*2.2. An Extension of the ARAS Method Based on the Use of Interval-Valued Fuzzy Numbers*

Based on Stanujkic et al. [52], the computational procedure for selecting the most acceptable alternative by applying the IVTFN-ARAS method that includes only beneficial criteria could be demonstrated through the following steps:

Step 1. Determination of the optimal performance rating for each criterion.

$$\left[\overline{\mathbf{x}}\_{0j} = [(l\_{0j}, l'\_{0j}), m\_{0j'} (u'\_{0j'}, u\_{0j})] \right] \tag{5}$$

with

$$l\_{0j} = \max\_{i} l\_{ij\nu} \tag{6}$$

$$l'\_{0j} = \max\_{i} l'\_{ij'} \tag{7}$$

$$m\_{0j} = \max\_{i} m\_{ij} \tag{8}$$

$$
u\_{0j}' = \max\_{i} \boldsymbol{u}\_{ij}'\tag{9}$$

$$\mathfrak{u}\_{0\dot{\jmath}} = \max\_{\dot{\jmath}} \mathfrak{u}\_{\dot{\jmath}\dot{\jmath}} \tag{10}$$

where <sup>e</sup>*x*0*<sup>j</sup>* represents the interval-valued fuzzy optimal performance rating of criterion *<sup>j</sup>*.

Step 2. Calculation of the normalized decision matrix.

$$
\widetilde{r}\_{ij} = \left[ \left( \frac{a\_{ij}}{c\_j^{+}}, \frac{a\_{ij}'}{c\_j^{+}} \right) \frac{b\_{ij}}{c\_j^{+}}, \left( \frac{c\_{ij}'}{c\_j^{+}}, \frac{c\_{ij}}{c\_j^{+}} \right) \right] \tag{11}
$$

wheree*rij* represents the normalized interval-valued fuzzy performance rating of alternative *<sup>i</sup>* in relation to the criterion *j*, *c* + *j* = P*m i*=0 *cij*.

Step 3. Calculation of the weighted interval-valued normalized fuzzy decision matrix.

$$
\widetilde{w}\_{\text{ij}} = w\_{\text{j}} \cdot \widetilde{r}\_{\text{ij}} \tag{12}
$$

where <sup>e</sup>*vij* represents the weighted normalized interval-valued fuzzy performance rating of alternative *<sup>i</sup>* in relation to the criterion *j*.

Step 4. Calculation of the overall interval-valued fuzzy performance ratings.

$$\widetilde{S}\_{\vec{l}} = \sum\_{j=1}^{n} w\_{j} \widetilde{r}\_{\vec{l}\vec{\prime}} \tag{13}$$

where e*S<sup>i</sup>* represent overall interval-valued fuzzy performance rating of alternative *i*.

Step 5. Calculation of the degree of utility, for each alternative. As a result of performing the previous steps, the obtained overall performance ratings are IVFNs. Therefore, overall performance ratings have to be defuzzified before the calculation of the overall degree of utility. In this way, the same equation as in the ordinary ARAS method is used to determine the overall degree of utility.

Step 6. Ranking of alternative selections by the most efficient. This step is the same as in the original ARAS method.
