# An Evidential Prospect Theory Framework in Hesitant Fuzzy Multiple-Criteria Decision-Making

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Evidence Theory

_{i}(i = 1, 2, 3, …, n). The power set of Ω is denoted by

_{1}and m

_{2}denote two pieces of evidence whose belief structures are independent of each other. Then, the combination rule $m={m}_{1}\oplus {m}_{2}$ is [15]

_{i}and m

_{j}be two belief structures, where i, j = 1, 2, 3, …, n. Then, the Jousselme distance between m

_{i}and m

_{j}is as follows [33]:

_{i}is

_{i}(i.e., the credibility degree of each belief structure) is

_{i}denote the weighting factor of m

_{i}($i=1,2,\dots ,n$), where ${\sum}_{i=1}^{n}{w}_{i}}=1$. Then, the average belief structure $\overline{m}$ is

^{f}formed by combining ${m}_{1},{m}_{2},\dots ,{m}_{n}$ is

#### 2.2. Prospect Theory

_{0}is the reference point, x −x

_{0}≥ 0 represents the gains, and x −x

_{0}< 0 represents the losses. The exponent parameters α and β (0 ≤α, β ≤ 1) are the coefficients of risk aversion. The parameter λ is the coefficient of loss aversion, and λ > 1. For simplicity, let α = β = 0.88, λ = 2.25, r = 0.65, as derived from the empirical evidence provided by Tversky and Kahneman [34].

#### 2.3. The Concept of HFEs

**Definition**

**1**

**.**Let a reference set be denoted by Y, and let a hesitant fuzzy set (HFS) on Y be denoted by B. Then, the hesitant fuzzy set B is represented as

_{B}(y) is a set of some different values in [0, 1], indicating the possible memberships of y∈Y to B. h

_{B}(y) is called a hesitant fuzzy element (HFE) [36]. If B = {〈 y, h

_{B}(y) = {0}〉| y∈Y}, then the hesitant fuzzy set B is called the empty hesitant fuzzy set [35]. If B = {〈 y, h

_{B}(y) = {1}〉| y∈Y}, then the hesitant fuzzy set B is called the full hesitant fuzzy set [35]. Similarly, if h

_{B}(y) = {0}, the hesitant fuzzy element h

_{B}(y) is called a hesitant empty element [37]. If h

_{B}(y) = {1}, the hesitant fuzzy element h

_{B}(y) is called a hesitant full element [37].

_{1}, h

_{2}, h

_{3}be three HFEs. Then, the following operations of HFEs are defined [36]:

## 3. A Novel Score Function

**Definition**

**2**

**.**Let an HFE be denoted by $h=\langle {\gamma}_{1},{\gamma}_{2},\dots ,{\gamma}_{l\left(h\right)}\rangle $, $S(h)={\displaystyle {\sum}_{\tau =1}^{l\left(h\right)}{\gamma}_{\tau}/l(h)}$ is called the score function, where l(h) is the number of elements in the HFE. For two HFEs h

_{1}and h

_{2}, if S(h

_{1}) > S(h

_{2}), then ${h}_{1}\succ {h}_{2}$, i.e., h

_{1}is superior to h

_{2}. If S(h

_{1}) = S(h

_{2}), then ${h}_{1}\u2053{h}_{2}$, i.e., h

_{1}is indifferent to h

_{2}.

_{1}) = S(h

_{2}), since S(h) is the average value of all elements in h.

**Definition**

**3**

**.**Let $h=\langle {\gamma}_{1},{\gamma}_{2},\dots ,{\gamma}_{l\left(h\right)}\rangle $ be an HFE, where l(h) is the number of elements in h. A score function is

_{1}and h

_{2}, if S

^{′}(h

_{1}) > S

^{′}(h

_{2}), then ${h}_{1}\succ {h}_{2}$, which means that h

_{1}is superior to h

_{2}. If S

^{′}(h

_{1}) = S

^{′}(h

_{2}), then ${h}_{1}\u2053{h}_{2}$, which means that h

_{1}is indifferent to h

_{2}.

