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

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Neutrosophic Soft Sets

**Definition**

**1**

**[16] (p. 1).**Let$U$be the initial universal set, a neutrosophic set$A=\{<u:{T}_{A(u)},{I}_{A(u)},{F}_{A(u)}>,u\in U\}$consists of the truth-membership${T}_{A(u)}$, the indeterminacy-membership${I}_{A(u)}$, and false-membership${F}_{A(u)}$of element$u\in U$to set$A$, where$T,I,F:U\to {]}^{-}0,{1}^{+}[$. ${]}^{-}0,{1}^{+}[$ is a non-standard interval, and the left and right borders of it are imprecise. Between them, ${(}^{-}0)=\{0-x:x\in {R}^{*},x\text{}\mathrm{is}\text{}\mathrm{infinitesimal}\}$, and $({1}^{+})=\{1+x:x\in {R}^{*},x\text{}\mathrm{is}\text{}\mathrm{infinitesimal}\}$.

**Definition**

**2**

**[29].**Let$U$be the universal set, a single valued neutrosophic set$A$over$U$can be defined as$A=\{<u:{T}_{A(u)},{I}_{A(u)},{F}_{A(u)}>,u\in U\}$, where$T,I,F:U\to [0,1]$. Similarly, the values of T

_{A(u)}, I

_{A(u)}and ${F}_{A(u)}$ 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.

**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:

- If$s({u}_{1})>s({u}_{2})$,${u}_{1}$is superior to${u}_{2}$and it can be denoted by${u}_{1}\succ {u}_{2}$;
- If$s({u}_{1})=s({u}_{2})$,$a({u}_{1})>a({u}_{2})$,${u}_{1}$is superior to${u}_{2}$and is denoted by${u}_{1}\succ {u}_{2}$;
- If$s({u}_{1})=s({u}_{2})$,$a({u}_{1})=a({u}_{2})$and$c({u}_{1})>c({u}_{2})$,${u}_{1}$is superior to${u}_{2}$and is denoted by${u}_{1}\succ {u}_{2}$;
- If$s({u}_{1})=s({u}_{2})$,$a({u}_{1})=a({u}_{2})$and$c({u}_{1})=c({u}_{2})$,${u}_{1}$is equal to${u}_{2}$, denoted by${u}_{1}\succ {u}_{2}$.

**Example**

**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:

**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$F:E\to P(U)$.

**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${]}^{-}0,{1}^{+}[$, 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$(F,E)$is regarded as a neutrosophic soft set over$U$, where$F$refers to the mapping$F:E\to P(U)$.

**Example**

**2.**

#### 2.2. Prospect Theory

_{t}) is the decision weight function as defined follows:

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

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

**Definition**

**8.**

**Definition**

**9.**

**Example**

**3.**

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

**Definition**

**10.**

**Definition**

**11.**

**Example**

**4.**

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

#### 4.1. Problem Description

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

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

#### 4.3.1. Computing the Subjective Weights

#### 4.3.2. Obtaining the Objective Weights: Information Entropy Method

#### 4.3.3. Calculating the Comprehensive Weights

#### 4.4. Computing the Comprehensive Prospect Values

#### 4.4.1. Constructing the Prospect Decision Matrix

#### 4.4.2. Computing the Comprehensive Prospect Values

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

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:
$$(\stackrel{\u25b3}{{F}^{(t)}},E)=\{\begin{array}{cc}({F}_{T}^{(t)}({e}_{j})({x}_{i}),{F}_{I}^{(t)}({e}_{j})({x}_{i}),{F}_{F}^{(t)}({e}_{j})({x}_{i})),& {e}_{j}\text{}\mathrm{is}\text{}\mathrm{a}\text{}\mathrm{benefit}\text{}\mathrm{parameter}\\ ({F}_{F}^{(t)}({e}_{j})({x}_{i}),1-{F}_{I}^{(t)}({e}_{j})({x}_{i}),{F}_{T}^{(t)}({e}_{j})({x}_{i})),& {e}_{j}\text{}\mathrm{is}\text{}\mathrm{a}\text{}\mathrm{cos}\mathrm{t}\text{}\mathrm{parameter}\end{array}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(24)$$
Step 3: Compute the determinacy degree vector ${\psi}_{\mathrm{t}}=({\psi}_{1},{\psi}_{2},\dots ,{\psi}_{p})$ of decision makers by Equation (8); Step 4: Construct the prospect decision matrix based on Equation (20). Step 5: Obtain the comprehensive weight vector ${\varpi}_{j}=({\varpi}_{1},{\varpi}_{2},\dots ,{\varpi}_{n})$ by Equation (18); Step 6: Calculate the comprehensive prospect value V _{i} 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

#### 5.1. Example Analysis

#### 5.2. Comparative Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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$\mathbf{(}{\mathbf{F}}^{\mathbf{(}\mathbf{1}\mathbf{)}},\mathbf{E}\mathbf{)}$ | ||||

${e}_{1}$ | ${e}_{2}$ | … | ${e}_{n}$ | |

${x}_{1}$ | ${F}^{(1)}({e}_{1})({x}_{1})$ | ${F}^{(1)}({e}_{2})({x}_{1})$ | … | ${F}^{(1)}({e}_{n})({x}_{1})$ |

