## 4 Method of Determining Realistic Tolerance Solutions
As shown in Figure 6, the selection of the control value \(x\in X_{FR}=[1.5,2]\) provides fully robust tolerance control, because regardless of what value the disturbance \(a\) from the range \([\underline{a},\overline{a}]=[2,3]\) takes, the output \(b\) will always be included in the desired range \([\underline{b},\overline{b}]=[3,6]\). If we select the control value \(x\) in the partly robust zone \(1\), \(X_{PR1}=[1,1.5]\), there is no certainty that \(b\) will be included in the tolerance zone \([3,6]\). For example, if we choose \(x=1.2\), then the output \(b\) will be included in \([3,6]\) only when the disturbance \(a\) is included in the range \([2.5,3]\). If \(a\) is in the range \([2,2.5]\) then the output \(b<\underline{b}=3\). Hence the value \(x=1.2\) has the robustness \(r(1.2)=0.5\). It means that for this \(x\)-value one half of the disturbance range \(a\in[2.5,3]\) results in hitting the value \(ax\) in the tolerance corridor \([\underline{b},\overline{b}]=[3,6]\). On the other hand, the second half \([2,2.5]\) excludes such a hit. If the true disturbance value is in the range \([2.5,3]\), the hit will occur, if it is in the range \([2,2.5]\), there will be no hit and the control objective will not be met. The robustness degree \(r(x)\) of the selected \(x\)-value in zone \(1\) can be determined from Formula (51). \[b\to b(\gamma_{b})=\underline{b}+\gamma_{b}(\overline{b}- \underline{b}),\ \ \gamma_{b}\in[0,1]\]2  
It is easy to see that for zone \(1\), \(r(x)\in[0,1]\), has a fractional value denoting incomplete robustness for the disturbance \(a\). In the \(X_{FR}=[1.5,2]\) zone, every control value \(x\) is completely robust (\(r(x)=1\))for all possible disturbance values \(a\in[2,3]\) providing \(b\in[3,6]\). In the partly robust zone \(2\), \(X_{PR2}=[2,3]\) we have a partial robustness \(r(x)\) for the individual values of \(x\), as seen in Formula (52). \[b\to b(\gamma_{b})=\underline{b}+\gamma_{b}(\overline{b}- \underline{b}),\ \ \gamma_{b}\in[0,1]\]3  
As shown by the MIA solution of the interval equation \([\underline{a},\overline{a}]x=[\underline{b},\overline{b}]\) considered by [PERSON], there are no anomalies in this equation if it is solved by using the algorithm presented in this section. The middle zone \(x\in[1,3]\) shown in Figure 7 does not contain any empty and mysterious sub-zones. It comprises \(2\) zones of partial robustness and \(1\) zone of full robustness to disturbance (uncertainty) \(a\). The left outer zone \(x\leq 1\) is the zone of completely no tolerance control, as is the right zone \(x\geq 3\). Figure 6 showing the distribution of the robustness degree \(r(x)\) explains [PERSON]'s interval anomaly. The discovery of [PERSON]'s anomaly was caused by the fact that the studied equation \([2,3]X=[3,6]\) was solved with the commonly used assumption that the solution \(X\) of  
\[b\to b(\gamma_{b})=\underline{b}+\gamma_{b}(\overline{b}- \underline{b}),\ \ \gamma_{b}\in[0,1]\]4 Distribution of the robustness degree \(r(x)\) of control \(x\) as illustration explaining [PERSON]’s interval equation anomaly. this equation is the interval \([\underline{x},\overline{x}]\). In other words, the equation \([2,3][\underline{x},\overline{x}]=[3,6]\) was solved, which gave the result \([\underline{x},\overline{x}]=[1.5,2]\). Such an assumption is incorrect and for many interval equations it does not provide generally feasible solutions in the form of proper intervals. For example, in the case of the equation \([1,3][\underline{x},\overline{x}]=[4,6]\) we obtain the \"solution\" \([\underline{x},\overline{x}]=[4,2]\), which according to [PERSON] and [PERSON] [7] is \"empty\" solution set. **Example 2**. : _This example concerns the equation \([3,5]X=[7,9]\), in which, assuming that \(X\) is the interval \([\underline{x},\overline{x}]\), tolerance control cannot be realized at all, because the obtained solution is an improper interval \([\underline{x},\overline{x}]=[2.3,1.8]\). Next, it will be shown what solution can be obtained with the use of MIA and the previously presented algorithm.