## 1 Introduction  
The high-speed press is a kind of automatic punching machine tool equipped with a high-speed automatic feeding device for metal sheet that is capable of continuously punching at a high speed. It has been widely applied in the automotive, aerospace and instrument industries due to its advantage of clean and green production. With the increasing demand placed on the amount and quality of stamping parts, the development of a new large-tonnage ultra-precision high-speed press has become an urgent task for manufacturers and engineering designers in the field [1]. Structural optimization of the high-speed press is very important in the early design phase, for the purposes of achieving high punching precision, good operational reliability, and low manufacturing cost [2], which constitute a multiple-attribute decision-making (MADM) problem when a set of configurations are provided for evaluation against multiple attributes [3]. The technique for order preference by similarity to an ideal solution (TOPSIS) [4] is widely utilized to solve MADM problems, due to its advantages of simple calculation, reasonable evaluation results, and flexibility in application compared with other MADM methods [5; 6; 7; 8; 9]. The basic principle of TOPSIS is that the optimal solution to a MADM problem should have the shortest distance from the positive ideal solution (PIS), and the farthest distance from the negative ideal solution (NIS), and that the distance can be measured by the Euclidean distance [10]. Scholars have conducted a great deal of research on the development of TOPSIS for solving various MADM problems in recent years. [PERSON] et al. [11] proposed a new method for dealing with stochastic MADM problems based on the ideal and anti-ideal cumulative distribution functions. [PERSON] et al. [12] transformed the multi-objective decision making (MODM) problem into a bi-objective one based on TOPSIS, and proposed an integrated multi-objective framework to solve MODM problems including both tangible and intangible factors based on the extended efficient epsilon-constraint method. [PERSON] and [PERSON] [13] applied fuzzy TOPSIS to the selection of the best procedural approach for solving facility layout-planning problems. [PERSON] and [PERSON] [14] proposed an extended fuzzy TOPSIS by utilizing possibility theory to deal with MADM problems with fuzzy numbers. [PERSON] and [PERSON] [15] proposed a hybrid MADM method for modelling complex decision-making problems by integrating fuzzy TOPSIS and fuzzy cognitive maps. [PERSON] and [PERSON] [16] developed a new MADM approach by integrating the TOPSIS and maximizing deviation methods, in which the evaluation information provided by the decision-maker is described as hesitant fuzzy elements, and the information about attribute weights is incomplete. [PERSON] et al. [17] proposed an intuitionistic fuzzy TOPSIS for evaluating advanced manufacturing technology considering ergonomic compatibility attributes, and applied it to the selection of numerically controlled milling machines. [PERSON] et al. [18] evaluated suppliers according to the criteria arising from strengths, weaknesses, opportunities and threats analysis in the MADM of suppliers, and solved the resulting MADM problem by fuzzy TOPSIS. [PERSON] and [PERSON] [19] proposed a novel method by integrating the interval-valued intuitionistic fuzzy (IVIF) sets, linear programming, and TOPSIS to solve MADM problems in IVIF environments. All of the above approaches were proposed to solve MADM problems with attributes in uniform mathematical descriptions. However, the attributes entailed by structural optimization of the high-speed press are heterogeneous, because multiple objectives and multi-source uncertainties need to be considered in decision-making. Specifically, the high punching precision of a high-speed press is usually ensured by the ideal mechanical performance indices of its key components. The stiffness indicated by maximum deformation is a cost attribute that can be described as a real number.