*4.1. Problem Description*

In this section, we give a concise description of a stochastic multi-criteria group decision making problem under neutrosophic soft sets. Let *U* = {*<sup>x</sup>*1, *x*2, ... , *xm*} be a set of *m* alternatives, *E* = {*<sup>e</sup>*1,*e*2, ... ,*en*} be a set of *n* parameters and *DM* = -*Z*1,*Z*2, ... *Zp* be a set of *p* decision makers. Assume that ω(*t*) =< ω(*t*) *T* , ω(*t*) *I* , ω(*t*) *F* > (*t* = 1, 2, ... , *p*) is the neutrosophic weight of decision maker *Zt*, δ(*t*) *j* =< δ(*t*) *Tj* , δ(*t*) *Ij* , δ(*t*) *Fj* > is the neutrosophic subjective weight assigned for parameter *ej* by decision maker *Zt*, and the evaluation value of alternative *xi* related to parameter *ej* by decision maker *Zt* is expressed as *<sup>F</sup>*(*t*)(*ej*)(*xi*) =< *F*(*t*) *T* (*ej*)(*xi*), *F*(*t*) *I* (*ej*)(*xi*), *F*(*t*) *F* (*ej*)(*xi*) >. Given *p* neutrosophic soft sets (*F*(*t*), *E*) (*t* = 1, 2, ... , *p*) of alternatives evaluated by decision makers, and the tabular representation of (*F*(*t*), *E*) (*t* = 1, 2, ... , *p*) is shown in Table 1.

#### *4.2. Determining the Determinacy Degree of Decision Makers*

In stochastic multi-criteria group decision making problems, the weights of decision makers are stochastic and indeterminate. Therefore, how to obtain the weights as determinate values has become an important research topic. In this paper, we express the weights of decision makers as a neutrosophic number,andthencomputethedeterminacydegreeofdecisionmakerstoreplacetraditionalweights.

 Considering Definition 9, let ω*t* =< ω*Tt* , ω*It*, ω*Ft* > (*t* = 1, 2, ... , *p*) be the neutrosophic weight of decision maker *Zt*, then the determinacy degree of *Zt* can be computed as follows by Equation (8):

$$d^{\Delta}(t) = \frac{1 - \frac{1}{5}(|\omega\_t^T - 1| + \omega\_t^I + \omega\_t^F)}{\sum\_{t=1}^p 1 - \frac{1}{5}(|\omega\_t^T - 1| + \omega\_t^I + \omega\_t^F)} (t = 1, 2, \dots, p), \tag{12}$$


**Table 1.** Tabular representation of neutrosophic soft sets (*F*(*t*), *E*) of alternatives.

#### *4.3. Calculating the Comprehensive Weights of Parameters*

In this paper, the parameter weights are determined by combining subjective weights with objective weights. Among them, subjective weights are obtained by aggregating neutrosophic subjective weights provided by decision makers, which is more accurate than the way directly given by determinate values [25] (p. 2). The objective weights are calculated by the information entropy method [35]. Then, the principle of minimum information entropy [36] is employed to obtain comprehensive weights

of parameters by integrating subjective weights and objective weights. The system framework is presented in Figure 1.

**Figure 1.** The system framework of the computing comprehensive weights of parameters.

#### 4.3.1. Computing the Subjective Weights

Under the stochastic environment, the judgements of decision makers are full of hesitancies. Considering this situation, instead of giving determinate values, this paper firstly aggregates neutrosophic subjective weights of parameters to obtain subjective weights in the form of neutrosophic numbers. Based on this, subjective weights are computed by the score function as Equation (1).

