## 4 Method of Determining Realistic Tolerance Solutions
The degree of robustness \(r(x^{*})\) of the chosen control value \(x^{*}\) for the range \([\underline{a},\overline{a}]\) of possible values of the uncertain disturbance \(a\) is a fraction \((r(x^{*})\in[0,1])\) informing how large part of the range \([\underline{a},\overline{a}]\) for the value \(x^{*}\) will give the product \(ax^{*}\) value in the required tolerance corridor \([\underline{b},\overline{b}]\). The robustness \(r(x^{*})\) is simply the ratio of lengths of two segments marked in \[a\to a(\gamma_{a})=\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\ \ \gamma_{a}\in[0,1]\]1  
To determine the robustness degree, follow the steps. 1. For the \(x^{*}\) value, calculate the upper \(a^{U}(x^{*})=\overline{b}/x^{*}\) and the lower \(a^{L}(x^{*})=\underline{b}/x^{*}\) coordinates of the points on the upper \(ax=\overline{b}\) and lower \(ax=\underline{b}\) border of the tolerance corridor \([\underline{b},\overline{b}]\). 2. Determine the common part of the intervals \([a^{L}(x^{*}),a^{U}(x^{*})]\) and \([\underline{a},\overline{a}]\): \[a\to a(\gamma_{a})=\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\ \ \gamma_{a}\in[0,1]\]2 (45)
3. Calculate the robustness degree for the considered value of \(x^{*}\): \[a\to a(\gamma_{a})=\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\ \ \gamma_{a}\in[0,1]\]3 (46)  
\[a\to a(\gamma_{a})=\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\ \ \gamma_{a}\in[0,1]\]4 Illustration of the computation method of \(r(x^{*})\). In the following, the above algorithm will be applied to the solution of the example of [PERSON] and [PERSON] in the invited talk \"Uncertainties in mathematical analysis and their use in optimization\" presented at Congresso Brasileiro de Sistemas Fuzzy in Sorocaba, Brasil, 2010. **Example 1**. : _Let's consider a system with inputs \(a\) and \(x\) and the output \(b\) realizing the relationship \(ax=b\). We have the following knowledge about the real values of the input \(a\) : \(a\in[\underline{a},\overline{a}]=[2,3]\). The input \(a\) is a generator of uncertainty (disturbance of the control process), which we have no influence on. The output \(b\) is the control target for which a realistic, tolerant requirement \(b\in[\underline{b},\overline{b}]=[3,6]\) was made. Tolerance \([\underline{b},\overline{b}]\) is not a disturbance, but a facilitation of the control process. The task here is to determine one, or perhaps a set of control values \(x\), which will allow the control objective to be achieved in the best possible way. In summary, our knowledge of the system is given by (47)._  
\[a\to a(\gamma_{a})=\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\ \ \gamma_{a}\in[0,1]\]5  
_Step 1._  
\[a\to a(\gamma_{a})=\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\ \ \gamma_{a}\in[0,1]\]6 \[a\to a(\gamma_{a})=\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\ \ \gamma_{a}\in[0,1]\]7  
_Step 2._  
\[a\to a(\gamma_{a})=\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\ \ \gamma_{a}\in[0,1]\]8 \[a\to a(\gamma_{a})=\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\ \ \gamma_{a}\in[0,1]\]9  
_Step 3._  
\[b\to b(\gamma_{b})=\underline{b}+\gamma_{b}(\overline{b}- \underline{b}),\ \ \gamma_{b}\in[0,1]\]0  
_A minimum of \(x(\gamma_{a},\gamma_{b})\) is obtained for \(\gamma_{a}=1\), \(\gamma_{b}=0\) and a maximum of \(x(\gamma_{a},\gamma_{b})\) for \(\gamma_{a}=0\), \(\gamma_{b}=1\). Figure 6 illustrates the meaning of the bag \(BG_{X}\) of possible system states \((a,x=b/a,b)\) satisfying the control objective, and the meaning of the span \(SP_{BG_{X}}=X_{\text{pass}}=[\underline{x},\overline{x}]\)._  
\[b\to b(\gamma_{b})=\underline{b}+\gamma_{b}(\overline{b}- \underline{b}),\ \ \gamma_{b}\in[0,1]\]1 Illustration of solution zones (tolerant control zones) in the Lodwick anomaly problem (47).