# A Novel Integrated Approach for Green Supplier Selection with Interval-Valued Intuitionistic Uncertain Linguistic Information: A Case Study in the Agri-Food Industry

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Green Supplier Selection Methods

#### 2.2. Food Supply Chain Management

## 3. Basic Concepts

**Definition**

**1.**

**Definition**

**2.**

- (1)
- ${\tilde{a}}_{1}\oplus {\tilde{a}}_{2}=\langle \left[{s}_{\theta \left({\tilde{a}}_{1}\right)+\theta \left({\tilde{a}}_{2}\right)},{s}_{\tau ({\tilde{a}}_{1})+\tau \left({\tilde{a}}_{2}\right)}\right],\text{\hspace{0.17em}}[1-\left(1-{u}^{L}\left({\tilde{a}}_{1}\right)\right)\left(1-{u}^{L}\left({\tilde{a}}_{2}\right)\right),$$\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}1-\left(1-{u}^{U}\left({\tilde{a}}_{1}\right)\right)\left(1-{u}^{U}\left({\tilde{a}}_{2}\right)\right)],\left[{v}^{L}\left({\tilde{a}}_{1}\right){v}^{L}\left({\tilde{a}}_{2}\right),{v}^{U}\left({\tilde{a}}_{1}\right){v}^{U}\left({\tilde{a}}_{2}\right)\right]\rangle ;$
- (2)
- ${\tilde{a}}_{1}\otimes {\tilde{a}}_{2}=\langle \left[{s}_{\theta \left({\tilde{a}}_{1}\right)\times \theta \left({\tilde{a}}_{2}\right)},{s}_{\tau \left({\tilde{a}}_{1}\right)\times \tau \left({\tilde{a}}_{2}\right)}\right],\left[{u}^{L}\left({\tilde{a}}_{1}\right){u}^{L}\left({\tilde{a}}_{2}\right),{u}^{U}\left({\tilde{a}}_{1}\right){u}^{U}\left({\tilde{a}}_{2}\right)\right],$$\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\left[1-\left(1-{v}^{L}\left({\tilde{a}}_{1}\right)\right)\left(1-{v}^{L}\left({\tilde{a}}_{2}\right)\right),1-\left(1-{v}^{U}\left({\tilde{a}}_{1}\right)\right)\left(1-{v}^{U}\left({\tilde{a}}_{2}\right)\right)\right]\rangle ;$
- (3)
- $\lambda {\tilde{a}}_{1}=\langle \left[{s}_{\lambda \times \theta \left({\tilde{a}}_{1}\right)},{s}_{\lambda \times \tau \left({\tilde{a}}_{1}\right)}\right],\left[1-{\left(1-{u}^{L}\left({\tilde{a}}_{1}\right)\right)}^{\lambda},1-{\left(1-{u}^{U}\left({\tilde{a}}_{1}\right)\right)}^{\lambda}\right],\left[{\left({v}^{L}\left({\tilde{a}}_{1}\right)\right)}^{\lambda},{\left({v}^{U}\left({\tilde{a}}_{1}\right)\right)}^{\lambda}\right]\rangle ;$
- (4)
- ${\tilde{a}}_{1}{}^{\lambda}=\langle \left[{s}_{{\left(\theta \left({\tilde{a}}_{1}\right)\right)}^{\lambda}},{s}_{{\left(\tau \left({\tilde{a}}_{1}\right)\right)}^{\lambda}}\right],\left[{\left({u}^{L}\left({\tilde{a}}_{1}\right)\right)}^{\lambda},{\left({u}^{U}\left({\tilde{a}}_{1}\right)\right)}^{\lambda}\right],\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}\left[1-{\left(1-{v}^{L}\left({\tilde{a}}_{1}\right)\right)}^{\lambda},1-{\left(1-{v}^{U}\left({\tilde{a}}_{1}\right)\right)}^{\lambda}\right]\rangle .$

**Definition**

**3**.

**.**${\tilde{a}}_{1}=\langle \left[{s}_{\theta \left({\tilde{a}}_{}\right)},{s}_{\tau \left(\tilde{a}\right)}\right],\left[{u}^{L}\left(\tilde{a}\right),{u}^{U}\left(\tilde{a}\right)\right],\left[{v}^{L}\left({\tilde{a}}_{}\right),{v}^{U}\left(\tilde{a}\right)\right]\rangle $ is an IVIULN. The expected value of $\tilde{a}$ is expressed as

