# A Novel Decision-Making Model with Pythagorean Fuzzy Linguistic Information Measures and Its Application to a Sustainable Blockchain Product Assessment Problem

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## Abstract

**:**

## 1. Introduction

- A new concept of PFLSs is introduced, which we believe to be more reasonable and convenient to express uncertain evaluation information;
- Two axiomatic definitions of information measures for PFLVs are presented;
- With the help of logarithmic functions, two new information measure formulas were constructed;
- A novel Pythagorean fuzzy linguistic multi-attribute decision-making model was developed to derive reliable ranking of the alternatives.

## 2. Preliminaries

#### 2.1. LTSs and PTSs

**Example**

**1.**

**Definition**

**1.**

#### 2.2. Pythagorean Fuzzy Linguistic Sets (PFLSs)

**Definition**

**2.**

## 3. The Pythagorean Fuzzy Linguistic Entropy and Pythagorean Fuzzy Linguistic Similarity Measure

#### 3.1. Pythagorean Fuzzy Linguistic Entropy

**Definition**

**3.**

**(E1)**- $E(\alpha )=0$ if and only if $\alpha =\langle {s}_{2\tau},{s}_{0}\rangle $ or $\langle {s}_{0},{s}_{2\tau}\rangle $;
**(E2)**- $E(\alpha )=1$ if and only if ${\mu}_{\alpha}={\nu}_{\alpha}$;
**(E3)**- $E(\alpha )=E({\alpha}^{c})$;
**(E4)**- $E(\alpha )\le E(\beta )$, if ${\mu}_{\alpha}\le {\mu}_{\beta}$ and ${v}_{\alpha}\ge {v}_{\beta}$ when ${\mu}_{\beta}\le {\nu}_{\beta}$ or ${\mu}_{\alpha}\ge {\mu}_{\beta}$ and ${v}_{\alpha}\le {v}_{\beta}$ when ${\mu}_{\beta}\ge {\nu}_{\beta}$;

**Theorem**

**1.**

**Proof.**

**(E1)**If $\alpha =\langle {s}_{2\tau},{s}_{0}\rangle \text{\hspace{0.17em}}or\text{\hspace{0.17em}}\langle {s}_{0},{s}_{2\tau}\rangle $, then we have $\frac{{I}^{2}({\mu}_{\alpha})+1-{I}^{2}({v}_{\alpha})}{2}=1$ or $\frac{{I}^{2}({\mu}_{\alpha})+1-{I}^{2}({v}_{\alpha})}{2}=0$.

**(E2)**By using the above analysis of function $g(x)$ on [0,1], if $e(\alpha )=1$:

**(E3)**Because ${\alpha}^{c}=({\nu}_{\alpha},{\mu}_{\alpha})$, then:

**(E4)**Suppose that ${\mu}_{\alpha}\le {\mu}_{\beta}$ and ${v}_{\alpha}\ge {v}_{\beta}$ when ${\mu}_{\beta}\le {\nu}_{\beta}$, we can derive that

#### 3.2. Pythagorean Fuzzy Linguistic Similarity Measure

**Definition**

**4.**

**(S1)**- $S(\alpha ,\beta )=0$ if and only if $\alpha =\langle {s}_{2\tau},{s}_{0}\rangle ,\beta =\langle {s}_{0},{s}_{2\tau}\rangle $ or $\alpha =\langle {s}_{0},{s}_{2\tau}\rangle ,\beta =\langle {s}_{2\tau},{s}_{0}\rangle $;
**(S2)**- $S(\alpha ,\beta )=1$ if and only if ${\mu}_{\alpha}={\mu}_{\beta},{\nu}_{\alpha}={\nu}_{\beta}$;
**(S3)**- $S(\alpha ,\beta )=S(\beta ,\alpha )$;
**(S4)**- $S(\alpha ,\gamma )\le S(\alpha ,\beta ),S(\alpha ,\gamma )\le S(\beta ,\gamma ),$ if ${\mu}_{\alpha}\le {\mu}_{\beta}\le {\mu}_{\gamma}$ and ${v}_{\alpha}\ge {v}_{\beta}\ge {v}_{\gamma}$ or ${\mu}_{\alpha}\ge {\mu}_{\beta}\ge {\mu}_{\gamma}$ and ${v}_{\alpha}\le {v}_{\beta}\le {v}_{\gamma}$;

**Theorem**

**2.**

**Proof.**

**(S1)**If $\alpha =\langle {s}_{2\tau},{s}_{0}\rangle ,\beta =\langle {s}_{0},{s}_{2\tau}\rangle $ or $\alpha =\langle {s}_{0},{s}_{2\tau}\rangle ,\beta =\langle {s}_{2\tau},{s}_{0}\rangle $, then:

**(S2)**Owing to $\frac{{I}^{2}({\xi}_{\alpha})+1-{I}^{2}({\xi}_{\beta})}{2}\in [0,1]$, for each $\xi =\mu ,v$, then $0\le g\left(\frac{{I}^{2}({\xi}_{\alpha})+1-{I}^{2}({\xi}_{\beta})}{2}\right)\le 1,$$\xi =\mu ,v$; therefore,

**(S3)**

**(S4)**If ${\mu}_{\alpha}\le {\mu}_{\beta}\le {\mu}_{\gamma}$ and ${v}_{\alpha}\ge {v}_{\beta}\ge {v}_{\gamma}$, then $0\le {I}^{2}({\mu}_{\alpha})\le {I}^{2}({\mu}_{\beta})\le {I}^{2}({\mu}_{\gamma})\le 1$ and $1\ge {I}^{2}({v}_{\alpha})\ge {I}^{2}({v}_{\beta})\ge {I}^{2}({v}_{\gamma})\ge 0$, thus:

