*4.3. Practical Application and Analysis of the Implementation of the Third Algorithm for Checking the Stability*

First, the step-by-step calculation of the weights of the original matrix using Chang's method will be illustrated for a better understanding of the stability check of the FAHP method. This paper uses a matrix with triangular fuzzy numbers, which will be formed from the AHP matrix. In many papers, the authors prefer to use symmetric triangular fuzzy numbers, such as {1: [1, 1, 1], 2: [1, 2, 3], 3: [2, 3, 4], 4: [3, 4, 5], 5: [4, 5, 6], 6: [5, 6, 7], 7: [6, 7, 8], 8: [7, 8, 9], 9: [8, 9, 9]}. There are also different modifications of this scale, for example, 1: [0, 1, 1] or 9: [7, 9, 9], and others. Ishizaka et al. [50] describe different modifications of these scales in paper. Chang himself used non-symmetric triangular fuzzy numbers in his paper [49]. This paper will use both symmetric and asymmetric scales (Tables 13 and 14). The scales used do not go beyond the AHP scale proposed by Saaty, in the range from 1 to 9, so in the case of "Equally important" and "Absolutely important", the symmetry of the triangle is broken (Table 13).

**Table 13.** Symmetrical scale of triangular fuzzy numbers.



In previous papers [51,52], it was noted that when forming a triangular fuzzy matrix from separate AHP matrices, the *M* value is located closer to *U* than to *L*, so the asymmetric triangular fuzzy scale is formed with a large distance from *L* to *M* (Table 14).

An important point in Chang's proposed method for calculating the criteria weights is the comparison of *S*e *i* , the extension of the fuzzy synthesis values (5). The essence of calculating the value of *V* (*S*e *<sup>j</sup>* ≥ *S*e *i* ) is to find the ordinate of the intersection point of two triangular numbers (Figure 1). The triangles *S*<sup>1</sup> and *S*<sup>2</sup> do not intersect in the case when *L*<sup>1</sup> > *U*2. Then, with further calculation, using the formulas (4)–(8), the criterion weight is zero. That is, if a triangle does not intersect with at least one of the subsequent triangles, its weight is zero. This case is possible if the distance between the values *M*<sup>1</sup> and *M*<sup>2</sup> is large, and the triangles themselves are narrow (the distance from *L* to *M* and from *M* to *U* is 1), as in the case of the symmetric scale from Table 13. To give a better explanation of the case of zero weights when using narrow symmetric triangles, we will illustrate it with an example. zero. That is, if a triangle does not intersect with at least one of the subsequent triangles, its weight is zero. This case is possible if the distance between the values ଵ and ଶ is large, and the triangles themselves are narrow (the distance from to and from to is 1), as in the case of the symmetric scale from Table 13. To give a better explanation of the case of zero weights when using narrow symmetric triangles, we will illustrate it with an example.

**Figure 1.** Illustration of the value of *V* (ሚ <sup>ଶ</sup> ≥ ሚ <sup>ଵ</sup>). **Figure 1.** Illustration of the value of *V* (*S*e <sup>2</sup> ≥ *S*e 1 ).

In solving real problems, when forming a fuzzy matrix from several AHP matrices, the values are most often averaged [49,51]. Chang's work uses triangular numbers where the value does not exceed three [49]. In solving real problems, when forming a fuzzy matrix from several AHP matrices, the *M* values are most often averaged [49,51]. Chang's work uses triangular numbers where the *M* value does not exceed three [49].

In the following sections, symmetric and asymmetric triangular fuzzy numbers are used to form the AHP fuzzy matrix of pairwise comparisons. The implementation of the third algorithm will be illustrated using different matrices of pairwise comparisons, depending on the scale of the triangular fuzzy numbers used. In the following sections, symmetric and asymmetric triangular fuzzy numbers are used to form the AHP fuzzy matrix of pairwise comparisons. The implementation of the third algorithm will be illustrated using different matrices of pairwise comparisons, depending on the scale of the triangular fuzzy numbers used.

