## 4 Case Study
### Validation of Evolution Design Rule for Electricaltoothbrushes
\begin{table}
\begin{tabular}{c c} \hline \hline
**Evolutionary Process** & **Customer Needs/Drive Elements** \\ \hline E421\(\rightarrow\)E421B & N2:1\(\rightarrow\)2; N3:1\(\rightarrow\)2; N6:1\(\rightarrow\)2; N9:1\(\rightarrow\)2; N21: lengthen \\ E421A\(\rightarrow\)E421B & N5:1\(\rightarrow\)2; N6:1\(\rightarrow\)2; N13:1\(\rightarrow\)2; N15: lighter; N24: shorter \\ E421B\(\rightarrow\)E421e & N2:1\(\rightarrow\)2; N3:1\(\rightarrow\)3; N10:1\(\rightarrow\)2; N141\(\rightarrow\)2; N23:2\(\rightarrow\)3 \\ E421B\(\rightarrow\)E421D & N7:1\(\rightarrow\)2; N13:1\(\rightarrow\)2; N19:1\(\rightarrow\)2; N22: lengthen; N23:2\(\rightarrow\)3; N24: shorter \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Evolution parameter of five types of electric toothbrush. the established analyzing network model for electrical toothbrush evolution. The detailed explanation of the evolution process is listed in Table 7. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Evolutionary Process** & **Node** & _P(X = 1)_ & _P(X = 2)_ & _P(X = 3)_ & _P(X = 4)_ \\ \hline E421\(\rightarrow\)E421B & N12 & 0.3205 & 0.6795 & - & - \\  & N18 & 0.2851 & 0.5204 & 0.1945 & - \\ \hline  & N11 & 0.203 & 0.797 & - & - \\ E421A\(\rightarrow\)E421B & N12 & 0.336 & 0.664 & - & - \\  & N17 & 0.2219 & 0.5313 & 0.0221 & 0.2247 \\ \hline  & N8 & 0.2051 & 0.7949 & - & - \\ E421B\(\rightarrow\)E421e & N15 & 0.6338 & 0.3662 & - & - \\ \hline  & N4 & 0.4375 & 0.5625 & - & - \\  & N8 & 0.2062 & 0.7938 & - & - \\ E421B\(\rightarrow\)E421D & N11 & 0.2378 & 0.7622 & - & - \\  & N16 & 0.4067 & 0.5933 & - & - \\  & N17 & 0.5046 & 0.1585 & 0.2783 & 0.0586 \\  & N18 & 0.246 & 0.2245 & 0.5295 & - \\ \hline \end{tabular}
\end{table} 
Table 6: Nodes with large changing probability. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Design Data** & **N1** & **N2** & **N3** & **N4** & **N5** & **N6** & **N7** \\ \hline \(P(X=1)\) & 0.1781 & 0.2123 & 0.3612 & 0.4375 & **1** & 0 & 0 \\ \(P(X=2)\) & 0.3215 & **0.7877** & 0.1474 & **0.5625** & 0 & **1** & **1** \\ \(P(X=3)\) & **0.5004** & - & **0.4914** & - & - & - & 0 \\ \(P(X=4)\) & - & - & - & - & - & - & - \\ E421D & 3 & 2 & 3 & 2 & 1 & 2 & 2 \\ \hline
**Design data** & **N8** & **N9** & **N10** & **N11** & **N12** & **N13** & **N14** \\ \hline \(P(X=1)\) & 0.2062 & 0.4231 & 0.3326 & 0.2378 & 0 & 0 & 0.4721 \\ \(P(X=2)\) & **0.7938** & **0.5769** & **0.6674** & **0.7622** & 1 & 1 & **0.5279** \\ \(P(X=3)\) & - & - & - & - & - & - & - \\ \(P(X=4)\) & - & - & - & - & - & - & - \\ E421D & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ \hline
**Design data** & **N15** & **N16** & **N17** & **N18** & **N19** & **N20** & **N21** \\ \hline \(P(X=1)\) & **1** & 0.4067 & **0.5046** & 0.246 & 0 & 0 & 0 \\ \(P(X=2)\) & 0 & **0.5933** & 0.1585 & 0.2245 & **1** & 0 & 0 \\ \(P(X=3)\) & - & - & 0.2783 & **0.5295** & - & **1** & **1** \\ \(P(X=4)\) & - & - & 0.0586 & - & - & - & - \\ E421D & 1 & 2 & 1 & 3 & 2 & 3 & 3 \\ \hline
**Design data** & **N22** & **N23** & **N24** & **N25** & **N26** & & \\ \hline \(P(X=1)\) & 0 & 0 & 0 & 0.3128 & 0.2594 & - & - \\ \(P(X=2)\) & **1** & 0 & 0 & **0.6872** & **0.7406** & & - \\ \(P(X=3)\) & 0 & **1** & **1** & - & - & - & - \\ \(P(X=4)\) & - & - & - & - & - & - & - \\ E421D & 2 & 3 & 3 & 2 & 2 & - & - \\ \hline \end{tabular}
\end{table} 
Table 5: Reasoning results in E421A design. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Evolutionary Process** & **Node** & _P(X = 1)_ & _P(X = 2)_ & _P(X = 3)_ & _P(X = 4)_ \\ \hline E421\(\rightarrow\)E421B & N12 & 0.3205 & 0.6795 & - & - \\  & N18 & 0.2851 & 0.5204 & 0.1945 & - \\ \hline  & N11 & 0.203 & 0.797 & - & - \\ E421A\(\rightarrow\)E421B & N12 & 0.336 & 0.664 & - & - \\  & N In the derivation model, different customers' demands can be used as evidence to infer from the network, and the result would propose reasonable change (probability) of the function or the design, therefore providing the theoretical guidance to the designer in product design process.