Assume parameter set *E* = *e*1,*e*2, ... ,*ej*is the initial universal set, the set of decision makers *Z* = {*<sup>z</sup>*1, *z*2, ... , *zt*} is the parameter set, and *P*(*Z*) is the set of all neutrosophic subsets of *E*. The neutrosophic soft set (*<sup>F</sup>*,*<sup>Z</sup>*) over *E* can be integrated by the weighted geometric aggregation rule as (*<sup>F</sup>*,*<sup>Z</sup>*)<sup>Θ</sup> = -*<sup>F</sup>*<sup>Θ</sup>(*<sup>e</sup>*1), *<sup>F</sup>*<sup>Θ</sup>(*<sup>e</sup>*2), ... , *<sup>F</sup>*<sup>Θ</sup>(*em*), and

$$F^{\Theta}(\boldsymbol{\varepsilon}\_{j}) = \prod\_{t=1}^{p} \delta\_{j}^{(t)} \quad = \epsilon \prod\_{t=1}^{p} \delta\_{Tj}^{(t)} \; , 1 - \prod\_{t=1}^{p} \left(1 - \delta\_{lj}^{(t)}\right) \; , 1 - \prod\_{t=1}^{p} \left(1 - \delta\_{Fj}^{(t)}\right) \; > \tag{13}$$

where δ(*t*) *j* =< δ(*t*) *jT* , δ(*t*) *Ij* , δ(*t*) *Fj* > (*j* = 1, 2, ... , *n*) is the neutrosophic subjective weight assigned for parameter *ej* by *Zt*, and ψt is the determinacy degree of *Zt*.

Then, the subjective weights can be computed by the score function as shown below:

$$\text{LSW}\_{j} = \frac{2 + \prod\_{t=1}^{p} \delta\_{Tj}^{(t)} - (1 - \prod\_{t=1}^{p} (1 - \delta\_{Ij}^{(t)})^{\quad \text{\textquotedbl{}}t}) - (1 - \prod\_{t=1}^{p} (1 - \delta\_{Fj}^{(t)})^{\quad \text{\textquotedbl{}}t})}{3}. \tag{14}$$

4.3.2. Obtaining the Objective Weights: Information Entropy Method

Considering that the computation of objective weights is not the focus of this paper, we obtain objective weights by the information entropy method. The information entropy is used to measure the uncertainty of events. The greater the information entropy is, the greater the uncertainty degree. That is, *Symmetry* **2019**, *11*, 1085

the smaller the amount of information it carries, the smaller the weight is. Note that the uncertainty of neutrosophic numbers consists of two factors, one is the truth-membership and false-membership, and the other is the indeterminacy-membership.

Based on the information entropy method, we can obtain that the information entropy of parameter *ej* given by decision maker *Zt* is defined as:

$$E\_j^t = 1 - \frac{1}{m} \sum\_{i=1}^m \left( F\_T^{(t)} \{ \mathbf{e}\_j \} (\mathbf{x}\_i) + F\_F^{(t)} \{ \mathbf{e}\_j \} (\mathbf{x}\_i) \right) |F\_I^{(t)} \{ \mathbf{e}\_j \} (\mathbf{x}\_i) - F\_I^{(t) \subset} \{ \mathbf{e}\_j \} (\mathbf{x}\_i)|(j = 1, 2, \dots, n). \tag{15}$$

Then, the comprehensive information entropy of parameter *ej* is defined as follows:

$$E\_j = \sum\_{t=1}^{p} q\_t E\_j^t (j = 1, 2, \dots, n) \tag{16}$$

where ϕ*t* is the determinacy degree of decision maker *Zt* computed by Equation (8).

So, the objective weights are obtained as:

$$OW\_j = \frac{1 - E\_j}{\sum\_{j=1}^{n} 1 - E\_j} (j = 1, 2, \dots, n). \tag{17}$$

#### 4.3.3. Calculating the Comprehensive Weights

Based on the principle of the minimum information entropy, the comprehensive weight of parameter *j* can be calculated as follows:

$$\alpha\_{\bar{j}} = \frac{\sqrt{\text{OW}\_{\bar{j}} \cdot \text{SW}\_{\bar{j}}}}{\sum\_{j=1}^{n} \sqrt{\text{OW}\_{\bar{j}} \cdot \text{SW}\_{\bar{j}}}} \tag{18}$$

where *SWj* and *OWj* represent the subjective weight and objective weight of parameter *ej*, respectively.

#### *4.4. Computing the Comprehensive Prospect Values*

The comprehensive prospect values of alternatives are determined by the prospect decision matrix and the comprehensive weights of parameters. Next, we expound how to generate the prospect decision matrix and obtain comprehensive values of alternatives, respectively.

