# A Novel Approach for Green Supplier Selection under a q-Rung Orthopair Fuzzy Environment

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Green Supplier Selection Approaches

#### 2.2. Q-ROFS

#### 2.3. Consensus Model

#### 2.4. The PA Operator

## 3. Preliminaries

#### 3.1. q-ROFS

**Definition**

**1 [17].**

**Definition**

**2 [17].**

**Example**

**1.**

- (1)
- ${\left({a}_{1}\right)}^{c}=\left(0.8298,0.6500\right)$, ${\left({a}_{2}\right)}^{c}=\left(0.7500,0.5000\right);$
- (2)
- ${a}_{1}\oplus {a}_{2}=\left(0.7149,0.6224\right);$
- (3)
- ${a}_{1}\otimes {a}_{2}=\left(0.3250,0.9094\right);$
- (4)
- $\lambda {a}_{1}=\left(0.7796,0.6886\right)$, $\lambda {a}_{2}=\left(0.6166,0.5625\right);$
- (5)
- ${\left({a}_{1}\right)}^{\lambda}=\left(0.4225,0.9346\right),{\left({a}_{2}\right)}^{\lambda}=\left(0.2500,0.8732\right).$

**Definition**

**3 [18,19].**

**Definition**

**4 [19].**

- (1)
- If$s\left({a}_{1}\right)<s\left({a}_{2}\right)$, then${a}_{1}<{a}_{2}$;
- (2)
- If$s\left({a}_{1}\right)=s\left({a}_{2}\right)$, then
- a.
- $h\left({a}_{1}\right)<h\left({a}_{2}\right)$, then${a}_{1}<{a}_{2}$;
- b.
- $h\left({a}_{1}\right)=h\left({a}_{2}\right)$, then${a}_{1}={a}_{2}$.

**Definition**

**5 [72].**

**Example**

**2.**

#### 3.2. The q-ROFPWA Operator

**Definition**

**6 [26].**

**Definition**

**7 [27].**

**Theorem**

**1 [27].**

## 4. Green Supplier Selection Method under q-ROF Environment

#### 4.1. Obtain the Normalized Evaluation Matrices of Decision Makers

**Step 1.1:**After the primary evaluation of the green supplier selection problem, decision makers can identify the potential green supplier ${A}_{i}\left(i=1,2,\dots ,m\right)$ and a collection of criteria ${C}_{j}\left(j=1,2,\dots ,n\right)$.

**Step 1.2:**Combined with the q-ROFS, the evaluation information of green suppliers can be expressed by q-ROF evaluation matrix ${\mathit{F}}^{\mathit{k}}={\left({\tilde{a}}_{ij}^{k}\right)}_{m\times n}$, where ${\tilde{a}}_{ij}^{k}=\left({\tilde{\mu}}_{ij}^{k},{\tilde{v}}_{ij}^{k}\right)$ indicates the q-ROF evaluation information of green supplier ${A}_{i}$ concerning criteria ${C}_{j}$ given by decision maker ${D}_{k}$. Moreover, decision makers also evaluate the weights of criteria using q-ROFNs; subsequently, the q-ROF evaluation matrix ${\mathit{W}}^{\mathit{k}}={\left({a}_{j}^{k}\right)}_{1\times n}$ was obtained, where ${a}_{j}^{k}=\left({\mu}_{j}^{k},{v}_{j}^{k}\right)$ represents the importance degree of criteria ${C}_{j}$ given by decision maker ${D}_{k}$.

**Step 1.3:**Generally, the criteria of green supplier selection can be divided into two types, namely, cost type and benefit type; thus, we should transform the information with respect to cost type criteria into the information with respect to benefit type criteria to determine the normalized q-ROF evaluation matrix ${\mathit{Q}}^{\mathit{k}}={\left({a}_{ij}^{k}\right)}_{m\times n}$ as:

#### 4.2. Consensus-Reaching Process

**Definition**

**8.**

**Definition**

**9.**

**Input:**The original individual evaluation matrix ${\mathit{Q}}^{\mathit{k}}$, the ideal consensus threshold $\epsilon $, and the maximum permission iterative number of times ${r}_{\mathrm{max}}$.

**Output:**The revised individual q-ROF evaluation matrix ${\overline{\mathit{Q}}}^{\mathit{k}}$ and the global consensus measure $ce$.

