## 4 Case Study
### Influence of Parameter Uncertainty  
Since many uncertainties influence the product degradation process, the relationship between parameter uncertainty and batch storage reliability can be quantified to guide subsequent product optimization. Therefore, this section will analyze the impact on storage reliability by considering the magnitude of uncertainty of different parameters. Relays are usually made of consistent materials and the storage environment is relatively stable, so the uncertainty effects of model parameters \(a^{\prime}\) and \(b\) are not analyzed, and their distribution is kept fixed. Instead, parameters \(P_{r}(0)\), \(c\), and \(\omega\) affect the batch product reliability assessment and are considered separately. Firstly, the influence of the uncertainty of parameter \(P_{r}(0)\) on the batch product is considered, and the distribution parameters \(d\) of \(P_{r}(0)\) are empirically made equal to 21.7, 13.1, and 8.1, respectively, while the other parameters keep the initial distribution unchanged. The evaluation results are shown in Figure 9. When \(d\) is equal to 21.7, 13.1, and 8.1, the MAEs between it and the reliability curves based on the test data are 0.081,  
Figure 8: The contrast of M1, M2, and M3. Figure 7: The curve of reliability. 0.098, and 0.109, respectively. This can be seen from the figure that the prediction accuracy gradually decreases as the uncertainty decreases. The main reason is that the uncertainty of the parameters decreases, which makes the distribution of \(P_{r}(0)\) unable to cover the batch products, thus decreasing the accuracy of the evaluation results. However, from the perspective of improving the AEMR's reliability, the uncertainty of \(P_{r}(0)\) is smaller, which means the consistency of the initial state of the batch product will be better, and the distribution of \(P_{r}(0)\) is mainly related to the assembly error and the manufacturing process data. If we want to improve the storage reliability of the batch product, we have to modify and optimize the assembly process, the assembly technology and the external characteristics of the product so that the reliability of the batch products can be effective. The model parameter \(c\) mainly reflects the individual differences of the same batch of relay products. This part was assessed by fixing the distribution of other parameters and changing the distribution parameter \(d\) of \(c\) to 25.7, 15.4, and 9.6, respectively, for reliability assessment. The assessment results are shown in Figure 10a. According to the observation curve in the figure, on the one hand, it can be seen that the uncertainty of parameter \(c\) is reduced to a certain degree. Then it will not further expand with the uncertainty reduction on the assessment results. On the other hand, it can be seen that the assessment results, obtained by reducing the uncertainty of the parameter \(c\), aggravate the reliability degradation in the early rapid degradation stage, which dramatically affects the calculation accuracy of the reliability assessment work performed in the early stage of product life. The most important thing for a product is the accuracy of the pre-life prediction, thus showing that the influence of the uncertainty of parameter \(c\) cannot be ignored. Figure 10: (**a**) Influence of reliability assessment by considering the uncertainty of \(c\); (**b**) Influence of reliability assessment by considering the uncertainty of \(\omega\). Figure 9: Influence of reliability assessment by considered the uncertainty of \(P_{r}(0)\). For the product failure threshold \(\omega\), when evaluating the same batch of products of a given model, the engineer usually gives a fixed value for calculation. However, in the actual product production process, due to the different distribution of the initial state of the product, heterogeneity will arise between batches of products, and thus, the failure threshold will vary from product to product.