4.3.1. Using the Symmetric Scale for the Fuzzy Triangle

4.3.1. Using the Symmetric Scale for the Fuzzy Triangle The fuzzy matrix (Table 15) is formed in such a way that the most likely estimate of corresponds to the estimate of the AHP method. A 6 × 6 matrix with a good consistency index is used (Table 1). Fuzzy numbers replaces natural numbers, using the scales specified The fuzzy matrix (Table 15) is formed in such a way that the most likely estimate of *M* corresponds to the estimate of the AHP method. A 6 × 6 matrix with a good consistency index is used (Table 1). Fuzzy numbers replaces natural numbers, using the scales specified in Tables 13 and 14, The remaining matrix estimates are calculated using the formula (4).

in Tables 13 and 14, The remaining matrix estimates are calculated using the formula (4).


**Table 15.** Pairwise comparison matrix using a symmetric triangular fuzzy numbers scale.

The matrix uses narrow symmetric triangular fuzzy numbers and a full scale of Saaty scores from 1 to 9. Systematically, we illustrate the calculation of weights using the Chang method. The matrix uses narrow symmetric triangular fuzzy numbers and a full scale of Saaty scores from 1 to 9. Systematically, we illustrate the calculation of weights using the Chang method.

Consider the summed values of = ∑ ∑ <sup>୫</sup> ୀଵ <sup>୫</sup> ୀଵ (Table 16). The values of the number *L* of the second and fourth criteria (ଶ,ସ) are very different from the other values of the *L* criteria. Note that the *L* value of the second criterion is greater than the *U* values of the first, third, fifth, and sixth criteria, and the *L* value of the fourth criterion is greater than the *U* values of the first, fifth and sixth criteria. Consider the summed values of *S<sup>i</sup>* = m ∑ *i*=1 m ∑ *j*=1 *<sup>p</sup>*e*ij* (Table 16). The values of the number *L* of the second and fourth criteria (*S*2,*S*4) are very different from the other values of the *L* criteria. Note that the *L* value of the second criterion is greater than the *U* values of the first, third, fifth, and sixth criteria, and the *L* value of the fourth criterion is greater than the *U* values of the first, fifth and sixth criteria.


**Table 16.** Summed values of *S<sup>i</sup>* .

Further normalization, proposed by Chang, increases the values of *U* (dividing their values by the sum of the *L* numbers) and decreases the values of *L* (dividing the values by the sum of the *U* numbers). The values of *M* are normalized by dividing all elements by their sum, so the sum of the normalized values of *M* by all criteria is 1 (Table 17).



For each *i*-th criterion, the value of *S*e *i*—the extension of the fuzzy synthesis—is calculated using the formula (5), and the specified answers are rounded.

$$\begin{aligned} \widetilde{S}\_{1} &= (5.7, 7.95, 10, 58) \odot \left(\frac{1}{91.54}, \frac{1}{74.5}, \frac{1}{58, 33}\right) = (0.06, 0.11, 0.18), \\ \widetilde{S}\_{2} &= (23, 28, 32) \odot \left(\frac{1}{91.54}, \frac{1}{74.5}, \frac{1}{58, 33}\right) = (0.25, 0.38, 0.55), \\ \widetilde{S}\_{3} &= (9.53, 12.75, 16, 33) \odot \left(\frac{1}{91.54}, \frac{1}{74.5}, \frac{1}{58, 33}\right) = (0.1, 0.17, 0.28), \\ \widetilde{S}\_{4} &= (15.33, 19.5, 24) \odot \left(\frac{1}{91.54}, \frac{1}{74.5}, \frac{1}{58, 33}\right) = (0.17, 0.26, 0.41), \\ \widetilde{S}\_{5} &= (2.83, 4.09, 5, 75) \odot \left(\frac{1}{91.54}, \frac{1}{74.5}, \frac{1}{58, 33}\right) = (0.03, 0.05, 0.1), \\ \widetilde{S}\_{6} &= (1.94, 2.2, 2.88) \odot \left(\frac{1}{91.54}, \frac{1}{74.5}, \frac{1}{58, 33}\right) = (0.02, 0.03, 0.05). \end{aligned}$$

