \tag{12}\]  
\[d_{i}^{-}=\sqrt{\sum\limits_{j=1}^{n}\left(d_{ij}^{-}\right)^{2}}\ (i=1,2,3, \cdots,m), \tag{13}\]  
where \(d_{ij}^{*}=\left|g_{ij}-g_{j}^{*}\right|\), \(d_{ij}^{-}=\left|g_{ij}-g_{j}^{-}\right|\) when the \(j\)th attribute value is described as a real number; when the \(j\)th attribute value is described as an interval number; and \(d_{ij}^{-}=\sqrt{\left(\left(g_{ij}^{L}-g_{j}^{L}\right)^{2}+\left(g_{ij}^{M} -g_{j}^{M}\right)^{2}+\left(g_{ij}^{U}-g_{j}^{U}\right)^{2}\right)/3}\) when the \(j\)th attribute is described as a TFN.  
Then the relative closeness coefficient of the \(i\)th alternative structural configuration is determined as  
\[CC_{i}^{*}=\frac{d_{i}^{-}}{d_{i}^{*}+d_{j}^{-}}\ (i=1,2,3,\ldots,m). \tag{14}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Attributes** & **Real Number** & **Interval Number** & **TFN** \\  & \(g_{j}^{*}\) & \(g_{j}^{*}=\left[g_{j}^{L},g_{j}^{U}\right]\) & \(g_{j}^{*}=\left(g_{j}^{L},g_{j}^{M*},g_{j}^{U*}\right)\) \\ \hline Benefit attribute & \(\max_{i}(g_{ij})\) & \(\left[\max_{i}(g_{ij}^{L}),\max_{i}(g_{ij}^{U})\right]\) & \(\left(\max_{i}(g_{ij}^{L}),\max_{i}(g_{ij}^{M}),\max_{i}(g_{ij}^{U})\right)\) \\ Cost attribute & \(\min_{i}(g_{ij})\) & \(\left[\min_{i}(g_{ij}^{L}),\min_{i}(g_{ij}^{U})\right]\) & \(\left(\min_{i}(g_{ij}^{L}),\min_{i}(g_{ij}^{M}),\min_{i}(g_{ij}^{U})\right)\) \\ Deviation attribute & \(\max_{i}(g_{ij})\) & \(\left[\max_{i}(g_{ij}^{L}),\max_{i}(g_{ij}^{U})\right]\) & \(\left(\max_{i}(g_{ij}^{L}),\max_{i}(g_{ij}^{U}),\max_{i}(g_{ij}^{U})\right)\) \\ Fixation attribute & 0 & [0,0] & (0,0,0) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Determination of positive ideal solutions (PISs) for heterogeneous attributes in different mathematical descriptions. TFN: triangular fuzzy number.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Attributes** & **Real Number** & **Interval Number** & **TFN** \\  & \(g_{j}^{-}\) & \(g_{j}^{-}=\left[g_{j}^{L},g_{j}^{U}\right]\) & \(g_{j}^{-}=\left(g_{j}^{L},g_{j}^{M-},g_{j}^{U}\right)\) \\ \hline Benefit attribute & \(\min_{i}(g_{ij})\) & \(\left[\min_{i}(g_{ij}^{L}),\min_{i}(g_{ij}^{U})\right]\) & \(\left(\min_{i}(g_{ij}^{L}),\min_{i}(g_{ij}^{M}),\min_{i}(g_{ij}^{U})\right)\) \\ Cost attribute & \(\max_{i}(g_{ij})\) & \(\left[\max_{i}(g_{ij}^{L}),\max_{i}(g_{ij}^{U})\right]\) & \(\left(\max_{i}(g_{ij}^{L}),\max_{i}(g_{ij}^{M}),\max_{i}(g_{ij}^{U})\right)\) \\ Deviation attribute & 0 & [0,0] & (0,0,0) \\ Fixation attribute & \(\max_{i}(g_{ij})\) & \(\left[\max_{i}(g_{ij}^{L}),\max_{i}(g_{ij}^{U})\right]\) & \(\left(\max_{i}(g_{ij}^{L}),\max_{i}(g_{ij}^{M}),\max_{i}(g_{ij}^{U})\right)\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Determination of negative ideal solutions (NISs) for heterogeneous attributes in different mathematical descriptions.  
Finally, the alternative structural configurations can be ranked according to their corresponding values of \(CC_{i}^{*}\) and the design corresponding to the largest \(CC_{i}^{*}\) value is selected as the optimal.