## 4 Design Example: The High-Speed Press
### HMADM Modeling of the Optimization Problem  
The press slider and upper beam are the key components of the high-speed press. Their mechanical performance indices, such as stiffness and strength, greatly influence the punching precision, and thus should be included as attributes for evaluating alternative structural configurations. The stiffness of the press slider is usually evaluated based on its maximum deformation during the punching process, which can be regarded as a cost attribute, since a lower maximum deformation means higher stiffness. The maximum deformation of the press slider is obviously a real number. The stress concentration and the effect of alternate loading over a long period of time may cause fatigue cracks and damage in the upper beam. Thus, it is necessary to ensure the strength of the upper beam in the design phase. Based on experience, the maximum equivalent stress of the upper beam can satisfy strength requirements, and will not be too conservative when it is close to [60,61] MPa for the investigated press. Hence, maximum equivalent stress developed in the upper beam can be regarded as a fixation attribute, which should be described as an interval number, taking into account its fluctuation under the influence of uncertain material properties, manufacturing errors, etc. The press slider vertically reciprocates along the guide columns at 80-240 spm (namely, 1.33-4 Hz) during the punching process. The mechanical resonance of press slider will occur when any of its natural frequencies is close to the punching frequency, which will cause damage and even breakdown of the high-speed press. Resonance will also occur when any natural frequency of the press slider is close to the working frequency of 50 Hz (in China). Therefore, the lower order natural frequencies of the press slider closest to the punching frequency of 1.33-4 Hz and working frequency of 50 Hz should be chosen as the attributes  
Figure 3: Schematic of the high-speed press and matching molds. for evaluating the operational reliability of press. These are deviation attributes, since the farther their values deviate from the punching and working frequencies, the better they are. The natural frequencies of the press slider should also be described by interval numbers, due to the uncertainties in its material properties, etc. Manufacturability is chosen to evaluate the manufacturing cost of the high-speed press, which can be described as a fuzzy linguistic term, and further transformed into TFN due to its cognitive fuzziness in the design phase. Manufacturability is a benefit attribute, because a better manufacturability will lead to a lower manufacturing cost for the press. In summary, a total of five heterogeneous attributes are chosen for evaluating the structural configurations of the high-speed press. They are the maximum deformation of the press slider \(a_{1}\), the maximum equivalent stress of the upper beam \(a_{2}\), the natural frequency of press slider closest to the punching frequency of 1.33-4 Hz \(a_{3}\), the natural frequency of press slider closest to the working frequency of 50 Hz \(a_{4}\), and the manufacturability of press \(a_{5}\).