## 2 Research Methods and Relevant Theories
### SEM-Grey Comprehensive Evaluation Calculation  
The GRA method is extensively employed for assessing the degree of closeness between main factors and sub-factors within a system, thereby determining the primary and secondary factors contributing to the system's development changes [34]. The specific steps are as follows:  
Step 1: A data comparison matrix was established to collect and analyze relevant data. Simultaneously, a reference data column was determined, reflecting idealized reference values. The reference data column is:  
\[\text{X}_{0}=(\text{X}_{01},\text{X}_{02},\cdots,\text{X}_{0n}) \tag{5}\]  
Step 2: Variables were treated using the dimensionless method. Commonly used methods include initial-value and averaging methods. In this case, the averaging method was applied for the transformation. \[\text{x}_{i}(\text{k})=\frac{\text{x}_{i}(\text{k})}{\bar{\text{x}}_{i}} \tag{6}\]  
Step 3: Calculation of correlation coefficients. \[\zeta_{\text{ik}}=\frac{\text{min}_{i}\text{min}_{\text{k}}|\text{X}_{0k}- \text{X}_{\text{ik}}|+\rho\text{max}_{i}\text{max}_{k}|\text{X}_{0k}-\text{X}_ {\text{ik}}|}{|\text{X}_{0k}-\text{X}_{\text{ik}}|+\rho\text{max}_{i}\text{max }_{k}|\text{X}_{0k}-\text{X}_{\text{ik}}|} \tag{7}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Product Family** & **Explanation of Concepts** & **Supporting Element** & **Refs.** \\ \hline \multirow{4}{*}{X1: Functional domain} & \multicolumn{2}{c}{The key aspect in the functional domain lies in meeting the production requirements of} & X11 main function & [25] \\  & customers and translating them into the functional characteristics of a product. During this & X12 analysis of function and structure & [26; 27] \\  & process, in-depth research is required on the primary production objectives and functional needs & & structure \\  & of the product. In the machine tool design phase, the functional domain emphasizes meeting the functional features required by the product. & X14 productivity & [4; 26] \\  & functional features required by the product. & X12 modularization & [4; 29] \\ \hline \multirow{4}{*}{X2: Technological domain} & The technological domain encompasses manufacturing solutions and modular design for a product, highlighting the operating principles and implementation methods of the product. It focuses on the technical feasibility of the product, specifically how the chosen technical solution & X23 working area allocation & [30] \\  & to be integrated into the actual manufacturing and utilization processes of the product. & X24 manufacturing method & [4] \\ \hline \multirow{4}{*}{X3: Physical domain} & The physical domain encompasses the physical components and assembly of a product, reflecting the actual construction of the product. The physical domain concentrates on manufacturing, structural assembly, and man-machine interaction, ensuring the precise realization of the physical structure and operational efficiency of the product. & X31 internal component & [27] \\  & & realization of the physical structure and operational efficiency of the product. & X32 assembly modes & [26; 27] \\ \hline \multirow{4}{*}{X4: Design domain} & The design domain highlights the consideration of the external characteristics and the aesthetic expression of a product. By imparting a unique aesthetic and design sense through industrial styling products can better capture the user’s attention and visually enhance their attractiveness. & X41 modeling shape & [30; 31] \\  & & x42 color design & [32] \\ \cline{1-1}  & & X43 overall size & [27] \\ \cline{1-1}  & & x44 material and techniques & [33] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Demand analysis platform based on PFA theory. In the equation, the distinguishing coefficient \(\rho\) is a parameter ranging from 0 to 1, used to represent the differentiation between correlation coefficients. A smaller \(\rho\) value enhances the differentiation between correlation coefficients. Typically, \(\rho\) is set to 0.5. Step 4: The relational sequence was computed.