# A Time Correlation Based Clustering Method for a Design of a Transformable Product
## 4 Case Study
(**a**) function chain \(G_{1}\), (**b**) function chain \(G_{2}\), (**c**) function chain \(G_{3}\), (**d**) function chain \(G_{4}\), (**e**) function chain \(G_{5}\), (**f**) function chain \(G_{6}\). \[\omega_{S_{1}S_{q}}= \begin{array}{c}S_{1}\\ S_{2}\\ S_{3}\\ S_{4}\\ S_{5}\\ S_{6}\end{array} \left[ \begin{array}{cccccccc}0&0.106&0.044&0.034&0.066&0.062\\ 0&0.094&0.084&0.116&0.112\\ 0&0&0.022&0.054&0.050\\ 0&0&0.044&0.040\\ 0&0&0.072\\ 0\\ S_{1}&S_{2}&S_{3}&S_{4}&S_{5}&S_{6}\end{array} \right]. \tag{20}\]  
According to Equation (12), the weighted similarity is obtained as shown in Equation (21):  
\[WSim[G_{p},G_{q}]= \begin{array}{c}S_{1}\\ S_{2}\\ S_{3}\\ S_{4}\\ S_{5}\\ S_{6}\end{array} \left[ \begin{array}{cccccccc}0&0.48&0.24&0.47&0.11&0.09\\ 0&0.14&0.14&0.18&0.16\\ 0&0.18&0.12&0.10\\ 0&0.10&0.08\\ 0&0.53\\ 0\\ S_{1}&S_{2}&S_{3}&S_{4}&S_{5}&S_{6}\end{array} \right]. \tag{21}\]  
An undirected graph of the matrix in Equation (21) is then established as shown in Figure 10.  
For a given threshold value R = 0.4, the resulting segmentation of the graph is formed in Figure 11.  
Therefore, two final concepts can be formed for the wheelchair based on two segments formed in Figure 11, and the first concept has three states S\({}_{1}\), S\({}_{2}\) and S\({}_{4}\). Its function chain is shown in Figure 12.  
Figure 11: Segmentations of the undirected graph based on the weighted similarity. Figure 10: The undirected graph of the weighted similarity. Following the mapping process in Figure 6, the relationship between functions and components of the first concept can be formed as shown in Table 7. Based on corresponding components to meet the required functions, a conceptual model of the wheelchair can be proposed. The computer-aided design(CAD) model of the first concept is shown in Figure 13. \begin{table}
\begin{tabular}{c c c} \hline \hline
**States** & **Functions** & **Components** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Relationship between functions and components of the first concept. Figure 12: Function chain of the first concept. The second concept consists of S\({}_{5}\) and S\({}_{6}\), its function chain is shown in Figure 14.  
According to the mapping relationship of functions-solutions-components in Figure 6, the relationship between functions and components of the second concept is mapped as shown in Table 8.  
Figure 14: Function chain of the second design concept. Figure 13: The first conceptual design of the wheelchair. (**a**) in climbing state, (**b**) in working state, (**c**) in sleeping state, (**d**) in navigation state. The CAD model of the second conceptual design is shown in Figure 15.  
According to Equation (13), the proportion of shared components in the first concept can be calculated as 70%. The proportion of shared components in the second concept can be calculated as 75%. More than half of the parts are reused, which saves a lot of material and space resources compared to the single-function wheelchairs. The number of useful functions and number of components are shown in Table 9. \begin{table}
\begin{tabular}{c c c} \hline \hline
**States** & **Functions** & **Components** \\ \hline \hline  & Brake wheelchair & \\  & Support passenger & \\  & Take off clothes & \\  & Collect dirt & \\  & Toilet state & Clean body & \\  & Support passenger & \\  & Wear clothes & \\  & Drive the wheelchair forward & \\  & Brake wheelchair & \\  & Support passenger & \\  & Take off clothes & \\  & Clean body & \\  & Clean body & \\  & Support passenger & \\  & Wear clothes & \\  & Drive the wheelchair forward & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Relationship between functions and components of the second concept. According to Equations (14) and (15), the change degree of the idealization level can be calculated as 1.43 in the first concept, and 1.33 in the second concept. The results show that our proposed method increases the level of the product idealization and conforms to the trend of technological system evolutions in the design for transformable products.