**Definition**

**4.**

**Definition**

**5.**

_{1}and h

_{2}, if S

^{″}(h

_{1}) > S

^{″}(h

_{2}), then ${h}_{1}\succ {h}_{2}$, i.e., h

_{1}is superior to h

_{2}. If S

^{″}(h

_{1}) = S

^{″}(h

_{2}), then ${h}_{1}\u2053{h}_{2}$, i.e., h

_{1}is indifferent to h

_{2}.

**Proposition**

**1.**

^{″}(h) lies in [0, 1] for any HFE h.

**Proof.**

_{τ}∈[0, 1] for τ = 1, 2, …, l(h),

^{″}(h) ≤ 1, i.e., S

^{″}(h)∈[0, 1]. □

**Example**

**1.**

_{1}= {0.3, 0.5}, h

_{2}= {0.4}, and h

_{1}= {0.2, 0.4, 0.6} be three HFEs. It follows from Definition 2 that S(h

_{1}) = S(h

_{2}) = S(h

_{3}) = 0.4, which means ${h}_{1}\u2053{h}_{2}\u2053{h}_{3}$. By applying Definition 3, we can obtain S

^{′}(h

_{1}) = 0.383, S

^{′}(h

_{2}) = 0.4, and S

^{′}(h

_{3}) = 0.467, which means ${h}_{1}\prec {h}_{2}\prec {h}_{3}$. By applying the proposed score function, we have S

^{″}(h

_{1}) = 0.1719, S

^{″}(h

_{2}) = 0.4, and S

^{″}(h

_{3}) = 0.1837, which means ${h}_{1}\prec {h}_{3}\prec {h}_{2}$.

## 4. The Evidential Prospect Theory Framework

#### 4.1. Evidential Decision-Making Problems

_{i}and finite criteria C

_{j}, where i = 1, 2, 3,…, t and j = 1, 2, 3, …, n.

_{ij}is used to measure the rating of the alternative A

_{i}based on the criterion C

_{j}. $w=({w}_{1},{w}_{2},\dots ,{w}_{n})$ is a weight vector on the criteria satisfying ${\sum}_{j=1}^{n}{w}_{j}}=1$ and $0\le {w}_{j}\le 1$.

_{j}for alternative A

_{i}, where the focal element ${B}_{ij}^{k}$ given by expert k is a set of linguistic terms with respect to C

_{j}for A

_{i}. Evaluation values of the criteria for all alternatives given by all experts are shown in Table 2.

#### 4.2. Combining Belief structures

_{i}is

#### 4.3. Applying Prospect Theory

_{i}

_{i}. Thus, from the increasing order of ${u}_{i}^{EP}$ with respect to the alternative A

_{i}, the ranking order of all alternatives is shown. Suitable alternatives can be chosen according to the ranking order.

## 5. A Case Study

#### 5.1. A Description of a Multiple-Criteria Decision-Making Problem

_{1}, A

_{2}, A

_{3}, A

_{4}, and A

_{5}, respectively. Since different choices might have different benefits and different development prospects, a suitable choice must be made among the five alternatives. To make the choice, we consider the performance of each city based on three different criteria, i.e., status quo (C

_{1}), future development (C

_{2}), and benefits (C

_{3}). Because this company pays more attention to potential development prospects, next to benefits, future development is the most important factor for the company. To reflect the importance of these criteria, a weight vector w = (0.25, 0.40, 0.35) was given by the renewable energy company. To choose the most desirable alternative, we introduced a group of linguistic terms, as shown in Table 1. The utilities of each linguistic term were characterized as an HFS, as shown in Table 4.

#### 5.2. Evaluations of Each Criterion for Each City

_{1}given by Expert 1, wherein criterion C

_{3}for A

_{1}is medium good at a probability of 0.7 and good at a probability of 0.3.

#### 5.3. Fusion of Evaluations

_{2}with respect to criterion C

_{2}, the belief structures were

_{2}were

_{2}were

_{21}= 0.25, w

_{22}= 0.40, w

_{23}= 0.35.

_{2}was as follows:

#### 5.4. Decision-Making Based on Prospect Theory

_{4}(i.e., Gansu Jiuquan) is the most desirable one.