**Step 2.1:**Let the initial iterative number be $r=1$, and the individual evaluation matrix in the first round be ${\mathit{Q}}_{\mathbf{1}}^{\mathit{k}}={\left({a}_{ij,1}^{k}\right)}_{m\times n}={\left({a}_{ij}^{k}\right)}_{m\times n}$.

**Step 2.2:**Calculate the similarity matrix $\mathit{S}{\mathit{M}}^{\mathit{kp}}\left(k=1,2,\dots ,l-1;p=k+1,k+2,\dots ,l\right)$ and aggregate them to obtain the consensus matrix $\mathit{CM}$; thus, the consensus measures $c{c}_{ij}$, $c{a}_{i}$, and $ce$ in round $r$ are computed. If $ce\ge \epsilon $ or $r>{r}_{\mathrm{max}}$, then proceed to Step 1.5; otherwise, proceed to the next step.

**Step 2.3:**Obtain the identification rules as in the following:

**Step 2.4:**Aggregate the individual evaluation matrix ${\mathit{Q}}_{\mathit{r}}^{\mathit{k}}$ using the q-ROFAA operator that is reduced by the q-ROFWA operator [18], then, the collective evaluation matrix ${\mathit{Q}}_{\mathit{r}}={\left({a}_{ij,r}\right)}_{m\times n}$ can be obtained as:

**Step 1.2**.

**Step 2.5:**Let ${\overline{\mathit{Q}}}^{\mathit{k}}={\mathit{Q}}_{\mathit{r}}^{\mathit{k}}={\left({\overline{a}}_{ij}^{k}\right)}_{m\times n}={\left({\overline{\mu}}_{ij}^{k},{\overline{v}}_{ij}^{k}\right)}_{m\times n}$. Output ${\overline{\mathit{Q}}}^{\mathit{k}}$ and $ce$ in this round.

#### 4.3. Aggregation of Individual Acceptable Consensus Evaluation Matrices

**Step 3.1:**Compute the support degree:

**Step 3.2:**Combined with the subjective weight vector of decision makers $\mathit{w}={\left({w}_{1},{w}_{2},\dots ,{w}_{l}\right)}^{T}$ that is provided by the enterprise, the weighted support degree of ${\overline{a}}_{ij}^{k}$ can be calculated as:

**Step 3.3:**Use the q-ROFPWA operator to fuse the evaluation matrix ${\overline{\mathit{Q}}}^{\mathit{k}}$ to obtain the collective evaluation matrix $\mathit{Q}$ as:

#### 4.4. Determine the Weights of Criteria

**Step 4.1:**Combined with the evaluation matrix ${\mathit{W}}^{\mathit{k}}$ and the similar steps in Section 4.3 and Section 4.4, we can obtain the collective evaluation matrix $\mathit{W}={\left({a}_{j}\right)}_{1\times n}$. The larger the score value of ${a}_{j}$, which means the criteria ${C}_{j}$ is more important, the higher the weight of criteria ${C}_{j}$ and vice versa. Then, the subjective weight vector of criteria ${\mathit{\lambda}}^{\mathit{S}}={\left({\lambda}_{1}^{S},{\lambda}_{2}^{S},\dots ,{\lambda}_{n}^{S}\right)}^{T}$ can be determined as:

**Step 4.2:**Let $\sum}_{h=1,h\ne i}^{m}d\left({a}_{ij},{a}_{hj}\right){\lambda}_{j}^{O$ be the deviation between the collective evaluation information on green supplier ${A}_{i}$ and other green suppliers concerning ${C}_{j}$, where $d\left({a}_{ij},{a}_{hj}\right)$ is the Minkowski distance between ${a}_{ij}$ and ${a}_{hj}$; then, the total deviation is obtained as $\sum}_{j=1}^{n}{\displaystyle {\sum}_{i=1}^{m}{\displaystyle {\sum}_{h=1,h\ne i}^{m}d\left({a}_{ij},{a}_{hj}\right){\lambda}_{j}^{O}}$. According to the information theory, if all green suppliers have similar evaluation information concerning one of criteria, a small weight value should be assigned to the criteria as it contributes less to differentiate green suppliers [73]. Subsequently, a deviation maximization model can be developed as:

**Step 4.3:**Determine the comprehensive weight vector of criteria $\mathit{\lambda}={\left({\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{n}\right)}^{T}$ as:

#### 4.5. Rank the Green Suppliers Using the TODIM Method under q-ROF Environment

**Step 5.1:**Compute the relative weight ${\lambda}_{jr}$ of criteria ${C}_{j}$ with respect to the reference criteria ${C}_{r}$ as:

**Step 5.2:**Calculate the dominance degree of green supplier ${A}_{i}$ over each green supplier ${A}_{h}\left(h=1,2,\dots ,m\right)$ by the following equation:

**Step 5.3:**Compute the global value of green supplier ${A}_{i}$ by:

**Step 5.4:**Determine the ranking of potential green suppliers based on their global values; the larger the global value $\Phi \left({A}_{i}\right)$, the higher the ranking of green supplier ${A}_{i}$.

## 5. Numerical Example

#### 5.1. Implementation

**Step 1:**Obtain the normalized evaluation matrices of decision makers.

**Step 1.1:**After a preliminary evaluation, four potential green suppliers ${A}_{i}\left(i=1,2,3,4\right)$ are determined by a group of decision makers ${D}_{k}\left(k=1,2,3\right)$. Decision makers evaluate the four green suppliers concerning six criteria ${C}_{j}\left(j=1,2,3,4,5,6\right)$, namely, environmental costs (${C}_{1}$), remanufacturing activity (${C}_{2}$), energy assumption (${C}_{3}$), reverse logistics program (${C}_{4}$), hazardous waste management (${C}_{5}$), and environmental certification (${C}_{6}$), where ${C}_{1}$ and ${C}_{3}$ are the cost type criteria, and the others are the benefit type criteria.

**Step 1.2:**According to the relationships between the linguistic terms and interval-valued Pythagorean fuzzy numbers [74], we can construct the transformation between the linguistic terms and the corresponding q-ROFNs ($q=3$) as shown in Table 1. Then, decision makers use the linguistic terms to assess the green suppliers as shown in Table 2; thus, the q-ROF evaluation matrix ${\mathit{F}}^{\mathit{k}}={\left({\tilde{a}}_{ij}^{k}\right)}_{4\times 6}$ is obtained. It is noteworthy that we adopt the subjective weights of criteria obtained in the literature [16], i.e., ${\mathit{\lambda}}^{\mathit{S}}={\left(0.180,0.090,0.130,0.130,0.310,0.160\right)}^{T}$.

**Step 1.3:**After the normalization step according to the different types of criteria, the normalized q-ROF evaluation matrix ${\mathit{Q}}^{\mathit{k}}={\left({a}_{ij}^{k}\right)}_{4\times 6}$ can be obtained by using Equation (13).

**Step 2:**Consensus-reaching process ($\epsilon =0.85,{r}_{\mathrm{max}}=5$).

**Step 2.1:**Let the initial iterative number be $r=1$, and ${\mathit{Q}}_{1}^{\mathit{k}}={\left({a}_{ij,1}^{k}\right)}_{4\times 6}={\left({a}_{ij}^{k}\right)}_{4\times 6}$.

**Step 2.2:**Calculate the similarity matrix $\mathit{S}{\mathit{M}}^{\mathit{kp}}$ between the q-ROF evaluation matrices ${\mathit{Q}}_{1}^{\mathit{k}}$ and ${\mathit{Q}}_{1}^{\mathit{p}}$ as

**Step 2.3:**Obtain the identification rules as in the following:

**Step 2.4:**Aggregate the individual evaluation matrix ${\mathit{Q}}_{1}^{\mathit{k}}$ in round one using the q-ROFAA operator to obtain collective evaluation matrix ${\mathit{Q}}_{1}={\left({a}_{ij,1}\right)}_{4\times 6}$; then, the direction rules can be put forward to revise the non-consensus evaluation information in set $IR$ as shown in Table 3. Set $r=2$ and proceed to Step 1.2.

**Steps 2.2~2.4**, we can obtain the global consensus measure in round four $ce=0.8556>\epsilon $, which means that a high consensus level between decision makers has been achieved; the individual acceptable consensus q-ROF evaluation matrix ${\overline{\mathit{Q}}}^{\mathit{k}}$ are determined as shown in Table 4.

