## 5 Application Discussion
### A Micro-Force Sensor
That is, the nominal values of the uncertain variables are used to calculate \(\eta\), \(\delta\) and \(\sigma\). In summary, the EBRO problem is formulated as follows:  
\[ \begin{array}{l}\max U_{0},\ \max Bel\left(U(\mathbf{X},\mathbf{P})\geq U_{0} \right)\\ \mathrm{s.t.}\ \frac{w}{h}\leq 0.83,\ \delta\left(\hat{\mathbf{X}}\right)\leq 2.5 \upmu,\ \sigma\left(\hat{\mathbf{X}}\right)\leq 32.0\mathrm{MPa}\\ 0.40\mathrm{mm}\leq\overline{L}\leq 1.20\mathrm{mm},0.06\mathrm{mm}\leq \overline{w}\leq 0.10\mathrm{mm},0.04\mathrm{mm}\leq\overline{h}\leq 0.10 \mathrm{mm}\end{array}  \tag{21}\]  
where \(\mathbf{X}=(L,\,w,\,h),\ \mathbf{P}=\left(d_{31}^{\mathrm{P}},e_{33}^{\mathrm{P}}\right)\); \(U_{0}\) represents the performance threshold, which is set as the deterministic design variable. The steps to solve this problem using the proposed method are detailed below. Firstly, according to the marginal BPAs of the five variables in Figure 5, the joint BPAs of the focal elements (\(8^{5}=32768\)) are calculated by Equation (4). Secondly, Equation (21) is converted into a series of sub-problems, which are expressed as:  
\[ \begin{array}{l}\max U_{0}\\ \mathrm{s.t.}\ Bel\left(U(\mathbf{X},\mathbf{P})\geq U_{0}\right)\geq Bel_{j}^{T}\\ \frac{w}{h}\leq 0.83,\ \delta\left(\hat{\mathbf{X}}\right)\leq 2.5\upmu,\ \sigma\left(\hat{\mathbf{X}} \right)\leq 32.0\mathrm{MPa}\\ 0.40\mathrm{mm}\leq\overline{L}\leq 1.20\mathrm{mm},0.06\mathrm{mm}\leq \overline{w}\leq 0.10\mathrm{mm},0.04\mathrm{mm}\leq\overline{h}\leq 0.10 \mathrm{mm}\\ Bel_{j}^{T}=(80\%,\,85\%,\,90\%,\,95\%,\,99.9\%)\end{array}  \tag{22}\]  
where \(Bel_{j}^{T}\) represent a series of target \(Bel\) for the proposition of \(U\geq U_{0}\), which are given by the designer according to engineering experience or quality standards. Thirdly, the iteration starts from the initial point of \(\left(\overline{L}^{(0)},\overline{w}^{(0)},\overline{h}^{(0)},U_{0}^{(0)} \right)=\left(0.60\,\mathrm{mm},0.08\,\mathrm{mm},0.06\,\mathrm{mm},35.6\, \mathrm{mV}\right)\), where \(L^{(0)},w^{(0)},h^{(0)}\) are selected by the designer and \(U_{0}^{(0)}\) is calculated by Equation (18). At each iteration step, the approximate function of \(U\) is established as Equation (13), and then 10 numbers of \(\Delta\mathbf{S}_{j}^{(k)}\) are obtained through Equation (16). Correspondingly, the 10 optimization problems as Equation (14) are updated. By solving them, the optimal set in the current iteration is obtained. After four iteration steps, the optimal set is converged as listed in Table 1. The results show that the performance threshold decreases gradually with increase in \(Bel\). In engineering, a designer can intuitively select the optimal design option from the optimal set by balancing the product performance and robustness. In term of accuracy, the solutions of the proposed method are very close to the corresponding reference solutions, and the maximal error is only 2.5% under the condition of \(Bel_{j}^{T}=95\%\). In efficiency, the proposed method calculates performance function only 248 times, and the computational cost is much less than that of evolutionary algorithms [31]. From a mathematical point of view, it is unfair to compare the efficiency of the proposed method with the evolutionary algorithms. From the view of engineering practicality, however, the solutions of the proposed method may help the designers create a relatively clear picture of the problem with high efficiency and acceptable accuracy.