#### 4.4.1. Constructing the Prospect Decision Matrix

The core of constructing the prospect decision matrix is to compute the value function and decision weight function. In terms of the value function, we need to analyze the distance between the reference point and the actual value. This paper regards the maximum conflict neutrosophic number as the reference point, then the distance can be treated as the conflict degree of the actual value. Additionally, actual values refer to the alternative evaluation values with respect to the parameters. As for the decision weight function, the objective possibility is seen as the determinacy degree of the decision makers. The system framework of constructing the prospect decision matrix is shown in Figure 2.

**Figure 2.** The system framework of constructing the prospect decision matrix.

We assume that the neutrosophic soft sets of alternatives and neutrosophic subjective weights of parameters are both provided by decision makers. So, the conflict degree of the alternative evaluation values with respect to the parameters should take the neutrosophic subjective weights of parameters into account. Based on the conflict degree measure given by Definition 9, we develop a modified conflict degree measure by introducing the neutrosophic subjective weights of parameters.

Assume *<sup>F</sup>*(*ej*)(*xi*) =< *FT*(*ej*)(*xi*), *FI*(*ej*)(*xi*), *FF*(*ej*)(*xi*) > is a neutrosophic number, which represents the value of alternative *xi* related to parameter*ej*, and α*j* =< <sup>α</sup>*jT*, <sup>α</sup>*jI*, <sup>α</sup>*jF* > is the neutrosophic subjective weight of parameter *ej*. Considering the sum of <sup>α</sup>*jT*, <sup>α</sup>*jI* and <sup>α</sup>*jF* may not be one, this paper normalizes them to be more consistent with the reality. Therefore, the measure of the modified conflict degree of *<sup>F</sup>*(*ej*)(*xi*) is defined as follows:

$$mc^{\Delta}(F(\boldsymbol{\varepsilon}\_{\text{j}})(\mathbf{x}\_{i})) = \frac{a\_{\overline{\boldsymbol{\varepsilon}}\overline{\boldsymbol{\Gamma}}} \cdot \left| \operatorname{F} \mathbf{r}(\boldsymbol{\varepsilon}\_{\text{j}})(\mathbf{x}\_{i}) - 0.5 \right|}{a\_{\overline{\boldsymbol{\varepsilon}}\overline{\boldsymbol{\Gamma}}} + a\_{\overline{\boldsymbol{\varepsilon}}\overline{\boldsymbol{\Gamma}}} + \frac{a\_{\overline{\boldsymbol{\varepsilon}}\overline{\boldsymbol{\Gamma}}} \cdot \left| \operatorname{F} \mathbf{r}(\boldsymbol{\varepsilon}\_{\text{j}})(\mathbf{x}\_{i}) - 1 \right|}{a\_{\overline{\boldsymbol{\varepsilon}}\overline{\boldsymbol{\Gamma}}} + a\_{\overline{\boldsymbol{\varepsilon}}\overline{\boldsymbol{\Gamma}}} + a\_{\overline{\boldsymbol{\varepsilon}}\overline{\boldsymbol{\Gamma}}} + \frac{a\_{\overline{\boldsymbol{\varepsilon}}\overline{\boldsymbol{\Gamma}}} \cdot \left| \operatorname{F} \mathbf{r}(\boldsymbol{\varepsilon}\_{\text{j}})(\mathbf{x}\_{i}) - 0.5 \right|}{a\_{\overline{\boldsymbol{\varepsilon}}\overline{\boldsymbol{\Gamma}}} + a\_{\overline{\boldsymbol{\varepsilon}}\overline{\boldsymbol{\Gamma}}} + a\_{\overline{\boldsymbol{\varepsilon}}\overline{\boldsymbol{\Gamma}}} \right|}. \tag{19}$$