#### 3.3. Relationship Between the Pythagorean Fuzzy Linguistic Entropy and Similarity Measure

**Theorem**

**3.**

**Proof.**

**(E1)**- $E(\alpha )=0\iff S(\alpha ,{\alpha}^{c})=0\iff \alpha =\langle {s}_{2\tau},{s}_{0}\rangle ,{\alpha}^{c}=\langle {s}_{0},{s}_{2\tau}\rangle $ or $\alpha =\langle {s}_{0},{s}_{2\tau}\rangle ,{\alpha}^{c}=\langle {s}_{2\tau},$ ${s}_{0}\rangle $, i.e.,$$\alpha =\langle {s}_{2\tau},{s}_{0}\rangle \text{\hspace{0.17em}}or\text{\hspace{0.17em}}\langle {s}_{0},{s}_{2\tau}\rangle .$$
**(E2)**- $E(\alpha )=1\iff S(\alpha ,{\alpha}^{c})=1\iff {\mu}_{\alpha}={\mu}_{{\alpha}^{c}},{\nu}_{\alpha}={\nu}_{{\alpha}^{c}}\iff {\mu}_{\alpha}={\nu}_{\alpha}$.
**(E3)**- $E({\alpha}^{c})=S({\alpha}^{c},({\alpha}^{c}{)}^{c})=S({\alpha}^{c},\alpha )=S(\alpha ,{\alpha}^{c})=E(\alpha )$.
**(E4)**- Let $\beta =\langle {\mu}_{\beta},{\nu}_{\beta}\rangle $ be a PFLV, if ${\mu}_{\alpha}\le {\mu}_{\beta}$ and ${v}_{\alpha}\ge {v}_{\beta}$ when ${\mu}_{\beta}\le {\nu}_{\beta}$, then$$0\le I({\mu}_{\alpha})\le I({\mu}_{\beta})\le I({\nu}_{\beta})\le I({v}_{\alpha})\le 1.$$

## 4. The MADM Model with Pythagorean Fuzzy Linguistic Information Measures

#### 4.1. Step 1: Constructing the Initial Pythagorean Fuzzy Linguistic Decision-Making Matrix

#### 4.2. Step 2: Normalization of the Pythagorean Fuzzy Linguistic Decision-Making Matrix

#### 4.3. Step 3: Determining the Attribute Weights with Pythagorean Fuzzy Linguistic Entropy

#### 4.4. Step 4: Obtaining the Weighted Similarity Degree for an Alternative with the Pythagorean Fuzzy Linguistic Similarity Measure

#### 4.5. Step 5: Deriving the Closeness Degrees of Alternatives

#### 4.6. Step 6: Ranking the Alternatives

## 5. Illustrative Example and Comparative Analysis

#### 5.1. Application to Sustainable Blockchain Product Assessment

#### 5.1.1. Step 1

#### 5.1.2. Step 2

#### 5.1.3. Step 3

#### 5.1.4. Step 4

#### 5.1.5. Step 5

#### 5.2. Comparative Analysis and Discussion

#### 5.2.1. Step 1

#### 5.2.2. Step 2

#### 5.2.3. Step 3

_{i}:

#### 5.2.4. Step 4

_{5}.

#### 5.3. The decision-making process with the method in Liang et al.

#### 5.3.1. Steps 1’ and 2’

#### 5.3.2. Step 3’

#### 5.3.3. Step 4’

#### 5.3.4. Step 5’

_{5}.

#### 5.4. The decision-making process with the method in Garg

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Approaches | The Ranking Results of the Sustainable Blockchain Products | The Most Desirable Sustainable Blockchain Product |
---|---|---|

Our model | ${x}_{5}\succ {x}_{4}\succ {x}_{2}\succ {x}_{1}\succ {x}_{3}$ | ${x}_{5}$ |

Wei and Wei’s [41] method | ${x}_{5}\succ {x}_{2}\succ {x}_{4}\succ {x}_{1}\succ {x}_{3}$ | ${x}_{5}$ |

Liang et al.’s [42] method | ${x}_{5}\succ {x}_{4}\succ {x}_{2}\succ {x}_{1}\succ {x}_{3}$ | ${x}_{5}$ |

Garg’s [43] method | ${x}_{5}\succ {x}_{4}\succ {x}_{1}\succ {x}_{2}\succ {x}_{3}$ | ${x}_{5}$ |

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## Share and Cite

**MDPI and ACS Style**

Jin, F.; Pei, L.; Chen, H.; Langari, R.; Liu, J. A Novel Decision-Making Model with Pythagorean Fuzzy Linguistic Information Measures and Its Application to a Sustainable Blockchain Product Assessment Problem. *Sustainability* **2019**, *11*, 5630.
https://doi.org/10.3390/su11205630

**AMA Style**

Jin F, Pei L, Chen H, Langari R, Liu J. A Novel Decision-Making Model with Pythagorean Fuzzy Linguistic Information Measures and Its Application to a Sustainable Blockchain Product Assessment Problem. *Sustainability*. 2019; 11(20):5630.
https://doi.org/10.3390/su11205630

**Chicago/Turabian Style**

Jin, Feifei, Lidan Pei, Huayou Chen, Reza Langari, and Jinpei Liu. 2019. "A Novel Decision-Making Model with Pythagorean Fuzzy Linguistic Information Measures and Its Application to a Sustainable Blockchain Product Assessment Problem" *Sustainability* 11, no. 20: 5630.
https://doi.org/10.3390/su11205630