## 4 Method of Determining Realistic Tolerance Solutions  
A static multiplicative system is given with inputs \(a\) and \(x\) and with output \(b\) realizing the operation \(ax=b\). We have an approximate knowledge about the true value \(a\): \(a\in[\underline{a},\overline{a}]\), \(\underline{a}\leq\overline{a}\), and about \(b\): \(b\in[\underline{b},\overline{b}]\), \(\underline{b}\leq\overline{b}\). We have no knowledge about the value of \(x\) and we want to gain this knowledge. In this situation, the variables \(a\) and \(b\) become information inputs and \(x\) information output. The knowledge we can get about \(x\) depends on the knowledge about \(a\) and \(b\). We can present the whole problem in the form (39). \[ax=b,\ \ a\in[\underline{a},\overline{a}],\ \ b\in[ \underline{b},\overline{b}],\ \ x=? \tag{39}\] \[a\to a(\gamma_{a})=\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\ \ \gamma_{a}\in[0,1]\] \[b\to b(\gamma_{b})=\underline{b}+\gamma_{b}(\overline{b}- \underline{b}),\ \ \gamma_{b}\in[0,1]\] \[ax=b\to a(\gamma_{a})x(\gamma_{a},\gamma_{b})=b(\gamma_{b})\]  
With the use of MIA, it is possible to determine the knowledge about the output \(x\), (40). \[x(\gamma_{a},\gamma_{b})=\frac{b(\gamma_{b})}{a(\gamma_{a})}=\frac{\underline{ b}+\gamma_{b}(\overline{b}-\underline{b})}{\underline{a}+\gamma_{a}(\overline{a}- \underline{a})},\ \ \gamma_{a},\gamma_{b}\in[0,1] \tag{40}\]  
For one selected numerical \(\gamma_{a}\) and \(\gamma_{b}\) value, Formula (40) determines one value of \(a(\gamma_{a})\) and \(b(\gamma_{b})\) and one value \(x(\gamma_{a},\gamma_{b})\) associated with them. In this sense, Formula (40) is a mathematical model of the true values of \(a\), \(b\) and \(x\). However, for all values of \(\gamma_{a}\in[0,1]\) and \(\gamma_{b}\in[0,1]\), this formula determines \(BG_{X}\), that is the bag of all possible values of \(x\) corresponding to all combinations of values \(\gamma_{a}\) and \(\gamma_{b}\), Formula (41). \[BG_{X}=\{x(\gamma_{a},\gamma_{b})=b(\gamma_{b})/a(\gamma_{a})\ |\ \forall\gamma_{a}\in[0,1],\forall\gamma_{b}\in[0,1]\} \tag{41}\]  
Given the formula of \(BG_{X}\), we can determine the set \(X=[\underline{x},\overline{x}]\) of all possible values of \(x\), which does not contain the repetitions of identical values, (42). \[SP_{BG_{X}}=X=[\underline{x},\overline{x}]=\left[\min_{\gamma_{a},\gamma_{b}} x(\gamma_{a},\gamma_{b}),\max_{\gamma_{a},\gamma_{b}}x(\gamma_{a},\gamma_{b}) \right],\ \gamma_{a},\gamma_{b}\in[0,1] \tag{42}\]  
Because the set \(X\) is a set in 1D-space, it can be represented in RDM terms as (43). \[X=X_{\text{poss}}=\{x(\gamma_{x})=\underline{x}+\gamma_{x}(\overline{x}- \underline{x})\ |\ \forall\gamma_{x}\in[0,1]\} \tag{43}\]In general, the set \(X_{\text{pass}}\) consists of 2 subsets: \(X_{FR}\) - set of fully robust (adjusted) \(x\)-values) and \(X_{PR}\) - set of \(x\)-values that are only partly robust, partly adjusted to the disturbance \(a\in[\underline{a},\overline{a}]\). Both subsets are expressed by (44). \[X_{\text{poss}}=X_{FR}\cup X_{PR} \tag{44}\] \[X_{FR}=\{x\mid\forall a\in[\underline{a},\overline{a}]:\;ax=b \in[\underline{b},\overline{b}]\}\] \[a\to a(\gamma_{a})=\underline{a}+\gamma_{a}(\overline{a}- \underline{a}),\ \ \gamma_{a}\in[0,1]\]0  
The authors' research shows that \(X_{FR}\) sets are rather rare in real problems and often only \(X_{PR}\) sets occur. Now, the algorithm of solving the tolerance control problem will be given. 1. Formulate the interval sets \([\underline{a},\overline{a}]\) and \([\underline{b},\overline{b}]\) in terms of MIA. 2. Determine the bag model \(BG_{X}=\{x(\gamma_{a},\gamma_{b})\mid\forall\gamma_{a}\in[0,1],\forall\gamma _{b}\in[0,1]\}\). 3. Specify bag span \(SP_{BG_{X}}=X_{\text{poss}}=[\underline{x},\overline{x}]\). 4. Determine the degrees of robustness \(r(x)\) for the individual \(x\)-values of the \(X_{\text{poss}}\) set to the disturbance \(A=[\underline{a},\overline{a}]\). 5. Determine the optimal tolerance control \(x_{opt}\) value based on the selected decision evaluation criterion.