*3.2. Numerical Example*

In this example, we have three criteria (C1, C2, and C3) with the importance weights w1 = 0.25, w2 = 0.45, and w3 = 0.30. We must choose between two alternatives, x and y. The two alternatives satisfy the criteria as follows, where the induced variables *μ* are *<sup>C</sup>*1(*μ*) = (5, <sup>6</sup>), *<sup>C</sup>*2(*μ*) = (8, 9) and *<sup>C</sup>*3(*μ*) = (2, <sup>3</sup>).

$$\mathbf{x} \quad \mathsf{C}\_1(\mathbf{x}) = (0.7, 0.2), \; \mathsf{C}\_2(\mathbf{x}) = (0.2, 0.9), \; \mathsf{C}\_3(\mathbf{x}) = (0.3, 0.5)$$

$$y \quad \mathbb{C}\_1(y) = (0.7, 0.4), \; \mathbb{C}\_2(y) = (0.5, 0.8), \; \mathbb{C}\_3(y) = (0.3, 0.1)$$

By using Equations (7)–(9), we calculate the following:

$$PMIGWA(\mathbf{x}) = \left(\sum\_{j=1}^{q} w\_j B\_j(\mathbf{x}), \sum\_{j=1}^{q} w\_j B'\_j(\mathbf{x})\right) = (0.38, 0.66).$$

$$PMIGOWWA(\mathbf{y}) = \left(\sum\_{j=1}^{q} w\_j B\_j(\mathbf{y}), \sum\_{j=1}^{q} w\_j B'\_j(\mathbf{y})\right) = (0.51, 0.51).$$

Now, *r*(*x*)<sup>2</sup> and *r*(*y*)<sup>2</sup> can be obtained as: *r*(*x*)<sup>2</sup> = (0.38)<sup>2</sup> + (0.66)<sup>2</sup> = 0.57 and *r*(*y*)<sup>2</sup> = (0.51)<sup>2</sup> + (0.51)<sup>2</sup> = 0.515. Here, *r*(*x*) = 0.75 and *r*(*y*) = 0.72. Additionally, if trigonometric values are used, it is possible to apply the function *<sup>F</sup>*(*r*(*x*), *<sup>θ</sup>*(*x*)). In this case, the results are as follows: cos (*θ*(*x*)) = 0.38 0.75 = 0.5236 (*rad*) and cos (*θ*(*y*)) = 0.51 0.72 = 0.7679 (*rad*); *<sup>F</sup>*(*r*(*x*), *<sup>θ</sup>*(*x*)) = 12 + *r*12 − 2*θπ* = 12 + 0.75 ∗ 12 − 2(0.5236) *π* = 0.6258 and *<sup>F</sup>*(*r*(*y*), *<sup>θ</sup>*(*y*)) = 1 2 + *r*12 − 2*θπ* = 12 + 0.72 ∗ 12 − 2(0.7679) *π* = 0.5080. By following the same process, we can obtain the results of Equations (8) and (9). In PMGOWMA, the moving average is calculated as: *B* = (((0.7 × 0.45) + (0.3 × 0.3)/2) + (0.2 × 0.25)/2). The same process can

**PMGIOWA PMGOWMA PMGIOWMA** *xyxyxy B* 0.38 0.51 0.13 0.15 0.11 0.15 *B* 0.66 0.51 0.16 0.13 0.18 0.12 *r* 0.75 0.72 0.21 0.20 0.21 0.19 *r*2 0.57 0.52 0.04 0.04 0.05 0.04 cos (*θ*(*x*)) 0.5236 0.7679 0.6632 0.6981 0.5585 0.6807 *<sup>F</sup>*(*<sup>r</sup>*, *θ*) 0.6258 0.5080 0.5161 0.5113 0.5310 0.5127

PMGOWMA

 and

PMGIOWMA.

 In all cases, the best

(see**Table 1.** Consolidated results.

 Table 1).

be used to obtain B and B for each

option is *x*

As can be seen in Table 1, even when different aggregation operators were used, the results were the same, but an interesting finding is that the results were not the same for all the formulations, and in some the difference between *x* and *y* was smaller, but in others it was larger. Therefore, analyzing the information with the use of more data is important for better understanding of the phenomenon under study.
