# Shadowed Sets-Based Linguistic Term Modeling and Its Application in Multi-Attribute Decision-Making

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Pythagorean Fuzzy Set (PFS)

**Definition**

**1**

**.**Suppose X is a fixed set. A PFS takes the form of:

**Definition**

**2**

**.**Assume $S=\left\{{s}_{i}|i=0,\cdots ,t,t\in R\right\}$ is a linguistic term set, ${s}_{i}$ is the linguistic evaluation value, $t$ is the granularity of $S$. Take the seven-level linguistic term as an example: S = {s

_{0}= Extremely low, s

_{1}= Very low, s

_{2}= Low, s

_{3}= Fair, s

_{4}= High, s

_{5}= Very high, s

_{6}= Extremely high}.

- (1)
- There is a negation operator: $neg\left({s}_{i}\right)={s}_{t-i}$;
- (2)
- If $i<j$ then ${S}_{i}<{S}_{j}$;

**Definition**

**3**

**.**Based on the definition of linguistic term set and PFS, the Pythagorean fuzzy linguistic set (PFLS) takes the form of $D=\left\{\langle {s}_{\tau \left(x\right)},{u}_{p}\left(x\right),{v}_{p}\left(x\right)\rangle |x\in X\right\}$, and the Pythagorean fuzzy linguistic number (PFLN) is denoted as $\langle {s}_{\tau \left(x\right)},{u}_{p}\left(x\right),{v}_{p}\left(x\right)\rangle $, where ${s}_{\tau \left(x\right)}$ is the linguistic evaluation value.

#### 2.2. Shadowed Set

**Definition**

**4**

**Definition**

**5.**

#### 2.3. Pythagorean Shadowed Set (PSS)

**Definition**

**6.**

**Definition**

**7.**

- (1)
- ${V}_{1}+{V}_{2}=\langle \text{}\left[{a}_{1}+{a}_{2},{b}_{1}+{b}_{2},{c}_{1}+{c}_{2},{d}_{1}+{d}_{2}\right],\text{}P\left(\sqrt{{\left({u}_{p}\left({x}_{1}\right)\right)}^{2}+{\left({u}_{p}\left({x}_{2}\right)\right)}^{2}-{\left({u}_{p}\left({x}_{1}\right)\right)}^{2}{\left({u}_{p}\left({x}_{2}\right)\right)}^{2}},{v}_{p}\left({x}_{1}\right){v}_{p}\left({x}_{2}\right)\right)\rangle $
- (2)
- ${V}_{1}\times {V}_{2}=\langle \text{}\left[{a}_{1}\times {a}_{2},{b}_{1}\times {b}_{2},{c}_{1}\times {c}_{2},{d}_{1}\times {d}_{2}\right],\text{}P\left({u}_{p}\left({x}_{1}\right){u}_{p}\left({x}_{2}\right),\sqrt{{\left({\nu}_{p}\left({x}_{1}\right)\right)}^{2}+{\left({\nu}_{p}\left({x}_{2}\right)\right)}^{2}-{\left({\nu}_{p}\left({x}_{1}\right)\right)}^{2}{\left({\nu}_{p}\left({x}_{2}\right)\right)}^{2}}\right)\rangle $
- (3)
- $\lambda {V}_{1}=\langle \text{}\left[\lambda {a}_{1},\lambda {b}_{1},\lambda {c}_{1},\lambda {d}_{1}\right],\text{}P\left(\sqrt{1-{\left(1-{\left({u}_{p}\left({x}_{1}\right)\right)}^{2}\right)}^{\lambda}},{\left({\nu}_{p}\left({x}_{1}\right)\right)}^{\lambda}\right)\rangle ,\lambda \ge 0$
- (4)
- ${V}_{1}{}^{\lambda}=\langle \text{}\left[{a}_{1}^{\lambda},{b}_{1}^{\lambda},{c}_{1}^{\lambda},{d}_{1}^{\lambda}\right],\text{}P\left({\left({u}_{p}\left({x}_{1}\right)\right)}^{\lambda},\sqrt{1-{\left(1-{\left({\nu}_{p}^{}\left({x}_{1}\right)\right)}^{2}\right)}^{\lambda}}\right)\rangle ,\lambda \ge 0$

