## 2 Research Methods and Relevant Theories  
### Research Process  
This study first summarizes the design factors in previous mature paradigms based on product family architecture theory and product family DNA theory [4; 5; 6; 7]. It then conducts research by combining these findings with the characteristics of the machine tool industry in the case. The research is divided into three main parts: model construction, model validation, and system evaluation. In the model construction phase, a framework and element system for evaluating the model are proposed. During the model validation, the structural equation modeling (SEM) method is used to validate the relationships between various elements and examine the model's validity. Finally, in the system evaluation phase, based on the model results, a combination of subjective and objective weighting methods is used to rank and analyze the various elements, providing guidance for subsequent design optimization. This sequence of steps not only helps in constructing an evaluation model but also validates the model and applies it to real-world situations, making the research more comprehensive. The specific steps are illustrated in Figure 1.  
Step one: Model construction based on PFA  
In this study, the first step involved a systematic literature review method, where we searched for English-language publications in the fields of product family design and machine tool design over the past 10 years. We used keywords such as \"product family design\" and \"product family architecture\" for our search. During the selection process, we relied on the relevance of titles and abstracts, ultimately selecting 18 high-quality papers for analysis. These papers provided a solid theoretical foundation for the research. Through the literature review method, we identified and summarized various factors influencing the exterior design of machine tools. This process formed the initial framework for the elements required in a machine tool product family. Specifically, by integrating product family architecture (PFA) theory and product family DNA theory, we identified and summarized various elements influencing the design of machine tool exteriors, thereby providing a solid foundation for the construction of the subsequent evaluation model. Step two: SEM model validation  
This step involved model validation with the establishment and verification of the PFA model. Distinguishing itself from decision-making methods that directly establish models based on expert opinions, building upon step one, this step incorporated the SEM method, proposed the theoretical framework and specific hypotheses of the model, defined the relationships between multiple factors, verified these relationships, and then conducted a system comprehensive evaluation to confirm objective correlations between different domains and the overall product competitiveness. Figure 1: PFA-SEM-based design flowchart. Step three: System comprehensive evaluation  
After using the SEM model to calculate and verify the influence of various factors, a combination of normalization algorithms and GRA was employed. This allowed for calculating and ranking the weights of different factors from both subjective and objective perspectives, producing the priority ranking of design elements for the product family of machine tools, which served as guidance for subsequent design practices and optimization. Step four: Systematic comprehensive evaluation  
After using structural equation modeling to calculate and verify the influence of various factors, a combination of normalization algorithms and grey relational analysis was employed. This combination approach was used to calculate and rank the weights of these factors from both subjective and objective perspectives. The result was the prioritization of product family machine tool design elements. Various weighting methods were explored, including analytic hierarchy process (AHP), entropy weighting, and structural equation model (SEM) path coefficient normalization weights.