*2.2. Prospect Theory*

The prospect theory [27] (p. 2), proposed by Tversky and Kahneman, is a mainstream theory of behavioral science, and it studies human judgments or decision making behaviors under uncertain environments. The prospect theory mainly considers the value function and decision weight function. It implies three characteristics: Reference dependence, diminishing sensitivity and lose aversion. Reference dependence refers to the change of people's perception depending on the change of the relative value. Diminishing sensitivity means that utility decreases as income increases. Additionally, loss aversion signifies that people value losses more than gains.

The prospect theory states that decision makers choose the optimal alternative based on the prospect value, which is determined by the value function and decision weight function. The prospect value can be obtained as follows:

$$V = \sum v(\mathbf{x} - r)w(p\_t). \tag{5}$$

*v*(*x* − *r*) is the value function as defined follows:

$$w(\mathbf{x} - r) = \begin{cases} (\mathbf{x} - r)^{\alpha}, & \mathbf{x} \ge r \\ -\lambda(\mathbf{x} - r)^{\beta}, & \mathbf{x} < r' \end{cases} \tag{6}$$

where *x* is the evaluation value of an object, *r* is the reference point, then (*x* − *r*) represents losses or gains. *x* ≥ *r* means gains, and the value function is concave; *x* < *r* means losses, and the value function is convex. So α, β stand for the concave degree and convexity degree of the value function, respectively. λ is the risk aversion coefficient, and λ > 1 indicates that decision makers value risk more. By experimental verification, Tversky and Kahneman took the value of parameters as follows: α = β = 0.88, λ = 2.25.

ω(*Pt*) is the decision weight function as defined follows:

$$\omega(p\_l) = \frac{p\_l^{\mathcal{V}}}{((p\_l^{\mathcal{V}}) + ((1 - p\_l)^{\mathcal{V}}))^{\frac{1}{\mathcal{V}}}} \, \tag{7}$$

where *pt* is the objective possibility, and Tversky and Kahneman took the value of parameter γ as 0.61.

#### **3. The Measures of Determinacy Degree and Conflict Degree and Neutrosophic Soft Set Aggregation Rules**

In this section, we initiate the determinacy degree measure and conflict degree measure of neutrosophic numbers, and then develop two kinds of aggregation rules of a neutrosophic soft set.

#### *3.1. The Measures of Determinacy Degree and Conflict Degree*

This paper employs the Hamming distance of information theory, which is a well-known measure designed to provide insights into the similarity of information [33,34] and has been widely employed in distance measures [26,35], to measure the determinacy degree and conflict degree. Before this, we present the concept of a minimum conflict neutrosophic number and maximum conflict neutrosophic number.

**Definition 8.** *Let Minc* =< 1, 0, 0 > *be the minimum conflict neutrosophic number, which means that the belongingness degree of an object is 1, and the non-belongingness degree and the neutrality degree of an object be zero, respectively. That is, the conflict degree of information is the smallest.*

*Additionally, let Maxc* =< 0.5, 1, 0.5 > *be the maximum conflict neutrosophic number. That is, the neutrosophic number, whose neutrality degree is one, and the belongingness degree and non-belongingness degree is 0.5. In order words, the conflict degree of information is the greatest.*

**Definition 9.** *Let u* =< *T*, *I*, *F* > *be a neutrosophic number, the determinacy degree of u based on Equation (4) can be defined as follows:*

$$d^{\Delta}(u) = \frac{(|T - 1| + I + F)}{3},\tag{8}$$

*which measures the normalized Hamming distance between u and the minimum conflict neutrosophic number.*

*Similarly, the conflict degree of u is determined by the normalized Hamming distance between u and the maximum conflict neutrosophic number, and defined as follows:*

$$\mathcal{L}^{\Lambda}(\mu) = \frac{(|T - 0.5| + |I - 1| + |F - 0.5|)}{3} \tag{9}$$

**Example 3.** *Considering Example 1, the determinacy degree and conflict degree of u*1 *can be computed as follows: <sup>d</sup>*<sup>Δ</sup>(*<sup>u</sup>*1) = 0.8/3*, <sup>c</sup>*<sup>Δ</sup>(*<sup>u</sup>*1) = 1.2/3.

#### *3.2. Aggregation Rules of a Neutrosophic Soft Set*

In this subsection, we define two kinds of aggregation rules of a neutrosophic soft set, namely the weighted average aggregation rule and weighted geometric aggregation rule.

