## 1 Introduction
The strength indicated by maximum equivalent stress can be regarded as a fixation attribute, since the optimization of the other mechanical performance indices usually moves its value to the allowable stress, which can be described as an interval number considering the uncertain material properties and manufacturing errors of the key components [20]. The good operational reliability of the high-speed press can be achieved by moving the natural frequencies of its key components away from the punching and working frequencies [21]. Thus, the natural frequencies of the key components closest to the punching and working frequencies can be chosen as the attributes for evaluating the operational reliability of the high-speed press, which are deviation attributes that can be described as interval numbers due to material uncertainties and manufacturing errors. The low manufacturing cost of the high-speed press is usually ensured by its good manufacturability, which is a benefit attribute, and should be described as a fuzzy linguistic term due to the cognitive fuzziness in the early design phase. Therefore, structural optimization of the high-speed press is a heterogeneous MADM (HMADM) problem involving benefit, cost, deviation and fixation attributes, the values of which are described in various mathematical forms such as real numbers, interval numbers, fuzzy linguistic terms, triangular fuzzy numbers (TFNs) and so on. None of the TOPSIS methods mentioned above are able to solve HMADM problems with deviation and fixation attributes. Although the heterogeneous axiomatic design (HAD) method proposed in our previously work [22] can solve HMADM problems with deviation and fixation attributes, the application of HAD has the strict prerequisite that all the attributes in a HMADM problem be independent of each other [23]. It is obvious that such a prerequisite cannot be satisfied in the structural optimization of a high-speed press, since mechanical performance indices,such as the stiffness and strength of the same component, are obviously interdependent. Moreover, the computation of information content based on distance measure for ranking different competing designs may be tedious and inefficient in the implementation of HAD. This work presents a new HMADM method for dealing with deviation and fixation attributes entailed by high-speed press optimization, which has no prerequisite of independency among heterogeneous attributes, and is also applicable to other HMADM problems. The proposed HMADM method relies on TOPSIS, considering its advantages over other MADM methods, and is therefore named heterogeneous TOPSIS (HTOPSIS) for the sake of brevity. A new approach for normalizing heterogeneous attributes is proposed based on vector transformation by integrating the possibility degree method, relative preference relation and the attribute transformation technique. Expressions for determining the positive and negative ideal solutions corresponding to heterogeneous attributes are also developed. The rest of the paper is organized as follows. Section 2 constructs the HMADM model of the high-speed press structural optimization. Section 3 describes the new HTOPSIS method for solving the resulting HMADM model. Section 4 demonstrates the implementation of the proposed HTOPSIS, as well as its validity and superiority in a complex design problem. Conclusions are made in Section 5.