Subsequently, calculate the prospect value of each alternative with respect to the parameters as follows:

$$\mathcal{V}\_{ij} = \sum\_{t=1}^{p} w(z\_t) \upsilon(F^{(t)}(\mathbf{e}\_j)(\mathbf{x}\_i) - \mathbf{x}\_0),\tag{20}$$

where

$$w(F^{(t)}(\mathbf{e}\_{\dot{\boldsymbol{\gamma}}})(\mathbf{x}\_{i}) - \mathbf{x}\_{0}) = \begin{cases} \left(mc^{\Lambda}(F^{(t)}(\mathbf{e}\_{\dot{\boldsymbol{\gamma}}})(\mathbf{x}\_{i}), \mathbf{x}\_{0})\right)^{0.88}, & F^{(t)}(\mathbf{e}\_{\dot{\boldsymbol{\gamma}}})(\mathbf{x}\_{i}) \ge \mathbf{x}\_{0} \\ -2.25(mc^{\Lambda}(F^{(t)}(\mathbf{e}\_{\dot{\boldsymbol{\gamma}}})(\mathbf{x}\_{i}), \mathbf{x}\_{0}))^{0.88}, & F^{(t)}(\mathbf{e}\_{\dot{\boldsymbol{\gamma}}})(\mathbf{x}\_{i}) < \mathbf{x}\_{0} \end{cases} \tag{21}$$

$$\wp(Z\_t) = \frac{\left(\psi\_t\right)^{0.61}}{\left(\left(\psi\_t\right)^{0.61} + \left(1 - \psi\_t\right)^{0.61}\right)^{\frac{1}{0.61}}}.\tag{22}$$

Then, we can obtain the prospect decision matrix.

#### 4.4.2. Computing the Comprehensive Prospect Values

Based on comprehensive weights of parameters and the prospect decision matrix, we can compute the comprehensive prospect values for alternatives as follows:

$$V\_i = \sum\_{j=1}^{n} \alpha\_j V\_{ij}.\tag{23}$$

The system framework of computing the comprehensive prospect values is shown in Figure 3.

**Figure 3.** The system framework of computing the comprehensive prospect values of alternatives.

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

In this section, a novel algorithm for neutrosophic soft sets in stochastic multi-criteria group decision making based on the prospect theory is proposed. The detailed operation steps of Algorithm 1 are presented below.

**Algorithm 1:** Neutrosophic soft sets in stochastic multi-criteria group decision making based on the prospect theory

Step 1: Input a neutrosophic set, which represents neutrosophic weights of decision makers and two neutrosophic soft sets, including alternatives description as shown in Table 1 and neutrosophic subjective weights of parameters evaluated by decision makers.

Step 2: Normalize the neutrosophic soft sets of alternatives as follows:

$$(F^{(t)}, E) = \begin{cases} (F\_T^{(t)}(\mathbf{e}\_j)(\mathbf{x}\_l), F\_I^{(t)}(\mathbf{e}\_j)(\mathbf{x}\_l), F\_F^{(t)}(\mathbf{e}\_j)(\mathbf{x}\_l)), & \mathbf{e}\_j \text{ is a benefit parameter} \\ (F\_F^{(t)}(\mathbf{e}\_j)(\mathbf{x}\_l), 1 - F\_I^{(t)}(\mathbf{e}\_j)(\mathbf{x}\_l), F\_T^{(t)}(\mathbf{e}\_j)(\mathbf{x}\_l)), & \mathbf{e}\_j \text{ is a cost parameter} \end{cases} (24)$$

Step 3: Compute the determinacy degree vector ψt = (ψ1,ψ2, ... ,ψ*p*) of decision makers by Equation (8); Step 4: Construct the prospect decision matrix based on Equation (20).

Step 5: Obtain the comprehensive weight vector *j* = ( 1, 2, ... , *n*) by Equation (18);

Step 6: Calculate the comprehensive prospect value *Vi* for each alternative through Equation (23).

Step 7: Make a decision by ranking alternatives based on comprehensive prospect values.

#### **5. An Application of the Proposed Algorithm**

In order to verify the feasibility of the proposed algorithm, we discuss the investment decision of a finance institution. Meanwhile, the existing five methods [17,25,37] (pp. 1–2) are employed for a comparative analysis to prove the feasibility and superiority of the proposed algorithm.
