## 2 Materials and Methods
In this approach, the values of the input parameters \(x_{i},\ i=\overline{1,n}\) are quantitatively measurable, and the values of the output parameters of the object \(\tilde{y}_{j}^{M},\ j=\overline{1,m}\) are immeasurable, they can be evaluated by a specialist, a decision-maker (DM) unclearly based on their experience, knowledge and intuition. Under these conditions, the mathematical relationship between input and output parameters, i.e., the model is obtained in the form of fuzzy regression equations:  
\[\tilde{y}_{j}^{M}=\tilde{f}_{j}^{M}(x_{1},\ldots,x_{n}) \tag{1}\]  
where ----fuzzy operator. Suppose that as a result of observing the operation of the object and the experiments carried out, \(k\) values of the input parameters \(x_{i}\Big{(}x_{1},\ i=\overline{1,n},\ 1=\overline{1,k}\Big{)}\) were obtained, and the corresponding fuzzy values of the output parameters \(\tilde{y}_{j}^{M}\Big{(}\tilde{y}_{j1}^{M},\ j=\overline{1,m},\ 1=\overline{1,k}\Big{)}\) were estimated by experts. Then, to build fuzzy models of this object, it is necessary to carry out the following two stages of solving the identification problem. Stage 1. Choose the structure of the function (structural identification):  
\[\tilde{y}_{j}^{M}=\tilde{f}_{j}^{M}(x_{1},\ldots,x_{n},\ \tilde{\alpha}_{0}, \ldots,\tilde{\alpha}_{n}),\ j=\overline{1,\ k} \tag{2}\]  
which corresponds to Equation (1). At this stage, the qualitative analysis of the object is of decisive importance, as a result of which the main parameters affecting the functioning of the object, their interrelationships, the influence of these parameters on the optimization criteria are revealed. To determine the structure of the model, you can use a fuzzy analog of the method of sequential inclusion of regressors [28]. Stage 2. Determine the estimates of the parameters of the selected function (2), for example, the values of fuzzy coefficients \(\tilde{\alpha}_{0},\ldots,\tilde{\alpha}_{n}\), i.e., the problem of parametric identification is being solved. For such an assessment, you can use the criterion of minimizing the deviation of the fuzzy values of the output parameter \(\tilde{y}_{j}^{M}\) obtained by the model from its sample fuzzy values obtained on the basis of expert judgment \(y_{j}^{c}\):  
\[\widetilde{R}_{j}=\min\sum_{l=1}^{k}\Big{(}\tilde{y}_{j1}^{c}-\tilde{y}_{j1}^{ M}\Big{)}^{2}=\min\sum_{l=1}^{k}\Big{(}\tilde{y}_{j1}^{c}-\tilde{y}_{j1}^{M}(x_{1 },\ldots,x_{n1},\ \tilde{\alpha}_{01},\ldots,\tilde{\alpha}_{n1})\Big{)}^{2} \tag{3}\]  
Note that when calculating criterion (3), operations are performed on fuzzy sets and numbers [4,6,9]. At this stage, the main questions are the choice of a method for estimating unknown parameters that ensure the adequacy of the model. In the general case, the fuzzy models obtained on the basis of this approach have the form:  
\[\tilde{y}_{j}^{M}=\tilde{\alpha}_{oj}+\sum_{i=1}^{n}\tilde{\alpha}_{ij}x_{ij} +\sum_{i=1}^{n}\sum_{k=i}^{n}\tilde{\alpha}_{ikj}x_{ij}+\ldots,\ j=\overline{1,m} \tag{4}\]  
Based on the set of level \(\alpha\ (\alpha\in[0,1],\ A_{\alpha}=\{x:x\in X,\ \mu_{A}(x)\geq\alpha\})\), the problem of estimating the fuzzy coefficients of the model can be reduced to the classical problemsof estimating the parameters of multiple regression equations using the idea of the least squares method. As a result of this approach, we obtain the values of the estimated coefficients of various levels \(\alpha_{q},\ q=\overline{1,L}-\alpha_{ij}^{aq}\). To obtain the coefficients \(\widetilde{\alpha}_{ij}\), the obtained values \(\alpha_{ij}^{aq}\) are combined:  
\[\widetilde{\alpha}_{ij}=\underbrace{U}_{\alpha\in[0.5+1]}\alpha_{ij}^{\alpha} \text{ or }\mu_{\widetilde{\alpha}_{ij}}\big{(}\alpha_{ij}\big{)}=\sup_{\alpha\in[0.1 +1]}\min\Big{\{}\alpha_{q},\ \mu_{\widetilde{\alpha}_{ij}}\big{(}\alpha_{ij}\big{)}\Big{\}}.\]  
The second approach is based on the use of logical rules for conditional inference.