**Definition**

**4**.

- (1)
- If $E\left({\tilde{a}}_{1}\right)>E\left({\tilde{a}}_{2}\right)$, then ${\tilde{a}}_{1}>{\tilde{a}}_{2}$;
- (2)
- If $E\left({\tilde{a}}_{1}\right)=E\left({\tilde{a}}_{2}\right)$, then
- (a)
- If $T\left({\tilde{a}}_{1}\right)>T\left({\tilde{a}}_{2}\right)$, then ${\tilde{a}}_{1}>{\tilde{a}}_{2}$;
- (b)
- If $T\left({\tilde{a}}_{1}\right)=T\left({\tilde{a}}_{2}\right)$, then ${\tilde{a}}_{1}={\tilde{a}}_{2}$.

**Definition**

**5**.

**Definition**

**6.**

## 4. The Proposed Green Supplier Selection Approach

**Step 1:**Construct the collective evaluation matrix $\tilde{P}$

**Step 2:**Acquire the collective criteria weight vector $\tilde{w}$

**Step 3:**Determine the PIS and the NIS

**Step 4:**Calculate the grey relation coefficients to the PIS and the NIS

**Step 5:**Compute the grey relation grades to the PIS and the NIS

**Step 6:**Calculate the relative closeness degrees of alternatives

## 5. Case Illustration

#### 5.1. Implementation

_{1}, O

_{2}, O

_{3}, and O

_{4}), three palm oil suppliers (P

_{1}, P

_{2}, and P

_{3}) and three sunflower-soybean oil suppliers (SS

_{1}, SS

_{2}, and SS

_{3}).

**Step 1:**By Equation (8), the five individual evaluation matrices ${\tilde{P}}^{k}\left(k=1,2,\dots ,5\right)$ are aggregated to obtain the collective evaluation matrix $\tilde{P}={\left[{p}_{ij}\right]}_{10\times 4}$, as shown in Table 3.

**Step 2:**Opinions of the five decision makers on criteria importance are aggregated based on Equation (9) and the collective weights of the four criteria are derived as follows:

**Step 3:**Since service level, product quality and environmental management system are benefit criteria, ${J}_{1}=\left\{{C}_{1},{C}_{2},{C}_{4}\right\}$ and price is a cost criterion, ${J}_{2}=\left\{{C}_{3}\right\}$, the PIS and the NIS of the olive oil suppliers are determined as:

**Step 4:**Based on Equations (24) and (25), the grey relation coefficient matrices of the four olive oil suppliers to the PIS and the NIS are computed are shown below:

**Step 5:**The grey relation grades of each supplier to the PIS and the NIS are calculated by using Equations (28) and (29), and the results are listed as follows:

**Step 6:**By utilizing Equation (33), the relative closeness degrees ${\tilde{c}}_{i}\left(i=1,2,3,4\right)$ of the four olive oil suppliers are calculated as shown below:

_{3}$\succ $ O

_{1}$\succ $ O

_{2}$\succ $ O

_{4}. Therefore, O

_{3}is the most appropriate green supplier among the alternative olive oil suppliers.

_{1}$\succ $ P

_{2}$\succ $ P

_{3}and SS

_{1}$\succ $ SS

_{2}$\succ $ SS

_{3}, respectively. Thus, the company can select P

_{1}and SS

_{1}as the palm oil supplier and the sunflower-soybean oil supplier to them for procurement.

#### 5.2. Comparisons and Discussion

_{3}, P

_{1}and SS

_{1}are respectively the most suitable green suppliers of olive oil, palm oil and sunflower-soybean oil. This reveals the effectiveness of the green supplier selection model proposed in this study. In addition, there are still some differences between the ranking results acquired by the proposed approach and the three comparative methods. The least optimal green suppliers are O

_{4}, P

_{3}and SS

_{3}of olive oil, palm oil and sunflower-soybean oil by the proposed approach. According to the three comparative methods, the least optimal green suppliers are O

_{4}, P

_{2}and SS

_{2}, correspondingly. The reasons that bring the inconsistence mainly lie in the characteristics of the three comparative methods. First, triangular fuzzy numbers are applied in the three comparative methods. In contrast, the IVIULSs used in the proposed approach can better reflect the uncertainty and vagueness of decision makers’ assessments. Second, GRA, TOPSIS, VIKOR are utilized to rank alternatives in the three comparative methods, respectively. But the GRA-TOPSIS can reflect the similarity between case data curves and the relationships of these curves simultaneously as compared with the GRA and TOPSIS; the GRA-TOPSIS is more convenient and rapid in determining the best supplier by comparing with the VIKOR. Therefore, the ranking result of the alternative suppliers produced by the proposed approach is more accurate and reasonable.

