4.3.2. Using the Asymmetric Scale for the Fuzzy Triangle

A fuzzy matrix using an asymmetric triangular scale is formed in the same way as a symmetric one, only using the scale from Table 14. The same 6 × 6 matrix with a good consistency index is used (Table 18). The values of the weights of the initial fuzzy matrix criteria are 0.171, 0.253, 0.208, 0.095, 0.179, and 0.093. In contrast to the symmetric scale, zero values of weights are extremely rare, 1–3%. The values of the criteria weights when using an asymmetric scale differ from the values obtained when using a symmetric one. However, the correlation coefficient of the weights obtained using symmetric and asymmetric scales is significant and is equal to 0.9987. The third matrix criteria weights were obtained using symmetric and asymmetric fuzzy scales from Tables 13 and 14, shown in Figure 2.

The last check step was implemented to avoid zero weights using a symmetric scale using a third matrix with scores not exceeding 3. The result showed that the correlation of weights when using symmetric and asymmetric scales is high. Small changes to the symmetric scale do not significantly affect the scores, but they avoid zero weights.

Analyzing the results of 15 repetitions (100 iterations) (Table 20), the values of the largest relative errors of the criteria weights of the second and fourth criteria are high, which means that the weights in some cases are more than doubled.

iterations, 10 repetitions.

**Figure 2.** Comparison of the weights of the third matrix criteria using symmetric and asymmetric fuzzy scales. **Figure 2.** Comparison of the weights of the third matrix criteria using symmetric and asymmetric fuzzy scales.

The last check step was implemented to avoid zero weights using a symmetric scale using a third matrix with scores not exceeding 3. The result showed that the correlation of **Table 20.** The largest relative errors of the criteria weights *δ ξ* , when checking the stability of the third algorithm, 100 iterations, 10 repetitions.


1.0924 1.1216 0.5863 1.0189 1.0446 0.6028 0.8377 0.6344 1.0367 0.5724

**Table 20.** The largest relative errors of the criteria weights క, when checking the stability of the third algorithm, 100 **1 2 3 4 5 6 7 8 9 10**  0.8933 0.7878 0.8397 0.9953 0.8933 0.6525 0.6052 0.6974 0.6496 0.7137 The results of the relative error intervals for different numbers of iterations are presented in Table 21. As the number of iterations increases, the largest relative errors also increase. The weights increase and decrease by almost a factor of two. The largest changes are for the weight value of the second criterion, by almost four times.

0.744 0.6998 0.7089 0.7983 0.7305 0.5985 0.6724 0.561 0.7166 0.6734 **Table 21.** The third algorithm, the largest *δ ξ* relative errors of the criteria weights, 10 repetitions.


cr2 0.5724–1.1216 2.3281–2.9479 2.9479 cr3 0.561–0.7983 1–1.0442 1.0442 cr4 0.6808–1.0571 1.333–1.8541 1.8541 cr5 0.5914–0.891 1–1.332 1.332 cr6 0.5921–0.9936 1.0 1.0 The use of an asymmetric scale of triangular numbers shows an improvement in the result compared to the results for the symmetric scale. There are practically no zero values of the weights, but even with a small fluctuation in the values of the matrix, the weights vary greatly. At the same time, the scale used in the AHP matrix varies from 1 to 4 or, respectively, fuzzy numbers from (1, 1, 1) to (3, 4, 5).

To further test the algorithm, the triangular values of fuzzy numbers are extended. We use the asymmetric scale from Table 22, in order to avoid zero weights for the criteria of the first matrix (Table 1). vary greatly. At the same time, the scale used in the AHP matrix varies from 1 to 4 or, respectively, fuzzy numbers from (1, 1, 1) to (3, 4, 5). To further test the algorithm, the triangular values of fuzzy numbers are extended.

We use the asymmetric scale from Table 22, in order to avoid zero weights for the criteria

The use of an asymmetric scale of triangular numbers shows an improvement in the result compared to the results for the symmetric scale. There are practically no zero values of the weights, but even with a small fluctuation in the values of the matrix, the weights


**Table 22.** Asymmetrical scale of triangular fuzzy numbers. of the first matrix (Table 1).

*Symmetry* **2021**, *13*, x FOR PEER REVIEW 20 of 25

The initial values of *M* are used from the first matrix, which has a high consistency score (Table 1). The weights of the criteria of the fuzzy matrix that is formed when using triangular fuzzy numbers with the new scale in Table 22 are 0.159, 0.286, 0.205, 0.247, 0.088, and 0.017. The initial values of *M* are used from the first matrix, which has a high consistency score (Table 1). The weights of the criteria of the fuzzy matrix that is formed when using triangular fuzzy numbers with the new scale in Table 22 are 0.159, 0.286, 0.205, 0.247, 0.088, and 0.017.

Figure 3 shows a greater degree of consistency of weight estimates by the AHP and FAHP methods using Chang's algorithm. This result could not be predicted because Chang's algorithm uses both different rating scales and the theory of comparing fuzzy numbers. The most significant difference in the weights of the first-matrix criteria obtained by the AHP and FAHP methods is observed in the first-third criterion. Figure 3 shows a greater degree of consistency of weight estimates by the AHP and FAHP methods using Chang's algorithm. This result could not be predicted because Chang's algorithm uses both different rating scales and the theory of comparing fuzzy numbers. The most significant difference in the weights of the first-matrix criteria obtained by the AHP and FAHP methods is observed in the first-third criterion.

**Figure 3.** Comparison of the weights of the first matrix criteria using Analytic Hierarchy Process (AHP) and fuzzy Analytic Hierarchy Process (FAHP) methods. **Figure 3.** Comparison of the weights of the first matrix criteria using Analytic Hierarchy Process (AHP) and fuzzy Analytic Hierarchy Process (FAHP) methods.

An analysis of the values of *δ ξ* for ten repetitions shows good results for the algorithm for criteria 1 to 5 (Table 23). The large relative error values for the sixth criterion are explained by the fact that the weight in the original matrix is not large (*ω*<sup>6</sup> = 0.017).


**Table 23.** The third algorithm, *δ ξ* the relative errors of the criteria weights, 100 iterations, 10 repetitions.

When checking the stability of the algorithm with a large number of iterations, the result remains the same, but the interval is narrowed to one value (Table 24).


**Table 24.** The third algorithm, *δ ξ* relative errors of the criteria weights, 10 repetitions.

Despite the results for criterion 6 (*δ* = 4.6606), the weight of which varies from 0.017 to 0.08, the method using the new scale for forming triangular numbers shows a good result, comparable to the results of algorithms 1 and 2.
