**2. Preliminaries**

In this part of the manuscript, some fundamental definitions and notations of neutrosophic sets (NS), single-valued neutrosophic set (SVNS), and single-valued neutrosophic numbers (SVNN) are given.

**Definition 1.** *[53] Let X denote the universe of discourse. The NS A in X has the following form*

$$A = \{ \langle \mathbf{x}, \mu\_A(\mathbf{x}), \pi\_A(\mathbf{x}), \nu\_A(\mathbf{x}) \rangle \mid \mathbf{x} \in \mathcal{X} \}, \tag{1}$$

*where* µ*A*(*x*) *denotes the truth–membership function;* µ*<sup>A</sup>* ∈ ] <sup>−</sup>0, 1+[*;* π*A*(*x*) *denotes the falsity-membership function;* π*<sup>A</sup>* ∈ ] <sup>−</sup>0, 1+[*; and* <sup>ν</sup>*A*(*x*) *denotes the falsity–membership function,* <sup>ν</sup>*<sup>A</sup>* <sup>∈</sup> ] <sup>−</sup>0, 1+[.

*These membership functions must satisfy the following constraint* <sup>−</sup>0 ≤ µ*A*(*x*) + π*A*(*x*) + ν*A*(*x*) ≤ 3 +.

**Definition 2.** *[54] Let X be a nonempty set. The SVNS A in X has the following form*

$$A = \left| \{ \mathbf{x}, \,\mu\_A(\mathbf{x}), \,\pi\_A(\mathbf{x}), \,\nu\_A(\mathbf{x}) \} \right| \ge \mathbf{x} \,\text{[}\tag{2}$$

*wheremembership functions TA, IA, and F<sup>A</sup>* ∈ [ <sup>−</sup>0, 1+] *and satisfy the following constraint* <sup>0</sup> <sup>≤</sup> <sup>µ</sup>*A*(*x*) + π*A*(*x*) + ν*A*(*x*) ≤ 3.

**Definition 3.** *[54] A SVNN a* =< *ta*, *ia*, *f<sup>a</sup>* > *is a special case of a SVNS on the set of real numbers* <*, where ta*, *ia*, *f<sup>a</sup>* ∈ [0, 1] *and* 0 ≤ *t<sup>a</sup>* + *i<sup>a</sup>* + *f<sup>a</sup>* ≤ 3.

**Definition 4.** *[53] Let x*<sup>1</sup> =< *t*1, *i*1, *f*<sup>1</sup> > *be a SVNN and* λ > *0. The multiplication SVNNs and* λ *are as follows:*

$$
\lambda \mathfrak{x}\_1 = < 1 - \left(1 - t\_1\right)^{\lambda}, \mathfrak{i}\_1^{\lambda}, \mathfrak{f}\_1^{\lambda} > \tag{3}
$$

**Definition 5.** *Let X* = *(x1, x2, ..., xn) and Y* = *(y1, y2, ..., yn) be two n-dimensional vectors, x<sup>i</sup>* =< *txi*, *ixi*, *fxi* > *and y<sup>i</sup>* =< *tyi*, *iyi*, *fyi* >*. The Hamming distance between X and Y is defined as*

$$h\_{(X,Y)} = \frac{1}{3n} \sum\_{i=1}^{n} \left( |t\_{xi} - t\_{yi}| + |i\_{xi} - i\_{yi}| + |f\_{xi} - f\_{yi}| \right). \tag{4}$$

**Definition 6.** *Let X* = *(x1, x2, ..., x<sup>n</sup> ) and Y* = *(y1, y2, ..., yn) be two n-dimensional vectors, x<sup>i</sup>* =< *txi*, *ixi*, *fxi* > *and y<sup>i</sup>* =< *tyi*, *iyi*, *fyi* >*. The Euclidean distance between X and Y is defined as*

$$e\_{(X,Y)} = \frac{1}{3n} \sqrt{\sum\_{i=1}^{n} \left( \left( t\_{xi} - t\_{yi} \right)^2 + \left( i\_{xi} - i\_{yi} \right)^2 + \left( f\_{xi} - f\_{yi} \right)^2 \right)}. \tag{5}$$

**Definition 7.** *[55] Let x* =< *t*, *i*, *f* > *be a SVNN. The score function s of x is defined as*

$$s = (1 + t - 2i - f) / 2,\tag{6}$$

*where s* ∈ [−1, 1].

**Definition 8.** *[56] Let x* =< *t*, *i*, *f* > *be a SVNN. The cosine similarity measure of x is the expression*

$$c = \frac{t}{\sqrt{t^2 + t^2 + f^2}}\tag{7}$$

**Definition 9.** *[55] Let a<sup>j</sup>* =< *t<sup>j</sup>* , *i*j , *f<sup>j</sup>* > *be a collection of SVNSs and W* = (*w*1, *w*2, . . . , *wn*) *T be an associated weighting vector. The Single-Valued Neutrosophic Weighted Average (SVNWA) operator of a<sup>j</sup> is*

$$\text{SVINWA}(a\_1, a\_2, \dots, a\_n) = \sum\_{j=1}^n w\_j a\_j = \left(1 - \prod\_{j=1}^n (1 - t\_j)^{w\_j}, \prod\_{j=1}^n (i\_j)^{w\_j}, \prod\_{j=1}^n (f\_j)^{w\_j}\right) \tag{8}$$

*where w<sup>j</sup> is the element j of the weighting vector, w<sup>j</sup>* <sup>∈</sup> [0, 1] *and* <sup>P</sup>*<sup>n</sup> <sup>j</sup>*=<sup>1</sup> *w<sup>j</sup>* = 1.
