## 3 Results
As a result of processing experimental-statistical and expert information using the fuzzy modeling method described in Section 3.1, the following structure of fuzzy models was identified, describing the product quality of the R-1 hydrotreating reactor in the form of fuzzy multiple regression equations:  
\[\widetilde{y}_{j}=a_{0j}x_{ij}+\sum_{i=1}^{5}a_{ij}x_{ij}+\sum_{i=1}^{5}\sum_ {k=i}^{5}a_{ijk}x_{ij}x_{kj},\ j=\overline{2,4} \tag{7}\]  
where \(\widetilde{y}_{2}\) is the unsaturated hydrocarbons in the product, i.e., hydrogenated products, which are characterized by indistinctness (should be no more, i.e., \(\widetilde{\leq}1\%\)); \(\widetilde{y}_{3}\)--sulfur in the hydrogenated product (\(\widetilde{\leq}0.00005\%\)); \(\widetilde{y}_{4}\)--water-soluble acids and alkalis in the hydrogenated product (\(\widetilde{=}0\%\)); The admissible fuzzy values of the hydrogenated product quality indicators are indicated in brackets; \(x_{i},\ i=\overline{1,5}\)--input, operating parameters of the hydrotreating reactor, i.e., respectively, the volume of raw materials, pressure, temperature, space velocity and circulation of HCG; \(\widetilde{a}_{0j}\), \(\widetilde{a}_{ij}\), \(\widetilde{a}_{ikj}\), \(i=\overline{1,5}\)--parameters, fuzzy regression coefficients subject to regression identification. To identify unknown parameters (regression coefficients) of the model (7): \(\widetilde{a}_{ij}\) (\(i=\overline{0,5}\), \(j=\overline{2,5}\)) and \(\widetilde{a}_{ikj}\) (\(i,k=\overline{1,5}\), \(j=\overline{2,5}\)), the membership functions of fuzzy sets describing the qualities of the hydrogenate are divided into the following sets of level \(\alpha\): \(\alpha=0.5\); \(0.85\); \(1\). Since in our case the membership function has a bell-shaped form, the values of fuzzy parameters at five points \(\alpha=0.5\); \(0.85\) (left); \(1\); \(0.85\); \(0.5\) (right). The values of the input, mode \(x_{ij},\ i,j=\overline{1,5}\) and output \(\widetilde{y}_{2},\widetilde{y}_{3},\widetilde{y}_{4}\) parameters for each selected \(\alpha\) level are observed. Thus, we obtained models describing the quality of the product from the outlet of the R-1 reactor in the form of multiple regression for each \(\alpha\) level. Since the obtained equations have the form of regression equations, the problem of identifying their unknown coefficients \(a_{ij}^{a_{q}},i=\overline{0,5},\ j=\overline{2,4},\ \ q=\overline{1,3}\) can be solved using known methods of parametric identification, for example, using the least squares method. In this work, to identify the regression coefficients, the REGRESS program package was used, which, based on modified least squares methods, allows one to determine the regression coefficients of linear and nonlinear regression models with any number of input parameters \(x_{i},i=\overline{1,n}\). Thus, after parametric identification, mathematical models describing the influence of input, operating parameters \(x_{i},\ i=\overline{1,5}\) on the quality of the hydrogenated product, i.e., on the content of unsaturated hydrocarbons (\(\widetilde{y}_{2}\)), sulfur (\(\widetilde{y}_{3}\)) and water-soluble acids and alkalis (\(\widetilde{y}_{4}\)) for each \(\alpha\) level have the form:  
\[\begin{array}{l}y_{2}=\left(\frac{0.5}{0.05}+\frac{0.85}{0.07}+\frac{1}{0.0 8}+\frac{0.85}{0.09}+\frac{0.5}{0.095}\right)-\left(\frac{0.5}{0.0215}+\frac{0.85}{0.0029}+\frac{1}{0.00324}+\frac{0.85}{0.00375}+  
furnaces are divided into a limited number of zones with the same radiation properties. This approach in calculating the furnace can provide sufficient accuracy; to increase the accuracy, it is necessary to increase the number of zones. However, this method is rather complicated and the collection of the necessary information for its application in practice is also difficult. To simulate the operation of industrial furnaces in an interactive mode and to quickly obtain the necessary information and results, simple models are required.