- The approach can well reflect the uncertainty and fuzziness of decision makers’ subjective data by utilizing IVIULSs. This enables decision makers to express their judgments more realistically and makes the assessment easier to be carried out.
- Both quantitative and qualitative criteria can be considered in the green supplier selection which makes the developed model more reasonable. The proposed approach is a general method and not limited to the four criteria listed in the case study, but applicable to any number of criteria.
- By utilizing the GRA-TOPSIS method, a more precise and reasonable ranking of alternative suppliers can be obtained based on the basic principles of GRA and TOPSIS methods, which facilitates the company to choose the most appropriate green supplier.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Kind of Suppliers | Suppliers | Criteria | |||
---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{4} | ||

Olive oil | O_{1} | $\langle \left[{s}_{4},{s}_{5}\right],\left[0.7,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{3},{s}_{4}\right],\left[0.6,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{3},{s}_{3}\right],\left[0.6,0.7\right],\left[0.2,0.3\right]\rangle $ | $\langle \left[{s}_{6},{s}_{7}\right],\left[0.7,0.8\right],\left[0.1,0.1\right]\rangle $ |

O_{2} | $\langle \left[{s}_{6},{s}_{7}\right],\left[0.6,0.7\right],\left[0.2,0.3\right]\rangle $ | $\langle \left[{s}_{4},{s}_{5}\right],\left[0.6,0.6\right],\left[0.2,0.2\right]\rangle $ | $\langle \left[{s}_{4},{s}_{4}\right],\left[0.8,0.9\right],\left[0.1,0.1\right]\rangle $ | $\langle \left[{s}_{5},{s}_{6}\right],\left[0.7,0.7\right],\left[0.1,0.2\right]\rangle $ | |

O_{3} | $\langle \left[{s}_{3},{s}_{4}\right],\left[0.7,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{7},{s}_{8}\right],\left[0.7,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{5},{s}_{6}\right],\left[0.6,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{6},{s}_{7}\right],\left[0.7,0.7\right],\left[0.1,0.2\right]\rangle $ | |

O_{4} | $\langle \left[{s}_{1},{s}_{2}\right],\left[0.6,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{0},{s}_{1}\right],\left[0.7,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{2},{s}_{2}\right],\left[0.8,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{3},{s}_{4}\right],\left[0.6,0.7\right],\left[0.1,0.1\right]\rangle $ | |

Palm oil | P_{1} | $\langle \left[{s}_{6},{s}_{7}\right],\left[0.6,0.6\right],\left[0.2,0.3\right]\rangle $ | $\langle \left[{s}_{5},{s}_{6}\right],\left[0.7,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{5},{s}_{5}\right],\left[0.7,0.8\right],\left[0.1,0.1\right]\rangle $ | $\langle \left[{s}_{5},{s}_{6}\right],\left[0.7,0.8\right],\left[0.1,0.1\right]\rangle $ |

P_{2} | $\langle \left[{s}_{2},{s}_{3}\right],\left[0.6,0.7\right],\left[0.2,0.2\right]\rangle $ | $\langle \left[{s}_{3},{s}_{3}\right],\left[0.7,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{3},{s}_{4}\right],\left[0.6,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{1},{s}_{2}\right],\left[0.7,0.7\right],\left[0.1,0.2\right]\rangle $ | |

P_{3} | $\langle \left[{s}_{5},{s}_{6}\right],\left[0.7,0.8\right],\left[0.1,0.1\right]\rangle $ | $\langle \left[{s}_{4},{s}_{4}\right],\left[0.7,0.8\right],\left[0.1,0.1\right]\rangle $ | $\langle \left[{s}_{4},{s}_{5}\right],\left[0.7,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{4},{s}_{5}\right],\left[0.6,0.7\right],\left[0.1,0.3\right]\rangle $ | |