**Step 3:**Aggregation of individual acceptable consensus evaluation matrices.

**Steps 3.1~3.2:**Suppose that the subjective weight values of decision makers are equal, i.e., $\mathit{w}={\left(1/3,1/3,1/3\right)}^{T}$; we can use Equations (21)~(23) to calculate the weighted support degree of ${\overline{a}}_{ij}^{k}$ as:

**Step 3.3:**Use the q-ROFPWA operator to fuse the evaluation matrix ${\overline{\mathit{Q}}}^{\mathit{k}}$ to obtain the collective evaluation matrix $\mathit{Q}$ as shown in Table 5.

**Step 4:**Determine the weights of the criteria.

**Step 4.1:**Because the subjective weights of criteria were determined in the literature [16], we adopt the subjective weight vector of criteria as ${\mathit{\lambda}}^{\mathit{S}}={\left(0.180,0.090,0.130,0.130,0.310,0.160\right)}^{T}$.

**Step 4.2:**Based on the collective evaluation matrix $\mathit{Q}$, we can construct the programming model, i.e., Equation (26); then, the objective weight vector of criteria can be determined as ${\mathit{\lambda}}^{\mathit{O}}={\left(0.201,0.160,0.150,0.182,0.151,0.156\right)}^{T}$.

**Step 4.3:**Set the importance coefficient of subjective weights to $\phi =0.5$; we can obtain the comprehensive weights of criteria as $\mathit{\lambda}={\left(0.191,0.125,0.140,0.156,0.230,0.158\right)}^{T}$.

**Step 5:**Rank the green suppliers using the TODIM method under a q-ROF environment ($\theta =1$).

**Step 5.1:**Utilize Equation (31) to compute the relative weight ${\lambda}_{jr}$ of criteria ${C}_{j}$ concerning the reference criteria ${C}_{r}$ as:

**Step 5.2:**Compute the dominance degree of green supplier ${A}_{i}$ over each green supplier:

**Step 5.3:**Compute the global value of green supplier ${A}_{i}$ by Equation (34):

**Step 5.4:**Based on the global values of green suppliers, the ranking of potential green suppliers can be determined as ${A}_{4}>{A}_{3}>{A}_{1}>{A}_{2}$. The green supplier ${A}_{4}$ is the best choice for the electric automobile company.

#### 5.2. Comparison and Sensitivity Analysis

- (1)
- The q-ROFS is utilized to represent the evaluation information of decision makers, which can express the membership, non-membership, and indeterminacy membership degrees, simultaneously. Furthermore, with the increasing rung q, the space of acceptable orthopairs of q-ROFS is larger than IFS and PFS; as a generalized form of IFS and PFS, the proposed approach can also be transformed into other green supplier selection approaches under an IF and PF environment if necessary.
- (2)
- In practice, decision makers always differentiate from research fields and domain experiences; the non-consensus evaluation information of green suppliers will inevitably be given. Combined with an iteration-based consensus model under q-ROF environment, the non-consensus evaluation information of all the decision makers can be revised in each round. Therefore, a ranking of green suppliers accepted by decision makers or enterprises can be obtained using the proposed approach, and the efficiency of the consensus-reaching process is relatively high.
- (3)
- The q-ROFPWA operator is introduced to fuse the individual evaluation matrices; the weight vectors of decision makers can be determined by two aspects, namely, the subjective aspect and the objective aspect. Consequently, we can obtain a ranking of green suppliers that is closer to reality. Additionally, the determination of weights of decision makers is solved, which has been ignored by most existing approaches.
- (4)
- The weights of criteria are determined by a comprehensive weighting approach, which is composed of the subjective evaluation method and a deviation maximization model. Through changing the valve of coefficient $\phi $, the weights of the criteria can be determined; whether they are closer to subjective weights or objective weights depends on the choice of the decision makers or enterprises. Thus, the proposed approach is more able to cope with different scenarios.
- (5)
- During the green supplier evaluation process, the bounded rationality behavior of decision makers cannot be avoided. The TODIM method is a powerful tool to solve these MCGDM problems; in the proposed approach, the TODIM method is extended to the q-ROF environment to compute the ranking of green suppliers, which makes the evaluation result more realistic and accurate. In addition, the robustness of the proposed method is relatively strong.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Step 1:**Obtain the normalized evaluation matrices of decision makers

**Step 2:**Aggregation of individual evaluation matrices

**Steps 2.1~2.2:**According to the subjective weight of decision makers $\mathit{w}={\left(1/3,1/3,1/3\right)}^{T}$, we can utilize Equations (21)~(23) to calculate the weighted support degree of ${a}_{ij}^{k}$ as:

**Step 2.3:**Use the q-ROFPWA operator to fuse the evaluation matrix ${\mathit{Q}}^{\mathit{k}}$ to obtain the collective evaluation matrix $\mathit{Q}$ as shown in Table A1.