**Theorem**

**1.**

- (1)
- ${V}_{1}+{V}_{2}={V}_{2}+{V}_{1}$
- (2)
- ${V}_{1}\times {V}_{2}={V}_{2}\times {V}_{1}$
- (3)
- $\lambda \left({V}_{1}+{V}_{2}\right)=\lambda {V}_{1}+\lambda {V}_{2},\lambda \ge 0$
- (4)
- ${V}_{1}^{{\lambda}_{1}+{\lambda}_{2}}={V}_{1}^{{\lambda}_{1}}+{V}_{2}^{{\lambda}_{2}},{\lambda}_{1},{\lambda}_{2}\ge 0$
- (5)
- ${\lambda}_{1}{V}_{1}+{\lambda}_{2}{V}_{1}=\left({\lambda}_{1}+{\lambda}_{2}\right){V}_{1},{\lambda}_{1},{\lambda}_{2}\ge 0$
- (6)
- ${V}_{1}^{\lambda}\times {V}_{2}^{\lambda}={\left({V}_{1}\times {V}_{2}\right)}^{\lambda},\lambda \ge 0$

## 3. Shadowed Set Model of Linguistic Terms

#### 3.1. Interval Data Preprocessing

**Step 1:**Bad data processing. This aims to remove unreasonable results from the surveyed people, whose answers were beyond the range of the universe of discourse $U$. If the interval endpoints satisfy the following conditions, the interval data are acceptable. Otherwise, they will be rejected.

**Step 2:**Outlier Processing. By using the Box and Whisker test [28], the data that are extremely large or small, i.e., outliers, can be eliminated. Outlier tests can be applied to process the endpoints of interval data and the lengths of interval data ${L}_{k}={b}_{k}-{a}_{k}$, respectively. Consequently, only the interval endpoints and lengths satisfying the following conditions are kept:

**Step 3:**Tolerance limit processing. If the remaining intervals satisfy the following conditions, then they will be accepted; otherwise, they will be rejected.

**Step 4:**Reasonable-interval processing. If the intervals satisfy the following conditions, they will be kept; otherwise, they will be rejected.

#### 3.2. Shadowed Set Model of Seven-Level Language Terms

**Step 1:**Calculate the mean ${m}_{l}$ and standard deviation ${\sigma}_{l}$ of the remaining left-end points

**Step 2:**Determine the representative interval. Let $\left[{L}_{l},{L}_{r}\right]$ and $\left[{R}_{l},{R}_{r}\right]$ be the representative intervals of the left-end points and right-end points, respectively.

## 4. The Score Function of Pythagorean Shadowed Number

**Definition**

**8.**

**Example**

**1.**

**Definition**

**9.**

**Example**

**2.**

## 5. MADM Method Based on the Pythagorean Shadowed Set

**Step 1:**Standardized decision matrix. For PFLVs ${P}_{ij}=\langle {s}_{{\tau}_{ij}},P\left({u}_{p}\left({x}_{ij}\right),{v}_{p}\left({x}_{ij}\right)\right)\rangle $

**Step 2:**Collect the data by questionnaire and get the shadowed set of language terms by processing the data. Transform Pythagorean fuzzy linguistic numbers into PSNs using Figure 4.

**Step 3:**Transform the PFSN decision matrix into score function matrix based on Equation (6).

**Step 4:**By OWA operator, the attribute values ${r}_{ij}$ of each alternative ${a}_{i}$ are aggregated to obtain the comprehensive attribute values ${z}_{i}$.