Sunflower-soybean oil | SS_{1} | $\langle \left[{s}_{7},{s}_{8}\right],\left[0.7,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{4},{s}_{5}\right],\left[0.6,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{7},{s}_{7}\right],\left[0.7,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{4},{s}_{4}\right],\left[0.7,0.7\right],\left[0.1,0.2\right]\rangle $ |

SS_{2} | $\langle \left[{s}_{5},{s}_{6}\right],\left[0.6,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{5},{s}_{5}\right],\left[0.6,0.7\right],\left[0.2,0.2\right]\rangle $ | $\langle \left[{s}_{4},{s}_{5}\right],\left[0.7,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{5},{s}_{6}\right],\left[0.6,0.7\right],\left[0.1,0.2\right]\rangle $ | |

SS_{3} | $\langle \left[{s}_{2},{s}_{3}\right],\left[0.6,0.7\right],\left[0.2,0.2\right]\rangle $ | $\langle \left[{s}_{7},{s}_{8}\right],\left[0.6,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{4},{s}_{5}\right],\left[0.7,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{4},{s}_{5}\right],\left[0.7,0.8\right],\left[0.1,0.1\right]\rangle $ |

Decision Makers | Criteria | |||
---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{4} | |

DM_{1} | $\langle \left[{s}_{6}^{\prime},{s}_{7}^{\prime}\right],\left[0.8,0.9\right],\left[0.1,0.1\right]\rangle $ | $\langle \left[{s}_{6}^{\prime},{s}_{7}^{\prime}\right],\left[0.7,0.8\right],\left[0.1,0.1\right]\rangle $ | $\langle \left[{s}_{5}^{\prime},{s}_{6}^{\prime}\right],\left[0.6,0.7\right],\left[0.1,0.3\right]\rangle $ | $\langle \left[{s}_{3}^{\prime},{s}_{4}^{\prime}\right],\left[0.7,0.8\right],\left[0.2,0.2\right]\rangle $ |

DM_{2} | $\langle \left[{s}_{7}^{\prime},{s}_{8}^{\prime}\right],\left[0.7,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{5}^{\prime},{s}_{6}^{\prime}\right],\left[0.7,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{6}^{\prime},{s}_{6}^{\prime}\right],\left[0.7,0.7\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{4}^{\prime},{s}_{4}^{\prime}\right],\left[0.7,0.9\right],\left[0.1,0.1\right]\rangle $ |

DM_{3} | $\langle \left[{s}_{7}^{\prime},{s}_{7}^{\prime}\right],\left[0.8,0.8\right],\left[0.0,0.1\right]\rangle $ | $\langle \left[{s}_{7}^{\prime},{s}_{8}^{\prime}\right],\left[0.7,0.9\right],\left[0.1,0.1\right]\rangle $ | $\langle \left[{s}_{4}^{\prime},{s}_{5}^{\prime}\right],\left[0.7,0.8\right],\left[0.1,0.1\right]\rangle $ | $\langle \left[{s}_{4}^{\prime},{s}_{5}^{\prime}\right],\left[0.6,0.7\right],\left[0.1,0.2\right]\rangle $ |

DM_{4} | $\langle \left[{s}_{7}^{\prime},{s}_{8}^{\prime}\right],\left[0.6,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{7}^{\prime},{s}_{8}^{\prime}\right],\left[0.7,0.8\right],\left[0.0,0.2\right]\rangle $ | $\langle \left[{s}_{5}^{\prime},{s}_{5}^{\prime}\right],\left[0.6,0.9\right],\left[0.1,0.1\right]\rangle $ | $\langle \left[{s}_{5}^{\prime},{s}_{6}^{\prime}\right],\left[0.5,0.7\right],\left[0.0,0.2\right]\rangle $ |

DM_{5} | $\langle \left[{s}_{5}^{\prime},{s}_{6}^{\prime}\right],\left[0.6,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{5}^{\prime},{s}_{6}^{\prime}\right],\left[0.5,0.7\right],\left[0.1,0.3\right]\rangle $ | $\langle \left[{s}_{5}^{\prime},{s}_{6}^{\prime}\right],\left[0.6,0.8\right],\left[0.1,0.2\right]\rangle $ | $\langle \left[{s}_{5}^{\prime},{s}_{5}^{\prime}\right],\left[0.6,0.6\right],\left[0.1,0.3\right]\rangle $ |