Alternatives | ${\mathit{C}}_{\mathbf{1}}$ | ${\mathit{C}}_{\mathbf{2}}$ | ${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | ${\mathit{C}}_{\mathbf{5}}$ | ${\mathit{C}}_{\mathbf{6}}$ |
---|---|---|---|---|---|---|

${A}_{1}$ | (0.4590,0.7352) | (0.6344,0.5111) | (0.6660,0.4425) | (0.2747,0.8461) | (0.7904,0.3133) | (0.5221,0.5811) |

${A}_{2}$ | (0.3692,0.7456) | (0.4985,0.6341) | (0.3893,0.7155) | (0.6888,0.4143) | (0.5883,0.5149) | (0.4665,0.6449) |

${A}_{3}$ | (0.7225,0.3801) | (0.7690,0.3396) | (0.5221,0.5811) | (0.3893,0.7155) | (0.7552,0.5600) | (0.7225,0.3801) |

${A}_{4}$ | (0.6660,0.4425) | (0.7177,0.4399) | (0.3116,0.8097) | (0.6220,0.4807) | (0.6220,0.4807) | (0.6747,0.4890) |

**Step 3:**Determine the weights of criteria.

**Step 3.1:**We adopt the subjective weights of criteria in the literature [16] as ${\mathit{\lambda}}^{\mathit{S}}={\left(0.180,0.090,0.130,0.130,0.310,0.160\right)}^{T}$.

**Step 3.2:**Based on the collective evaluation matrix $\mathit{Q}$, we construct the programming model, i.e., Equation (26), then, the objective weights of criteria can be determined as ${\mathit{\lambda}}^{\mathit{O}}={\left(0.187,0.157,0.157,0.186,0.152,0.161\right)}^{T}$.

**Step 3.3:**Set the importance coefficient of subjective weights $\phi =0.5$; we can obtain the comprehensive weights of criteria as $\mathit{\lambda}={\left(0.183,0.124,0.143,0.158,0.231,0.161\right)}^{T}$.

**Step 4:**Rank the green suppliers using the TODIM method under the q-ROF environment ($\theta =1$).

**Step 4.1:**Utilize Equation (31) to compute the relative weight ${\lambda}_{jr}$ of criteria ${C}_{j}$ concerning the reference criteria ${C}_{r}$ as:

**Step 4.2:**Compute the dominance degree of green supplier ${A}_{i}$ over each green supplier as:

**Step 4.3:**Compute the global value of green supplier ${A}_{i}$ by Equation (34):

**Step 4.4:**Based on the global values of green suppliers, the ranking of potential green suppliers can be determined as ${A}_{3}>{A}_{4}>{A}_{1}>{A}_{2}$. The green supplier ${A}_{3}$ is the best choice for the electric automobile company.

## Appendix B

**Step 1:**According to the linguistic terms of decision makers in Table 2 and the relationships between linguistic terms and intuitionistic fuzzy numbers in the literature [16], we transform the linguistic terms into IF evaluation matri ces of decision makers; then, the intuitionistic fuzzy weighted average operator [77] is utilized to fuse the individual evaluation information to determine the collective evaluation matrix as presented in Table A2.

Alternatives | ${\mathit{C}}_{\mathbf{1}}$ | ${\mathit{C}}_{\mathbf{2}}$ | ${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | ${\mathit{C}}_{\mathbf{5}}$ | ${\mathit{C}}_{\mathbf{6}}$ |
---|---|---|---|---|---|---|

${A}_{1}$ | (0.4590,0.7352) | (0.6344,0.5111) | (0.6660,0.4425) | (0.2747,0.8461) | (0.7904,0.3133) | (0.5221,0.5811) |

${A}_{2}$ | (0.3692,0.7456) | (0.4985,0.6341) | (0.3893,0.7155) | (0.6888,0.4143) | (0.5883,0.5149) | (0.4665,0.6449) |

${A}_{3}$ | (0.7225,0.3801) | (0.7690,0.3396) | (0.5221,0.5811) | (0.3893,0.7155) | (0.7552,0.5600) | (0.7225,0.3801) |

${A}_{4}$ | (0.6660,0.4425) | (0.7177,0.4399) | (0.3116,0.8097) | (0.6220,0.4807) | (0.6220,0.4807) | (0.6747,0.4890) |

**Step 2:**According to the type of criteria, we can obtain the IF positive ideal solution ${a}^{+}$ and IF negative ideal solution ${a}^{-}$ as:

**Step 3:**Utilize the maximum average weighted distance method to construct a programming model as:

**Step 4:**Set the importance coefficient of subjective weights $\phi =0.5$, combined with the subjective weight vector of criteria ${\mathit{\lambda}}^{\mathit{S}}={\left(0.180,0.090,0.130,0.130,0.310,0.160\right)}^{T}$, we can obtain the comprehensive weights of criteria as $\mathit{\lambda}={\left(0.217,0.106,0.173,0.158,0.213,0.133\right)}^{T}$. Furthermore, the weighted IF evaluation matrix can be determined as presented in Table A3.

Alternatives | ${\mathit{C}}_{\mathbf{1}}$ | ${\mathit{C}}_{\mathbf{2}}$ | ${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | ${\mathit{C}}_{\mathbf{5}}$ | ${\mathit{C}}_{\mathbf{6}}$ |
---|---|---|---|---|---|---|

${A}_{1}$ | (1.0000,0.0000) | (0.0756,0.8983) | (0.0708,0.9139) | (0.0172,0.9699) | (0.2482,0.6757) | (0.0744,0.9193) |

${A}_{2}$ | (0.2365,0.6907) | (0.0470,0.9345) | (0.1744,0.7749) | (0.1478,0.8093) | (0.1508,0.8320) | (0.0549,0.9332) |

${A}_{3}$ | (0.0564,0.9153) | (0.1235,0.8346) | (0.1243,0.8612) | (0.0414,0.9376) | (1.0000,0.0000) | (0.1370,0.8219) |

${A}_{4}$ | (0.0880,0.8933) | (0.0946,0.8709) | (1.0000,0.0000) | (0.1246,0.8493) | (0.1642,0.8024) | (0.1013,0.8670) |

**Step 5:**Utilize the following equations to calculate the distances between each green supplier and the IF positive ideal solution ${a}^{+}$ and IF negative ideal solution ${a}^{-}$, respectively.

**Step 6:**According to the relative closeness coefficient value of each green supplier, we can determine the ranking of the green supplier as ${A}_{3}>{A}_{2}>{A}_{4}>{A}_{1}$; the green supplier ${A}_{3}$ is the best choice for the electric automobile company.

## Appendix C

**Step 1:**Because of the linguistic terms utilized in the literature [75] are divided into five grades, we reconstruct the relationships between linguistic terms and triangular fuzzy numbers as presented in Table A4 to implement the numerical example in this paper.

Linguistic Terms | Corresponding Triangular Fuzzy Numbers |
---|---|

Extremely High (EH) | (0.8,0.9,1.0) |

Very High (VH) | (0.6,0.7,0.8) |

High (H) | (0.5,0.6,0.7) |

Medium High (MH) | (0.4,0.5,0.6) |

Medium (M) | (0.3,0.4,0.5) |

Medium Low (ML) | (0.2,0.3,0.4) |

Low (L) | (0.1,0.2,0.3) |

Very Low (VL) | (0.0,0.1,0.2) |

Extremely Low (EL) | (0.0,0.0,0.1) |

**Step 2:**According to Table 2 and Table A4, we can transform the linguistic evaluation information of decision makers into the corresponding triangular fuzzy numbers. The weights of decision makers are considered equal in the literature [75]; thus, the collective evaluation matrix can be obtained as shown in Table A5.

