## 5 Conclusions  
The structural optimization problem of a high-speed press with multi-source uncertainties was defined as a HMADM model involving deviation, fixation, benefit and cost attributes. Attribute values can be described as real numbers, interval numbers, fuzzy linguistic terms and TFNs. To solve such a HMADM problem, a new HTOPSIS was proposed, which realized the normalization of heterogeneous attributes based on the possibility degree method for comparing interval numbers, the relative preference relation for comparing TFNs, and the attribute transformation technique for handling deviation and fixation attributes. Equations for determining the positive and negative ideal solutions of heterogeneous attributes and the ranking method of different alternative designs were also presented.  
A numerical example was utilized to illustrate the HMADM modeling process of the structural optimization problem and the implementation of the proposed HTOPSIS for solving the HMADM model. A cost attribute described as a real number, a fixation attribute described as an interval number, two deviation attributes described as interval numbers and a benefit attribute described as a fuzzy linguistic term were chosen for evaluating nine alternative structural configurations of an ultra-precision high-speed press with double drivers. The 4 th structural configuration, with a maximum deformation of the press slider of 0.132 mm, a maximum equivalent stress of the upper beam of [61.62,62.12] MPa, 1 st and 2 nd natural frequencies of the press slider of [18.53,21.98] Hz and [55.86,60.12] Hz, and a very good manufacturability of whole press, was chosen as the optimal configuration according to the HMADM results. The successful use of the high-speed press developed from the 4 th structural configuration demonstrated the validity and effectiveness of the proposed HTOPSIS in solving the HMADM problem with deviation and fixation attributes. The advantages of the proposed HTOPSIS over HAD were also discussed, which included its universal applicability in solving HMADM problems and its efficient ranking of different alternatives.  
This work was supported by the National Natural Science Foundation of China (No. 51775491), Science Fund for Creative Research Groups of National Natural Science Foundation of China (No. 51521064), National Key Research and Development Project of China (No. 2017 YFB0603704) and Key  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Alternative** & \(a_{1}\) & \(a_{2}\) & \(a_{3}\) & \(a_{4}\) & \(a_{5}\) & **Total Information Content** & **Ranking** \\ \hline
1 & 0.0623 & 0.1308 & 0.6580 & 0.0248 & 0.0184 & 0.8943 & 5 \\
2 & 0.0678 & 0.1232 & 0.1513 & 0.2330 & 0.0729 & 0.6482 & 3 \\
3 & 0.0653 & 0.0320 & 0.3886 & 0.3012 & 0.1854 & 0.9723 & 7 \\
4 & 0.0613 & 0.1760 & 0.0000 & 0.0567 & 0.0729 & 0.3669 & 1 \\
5 & 0.0643 & 0.0960 & 0.5426 & 0.0657 & 0.1854 & 0.9540 & 6 \\
6 & 0.0633 & 0.3011 & 0.0317 & 0.0000 & 0.0184 & 0.4145 & 2 \\
7 & 0.0663 & 0.0913 & 0.1306 & 0.3415 & 0.1854 & 0.8150 & 4 \\
8 & 0.0608 & 0.2749 & 0.3701 & 0.0301 & 0.3528 & 1.0887 & 9 \\
9 & 0.0593 & 0.3321 & 0.2314 & 0.0763 & 0.0729 & 0.7720 & 8 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Ranking of the nine alternative structural configurations obtained by heterogeneous axiomatic design (HAD).  
Laboratory of Micro-systems and Micro-structures Manufacturing of Ministry of Education, Harbin Institute of Technology (No. 2015 KM001).  
[PERSON] and [PERSON] proposed and realized the HTOPSIS method for the structural optimization of a high-speed press considering multi-source uncertainties, analyzed the data and wrote the paper; [PERSON] and [PERSON] provided professional guidance on the paper writing; [PERSON] provided professional guidance on the work reported.  
The authors declare no conflict of interest.