## 6 Application and Significance  
In this section, we explore the application of UNMPVC in truss topology optimization with uncertainty due to variations in temperature. We examine a truss of size \(2\times 1\) consisting of six nodes and ten potential bars, with the two leftmost nodes fixed. The cross-sectional area of each of the \(10\) bars (\(i=1,\dots,10\)) is denoted by \(\mathfrak{a}_{i}\). Moreover, the length of each bar is denoted by \(L_{i}\). Let \(\mathfrak{a}\in\mathbb{R}^{10}\), and \(y_{1},\ y_{2},\ y_{3}\in\mathbb{R}^{8}\) are three auxiliary vector variables as defined in [59]. The global stiffness matrix of the truss, \(\mathcal{K}(\mathfrak{a})\), is given by  
\[\mathcal{K}(\mathfrak{a})=\sum_{i=1}^{10}\frac{\text{E}\mathfrak{a}_{i}}{L_{i }}\gamma_{i}\gamma_{i}^{T}\in\mathbb{R}^{8\times 8},\]  
with some vectors \(\text{Y}_{i}\in\mathbb{R}^{8}\) and the [PERSON]'s modulus \(E\). It is worth noting that \(\mathcal{K}(\mathfrak{a})\) depends on the Young's modulus, which is influenced by factors such as temperature, pressure, and material composition. Suppose that there is uncertainty due to variation in temperature, i.e., \(t\in\Omega:=[T^{L},T^{U}]\), where \(T^{L}\) and \(T^{U}\) represent the minimum and maximum temperature of the environment, respectively. Therefore, we suppose that \(\mathcal{K}(\mathfrak{a},t)\), \(t\in\Omega\) represents the uncertain global stiffness matrix of the truss. Building on the above discussions, we now consider the following uncertain truss-optimization problem:  
* Minimize \[\sum_{i=1}^{10}L_{i}\mathfrak{a}_{i},\] subject to \[\mathcal{K}(\mathfrak{a},t)y_{j}-f_{j}=0,\ \forall j\in\{1,2,3\},\] \[f_{j}^{T}y_{j}-c\leq 0,\ \forall j\in\{1,2,3\},\] \[-\mathfrak{a}_{i}\leq 0,\ \forall i\in\{1,\ldots,10\},\] \[\mathfrak{a}_{i}-A_{\max}\leq 0,\ \forall i=1,\ldots,10,\] \[(\sigma_{i}(a,y_{j})^{2}-\sigma_{\max}^{2})\mathfrak{a}_{i}\leq 0,\ \forall i\in\{1,\ldots,10\},\ \forall j\in\{1,2,3\},\] \[(A_{\min}-\mathfrak{a}_{i})\mathfrak{a}_{i}\leq 0\ \ \ \ \forall i\in\{1,\ldots,10\},\]  
where the inequality \(f_{j}^{T}y_{j}\leq c\) bounds the work caused by force \(f_{j}\), \(c>0\) is a constant, and \(t\in\Omega\) is an uncertain parameter. Moreover, \(A_{\max}\) and \(A_{\min}\) represent the maximum and minimum cross-sectional area, respectively. Furthermore, for each \(i\in\{1,\ldots,10\}\), the function \(\sigma_{i}(a,y_{j})\) denotes the stress on each bar. Then, the robust counterpart of the considered problem (P5) is given by  
* Minimize \[\sum_{i=1}^{10}L_{i}\mathfrak{a}_{i},\] subject to \[\mathcal{K}(\mathfrak{a},t)y_{j}-f_{j}=0,\ \forall t\in\Omega,\ \forall j\in\{1,2,3\},\] \[\sum_{i=1}^{10}L_{i}\mathfrak{a}_{i},\]0 \[\sum_{i=1}^{10}L_{i}\mathfrak{a}_{i},\]1 \[\sum_{i=1}^{10}L_{i}\mathfrak{a}_{i},\]2 \[\sum_{i=1}^{10}L_{i}\mathfrak{a}_{i},\]3  
The problem (P5) can be effectively solved by employing the results established in this article. **Remark 5**. : _If \(\Omega\) is a singleton set, then the problem (P5) reduces to the well-known deterministic truss optimization problem considered by [PERSON] et al. [59]._  
To illustrate the importance of the results deduced in the present article, we consider the subsequent example:  
**Example 5**. : _We consider the following problem with vanishing constraints under data uncertainty:_  
* \(\mathcal{G}(\mathfrak{a})=(\mathcal{G}_{1}(\mathfrak{a}),\mathcal{G}_{2}( \mathfrak{a})):=(e^{-\mathfrak{a}_{1}},e^{-2\mathfrak{a}_{2}})\)_,_ subject to_ \[\sum_{i=1}^{10}L_{i}\mathfrak{a}_{i},\]4 \[\sum_{i=1}^{10}L_{i}\mathfrak{a}_{i},\]5 \[\sum_{i=1}^{10}L_{i}\mathfrak{a}_{i},\]6  
_where \(\mathcal{G}_{t}:\mathbb{R}^{2}\to\overline{\mathbb{R}},\ t\in\{1,2\}\), \(\Psi:\mathbb{R}^{2}\times\Omega_{1}\to\overline{\mathbb{R}}\), \(\mathbb{K}(\mathfrak{a},\mathfrak{u}_{2}):\mathbb{R}^{2}\times\Omega_{2}\to \overline{\mathbb{R}}\), \(\mathbb{L}(\mathfrak{a},\mathfrak{u}_{2}):\mathbb{R}^{2}\times\Omega_{2}\to \overline{\mathbb{R}}\) and \(z=(\mathfrak{a}_{1},\mathfrak{a}_{2})\in\mathbb{R}^{2}\).