#### 5.5. Comparative Analysis and Discussion

_{1}(i.e., Xinjiang Hami) is the most desirable city here which would be chosen to build a new manufactory.

_{4}(i.e., Gansu Jiuquan) based on the evidential prospect theory framework, while city A

_{1}(i.e., Xinjiang Hami) is the most desirable one based on expected utility theory. The main reason for this is that the result based on the evidential prospect theory framework is more in line with the actual experience of decision-makers, while the result based on expected utility theory fails to take into account DMs’ nonrational behavior. In summary, compared with these two approaches, the evidential prospect theory framework can obtain a better final decision result because it effectively captures DMs’ nonrational behavior.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Assessment Grade | VP | P | MP | M | MG | G | VG |
---|---|---|---|---|---|---|---|

Numerical rating | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Description | Very poor | Poor | Medium poor | Medium | Medium good | Good | Very good |

Alternative | Criterion | Expert 1 | … | Expert k | … | Expert l |
---|---|---|---|---|---|---|

A_{1} | C_{1} | ${m}_{11}^{1}({B}_{11}^{1})$ | … | ${m}_{11}^{k}({B}_{11}^{k})$ | … | ${m}_{11}^{l}({B}_{11}^{l})$ |

… | … | … | … | … | … | |

C_{j} | ${m}_{1j}^{1}({B}_{1j}^{1})$ | … | ${m}_{1j}^{k}({B}_{1j}^{k})$ | … | ${m}_{1j}^{l}({B}_{1j}^{l})$ | |

… | … | … | … | … | … | |

C_{n} | ${m}_{1n}^{1}({B}_{1n}^{1})$ | … | ${m}_{1n}^{k}({B}_{1n}^{k})$ | … | ${m}_{1n}^{l}({B}_{1n}^{l})$ | |

∶ | ∶ | ∶ | ∶ | ∶ | ∶ | ∶ |

A_{i} | C_{1} | ${m}_{i1}^{1}({B}_{i1}^{1})$ | … | ${m}_{i1}^{k}({B}_{i1}^{k})$ | … | ${m}_{i1}^{l}({B}_{i1}^{l})$ |

… | … | … | … | … | … | |

C_{j} | ${m}_{ij}^{1}({B}_{ij}^{1})$ | … | ${m}_{ij}^{k}({B}_{ij}^{k})$ | … | ${m}_{ij}^{l}({B}_{ij}^{l})$ | |

… | … | … | … | … | … | |

C_{n} | ${m}_{in}^{1}({B}_{in}^{1})$ | … | ${m}_{in}^{k}({B}_{in}^{k})$ | … | ${m}_{in}^{l}({B}_{in}^{l})$ | |

∶ | ∶ | ∶ | ∶ | ∶ | ∶ | ∶ |

A_{t} | C_{1} | ${m}_{t1}^{1}({B}_{t1}^{1})$ | … | ${m}_{t1}^{k}({B}_{t1}^{k})$ | … | ${m}_{t1}^{l}({B}_{t1}^{l})$ |

… | … | … | … | … | … | |

C_{j} | ${m}_{tj}^{1}({B}_{tj}^{1})$ | … | ${m}_{tj}^{k}({B}_{tj}^{k})$ | … | ${m}_{tj}^{l}({B}_{tj}^{l})$ | |

… | … | … | … | … | … | |

C_{n} | ${m}_{tn}^{1}({B}_{tn}^{1})$ | … | ${m}_{tn}^{k}({B}_{tn}^{k})$ | … | ${m}_{tn}^{l}({B}_{tn}^{l})$ |

VP | P | MP | M | MG | G | VG | |
---|---|---|---|---|---|---|---|

u | $\{{\gamma}_{1},\dots ,{\gamma}_{e}\}$ | $\{{\gamma}_{e+1},\dots ,{\gamma}_{f}\}$ | $\{{\gamma}_{f+1},\dots ,{\gamma}_{g}\}$ | $\{{\gamma}_{g+1},\dots ,{\gamma}_{h}\}$ | $\{{\gamma}_{h+1},\dots ,{\gamma}_{w}\}$ | $\{{\gamma}_{w+1},\dots ,{\gamma}_{y}\}$ | $\{{\gamma}_{y+1},\dots ,{\gamma}_{z}\}$ |