${x}_{2}$ | ${F}^{(1)}({e}_{1})({x}_{2})$ | ${F}^{(1)}({e}_{2})({x}_{2})$ | … | ${F}^{(1)}({e}_{n})({x}_{2})$ |

$\vdots $ | $\vdots $ | $\ddots $ | $\vdots $ | |

${x}_{m}$ | ${F}^{(1)}({e}_{1})({x}_{m})$ | ${F}^{(1)}({e}_{2})({x}_{m})$ | … | ${F}^{(1)}({e}_{n})({x}_{m})$ |

$\mathbf{(}{\mathbf{F}}^{\mathbf{(}\mathbf{2}\mathbf{)}}\mathbf{,}\mathbf{E}\mathbf{)}$ | ||||

${e}_{1}$ | ${e}_{2}$ | … | ${e}_{n}$ | |

${x}_{1}$ | ${F}^{(2)}({e}_{1})({x}_{1})$ | ${F}^{(2)}({e}_{2})({x}_{1})$ | … | ${F}^{(2)}({e}_{n})({x}_{1})$ |

${x}_{2}$ | ${F}^{(2)}({e}_{1})({x}_{2})$ | ${F}^{(2)}({e}_{2})({x}_{2})$ | … | ${F}^{(2)}({e}_{n})({x}_{2})$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\ddots $ | $\vdots $ |

${x}_{m}$ | ${F}^{(2)}({e}_{1})({x}_{m})$ | ${F}^{(2)}({e}_{2})({x}_{m})$ | … | ${F}^{(2)}({e}_{n})({x}_{m})$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

$\mathbf{(}{\mathbf{F}}^{\mathbf{(}\mathbf{p}\mathbf{)}}\mathbf{,}\mathbf{E}\mathbf{)}$ | ||||

${e}_{1}$ | ${e}_{2}$ | … | ${e}_{n}$ | |

${x}_{1}$ | ${F}^{(p)}({e}_{1})({x}_{1})$ | ${F}^{(p)}({e}_{2})({x}_{1})$ | … | ${F}^{(p)}({e}_{n})({x}_{1})$ |

${x}_{2}$ | ${F}^{(p)}({e}_{1})({x}_{2})$ | ${F}^{(p)}({e}_{2})({x}_{2})$ | … | ${F}^{(p)}({e}_{n})({x}_{2})$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\ddots $ | $\vdots $ |

${x}_{m}$ | ${F}^{(p)}({e}_{1})({x}_{m})$ | ${F}^{(p)}({e}_{2})({x}_{m})$ | … | ${F}^{(p)}({e}_{n})({x}_{m})$ |

Method | The Final Ranking | The Optimal Alternative |
---|---|---|

The proposed method | ||

Weighted geometric neutrosophic rule | ${x}_{5}\succ {x}_{3}\succ {x}_{4}\succ {x}_{2}\succ {x}_{1}$ | ${x}_{5}$ |

Weighted average neutrosophic rule | ${x}_{5}\succ {x}_{3}\succ {x}_{4}\succ {x}_{2}\succ {x}_{1}$ | ${x}_{5}$ |

The determinacy degree of decision makers${\psi}_{t}=\{0.3913,0.2826,0.3261\}$ | ||

Maji [17] | ${x}_{5}\succ {x}_{4}\succ {x}_{3}\succ {x}_{2}\succ {x}_{1}$ | ${x}_{5}$ |

EDAS [25] | ${x}_{5}\succ {x}_{3}\succ {x}_{4}\succ {x}_{2}\succ {x}_{1}$ | ${x}_{5}$ |

Similarity [25] | ${x}_{5}\succ {x}_{3}\succ {x}_{4}\succ {x}_{2}\succ {x}_{1}$ | ${x}_{5}$ |

Level soft set [25] | ${x}_{5}\succ {x}_{4}\succ {x}_{3}\succ {x}_{2}\succ {x}_{1}$ | ${x}_{5}$ |

TOPSIS [37] | ${x}_{5}\succ {x}_{3}\succ {x}_{4}\succ {x}_{2}\succ {x}_{1}$ | ${x}_{5}$ |

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**MDPI and ACS Style**

Dong, Y.; Hou, C.; Pan, Y.; Gong, K.
Algorithm for Neutrosophic Soft Sets in Stochastic Multi-Criteria Group Decision Making Based on Prospect Theory. *Symmetry* **2019**, *11*, 1085.
https://doi.org/10.3390/sym11091085

**AMA Style**

Dong Y, Hou C, Pan Y, Gong K.
Algorithm for Neutrosophic Soft Sets in Stochastic Multi-Criteria Group Decision Making Based on Prospect Theory. *Symmetry*. 2019; 11(9):1085.
https://doi.org/10.3390/sym11091085

**Chicago/Turabian Style**

Dong, Yuanxiang, Chenjing Hou, Yuchen Pan, and Ke Gong.
2019. "Algorithm for Neutrosophic Soft Sets in Stochastic Multi-Criteria Group Decision Making Based on Prospect Theory" *Symmetry* 11, no. 9: 1085.
https://doi.org/10.3390/sym11091085