# A Hybrid Approach of Interval Neutrosophic Vague Sets and DEMATEL with New Linguistic Variable

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition 1.**

**Definition 2.**

**Definition 3.**

**Definition 4.**

## 3. Proposed Method

#### 3.1. Construction of Linguistic Variable

**Step 1:**Convert linguistic variable of INS to INVS.

**Step 2:**Calculation of ${m}^{L+},{m}^{U-},{n}^{L+},{n}^{U-},{p}^{L+},{p}^{L-}$ is obtained by condition of INVS.

**Step 3**: verify the linguistic variable for INVS.

#### 3.2. The INVS DEMATEL Procedures

**Step 1:**Construct linguistic data using the new linguistic variable.

**Step 2:**Aggregate DM’s preferences using the mean operator of INVS.

**Step 3:**Deneutrosophication process to obtain crisp value.

**Step 4:**Normalizing the direct relation matrix.

**Step 5:**Constructing the INVS total relation matrix.

**Step 6:**Calculating the sum of the rows and columns.

**Step 7:**Construct a causal diagram.

**Step 8:**Set up the threshold value and the network relationship map.

## 4. Illustrative Example: Hospital Service Quality

**Step 1:**Construct the decision matrix with proposed INVS linguistic variable.

**Step 2:**Aggregate DM’s preferences using mean operator of INVS.

**Step 3:**Deneutrosophication process to obtain crisp value.

**Step 4:**Normalizing the INVS direct relation matrix.

**Step 5:**Construct the INVS total relation matrix.

**Step 6:**Calculating the sum of the rows and columns.

**Step 7:**Construct a causal diagram.

**Step 8:**Setup a threshold value and construct the network relationship map.

## 5. Comparison Analysis

#### 5.1. Comparative Analysis

#### 5.2. Comparison between INVS-DEMATEL and the Existing Models

## 6. Conclusions

- In this research, the neutrosophic environment was used to establish the linguistic variable of the DEMATEL. The new INVS linguistic variable considers more range of value while handling uncertainty, since a new parameter is added to INS. This is accordance with recommendations by Rodríguez et al. [53]. It is useful to include the complex linguistic variable to capture information in different forms and to manage uncertainties of different types within a single framework.
- The combination of INVS and DEMATEL can manage the complex interactions between criteria.
- The insertion of a vague set with a neutrosophic set gives a new result on the degree of importance and net impact.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Linguistic variable [39].

Linguistic Variable | Interval Neutrosophic Set |
---|---|

No Influence (NI) | $\left[0.1,0.2\right],\left[0.5,0.6\right],\left[0.7,0.8\right]$ |

Very Low Influence (LI) | $\left[0.2,0.4\right],\left[0.5,0.6\right],\left[0.5,0.6\right]$ |

Medium Influence (MI) | $\left[0.4,0.6\right],\left[0.4,0.5\right],\left[0.3,0.4\right]$ |

High Influence (HI) | $\left[0.6,0.8\right],\left[0.3,0.4\right],\left[0.2,0.4\right]$ |

Absolutely Influence (AI) | $\left[0.7,0.9\right],\left[0.2,0.3\right],\left[0.1,0.2\right]$ |

Linguistic Variable | Interval Neutrosophic Vague Set |
---|---|

No Influence (NI) | $\langle \left\{\left[0.1,0.3\right],\left[0.2,0.2\right]\right\},\left\{\left[0.5,0.65\right],\left[0.6,0.6\right]\right\},\left\{\left[0.7,0.9\right],\left[0.8,0.8\right]\right\}\rangle $ |

Very Low Influence (LI) | $\langle \left\{\left[0.2,0.5\right],\left[0.4,0.4\right]\right\},\left\{\left[0.5,0.55\right],\left[0.5,0.6\right]\right\},\left\{\left[0.5,0.8\right],\left[0.6,0.6\right]\right\}\rangle $ |

Medium Influence (MI) | $\langle \left\{\left[0.4,0.7\right],\left[0.6,0.6\right]\right\},\left\{\left[\mathrm{0.40.45}\right],\left[0.4,0.5\right]\right\},\left\{\left[0.3,0.6\right],\left[0.4,0.4\right]\right\}\rangle $ |

High Influence (HI) | $\langle \left\{\left[0.6,0.8\right],\left[0.6,0.8\right]\right\},\left\{\left[0.3,0.35\right],\left[0.3,0.4\right]\right\},\left\{\left[0.2,0.4\right],\left[0.2,0.4\right]\right\}\rangle $ |

Absolutely Influence (AI) | $\langle \left\{\left[0.7,0.9\right],\left[0.8,0.9\right]\right\},\left\{\left[0.2,0.25\right],\left[0.2,0.3\right]\right\},\left\{\left[0.1,0.3\right],\left[0.1,0.2\right]\right\}\rangle $ |