VP | P | MP | M | MG | G | VG | |
---|---|---|---|---|---|---|---|

u | {0.0,0.1} | {0.2,0.3} | {0.3,0.4} | {0.5,0.6} | {0.7,0.8} | {0.8,0.9} | {0.9,1.0} |

City | Criterion | Expert 1 | Expert 2 | Expert 3 | Expert 4 | Expert 5 |
---|---|---|---|---|---|---|

A_{1} | C_{1} | ({MG},1.0) | ({MG},1.0) | ({M},0.7;{MG},0.3) | ({MG},0.2;{G},0.8) | ({G},1.0) |

C_{2} | ({MG},0.8;{G},0.2) | ({M},1.0) | ({MG},0.5;{G},0.5) | ({MG},1.0) | ({MP},1.0) | |

C_{3} | ({M},0.7;{MG},0.3) | ({MG},1.0) | ({MP},0.6;{ M,MG},0.4) | ({P},1.0) | ({M},0.9;{MG},0.1) | |

A_{2} | C_{1} | ({P},1.0) | ({MP},1.0) | ({M},1.0) | ({MP},1.0) | ({P},1.0) |

C_{2} | ({M},0.6;{ MG},0.4) | ({M,MG},1.0) | ({VG},1.0) | ({M},1.0) | ({M},1.0) | |

C_{3} | ({VG},1.0) | ({G},1.0) | ({G},1.0) | ({M},1.0) | ({M},1.0) | |

A_{3} | C_{1} | ({M},0.3;{G},0.7) | ({M},1.0) | ({M},0.8;{MG},0.2) | ({M},1.0) | ({M},1.0) |

C_{2} | ({M,VG},1.0) | ({M},1.0) | ({M},0.2;{VG},0.8) | ({M},1.0) | ({MG, G },1.0) | |

C_{3} | ({MG},0.2;{G},0.8) | ({MG},1.0) | ({MG},1.0) | ({G},1.0) | ({MG},1.0) | |

A_{4} | C_{1} | ({G},1.0) | ({MG},1.0) | ({MG},1.0) | ({MG},1.0) | ({MG},1.0) |

C_{2} | ({MP},0.2;{M,MG},0.8) | ({G},1.0) | ({MG},0.2;{G},0.8) | ({MG,G},1.0) | ({M.MG},1.0) | |

C_{3} | ({G},1.0) | ({MG},1.0) | ({M},0.6;{ G},0.4) | ({G},1.0) | ({G},1.0) | |

A_{5} | C_{1} | ({M},1.0) | ({MG},1.0) | ({MG},0.5;{G},0.5) | ({G},1.0) | ({G},1.0) |

C_{2} | ({M},1.0) | ({M},0.3;{MG},0.7) | ({MG, G},1.0) | ({M},1.0) | ({M,MG},1.0) | |

C_{3} | ({G},1.0) | ({VG},1.0) | ({MG},0.8;{G},0.2) | ({G},1.0) | ({G},1.0) |

City | Criterion | Final Belief Structure |
---|---|---|

A_{1} | C_{1} | ({MG}, 1.0000) |

C_{2} | ({MG}, 0.9968;{G},0.0032) | |

C_{3} | ({MP}, 0.0008;{M}, 0.9095;{MG},0.0896;{G,MG}0.0001) | |

A_{2} | C_{1} | ({P}, 0.5000;{MP}, 0.5000) |

C_{2} | ({M},0.9975;{MG},0.0023;{M,MG}0.0002) | |

C_{3} | ({M}, 0.5000;{G}, 0.5000) | |

A_{3} | C_{1} | ({M}, 1.0000) |

C_{2} | ({M}, 0.9958;{ VG}, 0.0038; {M, MG}, 0.0001;{M,VG}0.0003) | |

C_{3} | ({MG}, 1.0000) | |

A_{4} | C_{1} | ({MG}, 1.0000) |

C_{2} | ({M,MG},0.0420;{MG},0.4845;{G},0.4791;{MG.G}0.0016) | |

C_{3} | ({G}, 1.0000) | |

A_{5} | C_{1} | ({MG}, 0.0049; {G},0.9951) |

C_{2} | ({M}, 0.9458;{MG}, 0.0532,{M,MG},0.0010) | |

C_{3} | ({G},1.0000) |

