## 3 Uncertain Design Parameters  
### Interval Analysis  
[PERSON] [5] presented the fundamentals of interval analysis. Any parameter \(p_{i}\) can be described as an interval number as follows:  
\[p_{i}^{l}=\Big{[}\underline{p}_{i},\ \overline{p}_{i}\Big{]}. \tag{18}\]  
This means that it can vary within certain limits, where \(\underline{p}_{i}\) is the lower bound and \(\overline{p}_{i}\) is the upper bound. The middle value of the interval parameter can be expressed as follows:  
\[p_{i}^{c}=\frac{1}{2}\Big{(}\underline{p}_{i}+\overline{p}_{i}\Big{)}, \tag{19}\]  
and the radius value is determined as follows:  
\[\Delta p_{i}=\frac{1}{2}\Big{(}\underline{p}_{i}-\overline{p}_{i}\Big{)}. \tag{20}\]  
Operations on interval numbers require calculations that consider all possible combinations of upper and lower limits of the considered quantities. The result is the interval whose bounds are the smallest and largest result values. The basic operations on sample interval numbers \(x^{l}=[\underline{x},\overline{x}]\) and \(y^{l}=\Big{[}\underline{y},\overline{y}\Big{]}\) are calculated as follows:  
\[x^{l}+y^{l}=\Big{[}\underline{x}+\underline{y},\overline{x}+\overline{y} \Big{]}, \tag{21}\]  
\[x^{l}-y^{l}=\Big{[}\underline{x}-\overline{y},\overline{x}-\underline{y} \Big{]}, \tag{22}\]  
\[x^{l}.y^{l}=\Big{[}\min\Big{(}\underline{x}\cdot\underline{y},\underline{x} \cdot\overline{y},\overline{x}\cdot\underline{y},\overline{x}\cdot\overline{y} \Big{)},\max\Big{(}\underline{x}\cdot\underline{y},\underline{x}\cdot \overline{y},\overline{x}\cdot\underline{y}\cdot\overline{y}\Big{)}\Big{]}, \tag{23}\]  
\[\frac{x^{l}}{y^{l}}=\frac{\big{[}\underline{x},\overline{x}\big{]}}{\big{[} \underline{y},\overline{y}\big{]}}=[\underline{x},\overline{x}]\cdot\Bigg{[} \frac{1}{\overline{y}},\frac{1}{\underline{y}}\Bigg{]},\ \Big{[}\overline{y},\underline{y}\Big{]}\ \
otin 0. \tag{24}\]  
The number of combinations needed to consider for selecting the smallest and largest values of the increments is 2r. However, when the number of parameters or their variability is high, the approximation of the limits of the objective function using a first-order Taylor series expansion may be insufficient. If a higher accuracy is required, an approximation using the first three terms of the Taylor series can be used (second-order Taylor series expansion):  
\[f(\mathbf{p})=f(\mathbf{p}^{c})+\sum_{i=1}^{r}\frac{\partial f(\mathbf{p}^{c}) }{\partial p_{i}}\Delta p_{i}+\frac{1}{2}\sum_{i=1}^{r}\sum_{j=1}^{r}\frac{ \partial^{2}f(\mathbf{p}^{c})}{\partial f(p_{i})\partial f(p_{j})}\Delta p_{i} \Delta p_{j} \tag{32}\]  
where \(\partial^{2}f(\mathbf{p}^{c})/\left(\partial p_{i}\partial p_{j}\right)\) is the second-order derivative with respect to parameters \(p_{i}\) and \(p_{j}\) (the so-called sensitivity of the second order with respect to the simultaneous change in two design parameters \(p_{i}\) and \(p_{j}\)). Derivatives of the second order form the following Hessian matrix:  
\[ \begin{bmatrix}\frac{\partial^{2}f(\mathbf{p}^{c})}{\partial f(p_{i}^{2})} \frac{\partial^{2}f(\mathbf{p}^{c})}{\partial p_{i}\partial p_{2}}&\cdots& \frac{\partial^{2}f(\mathbf{p}^{c})}{\partial p_{i}\partial p_{r}}\\ \frac{\partial^{2}f(\mathbf{p}^{c})}{\partial p_{2}p_{1}}&\frac{\partial^{2}f (\mathbf{p}^{c})}{\partial f(p_{2}^{c})}&\cdots&\frac{\partial^{2}f(\mathbf{p} ^{c})}{\partial p_{i}\partial p_{2}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial^{2}f(\mathbf{p}^{c})}{\partial p_{i}\partial p_{1}}&\frac{ \partial^{2}f(\mathbf{p}^{c})}{\partial p_{i}\partial p_{2}}&\cdots&\frac{ \partial^{2}f(\mathbf{p}^{c})}{\partial f/p_{2}^{c}}\\ \end{bmatrix} . \tag{33}\]  
In this case, increments in Equation (27) can be written as follows:  
\[ \begin{bmatrix}\Delta f\left(\mathbf{p}^{l}\right),\overline{\Delta f}\left( \mathbf{p}^{l}\right)\end{bmatrix} = \begin{bmatrix}\text{min}\Delta f\left( \mathbf{p}^{l}\right),\text{max}\Delta f\left(\mathbf{p}^{l}\right)\end{bmatrix} =