**Step 5:**Determine the order of all the alternatives in the light of the comprehensive attribute values ${z}_{i}$.

## 6. Numerical Study

#### 6.1. Supplier Selection Problem

**Step 1:**${c}_{1}$, ${c}_{2}$, ${c}_{3}$, ${c}_{4}$ are beneficial attributes. Therefore, the standardized decision matrix is the same with Table 2.

**Step 2:**Transform PFLNs into Pythagorean shadowed numbers using Figure 4, and the result is shown in Table 3.

**Step 3:**Transform the PFSNs decision matrix into score function matrix (shown in Table 4) based on Equation (6).

**Step 4:**By OWA operator, the attribute values ${r}_{ij}$ of each alternative ${a}_{i}$ are aggregated to obtain the comprehensive attribute values ${z}_{i}$.

**Step 5:**Rank the alternatives and obtain the best alternative(s) according to the comprehensive attribute values ${z}_{i}$ in the Step 4.

#### 6.2. Comparison Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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${\mathit{m}}^{*}$ | $1-\mathit{\gamma}\text{}=\text{}0.95$ | $1-\mathit{\gamma}\text{}=\text{}0.99$ | ||
---|---|---|---|---|

$1-\mathit{\alpha}$ | $1-\mathit{\alpha}$ | |||

0.90 | 0.95 | 0.90 | 0.95 | |

10 | 2.839 | 3.379 | 3.582 | 4.265 |

15 | 2.480 | 2.954 | 2.945 | 3.507 |

20 | 2.310 | 2.752 | 2.659 | 3.168 |

30 | 2.140 | 2.549 | 2.358 | 2.841 |

50 | 1.996 | 2.379 | 2.162 | 2.576 |

100 | 1.874 | 2.233 | 1.977 | 2.355 |

1000 | 1.709 | 2.036 | 1.736 | 2.718 |

$\infty $ | 1.645 | 1.960 | 1.645 | 1.960 |

Alternatives | Attributes | |||
---|---|---|---|---|

${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | |

${a}_{1}$ | $\langle {s}_{4},P\left(0.7,0.4\right)\rangle $ | $\langle {s}_{5},P\left(0.5,0.6\right)\rangle $ | $\langle {s}_{2},P\left(0.7,0.3\right)\rangle $ | $\langle {s}_{3},P\left(0.8,0.4\right)\rangle $ |

${a}_{2}$ | $\langle {s}_{5},P\left(0.6,0.4\right)\rangle $ | $\langle {s}_{3},P\left(0.7,0.4\right)\rangle $ | $\langle {s}_{3},P\left(0.6,0.4\right)\rangle $ | $\langle {s}_{4},P\left(0.7,0.5\right)\rangle $ |

${a}_{3}$ | $\langle {s}_{6},P\left(0.6,0.5\right)\rangle $ | $\langle {s}_{3},P\left(0.8,0.3\right)\rangle $ | $\langle {s}_{5},P\left(0.6,0.5\right)\rangle $ | $\langle {s}_{2},P\left(0.6,0.4\right)\rangle $ |

${a}_{4}$ | $\langle {s}_{3},P\left(0.7,0.3\right)\rangle $ | $\langle {s}_{4},P\left(0.6,0.5\right)\rangle $ | $\langle {s}_{3},P\left(0.7,0.4\right)\rangle $ | $\langle {s}_{6},P\left(0.7,0.6\right)\rangle $ |

${a}_{5}$ | $\langle {s}_{4},P\left(0.7,0.4\right)\rangle $ | $\langle {s}_{5},P\left(0.6,0.5\right)\rangle $ | $\langle {s}_{4},P\left(0.7,0.4\right)\rangle $ | $\langle {s}_{3},P\left(0.8,0.4\right)\rangle $ |

Alternatives | Attributes | |

${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | |

${a}_{1}$ | $\langle \left[5.77,6.48,7.21,7.96\right],P\left(0.7,0.4\right)\rangle $ | $\langle \left[7.51,7.61,8.60,8.97\right],P\left(0.5,0.6\right)\rangle $ |