City | Final Aggregated Evaluation |
---|---|

A_{1} | ({M}, 0.2730; {MG}, 0.1909; {G}, 0.5361) |

A_{2} | ({P}, 0.0089; {MP}, 0.1238; {M}, 0.8673) |

A_{3} | ({M}, 0.9641; {MG}, 0.0359) |

A_{4} | ({M}, 0.0938; {MG}, 0.8727; {G}, 0.0335) |

A_{5} | ({M}, 0.4625; {MG}, 0.1685; {G}, 0.3690) |

City | Assessment Grade | Weighting Factor | Utility Value | Defuzzified Value |
---|---|---|---|---|

A_{1} | {M} | 0.2730 | {0.5, 0.6} | 0.5595 |

{MG} | 0.1909 | {0.7, 0.8} | 0.7595 | |

{G} | 0.5361 | {0.8, 0.9} | 0.8595 | |

A_{2} | {P} | 0.0089 | {0.2, 0.3} | 0.2595 |

{MP} | 0.1238 | {0.3, 0.4} | 0.3595 | |

{M} | 0.8673 | {0.5, 0.6} | 0.5595 | |

A_{3} | {M} | 0.9641 | {0.5, 0.6} | 0.5595 |

{MG} | 0.0359 | {0.7, 0.8} | 0.7595 | |

A_{4} | {M} | 0.0938 | {0.5, 0.6} | 0.5595 |

{MG} | 0.8727 | {0.7, 0.8} | 0.7595 | |

{G} | 0.0335 | {0.8, 0.9} | 0.8595 | |

A_{5} | {M} | 0.4625 | {0.5, 0.6} | 0.5595 |

{MG} | 0.1685 | {0.7, 0.8} | 0.7595 | |

{G} | 0.3690 | {0.8, 0.9} | 0.8595 |

City | Assessment Grade | Weighting Factor | Gains or Losses | Prospect Value |
---|---|---|---|---|

A_{1} | {M} | 0.2730 | −0.2 | −0.5459 |

{MG} | 0.1909 | 0 | 0 | |

{G} | 0.5361 | 0.1 | 0.1318 | |

A_{2} | {P} | 0.0089 | −0.5 | −1.2226 |

{MP} | 0.1238 | −0.4 | −1.0046 | |

{M} | 0.8673 | −0.2 | −0.5459 | |

A_{3} | {M} | 0.9641 | −0.2 | −0.5459 |

{MG} | 0.0359 | 0 | 0 | |

A_{4} | {M} | 0.0938 | −0.2 | −0.5459 |

{MG} | 0.8727 | 0 | 0 | |

{G} | 0.0335 | 0.1 | 0.1318 | |

A_{5} | {M} | 0.4625 | −0.2 | −0.5459 |

{MG} | 0.1685 | 0 | 0 | |

{G} | 0.3690 | 0.1 | 0.1318 |

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**MDPI and ACS Style**

Xing, H.; Song, L.; Yang, Z.
An Evidential Prospect Theory Framework in Hesitant Fuzzy Multiple-Criteria Decision-Making. *Symmetry* **2019**, *11*, 1467.
https://doi.org/10.3390/sym11121467

**AMA Style**

Xing H, Song L, Yang Z.
An Evidential Prospect Theory Framework in Hesitant Fuzzy Multiple-Criteria Decision-Making. *Symmetry*. 2019; 11(12):1467.
https://doi.org/10.3390/sym11121467

**Chicago/Turabian Style**

Xing, Huahua, Lei Song, and Zongxiao Yang.
2019. "An Evidential Prospect Theory Framework in Hesitant Fuzzy Multiple-Criteria Decision-Making" *Symmetry* 11, no. 12: 1467.
https://doi.org/10.3390/sym11121467