## 2 Methodology
### Experimental Procedure
#### 2.3.4. Scheme Evaluation and Optimization  
This study utilized the key user requirements and design parameters mapped by AD to develop diversified elderly-friendly game products. To scientifically determine the optimal design scheme, the TOPSIS method was employed for objective evaluation and comprehensive assessment of the design alternatives. First, the demand topics extracted from the BTM model were adopted as evaluation indicators, and a seven-point Likert scale was applied to rate the extent to which each product met the corresponding indicator requirements. To improve the objectivity of the evaluation and minimize the influence of the designers' subjective preferences, rating data from different user groups were homogenized to construct the initial evaluation matrix. Subsequently, data normalization was performed to create a standardized matrix F. According to the standard computational steps of the TOPSIS method, the distances between each design scheme and the positive and negative ideal solutions were calculated, allowing for determination of the relative closeness and final ranking of each scheme. The scheme with the closest distance to the positive ideal solution and farthest distance from the negative ideal solution was identified as the optimal solution. The detailed TOPSIS calculation steps are shown in Equations (11)-(16). Step 1. The data obtained from the questionnaire were averaged to build an initial evaluation matrix, denoted as \(f\). Step 2. The initial evaluation matrix was normalized to obtain the standardized matrix \(R_{ij}\). \[R_{ij}=\frac{f_{ij}}{\sum_{i=1}^{m}f_{ij}^{2}}(i=1,2,\ldots,m;j=1,2,\ldots,n) \tag{11}\]  
where \(f_{ij}\) is derived from the feedback scores based on the design dimensions extracted from user comments using the BTM topic model, representing the value of the \(i\) alternative under the \(j\) criterion in the original decision matrix, and \(m\) denotes the total number of alternatives. Step 3. The weighted matrix \(u_{ij}\) was obtained based on the calculation of the target weights of each evaluation indicator. \[u_{ij}=W_{j}R_{ij}(i=1,2,\ldots,m;j=1,2,\ldots,n) \tag{12}\]  
where \(W_{j}\) represents the weight of the \(j\) criterion, which is obtained during the AHP phase as the importance weight of user needs. Step 4. The positive and negative ideal solutions, \(A^{+}\) and \(A^{-}\), respectively, were determined. Figure 6: Three typical design matrix types in Axiomatic Design. \[ \begin{split} M_{j}^{+}&=max\big{\{}u_{1j},u_{2j},u, \ldots,u_{nj}\big{\}}(j=1,2,\ldots,n)\\ M_{j}^{-}&=min\big{\{}u_{1j},u_{2j},u,\ldots,u_{mj} \big{\}}(j=1,2,\ldots,n)\end{split}  \tag{13}\]  
Thus,  
\[ \begin{split} A^{+}&=\big{(}M_{1}^{+},M_{2}^{+}, \ldots,M_{n}^{+}\big{)}\\ A^{-}&=\big{(}M_{1}^{-},M_{2}^{-},\ldots,M_{n}^{-} \big{)}\end{split}  \tag{14}\]  
Step 5. The Euclidean distance is solved to obtain the distance between each scheme and the ideal solution, serving as a basis for evaluating the overall performance of each alternative. \(Si^{+}\) and \(Si^{-}\) represent the distances to positive and negative ideal solutions, respectively. \[ \begin{split} S_{i}^{+}&=\sqrt{\sum\limits_{j=1}^{ n}\Big{(}u_{ij}-u_{j}^{+}\Big{)}^{2}}(i=1,2,\ldots,m;j=1,2,\ldots,n)\\ S_{i}^{-}&=\sqrt{\sum\limits_{j=1}^{n}\Big{(}u_{ij}-u _{j}^{-}\Big{)}^{2}}(i=1,2,\ldots,m;j=1,2,\ldots,n)\end{split}  \tag{15}\]  
Step 6. The relative closeness coefficient \(Ci\) is calculated for each game design alternative. A higher \(Ci\) value indicates that the alternative is closer to the ideal solution and has greater relative superiority. \[C_{i}=\frac{S_{i}^{-}}{S_{i}^{+}+S_{i}^{-}}(i=1,2,\ldots,m) \tag{16}\]  
The comprehensive evaluation and decision making of product design schemes is a complex multi-criteria process that integrates both subjective judgments and objective data, with outcomes that significantly influence subsequent design strategies.