${a}_{2}$ | $\langle \left[7.51,7.61,8.60,8.97\right],P\left(0.6,0.4\right)\rangle $ | $\langle \left[3.83,4.84,5.52,6.46\right],P\left(0.7,0.4\right)\rangle $ |

${a}_{3}$ | $\langle \left[8.61,9.22,9.62,9.89\right],P\left(0.6,0.5\right)\rangle $ | $\langle \left[3.83,4.84,5.52,6.46\right],P\left(0.8,0.3\right)\rangle $ |

${a}_{4}$ | $\langle \left[3.83,4.84,5.52,6.46\right],P\left(0.7,0.3\right)\rangle $ | $\langle \left[5.77,6.48,7.21,7.96\right],P\left(0.6,0.5\right)\rangle $ |

${a}_{5}$ | $\langle \left[5.77,6.48,7.21,7.96\right],P\left(0.7,0.4\right)\rangle $ | $\langle \left[7.51,7.61,8.60,8.97\right],P\left(0.6,0.5\right)\rangle $ |

Alternatives | Attributes | |

${\mathit{c}}_{\mathbf{3}}$ | ${\mathit{c}}_{\mathbf{4}}$ | |

${a}_{1}$ | $\langle \left[2.38,3.28,3.83,4.90\right],P\left(0.7,0.3\right)\rangle $ | $\langle \left[3.83,4.84,5.52,6.46\right],P\left(0.8,0.4\right)\rangle $ |

${a}_{2}$ | $\langle \left[3.83,4.84,5.52,6.46\right],P\left(0.6,0.4\right)\rangle $ | $\langle \left[5.77,6.48,7.21,7.96\right],P\left(0.7,0.5\right)\rangle $ |

${a}_{3}$ | $\langle \left[7.51,7.61,8.60,8.97\right],P\left(0.6,0.5\right)\rangle $ | $\langle \left[2.38,3.28,3.83,4.90\right],P\left(0.6,0.4\right)\rangle $ |

${a}_{4}$ | $\langle \left[3.83,4.84,5.52,6.46\right],P\left(0.7,0.4\right)\rangle $ | $\langle \left[8.61,9.22,9.62,9.89\right],P\left(0.7,0.6\right)\rangle $ |

${a}_{5}$ | $\langle \left[5.77,6.48,7.21,7.96\right],P\left(0.7,0.4\right)\rangle $ | $\langle \left[3.83,4.84,5.52,6.46\right],P\left(0.8,0.4\right)\rangle $ |

Alternatives | Attributes | |||
---|---|---|---|---|

${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | |

${a}_{1}$ | 31.33 | 18.39 | 25.29 | 26.76 |

${a}_{2}$ | 33.11 | 23.42 | 20.07 | 25.06 |

${a}_{3}$ | 40.97 | 35.68 | 26.49 | 16.26 |

${a}_{4}$ | 31.22 | 21.48 | 23.42 | 39.83 |

${a}_{5}$ | 31.33 | 26.49 | 31.33 | 26.76 |

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**MDPI and ACS Style**

Wang, H.; He, S.; Pan, X.; Li, C.
Shadowed Sets-Based Linguistic Term Modeling and Its Application in Multi-Attribute Decision-Making. *Symmetry* **2018**, *10*, 688.
https://doi.org/10.3390/sym10120688

**AMA Style**

Wang H, He S, Pan X, Li C.
Shadowed Sets-Based Linguistic Term Modeling and Its Application in Multi-Attribute Decision-Making. *Symmetry*. 2018; 10(12):688.
https://doi.org/10.3390/sym10120688

**Chicago/Turabian Style**

Wang, Huidong, Shifan He, Xiaohong Pan, and Chengdong Li.
2018. "Shadowed Sets-Based Linguistic Term Modeling and Its Application in Multi-Attribute Decision-Making" *Symmetry* 10, no. 12: 688.
https://doi.org/10.3390/sym10120688