*3.3. Stability Check Algorithm for the FAHP Method*

Various types of uncertainties influence the evaluation of the stability of the FAHP method. As in the deterministic case, the values of the elements of the criteria comparison matrix depend on the logic of the thinking of the experts, and their state at the moment of evaluation. Besides, the fuzzy method itself includes uncertainty in the evaluations—a triad of values is used instead of a one-point evaluation. It should also be kept in mind that the stability evaluation for the FAHP method refers solely to the weight evaluation algorithm used by us (4)–(8).

The stability check for the FAHP method for the calculation of the criteria weights can be represented in the form of the following steps.

Step 1. A fuzzy pairwise comparison matrix for the criteria is formed on the basis of the AHP matrix *M* = *Mij* : *P*e = *p*e*ij* = *Lij*, *Mij*, *Uij* . The values *Lij* and *Uij* vary depending on the selected symmetric or asymmetric scale of the fuzzy number.

Step 2. The consistency of evaluations (*CR* < 0.1) of the matrix *M* = *Mij* is verified. If the Consistency Ratio is *CR* ≥ 0.1, the pairwise comparison matrix is discarded.

Step 3. A sequence of random numbers *ξ<sup>r</sup>* (*r* = 1), uniformly distributed within the interval of [0, 1], is selected using the statistical simulation method (Monte Carlo). The values of all the *Mij* evaluations of the experts – the elements *p*e*ij* = *Lij*, *Mij*, *Uij* (*i* 6= *j*) of the matrix *P*e—are varied (increased or decreased) by 1. To do that, if 0 ≤ *ξ<sup>r</sup>* <0.5, the value increases. In the other case (0.5 < *ξ<sup>r</sup>* ≤1), the value decreases. In order to attain complete symmetry, we exclude the value of *ξ<sup>r</sup>* = 0.5, that is, we do not vary the elements of the matrix. The two options have equal probability. If at least one of the numbers *Lij*, *Mij*, *Uij* is equal to 1, the value is always increased. If at least one of the numbers *Lij*, *Mij*, *Uij* is equal to 9, the value is decreased. The elements of the main diagonal remain unchanged: (*p*e*ii*) <sup>=</sup> (*Lii*, *<sup>M</sup>ii*, *<sup>U</sup>ii*) <sup>=</sup> (1, 1, 1). The symmetric elements with respect to the main diagonal are (*p*e*ii*) <sup>=</sup> (*Lii*, *<sup>M</sup>ii*, *<sup>U</sup>ii*) <sup>=</sup> 1 *p*e*ii* = 1 *Uii* , 1 *Mii* , 1 *Lii* . A new random number from the sequence *ξ<sup>r</sup>* is used for each element *<sup>p</sup>*e*ij* of the matrix.

The matrix *P*e is formed from the values of the matrix *M* = *Mij* . The consistency of evaluation of the values of the matrix *M* (*CR* < 0.1) is verified. If the Consistency Ratio is *CR* ≥ 0.1, the pairwise comparison matrix is discarded and a new fuzzy matrix *P*e is formed.

The criteria weights Ω = *ω* (*r*) *j* , (*r* = 1) are calculated.

Step 4. New sequences of random numbers *ξ<sup>r</sup>* (*r* = 2, 3, . . . , *T*) are selected, where *T* is the number of repetitions (simulations). Step 3 is repeated. The criteria weights (*ω* (*r*) *j* ), (*r* = 2, 3, . . . , *T*) are calculated.

Step 5. The relative errors of the values of the weight for each of the *j*-th criterion of the AHP method are calculated for every simulation ξ: *δ* (*ξ*) *j* = *ω* (*ξ*) *<sup>j</sup>* −*ω<sup>j</sup> ωj* .

Step 6. The largest values of the relative errors *δ* (*ξ*) *j* of the criteria weights for all the criteria are calculated for every simulation ξ: *δ <sup>ξ</sup>* = max *j δ* (*ξ*) *j* .

Step 7. We take the largest value of the relative errors *δ <sup>ξ</sup>* of the criteria weights over all the simulations ξ as the FAHP method error for the given comparison matrix: δ = max *ξ δ ξ* .
