## 6 Application Examples  
### Example of Crank-Slider Mechanism  
The example of a crank-slider mechanism is used to demonstrate the effectiveness of the proposed algorithm. The mechanism is shown in Figure 2. The length of the crank \(x_{1}\), the length of the coupler \(x_{2}\), and the external force \(x_{3}\) are aleatoric uncertainty variables. Their uncertainties are listed in Table 1. The Young's modulus of the material \(y_{1}\) is an epistemic uncertainty variable, whose distribution type is Normal with a deterministic mean \(\xi_{1}=200\) GPa and an uncertain standard deviation \(\xi_{2}\). The insufficient data for \(\xi_{2}\) from Source 1 contain sparse sampling points \(\{0.36,0.51\}\) and multiple intervals \(\{[0.28,0.31],[0.35,0.39],[0.54,0.57]\}\). The sub-intervals with their corresponding BPAs from Source 2 are listed in Table 2. The coupler's internal diameter \(d_{1}=25\) mm, external diameter \(d_{2}=60\) mm, the offset \(z_{2}=0.18\) mm of the mechanism, and the coefficient of friction \(z_{1}=100\) acting on the slider are all deterministic values.  
Figure 1: The solution flowchart of the proposed algorithm.  
The probabilistic representation model for Source 2 is transfered from evidence theory using Equations (3) and (4), and the result is shown in Figure 3a. Then, the design domain of \(\xi_{2}\) is determined to be \([0.16,0.69]\) using Equations (7) and (8). The Gaussian interpolation model considering only insufficient data from Source 1 is constructed by maximizing the likelihood function in Equation (9). In the optimization design of the fusion model, the weight ratios \(\mathbf{w}\) are optimized through minimizing \(L(\mathbf{w},\mathbf{p})\) in Equation (11). The fusion Gaussian model is shown in Figure 3b.  
The product performance function is defined as follows  
\[G(\mathbf{x},\mathbf{y},\mathbf{z})=\frac{\pi^{3}y_{1}(d_{2}^{4}-d_{1}^{4})}{64 x_{2}^{2}}-\frac{x_{3}(x_{2}-x_{1})}{\sqrt{\left(x_{2}-x_{1}\right)^{2}-z_{2}^{2 }-z_{1}z_{2}}}. \tag{16}\]  
The hybrid reliability is analysed using the proposed algorithm, then the belief and plausibility results for different reliability limit states \(\psi\) are calculated. The reliability results comparison in Figure 4 shows that the results of the proposed method are identical to those of the Monte Carlo method, confirming the effectiveness of the proposed method. In order to further illustrate the effectiveness of the proposed algorithm, the reliability indices considering sampling information from Source 1 or Source 2 are calculated, which  
\begin{table}
\begin{tabular}{c c c} \hline \multirow{2}{*}{**Variable**} & \multirow{2}{*}{**Distribution Type**} & \multicolumn{2}{c}{**Distribution Parameters**} \\ \cline{3-4}  & & **Mean Value** & **Standard Deviation** \\ \hline \(x_{1}\) & Normal & 21.2 mm & 0.01 mm \\ \(x_{2}\) & Normal & 5.2 mm & 0.01 mm \\ \(x_{3}\) & Normal & 250 KN & 25 KN \\ \hline \end{tabular}
\end{table}
Table 1: Aleatoric uncertainty variables \(\mathbf{x}\).  
\begin{table}
\begin{tabular}{c c c} \hline \multirow{2}{*}{**Interval**} & **BPA** \\ \hline \([0.2,0.33]\) & 0.34 \\ \([0.31,0.46]\) & 0.22 \\ \([0.40,0.55]\) & 0.31 \\ \([0.55,0.59]\) & 0.13 \\ \hline \end{tabular}
\end{table}
Table 2: Intervals and basic probability assignments (BPAs) of \(\xi_{2}\) from Source 2.  
Figure 2: Crank-Slider mechanism.  
are also shown in Figure 4. The proposed algorithm can obtain more accurate reliability indices compared with results under sampling information from a single source, which can also indicate the effectiveness of the proposed method.  
Figure 3: Optimum distribution of uncertain distribution parameters: **(a)** Source 1. **(b)** Fusion model.