**5. Discussion**

Using the results of calculations of MCDM methods to choose the best alternative and to make the right decision makes practical sense if the models used are stable with respect to possible minor fluctuations in the initial data. Experts play a significant, and often crucial, role in the preparation of these MCDM methods. They form a set of criteria that characterize the process being evaluated. The criteria weights are usually calculated on the basis of their estimates, and often experts evaluate the values of the criteria themselves. Experts' estimates are characterized by uncertainty. Therefore, when using MCDM methods, it is very important to investigate the influence of incomplete certainty about the data on the results of the calculations, and to assess the stability of the methods themselves.

The stability of MCDM models and the stability of the results depend both on the methods used and on the problem data themselves. Therefore, for each specific problem solved by MCDM, the use of methods for calculating the criteria weights and the specific methods for evaluating alternatives can be selected after checking these methods and the data for stability.

This paper should be considered as an integral part of the general task of studying the stability of MCDM models. For each specific MCDM method, it is necessary to investigate the stability of the weight estimates (this paper), as well as to evaluate the stability of the MCDM method itself. After that, the total error of the calculations can be estimated and the model with the smallest errors accepted.

A change often expresses the assessment of MCDM methods' stability in the ranking of alternatives. In solving specific problems using many possible MCDM methods, the method's result with the lowest degree of change of rang estimates is used.

Various methods can establish the weights of the criteria. Still, the AHP and FAHP methods' peculiarity is in checking the consistency of expert assessments, which allows controlling the correctness of filling out the questionnaire.

The AHP method, as a mathematical method, shows a high degree of stability. This is to be expected: the method is mathematically justified, the elements of the matrix of the pairwise comparison of the criteria are ideally the ratio of the unknown criteria weights, and the weights themselves are normalized values of the eigenvector of the matrix.

Issues related to changes in the criteria weights that depend on ambiguous expert assessments have not received sufficient attention in the scientific literature. An expert, when filling out the same questionnaire again, usually fills it out a little differently. The errors in the calculated weights of the AHP method described in the second algorithm, which are related to the estimates of the experts themselves and the logic of their thinking, are significantly higher than the errors in the mathematical method itself. That, too, was to be expected. Reducing or increasing the comparative estimates of the experts by one significantly changes the values of the components of the eigenvector and the values of the criteria weights. The relative error of the weights increases. When comparing one criterion with all the others, the expert should also remember his previous assessments of the other criteria. With a large number of criteria, this task is not simple and often the comparison matrix is contradictory, not consistent. The expert is forced to fill in a new matrix, and his new estimates, of course, differ significantly from the original ones, as do the weights of the criteria. The relative error is quite large. It is possible to recommend that experts, before filling out an AHP matrix, rank the criteria according to their significance and, when filling out the matrix, constantly take into account the ranking results.

The greatest problems arise when evaluating the stability of the FAHP method weights. This is mainly due to the Chang algorithm used. Checking the AHP fuzzy method shows non-unambiguous results when using different scales of triangular numbers. In the analysis of the algorithm proposed by Chang for calculating weights, the possibility of zero weights for the criteria is emphasized. This is due to the possible excess of the value of *L* over *U* (*L* > *U*), which occurs because of the narrow scale of triangular numbers. Chang's proposed algorithm includes normalization, which partially solves the problem of decreasing *L* and increasing *U* values, by expanding the triangular numbers and increasing the probability of an intersection of the *S*e *<sup>i</sup>* values. This paper proposed an asymmetric scale of triangular numbers, which excluded the appearance of zero weights. At the same time, the calculated fuzzy weights correlated well with the weights of the AHP method. In subsequent work, the authors plan to study in more detail the influence on the final result of the scale used for the triangular numbers and suggest other ways of normalization, to avoid zero values for the criteria weights. Chang's algorithm is also sensitive to large estimates from the Saaty scale (close to 9), in which case the criteria weights can take zero values.

The criteria weights calculated using the AHP and FAHP methods are naturally different, but as the average *M* values of the FAHP matrix coincide with the values of the AHP matrix, the weights of the two methods should correlate with each other. However, when some of the criteria weights are zero, there is no need to talk about compliance. The situation is "corrected" by the use of the asymmetric scale proposed in this article.

The order of significance of the criteria weights of the first matrix calculated by the AHP and FAHP methods coincided: (cr2 cr cr3 cr1 cr5 cr6). The correlation coefficient of the AHP weights (0.0873, 0.4246, 0.149, 0.2585, 0.0496, 0.0311) and the FAHP scales ( 0.159, 0.286, 0.205, 0.247, 0.088, 0.017) is large enough and equal to 0.8894.

A review of the scientific literature confirms the relevance of the problems studied in this paper. Despite the presence of papers that use a stability check for the AHP method, the results of the algorithms could not be compared because of different interpretations of the results. Noted that despite the widespread use of the FAHP method, little attention has been paid to checking its stability, so the results of this paper are of scientific interest.

The theory of interval numbers is a universal approach for solving many applied problems. The FAHP method is useful for solving problems using linguistic scales that cannot be written down in a single number. In this case, the sensitivity test of the FAHP method to select the scale of triangular numbers is recommended. Otherwise, the AHP method is recommended.
