*2.2. Fuzzy Analytic Hierarchy Process (FAHP) Method*

In the case of weight calculations using the FAHP method, the experts evaluate the criteria using interval values. Therefore, the uncertainty is included in the evaluations themselves. Unlike the deterministic AHP case, each evaluation for triangular fuzzy numbers can be represented as (*L, M, U*). The most probable evaluation *M* corresponds to the evaluation provided by the AHP method. The number *L* shows the lowest possible boundary of the evaluation, and the number *U* the corresponding upper boundary [48].

The matrix *P*e for the pairwise comparison of the criteria by the expert (or total evaluation by the entire group of experts) has the following representation:

$$
\tilde{\mathbf{P}} = \begin{pmatrix} \tilde{p}\_{ij} \end{pmatrix} = \begin{pmatrix} L\_{ij}, M\_{1j}, \mathbf{L}\_{ij} \end{pmatrix} = \begin{pmatrix} (1, 1, 1) & (\mathbf{L}\_{12}, \mathbf{M}\_{12}, \mathbf{U}\_{12}) & \dots & (\mathbf{L}\_{1m}, \mathbf{M}\_{1m}, \mathbf{U}\_{1m}) \\ (\mathbf{1}/\mathbf{L}\_{12}, \mathbf{1}/\mathbf{M}\_{12}, \mathbf{1}/\mathbf{L}\_{12}) & (\mathbf{1}, \mathbf{1}, \mathbf{1}) & \dots & (\mathbf{L}\_{2m}, \mathbf{M}\_{2m}, \mathbf{U}\_{2m}) \\ \vdots & \vdots & \vdots & \vdots \\ (\mathbf{1}/\mathbf{L}\_{1m}, \mathbf{1}/\mathbf{M}\_{1m}, \mathbf{1}/\mathbf{L}\_{1m}) & (\mathbf{1}/\mathbf{L}\_{2m}, \mathbf{1}/\mathbf{M}\_{2m}, \mathbf{1}/\mathbf{L}\_{2m}) & \dots & (\mathbf{1}, \mathbf{1}, \mathbf{1}) \end{pmatrix} . \tag{4}
$$

The symmetric fuzzy numbers with respect to the main diagonal are *<sup>p</sup>*e*ji*=*p*e*ij* <sup>−</sup><sup>1</sup> = 1 *Uij* , 1 *Mij* , 1 *Lij* ; the main diagonal elements are *<sup>p</sup>*e*ii* <sup>=</sup> (1, 1, 1).

The Chang algorithm [49] is used for calculation of the criteria weights. The value *S*e *i* , called the extension of the fuzzy synthesis, is calculated for the *i*-th criterion using the following formula:

$$\widetilde{S}\_{i} = \sum\_{j=1}^{m} \widetilde{p}\_{ij} \otimes \left\{ \sum\_{i=1}^{m} \sum\_{j=1}^{m} \widetilde{p}\_{ij} \right\}^{-1}; i = 1, \dots, m \tag{5}$$

All the criteria are compared pairwise using the value *S*e *i* :

$$V\left(\breve{S}\_{\dot{j}} \ge \breve{S}\_{\dot{i}}\right) = \begin{cases} \frac{1, \text{ if } M\_{\dot{j}} \ge M\_{\dot{i}}}{L\_{\dot{i}} - \mathcal{U}\_{\dot{j}}}, & \text{ if } L\_{\dot{i}} \le \mathcal{U}\_{\dot{j}} \text{ }, i = 1, \dots, m; \; j = 1, \dots, m\\\ 0, \text{ in other cases} \end{cases} \tag{6}$$

The theory of the fuzzy numbers comparison is applied to the comparison of the values:

$$V\_{\tilde{\jmath}} = V\left(\tilde{S}\_{\tilde{\jmath}} \ge \tilde{S}\_1, \tilde{S}\_2, \dots, \tilde{S}\_{\tilde{\jmath}-1}, \tilde{S}\_{\tilde{\jmath}+1}, \dots, \tilde{S}\_m\right) = \min\_{i \in \{1, \dots, m; i \ne \tilde{\jmath}\}} V\left(\tilde{S}\_{\tilde{\jmath}} \ge \tilde{S}\_{\tilde{\jmath}}\right), i = 1, \dots, m. \tag{7}$$

The weight vector *w<sup>j</sup>* of the criteria is calculated using the following formula:

$$w\_j = \frac{V\_j}{\sum\_{j=1}^{m} V\_j}, j = 1, \dots, m. \tag{8}$$
