*3.5. MCDM WASPAS Method for Data Processing and Evaluation*

In the present research, WASPAS has been selected for the exploration of the problem related to the qualitative rating of aerial image lossy compression. The applications of the WASPAS method were presented in [67]. The WASPAS approach is used for the solution to a broad range MCDM problems. This method is popular due to stability and simplicity. WASPAS method initially was introduced in [72] and later was extended by single-valued neutrosophic sets (WASPAS-SVNS) [73]. Although this method has been already used to solve different MCDM tasks [74,75], we could not find any research where WASPAS was applied to assess the lossy compression in the satellite images.

WASPAS-SVNS approach can be decomposed into several steps [73] presented below:

1. For the construction of the decision matrix, we need to have the initial information that consists of the evaluations of lossy compression algorithms and compression ratios (as alternatives) according to the qualitative parameters of compressed aerial images (as criteria). When the decision matrix *X* is constructed, vector normalization is used to normalize the decision matrix *X*.

$$\widetilde{\mathfrak{X}}\_{ij} = \frac{\mathfrak{x}\_{ij}}{\sqrt{\sum\_{i=1}^{m} \left(\mathfrak{x}\_{ij}\right)^{2}}} \tag{18}$$

Here, *xij*, *i* = 1, . . . *m*; *j* = 1, . . . *n* is the value of the of *j th* variable for the *i th* ithalternative (criteria).


$$\widetilde{Q}\_{i}^{(1)} = \sum\_{j=1}^{L\_{\text{max}}} \widetilde{\boldsymbol{x}}\_{+ij}^{n} \cdot \boldsymbol{w}\_{+j} + \left(\sum\_{j=1}^{L\_{\text{min}}} \widetilde{\boldsymbol{x}}\_{-ij}^{n} \cdot \boldsymbol{w}\_{-j}\right)^{c} \tag{19}$$

Here, the values *<sup>x</sup>*<sup>e</sup> *n* <sup>+</sup>*ij* and *w*+*<sup>j</sup>* are associated with the criteria which are maximized; consequently, *<sup>x</sup>*<sup>e</sup> *n* <sup>−</sup>*ij* and *<sup>w</sup>*−*<sup>j</sup>* correspond to the criteria which are minimized. The weight of criteria is an arbitrary positive real number, the amount of the maximized criteria is *L*max, and the amount of the minimized criteria is *L*min. For the single-valued neutrosophic numbers (SVNNs), the following algebra operations should be applied:

$$
\hat{\mathfrak{X}}\_1^{\mathbb{II}} \oplus \hat{\mathfrak{X}}\_2^{\mathbb{II}} = \left(t\_1 + t\_2 - t\_1 t\_2, i\_1 i\_2, f\_1 f\_2\right) \tag{20}
$$

$$
\widetilde{\mathbf{x}}\_1^n \otimes \widetilde{\mathbf{x}}\_2^n = (t\_1 t\_2, i\_1 + i\_2 - i\_1 i\_2, f\_1 + f\_2 - f\_1 f\_2) \tag{21}
$$

$$w\hat{x}\_1^n = \left(1 - \left(1 - t\_1\right)^w, i\_1^w, f\_1^w\right), w > 0\tag{22}$$

$$\hat{\mathbf{x}}\_1^{uw} = \left( t\_1^w, 1 - \left( 1 - i\_1 \right)^w, 1 - \left( 1 - f\_1 \right)^w \right), w > 0 \tag{23}$$

$$
\widetilde{x}\_1^{n\varepsilon} = (f\_1, 1 - i\_1, t\_1) \tag{24}
$$

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

4. The second decision component based on the product of total relative importance in the *i* th alternative is calculated by the equation:

$$\widetilde{Q}\_{i}^{(2)} = \prod\_{j=1}^{L\_{\text{max}}} \left( \widetilde{\boldsymbol{x}}\_{+ij}^{\boldsymbol{\eta}} \right)^{\boldsymbol{w}\_{+j}} \cdot \left( \prod\_{i=1}^{L\_{\text{min}}} \left( \widetilde{\boldsymbol{x}}\_{-ij}^{\boldsymbol{\eta}} \right)^{\boldsymbol{w}\_{-j}} \right)^{\boldsymbol{c}} \tag{25}$$

5. The following equation calculates the weighted criteria:

$$
\tilde{Q}\_{\dot{i}} = 0.5 \tilde{Q}\_{\dot{i}}^{(1)} + 0.5 \tilde{Q}\_{\dot{i}}^{(2)} \tag{26}
$$

6 The final ranking of the alternatives is evaluated considering the descending order of the *S Q*e *i* . This is a score function (further referred to as utility function) for deneutrosophication of the joint generalized criteria and is calculated as follows:

$$S\left(\tilde{Q}\_{i}\right) = \frac{3 + t\_i - 2i\_i - f\_i}{4} \tag{27}$$