**Definition 10.** *Weighted average aggregation rule. Let U be the initial universal set, E be the set of parameters,* (*<sup>F</sup>*, *E*) *be a neutrosophic soft set over U, as represented by <sup>F</sup>*(*ej*)(*xi*) =< *FT*(*ej*)(*xi*), *FI*(*ej*)(*xi*), *FF*(*ej*)(*xi*) > (*i* = 1, 2, ... , *m*; *j* = 1, 2, ... , *<sup>n</sup>*)*. Then, the weighted average aggregation rule of* (*<sup>F</sup>*, *E*) *can be denoted by* (*<sup>F</sup>*, *E*)<sup>Γ</sup> = -*<sup>F</sup>*<sup>Γ</sup>(*<sup>x</sup>*1), *<sup>F</sup>*<sup>Γ</sup>(*<sup>x</sup>*2), ... , *<sup>F</sup>*<sup>Γ</sup>(*xm*)*, and defined as*

$$F^{\Gamma}(\mathbf{x}\_{i}) = \prod\_{j=1}^{n} F(\boldsymbol{\varepsilon}\_{j})(\mathbf{x}\_{i}) \boldsymbol{\omega}\_{j} = < 1 - \prod\_{j=1}^{n} \left( 1 - F\_{T}(\boldsymbol{\varepsilon}\_{j})(\mathbf{x}\_{i}) \right)^{\boldsymbol{\omega}\_{j}} \cdot \prod\_{j=1}^{n} \left( F\_{I}(\boldsymbol{\varepsilon}\_{j})(\mathbf{x}\_{i}) \right)^{\boldsymbol{\omega}\_{j}} \cdot \prod\_{j=1}^{n} \left( F\_{F}(\boldsymbol{\varepsilon}\_{j})(\mathbf{x}\_{i}) \right)^{\boldsymbol{\omega}\_{j}} > \tag{10}$$

*where the vector* ω = {<sup>ω</sup>1, ω2, ... , <sup>ω</sup>*n*} *stands for the weights of parameters, and n j*=1 <sup>ω</sup>*j* = 1.

*Based on Definition 10, the weighted geometric aggregation rule of a neutrosophic soft set is constructed.*

**Definition 11.** *Weighted geometric aggregation rule. Considering the neutrosophic soft set* (*<sup>F</sup>*, *E*) *in Definition 10, we define the weighted geometric aggregation rule as* (*<sup>F</sup>*, *E*)<sup>Θ</sup> = {*F*<sup>Θ</sup>(*<sup>x</sup>*1), *<sup>F</sup>*<sup>Θ</sup>(*<sup>x</sup>*2), ... , *<sup>F</sup>*<sup>Θ</sup>(*xm*)}*, and*

$$F^{\mathfrak{B}}(\mathbf{x}\_{i}) = \prod\_{j=1}^{n} \left( F(e\_{j})(\mathbf{x}\_{i}) \right)^{\omega\_{\parallel}} = \epsilon \prod\_{j=1}^{n} \left( F\_{T}(e\_{j})(\mathbf{x}\_{i}) \right)^{\omega\_{\parallel}}, \mathbf{1} - \prod\_{j=1}^{n} \left( 1 - \left( F\_{I}(e\_{j})(\mathbf{x}\_{i}) \right) \right)^{\omega\_{\parallel}}, \mathbf{1} - \prod\_{j=1}^{n} \left( 1 - \left( F\_{F}(e\_{j})(\mathbf{x}\_{i}) \right) \right)^{\omega\_{\parallel}} > \tag{11}$$

*where the vector* ω = {<sup>ω</sup>1, ω2, ... , <sup>ω</sup>*n*}*stands for the weights of parameters, and n j*=1 <sup>ω</sup>*j* = 1.

**Example 4.** *Consider Example 2. Assume that the weight vector of parameters is* ω = {0.4, 0.2, 0.3}*, then we can obtain the results of the weighted average aggregation and weighted geometric aggregation as follows, respectively.*

(*<sup>F</sup>*, *E*)<sup>Γ</sup> = {< *u*1, 0.6077, 0.5537, 0.3584 >,< *u*2, 0.7015, 0.4749, 0.2244 >,< *u*3, 0.3169, 0.6512, 0.6467 <sup>&</sup>gt;}. (*<sup>F</sup>*, *E*)<sup>Θ</sup> = {< *u*1, 0.6049, 0.9905, 0.9987 >,< *u*2, 0.6837, 0.9973, 0.9998 >,< *u*3, 0.3474, 0.9798, 0.9885 >}

#### **4. Algorithm for Neutrosophic Soft Sets in Stochastic Multi-Criteria Group Decision Making Based on Prospect Theory**
