**2.1 Representation of fuzzy if: then rules and possibility-measure-based inference algorithm**

The framework of the knowledge base relies on the fuzzy interpretation of production rules [20]:

$$\begin{array}{l} \text{ $R^k$ :} \, IF \, \mathbf{x\_1} \text{ is  $\tilde{\mathbf{A}}\_{\mathbf{k1}}$  and  $\mathbf{x\_2}$  is  $\tilde{\mathbf{A}}\_{\mathbf{k2}}$  and  $\dots$  and  $\mathbf{x\_m}$  is  $\tilde{\mathbf{A}}\_{\mathbf{km}}$  \text{ THEN} \\\\ \mathbf{y\_1} \text{ is  $\tilde{\mathbf{B}}\_{\mathbf{k1}}$  and  $\mathbf{y\_2}$  is  $\tilde{\mathbf{B}}\_{\mathbf{k2}}$  and  $\dots$  and  $\mathbf{y\_{kl}}$  is  $\tilde{\mathbf{B}}\_{\mathbf{kl}}$ ,  $k = \overline{\mathbf{1}, K}$  \end{array} \tag{1}$$

where *xi*, *i* ¼ 1, *m* and *yj* , *j* ¼ 1, *l* are input and output variables, rule antecedents

*A*~ *kj* and consequents *B*~*kj* are defined by using fuzzy sets, and *k* is the number of rules. Inputs and outputs of the rule are linguistic data.

Main steps of the applied fuzzy-measure-based reasoning algorithm are as follows:


If the sign is " <sup>¼</sup> " and *<sup>λ</sup><sup>k</sup>* <sup>¼</sup> <sup>1</sup> � *Poss* <sup>~</sup>*vk*j*a*~*jk*ÞÞ � *cf <sup>k</sup>*, then (2)

$$
\lambda\_{jk} = \left(\mathbf{1} - \text{Pos}\left(\tilde{v}\_k | \tilde{a}\_{jk}\right)\right) \cdot \mathbf{c} f\_k. \tag{3}
$$

If the sign is "6¼", then *Poss* is determined as

$$\operatorname{Pos}(\bar{v}|\bar{a})) = \max\_{\mathcal{Y}} \min(\mu\_{\bar{v}}(\mathcal{y}), (\mu\_{\bar{a}}(\mathcal{y})) \in [0, 1], \tau\_{\bar{\mathcal{Y}}} = \min(\lambda\_{\bar{\mathcal{Y}}}). \tag{4}$$

Here, one of the main elements of logistic inference is demonstration of the object. Value of each *wi* object consists of its linguistic value and the confidence degree of the linguistic value. Together, they constitute a pair, *vi*,*cfi* .

3.For each rule, the following computation is performed

$$R\_{\circ} = \left(\min\_{j} \mathbb{X}\_{\circ k}\right) \* \text{CF}\_{\circ} / \mathbf{100} \tag{5}$$

where CF is the confidence degree of a rule, *j* is the index of a rule, *k* is the index of the relation, and *λjk* is the truth degree of the *k*th elementary antecedent.


6.Calculation of the resulting value by using the fuzzy average value is performed as follows

$$\overline{v\_i} = \frac{\sum\_{n=1}^{S\_i} v\_i^n \cdot \mathcal{G}\_i^n}{\sum\_{n=1}^{S\_i} \mathcal{G}\_i^n} \tag{6}$$

IF *<sup>x</sup>*<sup>1</sup> <sup>¼</sup> *<sup>a</sup>*~*<sup>j</sup>* <sup>1</sup> AND *<sup>x</sup>*<sup>2</sup> <sup>¼</sup> *<sup>a</sup>*~*<sup>j</sup>* <sup>2</sup> AND ... THEN *<sup>y</sup>*<sup>1</sup> <sup>¼</sup> <sup>~</sup> *b j* <sup>1</sup> AND *<sup>y</sup>*<sup>2</sup> <sup>¼</sup> <sup>~</sup> *b j* <sup>2</sup> AND ... IF … THEN *Y*<sup>1</sup> ¼ *AVR y*<sup>1</sup> � � AND *<sup>Y</sup>*<sup>2</sup> <sup>¼</sup> *AVR y*<sup>2</sup> � � AND …

This model has a built-in function, AVRG, which calculates the average value.

#### **2.2 Fuzzy C-means algorithm**

Fuzzy C-Means algorithm attempts to minimize the sum of squared errors. The algorithm is based on the iterative minimization of the following objective function [22, 23]:

$$J(\mathcal{W}, \mathcal{C}) = \sum\_{j=1}^{k} \sum\_{i=1}^{n} w\_{ij}^p dist(\mathbf{x}\_i, c\_j) \tag{7}$$

The following condition is satisfied for the sum of degrees of membership of a given element xi to all clusters:

$$\sum\_{j=1}^{k} w\_{i\_j} = 1 \tag{8}$$

The following condition is satisfied for the sum of membership degrees of all elements in each cluster:

$$0 < \sum\_{i=1}^{n} w\_{i,j} < n \tag{9}$$

The corresponding *cj* centroid for a *Cj* cluster is defined as:

$$\mathbf{c}\_{j} = \frac{\sum\_{i=1}^{n} w\_{i,j}^{p} \mathbf{x}\_{i}}{\sum\_{i=1}^{n} w\_{i,j}^{p}} \tag{10}$$

The fuzzy partition update formula can be obtained by minimizing the objective function with the constraint that the sum of the weights equals 1:

$$w\_{i,j} = \frac{\left(\mathbf{1}/dist\left(\mathbf{x}\_i, c\_j\right)^2\right)^{\frac{1}{p-1}}}{\sum\_{q=1}^k \left(\mathbf{1}/dist\left(\mathbf{x}\_i, c\_q\right)^2\right)^{\frac{1}{p-1}}}\tag{11}$$
