## 4 Method of Determining Realistic Tolerance Solutions
The knowledge about the system is given by Equations (53)._  
\[b\to b(\gamma_{b})=\underline{b}+\gamma_{b}(\overline{b}- \underline{b}),\ \ \gamma_{b}\in[0,1]\]5  
_Step 1._  
\[b\to b(\gamma_{b})=\underline{b}+\gamma_{b}(\overline{b}- \underline{b}),\ \ \gamma_{b}\in[0,1]\]6 \[b\to b(\gamma_{b})=\underline{b}+\gamma_{b}(\overline{b}- \underline{b}),\ \ \gamma_{b}\in[0,1]\]7  
_Step 2._  
\[b\to b(\gamma_{b})=\underline{b}+\gamma_{b}(\overline{b}- \underline{b}),\ \ \gamma_{b}\in[0,1]\]8 \[b\to b(\gamma_{b})=\underline{b}+\gamma_{b}(\overline{b}- \underline{b}),\ \ \gamma_{b}\in[0,1]\]9  
_Step 3._  
\[ax=b\to a(\gamma_{a})x(\gamma_{a},\gamma_{b})=b(\gamma_{b})\]0  
_Note that \(SP_{BG_{X}}=[1.4,3]\) is a normal, proper (not improper) interval. Figure 8 shows the set of possible states of the system \((a,b,x=b/a)\) in the projection into the space \(A\times X\) with the values of the third variable \(b=ax\) plotted._  
\[ax=b\to a(\gamma_{a})x(\gamma_{a},\gamma_{b})=b(\gamma_{b})\]1 The set \(S\) of possible states \((a,b,x)\) of the system \(ax=b\), \(a\in[3,5]\), \(b\in[7,9]\), \(x\in[1.4,3]=X_{\text{poss}}\), in 2D-space \(A\times X\). In order to determine the \(x\)-value for which the robustness \(r(x)\) to the disturbance \(a\) is the greatest, it is enough to examine \(r(x)\) at 4 characteristic points (L-lower, U-upper) \(x_{LL}=\underline{b}/\underline{a}=2.33\), \(x_{LU}=\underline{b}/\overline{a}=1.4\), \(x_{UL}=\overline{b}/\underline{a}=3\), \(x_{UU}=\overline{b}/\overline{a}=1.8\). Then we rank these values, as shown in (57). \[ax=b\to a(\gamma_{a})x(\gamma_{a},\gamma_{b})=b(\gamma_{b})\]2  
Figure 8 shows that for the outer values \(x_{LU}=1.4\), \(x_{UL}=3\) robustness \(r(x)=0\). Hence the greatest robustness can have the inner characteristic values \(x_{LU}=1.8\), \(x_{LL}=2.33\). The robustness for these \(x\)-values can be calculated from the Formula (58), which applies to the inner, second \(x\)-control zone. \[ax=b\to a(\gamma_{a})x(\gamma_{a},\gamma_{b})=b(\gamma_{b})\]3  
The calculation gives the results \(r(x=1.8)=0.556\), \(r(x=2.33)=0.429\). The distribution of the robustness for individual possible controls \(x\) is shown in \[ax=b\to a(\gamma_{a})x(\gamma_{a},\gamma_{b})=b(\gamma_{b})\]4  
As the distribution of the robustness \(r(x)\) on Figure 9 shows, it is impossible to obtain full robustness for the disturbance change. This situation occurs in the case of many problems described by the interval equation \([\underline{a},\overline{a}]x=[\underline{b},\overline{b}]\). This is a frequent situation and hence realistic. Uncertainty problems where \(r(x)=1\) can be obtained are rare in practice because real problems usually have more than just 1 uncertainty generator. However, even when full robustness cannot be obtained, we should not wiring our hands, because the incomplete robustness can be maximized by appropriate selection of \(x\)-control. As shown in Example 2, the optimal value of \(x\) can be detected. In Examples 1 and 2, there are intervals \([\underline{a},\overline{a}]\) and \([\underline{b},\overline{b}]\) that do not contain zero. Examples with one unknown \(x\) with intervals containing zero will be shown in the following. In contrast, these intervals will be represented as \([\underline{a},0,\overline{a}]\) and \([\underline{b},0,\overline{b}]\). **Example 3**. : _In this example, an equation of the form \([\underline{a},\overline{a}]x=[\underline{b},0,\overline{b}]\) in which the tolerance corridor contains zero will be solved.