The normalized summed values of *S*e *<sup>i</sup>* are shown in Table 17. After normalization, the sum of the values of the triangular numbers is 0< Σ*L*ˆ <1, Σ*M*ˆ = 1, and Σ*U*ˆ > 1. Let us analyze how normalization changed the values of *L* and *U*, while expanding the values of the triangular numbers. Taking the example of the first criterion, the difference increased from *<sup>U</sup>*<sup>1</sup> *L*1 = 1.85 to *<sup>U</sup>*<sup>ˆ</sup> 1 *L*ˆ 1 = 3. Unfortunately, normalization did not solve all the problems of the second and fourth criteria (*L*ˆ <sup>2</sup> > *U*ˆ <sup>1</sup>, *L*ˆ <sup>2</sup> > *U*ˆ <sup>5</sup>, *L*ˆ <sup>2</sup> > *U*ˆ <sup>6</sup>, *L*ˆ <sup>4</sup> > *U*ˆ <sup>5</sup>, *L*ˆ <sup>4</sup> > *U*ˆ <sup>6</sup>). The inequalities show that the weights of the first, fifth, and sixth criteria will be zero. Further calculations will also show this.

To avoid this situation, it is recommended that, when using a narrow symmetric scale to fill in the matrix of pairwise comparisons, extreme/large values, such as (8, 9, 9), (7, 8, 9), and so on, are not used. Another possible way to avoid zero values is to use an asymmetric scale, which expands the triangle and the differences between the *L* and *U* values.

Then, using the extension of the fuzzy synthesis, all the criteria are compared in pairs using the formula (6):

$$\begin{array}{c} V\left(\widetilde{S}\_{1} \geq \widetilde{S}\_{2}\right) = 0, \ (\widehat{L}\_{2} > \widehat{\mathcal{U}}\_{1}),\\ V\left(\widetilde{S}\_{1} \geq \widetilde{S}\_{3}\right) = \frac{0.1 - 0.18}{\left(\begin{array}{c} 0.11 - 0.18 \ \big|\, -\operatorname{0.17} - \operatorname{0.1}\big|\, \end{array}\right)} = \mathbf{0.545},\\ V\left(\widetilde{S}\_{1} \geq \widetilde{S}\_{4}\right) = \frac{0.17 - 0.18}{\left(\begin{array}{c} 0.11 - 0.18 \ \big|\, -\operatorname{0.26} - \operatorname{0.17}\big|\, \end{array}\right)} = \mathbf{0.083},\\ V\left(\widetilde{S}\_{1} \geq \widetilde{S}\_{5}\right) = 1, \ (\widehat{M}\_{1} > \widehat{M}\_{5}),\\ V\left(\widetilde{S}\_{1} \geq \widetilde{S}\_{6}\right) = 1, \ (\widehat{M}\_{1} > \widehat{M}\_{6}).\end{array}$$

*V*<sup>1</sup> = min(0, 0.545, 0.083, 1, 1) = 0.

Then, a comparison is made in pairs of *S*e<sup>2</sup> with all the other extensions of the fuzzy synthesis:

