## 4 Case Study
### Validation of Evolution Design Rule for Electricaltoothbrushes  
In order to facilitate the network reasoning as well as obtaining accurate results, we conducted discretization of the continuous nodes before the analyzing the joint probability. According to Figures 6-9, the continuous nodes are divided into two classes, i.e., the second class includes the nodes 15, 16, 25, 26, while the third class contains nodes 18, 20, 21, 22, and 24. Then, we conducted the parameter learning of the network by utilizing the historical data. Then, the reasoning of the evolution design model of the electrical toothbrush is carried out in the absence of any evidence. In addition, conditional probabilities are obtained through two methods: (I) Network is trained (parameter training) using the product design and sales data, and then the conditional probabilities are obtained through network reasoning; (II) the conditional probabilities are determined by field experts. For example, let node A be conditioned on node B and C, and the three nodes have two states, respectively. The original probabilities of node A and node B are (0.4, 0.6) and (0.5, 0.5), respectively, and then the edge distribution probability of node C is (0.7, 0.3) after the network model reasoning. However, it was found that the discrepancy between this inferred probabilities (0.7, 0.3) and the values in actual situation is too large. So, these two probabilities are revised to (0.5, 0.5) by field experts based on their estimation of real cases. We obtained the edge distribution for partial network nodes, which are listed in Table 2.  
Figure 8: Joint probability densityof customer satisfaction and duration. Figure 9: Joint probability densityof price and customer satisfaction. By reference to Table 1, the result in Table 2 reveals that the customer satisfaction of the three classes of the node N3 are basically the same, i.e., \(\sim\)1/3. Regarding to the node N17, the larger probability (in Table 2, 0.3087 and 0.3373) is achieved when the length of the toothbrush is either 250 cm or 255 cm. As to the nodes N25 and N26, the marginal probability distribution of the product cost is close to that of the sale price, i.e., the ratio of high cost and low cost is nearly the same to the ratio of high sale price and low sale price. Moreover, according to the result, the percentage of low sale price with low cost is about 45%, while the percentage of high sale price with high cost is 55%, implying that the most of the demand is in the level of mid-to-high-end. This is consistent with the fact. On the other hand, in the node N20, the customer satisfaction is mostly in the middle level (0.3683), which shows a potential demand for enhancement. Moreover, the economic nature should also be improved (high cost accounted for 0.5478 on N26). However, high prices accounted for only about 0.5145 on N25, so the latter was significantly lower. Consequently, by training with the historical data, the correctness of the established Bayesian network reasoning model is proved in the absence of evidence. After above investigation, our evolution analyzing method is further validated under the premise of evidences, by the evolution design process of five types of electrical toothbrushes, as shown in Figure 10.  
According to Figure 10, the fundamental cause for the evolution of the electric toothbrush can be attributed as the change of customer demand and the progress in science and technology, which therefore results in the variation of the product structure design. On the basis of the research and analysis on the customer demand and enterprise product design, the evolution processes and the driven factors of five types of electric toothbrushes are shown in Table 3.