3.2.2. The Weighting of the Predefined Variables

Three experts working as the data analysts were introduced with the different aspects of the VAS matrix. Then they were asked to weight all the variables according to the SMARTS methodology. At first, all the experts found a consensus that both the cardinal and ordinal information is equally important for the final decision, therefore the sum of the weights for the variables V1–V5 should be equal to the variable V6.

At the next step, experts ranked all the cardinal variables according to their importance for the criteria weighting and the psychometric features of the VAS scales. Due to the positive skew that can be typically observed in the VAS based valuations, the lowest importance was set to the preference valuations where *rnl* ∈ [50 − 74]. The highest importance was determined for the VAS values when *rnl* ∈ [95 − 100]. Due to the tendency towards the positive assessment, critical opinions encountered in the variables V1 and V2 were considered more important than the positive ones (V3, V4). The final ranking order of the nominal variables was determined as V3 < V4 < V2 < V1 < V5. The relative scores were assigned to V4, V2, and V1 considering their trade-off to the variables V3 and V5.

3.2.3. Preference elicitation by the WASPAS-SVNS Approach

The WASPAS-SVNS approach can be deconstructed into several steps [54]:


$$\widetilde{\mathfrak{X}}\_{ij} = \frac{\mathfrak{x}\_{ij}}{\sqrt{\sum\_{i=1}^{m} \left(\mathfrak{x}\_{ij}\right)^2}} \,\tag{13}$$


$$\widetilde{Q}\_{i}^{(1)} = \sum\_{j=1}^{L\_{\text{max}}} \widetilde{\mathbf{x}}\_{+ij}^{n} \cdot w\_{+j} + \left(\sum\_{j=1}^{L\_{\text{min}}} \widetilde{\mathbf{x}}\_{-ij}^{n} \cdot w\_{-j}\right)^{c}.\tag{14}$$

The sum of the total relative importance of the *i th* alternative is used to calculate *<sup>Q</sup>*e(1) *i* . The <sup>e</sup>*<sup>x</sup> n* +*ij* and *<sup>w</sup>*+*<sup>j</sup>* are the values related with the criteria that should be maximized; <sup>e</sup>*<sup>x</sup> n* <sup>−</sup>*ij* and *<sup>w</sup>*−*<sup>j</sup>* are associated to the criteria that should be minimized. Criteria weights *w*+*<sup>j</sup>* and *w*−*<sup>j</sup>* are the arbitrary positive real numbers, *Lmax* and *Lmin* are the amount of the maximized and minimized criteria. The following algebra operations should be applied for the single-valued neutrosophic numbers:

$$
\widetilde{\mathbf{x}}\_1^n \oplus \widetilde{\mathbf{x}}\_2^n = (t\_1 + t\_2 - t\_1 t\_2, i\_1 i\_{2\prime} f\_1 f\_2),
\tag{15}
$$

$$(\overline{\mathbf{x}}\_1^n \otimes \overline{\mathbf{x}}\_2^n = (t\_1 t\_2 \, \dot{\mathbf{z}}\_1 + \dot{\mathbf{z}}\_2 - \dot{\mathbf{z}}\_1 \dot{\mathbf{z}}\_2 \, f\_1 + f\_2 - f\_1 f\_2), \tag{16}$$

$$w\overline{\chi}\_1^n = \left(1 - (1 - t\_1)^w, i\_1^w, f\_1^w\right) w > 0,\tag{17}$$

*Symmetry* **2020**, *12*, 1641

$$\overline{X}\_1^{uw} = \left( t\_1^w, 1 - (1 - i\_1)^w, 1 - (1 - f\_1)^w \right) w > 0,\tag{18}$$

$$
\widetilde{x}\_1^{nc} = (f\_1, 1 - i\_1, t\_1), \tag{19}
$$

here <sup>e</sup>*<sup>x</sup> n* 1 <sup>=</sup> (*t*1, *<sup>i</sup>*1, *<sup>f</sup>*1) and <sup>e</sup>*<sup>x</sup> n* <sup>2</sup> = (*t*2, *i*2, *f*2).

5. Calculation of the second decision component *<sup>Q</sup>*e(2) *i* is done by the formula:

$$\widetilde{Q}\_{i}^{(2)} = \prod\_{j=1}^{L\_{\text{max}}} \left( \widetilde{\mathbf{x}}\_{+ij}^{n} \right)^{w\_{+j}} \cdot \left( \prod\_{j=1}^{L\_{\text{min}}} \left( \widetilde{\mathbf{x}}\_{-ij}^{n} \right)^{w\_{-j}} \right)^{c}. \tag{20}$$

*<sup>Q</sup>*e(2) *i* value is based on the product of total relative importance in the ith alternative 6. Joint generalized criteria is computed by:

$$
\widetilde{Q}\_{i} = 0.5\widetilde{Q}\_{i}^{(1)} + 0.5\widetilde{Q}\_{i}^{(2)}.\tag{21}
$$

7. The final weights of the criteria importance are determined considering the descending order of the score function *S Q*e*i* , which is used for the deneutrosophication of the joint generalized criteria:

$$S(\widetilde{Q}\_i) = \frac{3 + t\_i - 2i\_i - f\_i}{4}.\tag{22}$$