$$\begin{array}{c} V\left(\widetilde{\boldsymbol{S}}\_{2} \geq \widetilde{\boldsymbol{S}}\_{1}\right) = 1, \left(\boldsymbol{\hat{M}}\_{2} > \boldsymbol{\hat{M}}\_{1}\right), \\ V\left(\widetilde{\boldsymbol{S}}\_{2} \geq \widetilde{\boldsymbol{S}}\_{3}\right) = 1, \left(\boldsymbol{\hat{M}}\_{2} > \boldsymbol{\hat{M}}\_{3}\right), \\ V\left(\widetilde{\boldsymbol{S}}\_{2} \geq \widetilde{\boldsymbol{S}}\_{4}\right) = 1, \left(\boldsymbol{\hat{M}}\_{2} > \boldsymbol{\hat{M}}\_{4}\right), \\ V\left(\widetilde{\boldsymbol{S}}\_{2} \geq \widetilde{\boldsymbol{S}}\_{5}\right) = 1, \left(\boldsymbol{\hat{M}}\_{2} > \boldsymbol{\hat{M}}\_{5}\right), \\ V\left(\widetilde{\boldsymbol{S}}\_{2} \geq \widetilde{\boldsymbol{S}}\_{6}\right) = 1, \left(\boldsymbol{\hat{M}}\_{2} > \boldsymbol{\hat{M}}\_{6}\right). \end{array}$$

$$V\_{2} = \min\left(1, \ 1, \ 1, \ 1, \ 1\right) = 1.$$

Then, we carry out a pairwise comparison of *S*e<sup>3</sup> with all the other values:

$$V\left(\tilde{S}\_3 \ge \tilde{S}\_1\right) = 1\,\left(\hat{M}\_3 > \hat{M}\_1\right),$$

$$\begin{split} V\left(\tilde{S}\_3 \ge \tilde{S}\_2\right) &= \frac{0.25 - 0.28}{\left(0.17 - 0.28\right) - \left(0.38 - 0.25\right)} = 0.123, \\ V\left(\tilde{S}\_3 \ge \tilde{S}\_4\right) &= \frac{0.17 - 0.28}{\left(0.17 - 0.28\right) - \left(0.26 - 0.17\right)} = 0.554, \\ V\left(\tilde{S}\_3 \ge \tilde{S}\_5\right) &= 1\,\left(\hat{M}\_3 > \hat{M}\_5\right), \\ V\left(\tilde{S}\_3 \ge \tilde{S}\_6\right) &= 1\,\left(\hat{M}\_3 > \hat{M}\_6\right). \end{split}$$

*V*<sup>3</sup> = min(1, 0.123, 0.554, 1, 1) = 0.123.

Subsequent values *V*4, *V*5, and *V*<sup>6</sup> are calculated in the same way. The following values are obtained: *V*<sup>4</sup> = min(1, 0.584, 1, 1, 1) = 0.584; *V*<sup>5</sup> = min(1, 0, 0.123, 0.584, 0) = 0; *V*<sup>6</sup> = min(0, 0, 0, 0, 0.42) = 0; The vector of non-normalized criteria weights is *V*= (0, 1, 0.123, 0.584, 0, 0).

The vector of normalized criteria weights is obtained by applying formula (8): *w*= (0, 0.586, 0.072, 0.342, 0, 0).

In our further study of the stability of the AHP fuzzy method, we use the matrix from Table 18, with a good consistency index (*CR* = 0.022).

**Table 18.** The third matrix of pairwise comparisons (6 × 6).


Using the third algorithm, the stability of the method is also set by different numbers of iterations: 100, 10,000, and 100,000. The value of the largest relative errors *δ <sup>ξ</sup>* of the values of the criteria weights is fixed. At least ten attempts are made to fix the interval of the largest relative errors, for non-zero values of the weights.

To analyze the stability with a symmetric scale of triangular numbers, we use the matrix presented in Table 18. When simulating an AHP matrix with 100 iterations, 60–70% of the subsequent matrices are consistent. The values of the weights of the initial fuzzy matrix criteria are 0.171, 0.288, 0.227, 0.071, 0.187, and 0.055. When calculating the criteria weights generated, and the matched pairwise comparison matrices, 70–80% of the matrices have zero weights, with the largest relative error being 1. The non-zero values of the weights and their largest relative errors are shown in Table 19. The results show a high maximum relative error δ = 0.9797.

**Table 19.** The criteria weights of 5 matrices and the errors of the criteria weights when using a symmetric scale.


In general, the results for the relative errors of the third algorithm (Table 19), using a symmetric scale of triangular numbers, are greater than those for the first and second algorithms.