Alternatives | ${\mathit{C}}_{\mathbf{1}}$ | ${\mathit{C}}_{\mathbf{2}}$ | ${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | ${\mathit{C}}_{\mathbf{5}}$ | ${\mathit{C}}_{\mathbf{6}}$ |
---|---|---|---|---|---|---|

${A}_{1}$ | (0.53,0.63,0.73) | (0.30,0.40,0.50) | (0.20,0.30,0.40) | (0.03,0.10,0.20) | (0.53,0.63,0.73) | (0.27,0.37,0.47) |

${A}_{2}$ | (0.50,0.60,0.70) | (0.20,0.30,0.40) | (0.47,0.57,0.67) | (0.43,0.53,0.63) | (0.33,0.43,0.53) | (0.20,0.30,0.40) |

${A}_{3}$ | (0.13,0.23,0.33) | (0.50,0.60,0.70) | (0.33,0.43,0.53) | (0.13,0.23,0.33) | (0.27,0.30,0.40) | (0.47,0.57,0.67) |

${A}_{4}$ | (0.20,0.30,0.40) | (0.37,0.43,0.53) | (0.60,0.70,0.80) | (0.37,0.47,0.57) | (0.37,0.47,0.57) | (0.33,0.40,0.50) |

**Step 3:**To obtain a more objective comparison result, we adopt the weights of criteria in the Section 5.1 as $\mathit{\lambda}={\left(0.191,0.125,0.140,0.156,0.230,0.158\right)}^{T}$.

**Step 4:**Rank the green suppliers using the fuzzy TODIM method ($\theta =1$); similar to the improved TODIM method in this paper, compute the relative weight ${\lambda}_{jr}$ of criteria ${C}_{j}$ concerning the reference criteria ${C}_{r}$ as

**Step 5:**Compute the dominance degree of green supplier ${A}_{i}$ over each green supplier:

**Step 6:**Compute the global value of green supplier ${A}_{i}$:

**Step 7:**Based on the global values of green suppliers, the ranking of potential green suppliers can be determined as ${A}_{3}>{A}_{4}>{A}_{1}>{A}_{2}$. The green supplier ${A}_{3}$ is the best choice for the electric automobile company.