Kind of Suppliers | Suppliers | Criteria | |||
---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{4} | ||

Olive oil | O_{1} | $\langle \left[{s}_{1.866},{s}_{2.958}\right],\left[0.673,0.738\right],\left[0.071,0.171\right]\rangle $ | $\langle \left[{s}_{3.866},{s}_{4.891}\right],\left[0.696,0.753\right],\left[0.122,0.190\right]\rangle $ | $\langle \left[{s}_{3.224},{s}_{4.084}\right],\left[0.678,0.751\right],\left[0.171,0.246\right]\rangle $ | $\langle \left[{s}_{4.704},{s}_{5.720}\right],\left[0.673,0.789\right],\left[0.100,0.146\right]\rangle $ |

O_{2} | $\langle \left[{s}_{5.598},{s}_{6.614}\right],\left[0.553,0.648\right],\left[0.171,0.271\right]\rangle $ | $\langle \left[{s}_{2.681},{s}_{3.852}\right],\left[0.600,0.643\right],\left[0.185,0.251\right]\rangle $ | $\langle \left[{s}_{4.830},{s}_{5.275}\right],\left[0.743,0.823\right],\left[0.115,0.176\right]\rangle $ | $\langle \left[{s}_{4.884},{s}_{5.638}\right],\left[0.655,0.718\right],\left[0.071,0.146\right]\rangle $ | |

O_{3} | $\langle \left[{s}_{3.458},{s}_{4.839}\right],\left[0.676,0.718\right],\left[0.071,0.171\right]\rangle $ | $\langle \left[{s}_{4.512},{s}_{5.554}\right],\left[0.676,0.800\right],\left[0.126,0.171\right]\rangle $ | $\langle \left[{s}_{5.821},{s}_{6.430}\right],\left[0.724,0.762\right],\left[0.126,0.236\right]\rangle $ | $\langle \left[{s}_{6.283},{s}_{6.957}\right],\left[0.706,0.753\right],\left[0.131,0.185\right]\rangle $ | |

O_{4} | $\langle \left[{s}_{1.722},{s}_{2.805}\right],\left[0.640,0.711\right],\left[0.071,0.187\right]\rangle $ | $\langle \left[{s}_{0.000},{s}_{2.806}\right],\left[0.610,0.687\right],\left[0.148,0.176\right]\rangle $ | $\langle \left[{s}_{3.512},{s}_{4.143}\right],\left[0.659,0.743\right],\left[0.126,0.185\right]\rangle $ | $\langle \left[{s}_{4.255},{s}_{5.275}\right],\left[0.653,0.709\right],\left[0.090,0.179\right]\rangle $ | |

Palm oil | P_{1} | $\langle \left[{s}_{6.382},{s}_{7.384}\right],\left[0.629,0.711\right],\left[0.146,0.196\right]\rangle $ | $\langle \left[{s}_{5.632},{s}_{6.382}\right],\left[0.700,0.738\right],\left[0.110,0.231\right]\rangle $ | $\langle \left[{s}_{5},{s}_{5.378}\right],\left[0.700,0.800\right],\left[0.075,0.131\right]\rangle $ | $\langle \left[{s}_{5.378},{s}_{6.058}\right],\left[0.678,0.809\right],\left[0.046,0.120\right]\rangle $ |

P_{2} | $\langle \left[{s}_{2.822},{s}_{3.657}\right],\left[0.653,0.743\right],\left[0.105,0.148\right]\rangle $ | $\langle \left[{s}_{2.470},{s}_{3.318}\right],\left[0.643,0.700\right],\left[0.156,0.246\right]\rangle $ | $\langle \left[{s}_{2.499},{s}_{3.112}\right],\left[0.648,0.709\right],\left[0.100,0.171\right]\rangle $ | $\langle \left[{s}_{1.899},{s}_{2.564}\right],\left[0.668,0.780\right],\left[0.085,0.146\right]\rangle $ | |

