## 2 Problem Statement  
### Interval Mathematics  
An \(n\)-dimensional interval vector is represented as \(\mathbf{a}\in\mathbf{a}^{I}=[a_{1}^{I},\cdots,a_{n}^{I}]\in\mathbb{R}^{N}\). The element in the vector denotes the interval parameter, which has the following range :  
\[a_{i}\in a_{i}^{I}=\left[\underline{a_{i}},\overline{a_{i}}\right]=\left\{a_{ i}\in\mathbb{R},\underline{a_{i}}\leq a_{i}\leq\overline{a_{i}}\right\} \tag{1}\]  
where the superscript I denotes that this parameter is an interval parameter, \(\underline{a_{i}}\) represents the lower bound of the interval parameter (LB), \(\overline{a_{i}}\) represents the upper bound of the interval parameter (UB), and \(\mathbb{R}\) denotes the domain of real numbers. For the interval parameter \(a_{i}^{I}\), \(a_{i}^{C}\) denotes its midpoint, \(a_{i}^{W}\) denotes its width, and \(a_{i}^{R}\) denotes its radius. \(a_{i}^{C}\), \(a_{i}^{W}\), and \(a_{i}^{R}\) can be determined from the following formula:  
\[a_{i}^{C}=\frac{a_{i}+\overline{a_{i}}}{2},a_{i}^{W}=\overline{a_{i}}-\underline {a_{i}},a_{i}^{R}=\frac{\overline{a_{i}}-\underline{a_{i}}}{2} \tag{2}\]  
In addition, the uncertainty level index \(\mu_{i}\) is introduced to describe the uncertainty degree of the interval parameter, which can be determined using the following formula:  
\[\mu_{i}=\frac{a_{i}^{R}}{|a_{i}^{C}|}\times 100\% \tag{3}\]  
Each interval parameter can also be expressed as follows:  
\[a_{i}^{I}=[1-\mu_{i},1+\mu_{i}]\times a_{i}^{C} \tag{4}\]  
The level of uncertainty can be classified into large-range uncertainty/small-range uncertainty based on the magnitude of \(\mu_{i}\). Note that \(a_{i}^{C}\
eq 0\). For interval calculations, the core is the interval arithmetic operators extended from scalars. For example, the addition (+), subtraction (-), multiplication (\(\times\)), and division (/) of two intervals \(x^{I}\) and \(y^{I}\) can be expressed as follows:  
\[ \begin{split} x^{I}+y^{I}&=\left[x+\underline{y}, \overline{x}+\overline{y}\right],\\ x^{I}-y^{I}&=\left[\underline{x}-\overline{y}, \overline{x}-\underline{y}\right],\\ x^{I}\times y^{I}&=\left[\min\{\underline{x}\cdot \underline{y},\underline{x}\cdot\overline{y},\overline{x}\cdot\underline{y}, \overline{x}\cdot\overline{y}\},\max\{\underline{x}\cdot\underline{y}, \underline{x}\cdot\overline{y},\overline{x}\cdot\underline{y},\overline{x} \cdot\overline{y}\}\right],\\ x^{I}/y^{I}&=x^{I}\times\left[\frac{1}{\overline{y }},\frac{1}{\underline{y}}\right],0\
otin\left[\underline{y},\overline{y} \right]\end{split}  \tag{5}\]  
However, it must be noted that in the classical interval algorithm (CIA), the result of interval operation will be expanded if an interval number occurs several times (\(\geq\)2). For example, bounds of the function \(f\left(x^{I}\right)=\sqrt{x^{2}-x+0.5}/\sqrt{x^{2}+0.5}\) predicted by CIA are shown in the blue bar chart in Figure 1a when the values of interval variable \(x^{I}\) are in different ranges. The range is much larger than the exact value due to the interval variable being treated independently for each occurrence. In practical engineering problems, the calculation will lose its application value if the interval result is exaggerated too much. To avoid this phenomenon, the concept of EUI was proposed. Based on this concept, the \(i\)-th interval parameter \(a_{i}^{I}\) can be expressed as follows:  
\[a_{i}^{I}=a_{i}^{C}+\Delta a_{i}^{I}=a_{i}^{C}+a_{i}^{R}e_{i}^{I} \tag{6}\]  
where \(e_{i}^{I}=[-1,1]\) is the EUI associated with the interval parameter \(a_{i}^{I}\), which is used to track the uncertainty of this parameter during the calculation to avoid interval expansion. For the same function \(f\left(x^{I}\right)=\sqrt{x^{2}-x+0.5}/\sqrt{x^{2}+0.5}\), the bounds predicted using the improved interval algorithm (IIA) are shown in the yellow bar chart in Figure 1b, which are consistent with the exact bounds. Because IIA takes into account the dependence of the interval parameter, the expansion phenomenon caused by the classical interval calculation method can be avoided effectively.