${\mathit{F}}_{1}$ | ${\mathit{F}}_{2}$ | ${\mathit{F}}_{3}$ | ${\mathit{F}}_{4}$ | ${\mathit{F}}_{5}$ | ${\mathit{F}}_{6}$ | ${\mathit{F}}_{7}$ | |
---|---|---|---|---|---|---|---|

${F}_{1}$ | 0 | HI, MI, MI | MI, MI, HI | LI, HI, MI | MI, LI, HI | MI, AI, HI | NI, HI, MI |

${F}_{2}$ | MI, LI, HI | 0 | MI, AI, LI | HI, HI, NI | HI, HI, MI | LI, MI, HI | LI, HI, MI |

${F}_{3}$ | M, VU, M | NI, MI, NI | 0 | HI, NI, LI | NI, MI, AI | NI, MI, HI | HI, MI, LI |

${F}_{4}$ | HI, HI, MI | MI, AI, HI | LI, MI, MI | 0 | MI, HI, AI | NI, HI, MI | NI, AI, HI |

${F}_{5}$ | HI, MI, MI | MI, NI, AI | MI, MI, HI | HI, MI, MI | 0 | NI, MI, HI | AI, HI, MI |

${F}_{6}$ | MI, NI, MI | HI, MI, MI | MI, NI, AI | LI, HI, MI | MI, MI, MI | 0 | LI, MI, AI |

${F}_{7}$ | HI, MI, LI | AI, HI, MI | LI, VI, LI | HI, HI, MI | AI, HI, NI | MI, HI, AI | 0 |

${\mathit{F}}_{1}$ | ${\mathit{F}}_{2}$ | ${\mathit{F}}_{3}$ | ${\mathit{F}}_{4}$ | ${\mathit{F}}_{5}$ | ${\mathit{F}}_{6}$ | ${\mathit{F}}_{7}$ | ||
---|---|---|---|---|---|---|---|---|

${F}_{1}$ | 0.0000 | 1.2461 | 1.2461 | 1.1022 | 1.1022 | 1.4383 | 0.9911 | 7.1261 |

${F}_{2}$ | 1.1022 | 0.0000 | 1.1789 | 1.0833 | 1.3519 | 1.1789 | 1.1022 | 6.9975 |

${F}_{3}$ | 0.9011 | 0.7169 | 0.0000 | 0.8756 | 1.0589 | 0.7169 | 1.1022 | 5.3717 |

${F}_{4}$ | 1.3519 | 1.4383 | 1.0044 | 0.0000 | 1.4383 | 0.9911 | 1.1528 | 7.3769 |

${F}_{5}$ | 1.2461 | 1.0589 | 1.2461 | 1.2461 | 0.0000 | 0.9911 | 1.4383 | 7.2267 |

${F}_{6}$ | 0.9011 | 1.2461 | 1.0589 | 1.1022 | 1.1425 | 0.0000 | 1.1789 | 6.6297 |

${F}_{7}$ | 1.1022 | 1.4383 | 1.0408 | 1.3519 | 1.1528 | 1.4383 | 0.0000 | 7.5244 |

${\mathit{F}}_{1}$ | ${\mathit{F}}_{2}$ | ${\mathit{F}}_{3}$ | ${\mathit{F}}_{4}$ | ${\mathit{F}}_{5}$ | ${\mathit{F}}_{6}$ | ${\mathit{F}}_{7}$ | |
---|---|---|---|---|---|---|---|

${F}_{1}$ | 0.0000 | 0.1656 | 0.1656 | 0.1465 | 0.1465 | 0.1912 | 0.1317 |

${F}_{2}$ | 0.1465 | 0.0000 | 0.1567 | 0.1440 | 0.1797 | 0.1567 | 0.1465 |

${F}_{3}$ | 0.1198 | 0.0953 | 0.0000 | 0.1164 | 0.1407 | 0.0953 | 0.1465 |

${F}_{4}$ | 0.1797 | 0.1912 | 0.1335 | 0.0000 | 0.1912 | 0.1317 | 0.1532 |

${F}_{5}$ | 0.1656 | 0.1407 | 0.1656 | 0.1656 | 0.0000 | 0.1317 | 0.1912 |

${F}_{6}$ | 0.1198 | 0.1656 | 0.1407 | 0.1465 | 0.1518 | 0.0000 | 0.1567 |

${F}_{7}$ | 0.1465 | 0.1912 | 0.1383 | 0.1797 | 0.1532 | 0.1912 | 0.0000 |

${\mathit{F}}_{1}$ | ${\mathit{F}}_{2}$ | ${\mathit{F}}_{3}$ | ${\mathit{F}}_{4}$ | ${\mathit{F}}_{5}$ | ${\mathit{F}}_{6}$ | ${\mathit{F}}_{7}$ | |
---|---|---|---|---|---|---|---|