P_{3} | $\langle \left[{s}_{4.394},{s}_{5.405}\right],\left[0.707,0.800\right],\left[0.071,0.126\right]\rangle $ | $\langle \left[{s}_{3.886},{s}_{4.276}\right],\left[0.684,0.762\right],\left[0.090,0.192\right]\rangle $ | $\langle \left[{s}_{4.657},{s}_{5.578}\right],\left[0.718,0.784\right],\left[0.056,0.146\right]\rangle $ | $\langle \left[{s}_{3.565},{s}_{4.573}\right],\left[0.609,0.709\right],\left[0.046,0.198\right]\rangle $ | |

Sunflower-soybean oil | SS_{1} | $\langle \left[{s}_{6.123},{s}_{6.804}\right],\left[0.723,0.792\right],\left[0.056,0.126\right]\rangle $ | $\langle \left[{s}_{3.867},{s}_{4.884}\right],\left[0.643,0.773\right],\left[0.120,0.200\right]\rangle $ | $\langle \left[{s}_{5.669},{s}_{6.118}\right],\left[0.678,0.698\right],\left[0.056,0.146\right]\rangle $ | $\langle \left[{s}_{4.128},{s}_{4.522}\right],\left[0.629,0.753\right],\left[0.076,0.156\right]\rangle $ |

SS_{2} | $\langle \left[{s}_{4.148},{s}_{5.169}\right],\left[0.578,0.701\right],\left[0.081,0.176\right]\rangle $ | $\langle \left[{s}_{3.337},{s}_{4.194}\right],\left[0.547,0.687\right],\left[0.126,0.200\right]\rangle $ | $\langle \left[{s}_{2.297},{s}_{2.692}\right],\left[0.612,0.680\right],\left[0.166,0.251\right]\rangle $ | $\langle \left[{s}_{2.326},{s}_{2.849}\right],\left[0.568,0.700\right],\left[0.056,0.148\right]\rangle $ | |

SS_{3} | $\langle \left[{s}_{3.047},{s}_{3.901}\right],\left[0.590,0.753\right],\left[0.136,0.200\right]\rangle $ | $\langle \left[{s}_{3.152},{s}_{4.239}\right],\left[0.587,0.687\right],\left[0.068,0.171\right]\rangle $ | $\langle \left[{s}_{3.365},{s}_{4.128}\right],\left[0.683,0.783\right],\left[0.111,0.156\right]\rangle $ | $\langle \left[{s}_{4.276},{s}_{4.884}\right],\left[0.706,0.828\right],\left[0.081,0.126\right]\rangle $ |

Kind of Suppliers | Suppliers | Fuzzy TOPSIS | Fuzzy VIKOR | Fuzzy GRA | The proposed Approach |
---|---|---|---|---|---|

Olive oil | O_{1} | 3 | 3 | 3 | 2 |

O_{2} | 2 | 2 | 2 | 3 | |

O_{3} | 1 | 1 | 1 | 1 | |

O_{4} | 4 | 4 | 4 | 4 | |

Palm oil | P_{1} | 1 | 1 | 1 | 1 |

P_{2} | 3 | 3 | 3 | 2 | |

P_{3} | 2 | 2 | 2 | 3 | |

Sunflower-soybean oil | SS_{1} | 1 | 1 | 1 | 1 |

SS_{2} | 3 | 3 | 3 | 2 | |

SS_{3} | 2 | 2 | 2 | 3 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shi, H.; Quan, M.-Y.; Liu, H.-C.; Duan, C.-Y. A Novel Integrated Approach for Green Supplier Selection with Interval-Valued Intuitionistic Uncertain Linguistic Information: A Case Study in the Agri-Food Industry. *Sustainability* **2018**, *10*, 733.
https://doi.org/10.3390/su10030733

**AMA Style**

Shi H, Quan M-Y, Liu H-C, Duan C-Y. A Novel Integrated Approach for Green Supplier Selection with Interval-Valued Intuitionistic Uncertain Linguistic Information: A Case Study in the Agri-Food Industry. *Sustainability*. 2018; 10(3):733.
https://doi.org/10.3390/su10030733

**Chicago/Turabian Style**

Shi, Hua, Mei-Yun Quan, Hu-Chen Liu, and Chun-Yan Duan. 2018. "A Novel Integrated Approach for Green Supplier Selection with Interval-Valued Intuitionistic Uncertain Linguistic Information: A Case Study in the Agri-Food Industry" *Sustainability* 10, no. 3: 733.
https://doi.org/10.3390/su10030733