*4.1. Practical Application and Analysis of the Implementation of the First Algorithm for Checking the Stability*

Using the first algorithm for checking the stability, a different percentage of deviation of all the elements is set. In the first case, the deviation is *q* = 5%, while in the second *q* = 10%. In each case, ten attempts are made to fix the interval of the largest relative errors.

In the first case, the matrix data with a good consistency index are used (Table 1). Table 3 shows the results of ten attempts to check the stability of the method, and their largest relative errors *δ ξ* , with the number of iterations equal to 100 and 10% deviation, using the matrix of data from Table 1. The results show small *δ <sup>ξ</sup>* values for the largest relative errors with a deviation of all elements by 10%.


**Table 3.** The largest relative errors of the criteria weights *δ ξ* , when checking the stability by the first algorithm, *q* = 10%, 100 iterations, 10 repetitions, 1st matrix.

During the checking process, the percentage of consistent matrices is fixed for the total number of simulated matrices. At 5% and 10% deviation and with the number of simulations from 100 to 100,000, all the matrices, that is, 100%, are consistent.

The intervals of the largest relative errors for different numbers of iterations are shown in Tables 4 and 5 (5% and 10% deviation, respectively). With an increase in the number of iterations, the range of values of the largest relative errors narrows. With a deviation *q* of 5%, the value of the relative errors *δ ξ* is less than with a deviation of 10%.

**Table 4.** The interval of the largest relative errors of the criteria weights *δ ξ* , when checking the stability by the first algorithm, *q* = 5%, 10 repetitions, 1st matrix.


**Table 5.** The interval of the largest relative errors of the criteria weights *δ ξ* , when checking the stability by the first algorithm, *q* = 10%, 10 repetitions, 1st matrix.


In both verification cases, the results show the stability of the method: *δ* = 0.0579 (at 5% deviation) and *δ* = 0.1149 (at 10% deviation).

A similar stability check is implemented with the data from the other matrix, for which data consistency is critical (Table 2). With the number of simulations equal to 100, the consistency interval is wider than with 10,000 and 100,000 simulations. In the general case, with a deviation of 5%, the percentage of consistent matrices ranges from 97% to 100%; with a deviation of 10%, the percentage of consistent matrices is, on average, 16% less and, more precisely, fluctuates in the range from 77% to 89% (Table 6).


**Table 6.** The percentage of consistent matrices when checking the stability of the first algorithm using the data of the second matrix (5%, 10% deviation).

The intervals for the largest relative errors for the different numbers of iterations are shown in Tables 7 and 8 (5% and 10% deviation, respectively). With a deviation of 5%, the value of the largest relative errors *δ ξ* is less than with a deviation of 10%. In both verification cases, the results show the stability of the method: *δ* = 0.0531 (at 5% deviation) and *δ* = 0.116 (at 10% deviation).

**Table 7.** The interval of the largest relative errors of the criteria weights *δ ξ* , when checking the stability by the first algorithm, *q* = 5%, 10 repetitions, 2nd matrix.


**Table 8.** The interval of the largest relative errors of the criteria weights *δ ξ* , when checking the stability by the first algorithm, *q* = 10%, 10 repetitions, 2nd matrix.


The results obtained for the largest relative errors of the second matrix differ a little from the results for the largest relative errors of the first matrix (Tables 4 and 5). Testing the AHP method with the first algorithm shows a good result.