${F}_{1}$ | 1.562 | 1.8104 | 1.8104 | 1.7226 | 1.8214 | 1.7509 | 1.768 |

${F}_{2}$ | 1.5936 | 1.5657 | 1.6528 | 1.6225 | 1.7405 | 1.627 | 1.6726 |

${F}_{3}$ | 1.3341 | 1.3978 | 1.3108 | 1.3572 | 1.4518 | 1.336 | 1.4209 |

${F}_{4}$ | 1.8383 | 1.9587 | 1.9587 | 1.7196 | 1.9871 | 1.83 | 1.9091 |

${F}_{5}$ | 1.7028 | 1.7904 | 1.7904 | 1.7354 | 1.6903 | 1.7048 | 1.8053 |

${F}_{6}$ | 1.6069 | 1.7427 | 1.7427 | 1.6584 | 1.757 | 1.5243 | 1.7163 |

${F}_{7}$ | 1.8336 | 1.9825 | 1.9825 | 1.8937 | 1.9824 | 1.8957 | 1.7983 |

$\mathit{R}$ | $\mathit{C}$ | $\mathit{R}+\mathit{C}$ | Rank of Importance | $\mathit{R}-\mathit{C}$ | Rank of Effect | Cause/Effect | |
---|---|---|---|---|---|---|---|

${F}_{1}$ | 12.2457 | 11.4713 | 23.717 | 5 | 0.7744 | 4 | Cause |

${F}_{2}$ | 11.4747 | 12.2482 | 23.7229 | 4 | −0.7735 | 6 | Effect |

${F}_{3}$ | 9.6086 | 12.2483 | 21.8569 | 7 | −2.6397 | 7 | Effect |

${F}_{4}$ | 13.2015 | 11.7094 | 24.9109 | 2 | 1.4921 | 1 | Cause |

${F}_{5}$ | 12.2194 | 12.4305 | 24.6499 | 3 | −0.2111 | 5 | Effect |

${F}_{6}$ | 11.7483 | 11.6687 | 23.417 | 6 | 0.0796 | 3 | Cause |

${F}_{7}$ | 13.3687 | 12.0905 | 25.4592 | 1 | 1.2782 | 2 | Cause |

${\mathit{F}}_{1}$ | ${\mathit{F}}_{2}$ | ${\mathit{F}}_{3}$ | ${\mathit{F}}_{4}$ | ${\mathit{F}}_{5}$ | ${\mathit{F}}_{6}$ | ${\mathit{F}}_{7}$ | |
---|---|---|---|---|---|---|---|

${F}_{1}$ | 0 | 1 | 1 | 1 | 1 | 1 | 1 |

${F}_{2}$ | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

${F}_{3}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${F}_{4}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${F}_{5}$ | 0 | 1 | 1 | 1 | 0 | 0 | 1 |

${F}_{6}$ | 0 | 1 | 1 | 0 | 1 | 0 | 1 |

${F}_{7}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Type of Assessment | Degree of Importance & Net Impact |
---|---|

INVS-DEMATEL with new linguistic variable (proposed method) | ${F}_{7}\succ {F}_{4}\succ {F}_{5}\succ {F}_{2}\succ {F}_{1}\succ {F}_{6}\succ {F}_{3}$ Cause criterion: ${F}_{1}$, ${F}_{4}$, ${F}_{6}$, ${F}_{7}$ Effect criterion: ${F}_{2}$, ${F}_{3}$, ${F}_{5}$ |

Neutrosophic DEMATEL | ${F}_{3}\succ {F}_{2}\succ {F}_{6}\succ {F}_{1}\succ {F}_{4}\succ {F}_{5}\succ {F}_{7}$ Cause criterion: ${F}_{2}$, ${F}_{3}$, ${F}_{6}$ Effect criterion: ${F}_{1}$, ${F}_{4}$, ${F}_{5}$, ${F}_{7}$ |

DEMATEL | ${F}_{4}\succ {F}_{2}\succ {F}_{6}\succ {F}_{7}\succ {F}_{5}\succ {F}_{3}\succ {F}_{1}$ Cause criterion: ${F}_{1}$, ${F}_{4}$, ${F}_{7}$ Effect criterion: ${F}_{2}$, ${F}_{3}$, ${F}_{5}$, ${F}_{6}$ |

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**MDPI and ACS Style**

Al-Quran, A.; Hashim, H.; Abdullah, L.
A Hybrid Approach of Interval Neutrosophic Vague Sets and DEMATEL with New Linguistic Variable. *Symmetry* **2020**, *12*, 275.
https://doi.org/10.3390/sym12020275

**AMA Style**

Al-Quran A, Hashim H, Abdullah L.
A Hybrid Approach of Interval Neutrosophic Vague Sets and DEMATEL with New Linguistic Variable. *Symmetry*. 2020; 12(2):275.
https://doi.org/10.3390/sym12020275

**Chicago/Turabian Style**

Al-Quran, Ashraf, Hazwani Hashim, and Lazim Abdullah.
2020. "A Hybrid Approach of Interval Neutrosophic Vague Sets and DEMATEL with New Linguistic Variable" *Symmetry* 12, no. 2: 275.
https://doi.org/10.3390/sym12020275