**2. Preliminaries**

In this section, we briefly recall some basic concepts of neutrosophic sets, soft sets, neutrosophic soft sets, and prospect theory. More detailed conceptual basics can be found in references [4,16,17,27] (pp. 1–2).

#### *2.1. Neutrosophic Soft Sets*

**Definition 1 [16] (p. 1).** *Let U be the initial universal set, a neutrosophic set A* = -< *u* : *TA*(*u*), *IA*(*u*), *FA*(*u*) >, *u* ∈ *U consists of the truth-membership TA*(*u*)*, the indeterminacy-membership IA*(*u*)*, and false-membership FA*(*u*) *of element u* ∈ *U to set A, where T*, *I*, *F* : *U* →]−0, <sup>1</sup>+[. ]−0, 1+[ is a non-standard interval, and the left and right borders of it are imprecise. Between them, (−<sup>0</sup>) = {0 − *x* : *x* ∈ *R*<sup>∗</sup>, *x* is infinitesimal}*, and* (1+) = {1 + *x* : *x* ∈ *R*<sup>∗</sup>, *x* is infinitesimal}.

For convenience, we employ *u* =< *T*,*I*, *F* > to represent the element *u* in the neutrosophic set *A*, and it can be called a neutrosophic number.

Considering that neutrosophic sets are proposed from the philosophical point of view, it is difficult to apply to practical problems, such as managemen<sup>t</sup> and engineering problems. Then, Haibin et al. [29] developed single valued neutrosophic sets.

**Definition 2 [29].** *Let U be the universal set, a single valued neutrosophic set A over U can be defined as A* = -< *u* : *TA*(*u*), *IA*(*u*), *FA*(*u*) >, *u* ∈ *U, where T*, *I*, *F* : *U* → [0, <sup>1</sup>]. *Similarly, the values of TA(u), IA(u)* and *FA*(*u*) stand for the truth-membership, indeterminacy-membership, and false-membership of *u to A*, respectively.

**Definition 3 [30].** *Let u* =< *T*, *I*, *F* > *be a neutrosophic number, then the score function, accuracy function and certainty function are defined as follows, respectively.*

$$s(u) = \frac{2 + T - I - F}{3},\tag{1}$$

$$a(u) = T - F,\tag{2}$$

$$c(u) = T,\tag{3}$$

The score function is an important index for evaluating neutrosophic numbers. For a neutrosophic number *R* =< *Tr*,*Ir*, *Fr* >, the truth-membership *Tr* is positively correlated with the score function, and the indeterminacy-membership *Ir* and false-membership *Fr* are negatively correlated with the score function. In terms of the accuracy function, the greater the difference between the truth-membership *Tr* and false-membership *Fr* is, the more affirmative the statement is. Additionally, in regard to the certainty function, it positively depends on the truth-membership *Tr*.

On the basis of Definition 3, the comparison method between two neutrosophic numbers is represented as follows.

**Definition 4 [30].** *Let u*1 =< *T*1,*I*1, *F*1 >, *u*2 =< *T*2,*I*2, *F*2 > *be two neutrosophic numbers, the comparison relationships between u*1 *and u*2 are as follows:


**Example 1.** *For two neutrosophic numbers u*1 =< 0.8, 0.2, 0.4 > *and u*2 =< 0.7, 0.4, 0.1 >*, we can obtain that <sup>s</sup>*(*<sup>u</sup>*1) = 2.2/3*, <sup>s</sup>*(*<sup>u</sup>*2) = 2.2/3*, <sup>a</sup>*(*<sup>u</sup>*1) = 1.2/3*, <sup>a</sup>*(*<sup>u</sup>*2) = 1.8/3*, <sup>c</sup>*(*<sup>u</sup>*1) = 2.4/3 *and <sup>c</sup>*(*<sup>u</sup>*2) = 2.1/3 *based on Definition 3. Considering Definition 4, we can infer that u*2 *is superior to u*1*, as denoted by u*2 *u*1.

**Definition 5 [31].** *Let u*1 =< *T*1,*I*1, *F*1 >, *u*2 =< *T*2,*I*2, *F*2 > *be two neutrosophic numbers, then the normalized Hamming distance between u*1 *and u*2 is defined as follows:

$$D^{\triangle}(\mu\_1, \mu\_2) = \frac{(|T\_1 - T\_2| + |I\_1 - I\_2| + |F\_1 - F\_2|)}{3}. \tag{4}$$

**Definition 6 [4] (p. 1).** *Let U be the set of initial universe, E be the parameter set, and P*(*U*) *be the power set of U. Then a pair (F, E)is called a soft set over U where F is a mapping given by F* : *E* → *<sup>P</sup>*(*U*).

**Remark 1 [32].** *On account of the single valued neutrosophic set is an instance of the neutrosophic set, it is natural to infer that a single valued neutrosophic soft set is an instance of the neutrosophic soft set. However, Maji only considers neutrosophic soft sets, which take value from the standard subset of* [0, 1] *rather than* ]−0, <sup>1</sup>+[*, so the definition of the single valued neutrosophic soft set is exactly the same as the concept of the neutrosophic soft set defined by Maji.*

**Definition 7 [17] (p. 1).** *Let U be the initial universal set, E be a set of parameters, and P*(*U*) *be the set of all neutrosophic subsets of U. The collection* (*<sup>F</sup>*, *E*) *is regarded as a neutrosophic soft set over U, where F refers to the mapping F* : *E* → *<sup>P</sup>*(*U*).

**Example 2.** *Assume U* = {*<sup>u</sup>*1, *u*2, *<sup>u</sup>*3} *is a set of three cars under consideration, and E* = *e*1 = cheap,*<sup>e</sup>*2 = equipment,*<sup>e</sup>*3 = fuel consumption *be the set of parameters for describing the three. In this case, we can define a function F* : *E* → *P*(*U*) *as a neutrosophic soft set* (*<sup>F</sup>*, *<sup>E</sup>*)*, and it is represented as follows:*

$$F(F,E) = \left\{ F(\mathfrak{e}\_2) = \left\{ F(\mathfrak{e}\_2) = \{ \begin{aligned} F(\mathfrak{e}\_1) &= \{ \times u\_1, 0.8, 0.4, 0.3 > \swarrow \omega\_2, 0.5, 0.7, 0.3 > \swarrow \omega\_3, 0.2, 0.5, 0.8 > \} \\ F(\mathfrak{e}\_2) &= \{ \times u\_1, 0.5, 0.7, 0.4 > \swarrow \omega\_2, 0.7, 0.3, 0.2 > \swarrow \omega\_3, 0.5, 0.8, 0.5 > \} \\ F(\mathfrak{e}\_3) &= \{ \times u\_1, 0.4, 0.6, 0.3 > \swarrow \omega\_2, 0.9, 0.3, 0.1 > \swarrow \omega\_3, 0.4, 0.7, 0.5 > \} \end{aligned} \right\} \right\}.$$
