## 4 Method of Determining Realistic Tolerance Solutions
The robustness distribution \(r(x)\) of possible controls \(x\), for \(x>0\), is given by Formula (75) and is shown in \[BG_{X}=\{x(\gamma_{a},\gamma_{b})=b(\gamma_{b})/a(\gamma_{a})\ |\ \forall\gamma_{a}\in[0,1],\forall\gamma_{b}\in[0,1]\} \tag{41}\]8  
\[BG_{X}=\{x(\gamma_{a},\gamma_{b})=b(\gamma_{b})/a(\gamma_{a})\ |\ \forall\gamma_{a}\in[0,1],\forall\gamma_{b}\in[0,1]\} \tag{41}\]9 \[SP_{BG_{X}}=X=[\underline{x},\overline{x}]=\left[\min_{\gamma_{a},\gamma_{b}} x(\gamma_{a},\gamma_{b}),\max_{\gamma_{a},\gamma_{b}}x(\gamma_{a},\gamma_{b}) \right],\ \gamma_{a},\gamma_{b}\in[0,1] \tag{42}\]0 \[SP_{BG_{X}}=X=[\underline{x},\overline{x}]=\left[\min_{\gamma_{a},\gamma_{b}} x(\gamma_{a},\gamma_{b}),\max_{\gamma_{a},\gamma_{b}}x(\gamma_{a},\gamma_{b}) \right],\ \gamma_{a},\gamma_{b}\in[0,1] \tag{42}\]1  
As shown by Formula (75) and Figure 15, the realization of the fully robust tolerance control for \(x>0\) is not possible. In this area, at most robustness \(r(x)=0.148\) can be obtained on condition that the disturbance \(a\) is contained only in its upper range \(a\in[\Delta,2]\). However, the full interval of this disturbance is \([-1,0,2]\). In order to decide on the choice of the optimal control value \(x\), it is necessary to consider the total distribution \(r(x)\) presented in \[SP_{BG_{X}}=X=[\underline{x},\overline{x}]=\left[\min_{\gamma_{a},\gamma_{b}} x(\gamma_{a},\gamma_{b}),\max_{\gamma_{a},\gamma_{b}}x(\gamma_{a},\gamma_{b}) \right],\ \gamma_{a},\gamma_{b}\in[0,1] \tag{42}\]2  
The interpretation of the results shown in Figure 16 is as follows. In the system under consideration, it is not possible to implement fully robust tolerance control. Only partly robust TC can be realized. PR-TC can be obtained by either negative or positive \(x\) controls excluding the range \(x\in[-7,0,3.5]\). The most preferable control is \(x=4.5\), giving the robustness of \(0.148\) to find \(b\) in the range of tolerance corridor \(b\in=[7,9]\). If disturbance \(a\) were \(a=0\), then any influence on output \(b\) would become impossible. However, the probability of such an event is infinitely small. **Example 5**. : _Real-life case. A doctor recommended to a 70-year old patient that his daily diet should contain [25; 40] \(\mu\)g of vitamin \(D\) (tolerance corridor \([\underline{b},\overline{b}]\)). To supply him with this vitamin, he recommended eating sea fish (such as: mackerel, herring) containing healthy fats. The content of vitamin \(D\) in such fish is not constant and varies in the range of [5; 10] \(\mu\)g for every \(100\) g of fish  
\[SP_{BG_{X}}=X=[\underline{x},\overline{x}]=\left[\min_{\gamma_{a},\gamma_{b}} x(\gamma_{a},\gamma_{b}),\max_{\gamma_{a},\gamma_{b}}x(\gamma_{a},\gamma_{b}) \right],\ \gamma_{a},\gamma_{b}\in[0,1] \tag{42}\]3 Robustness distribution \(r(x)\) of control \(x\) for possible changes of the disturbance \(a\in[-1,0,2]\) of the system \([-1,0,2]x=[7,9]\), together with the optimal control value \(x_{\text{\emph{opt}}}=4.5\). \[SP_{BG_{X}}=X=[\underline{x},\overline{x}]=\left[\min_{\gamma_{a},\gamma_{b}} x(\gamma_{a},\gamma_{b}),\max_{\gamma_{a},\gamma_{b}}x(\gamma_{a},\gamma_{b}) \right],\ \gamma_{a},\gamma_{b}\in[0,1] \tag{42}\]4 Robustness distribution \(r(x)\) of the tolerance control \(x\) for \(x>0\). (uncertainty \([\underline{a},\overline{a}]\)). Neither too little nor too much vitamin \(D\) is recommended for the patient for health reasons.