## References

- Sahu, N.K.; Datta, S.; Mahapatra, S.S. Establishing green supplier appraisement platform using grey concepts. Grey Syst.
**2012**, 2, 395–418. [Google Scholar] [CrossRef] - Rostamzadeh, R.; Govindan, K.; Esmaeili, A.; Sabaghi, M. Application of fuzzy VIKOR for evaluation of green supply chain management practices. Ecol. Indic.
**2015**, 49, 188–203. [Google Scholar] [CrossRef] - Vachon, S. Green supply chain practices and the selection of environmental technologies. Int. J. Prod. Res.
**2007**, 45, 4357–4379. [Google Scholar] [CrossRef] - Cabral, I.; Grilo, A.; Cruz-Machado, V. A decision-making model for lean, agile, resilient and green supply chain management. Int. J. Prod. Res.
**2012**, 50, 4830–4845. [Google Scholar] [CrossRef] - Beamon, B.M. Designing the green supply chain. Logist. Inf. Manag.
**2013**, 12, 332–342. [Google Scholar] [CrossRef] - Bai, C.; Sarkis, J. Green supplier development: Analytical evaluation using rough set theory. J. Clean. Prod.
**2010**, 18, 1200–1210. [Google Scholar] [CrossRef] - Wang, K.Q.; Liu, H.C.; Liu, L.; Huang, J. Green supplier evaluation and selection using cloud model theory and the QUALIFLEX method. Sustainability
**2017**, 9, 688. [Google Scholar] [CrossRef] - Qin, J.D.; Liu, X.W.; Pedrycz, W. An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment. Eur. J. Oper. Res.
**2017**, 258, 626–638. [Google Scholar] [CrossRef] - Kannan, D.; Govindan, K.; Rajendran, S. Fuzzy axiomatic design approach based green supplier selection: Acase study from singapore. J. Clean. Prod.
**2015**, 96, 194–208. [Google Scholar] [CrossRef] - Blome, C.; Hollos, D.; Paulraj, A. Green procurement and green supplier development: Antecedents and effects on supplier performance. Int. J. Prod. Res.
**2014**, 52, 32–49. [Google Scholar] [CrossRef] - Govindan, K.; Rajendran, S.; Sarkis, J.; Murugesan, P. Multi criteria decision making approaches for green supplier evaluation and selection: A literature review. J. Clean. Prod.
**2015**, 98, 66–83. [Google Scholar] [CrossRef] - Zhu, J.H.; Li, Y.L. Green supplier selection based on consensus process and integrating prioritized operator and Choquet integral. Sustainability
**2018**, 10, 2744. [Google Scholar] [CrossRef] - Wang, J.; Wei, G.W.; Wei, Y. Models for green supplier selection with some 2-tuple linguistic neutrosophic number Bonferroni mean operators. Symmetry
**2018**, 10, 131. [Google Scholar] [CrossRef] - Banaeian, N.; Mobli, H.; Fahimnia, B.; Nielsen, I.E.; Omid, M. Green supplier selection using fuzzy group decision making methods: A case study from the agri-food industry. Comput. Oper. Res.
**2017**, 89, 337–347. [Google Scholar] [CrossRef] - Ghorabaee, M.K.; Zavadskas, E.K.; Amiri, M.; Esmaeili, A. Multi-criteria evaluation of green suppliers using an extended WASPAS method with interval type-2 fuzzy sets. J. Clean. Prod.
**2016**, 137, 213–229. [Google Scholar] [CrossRef] - Cao, Q.W.; Wu, J.; Liang, C.Y. An intuitionsitic fuzzy judgement matrix and TOPSIS integrated multi-criteria decision making method for green supplier selection. J. Intell. Fuzzy Syst.
**2015**, 28, 117–126. [Google Scholar] [CrossRef] - Yager, R.R. Generalized orthopair fuzzy sets. IEEE Trans. Fuzzy Syst.
**2017**, 25, 1222–1230. [Google Scholar] [CrossRef] - Liu, P.D.; Wang, P. Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making. Int. J. Intell. Syst.
**2018**, 33, 259–280. [Google Scholar] [CrossRef] - Wei, G.W.; Gao, H.; Wei, Y. Some q-rung orthopair fuzzy Heronian mean operators in multiple attribute decision making. Int. J. Intell. Syst.
**2018**, 33, 1426–1458. [Google Scholar] [CrossRef] - Liu, P.D.; Liu, J.L. Some q-rung orthopair fuzzy Bonferroni mean operators and their application to multi-attribute group decision making. Int. J. Intell. Syst.
**2018**, 33, 315–347. [Google Scholar] [CrossRef] - Alonso, S. A web based consensus support system for group decision making problems and incomplete preferences. Inf. Sci.
**2010**, 180, 4477–4495. [Google Scholar] [CrossRef] - Chiclana, F.; Mata, F.; Martinez, L.; Herrera-Viedma, E.; Alonso, S. Integration of a consistency control module within a consensus model. Int. J. Uncertain. Fuzz. Knowl.-Based Syst.
**2008**, 16, 35–53. [Google Scholar] [CrossRef] - Herrera-Viedma, E.; Cabrerizo, F.J.; Kacprzyk, J.; Pedrycz, W. A review of soft consensus models in a fuzzy environment. Inf. Fusion
**2014**, 17, 4–13. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; Calle, R.D.A.; Cascón, J.M. On measures of cohesiveness under dichotomous opinions: Some characterizations of approval consensus measures. Inf. Sci.
**2013**, 240, 45–55. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; Calle, R.D.A.; Cascón, J.M. A unifying model to measure consensus solutions in a society. Math. Comput. Model.
**2013**, 57, 1876–1883. [Google Scholar] [CrossRef] - Yager, R.R. The power average operator. IEEE Trans. Syst. Man Cybern Part A Syst. Hum.
**2001**, 31, 724–731. [Google Scholar] [CrossRef] - Liu, P.; Chen, S.M.; Wang, P. Multiple-attribute group decision-making based on q-rung orthopair fuzzy power maclaurin symmetric mean operators. IEEE Trans. Syst. Man Cybern. Syst.
**2018**, 1–16. [Google Scholar] [CrossRef] - Gomes, L.F.A.M.; Lima, M.M.P.P. TODIM: Basic and application to multicriteria ranking of projects with environmental impacts. Found. Comput. Decis. Sci.
**1991**, 16, 113–127. [Google Scholar] - Kahneman, D.; Tversky, A. Prospect theory: An analysis of decision under risk. Econometrica
**1979**, 47, 263–291. [Google Scholar] [CrossRef] - Morente-Molinera, J.A.; Kou, G.; Crespo, R.G.; Corchado, J.M.; Herrera-Viedma, E. Solving multi-criteria group decision making problems under environments with a high number of alternatives using fuzzy ontologies and multi-granular linguistic modelling methods. Knowl.-Based Syst.
**2017**, 137, 54–64. [Google Scholar] [CrossRef] - García-Díaz, V.; Espada, J.P.; Crespo, R.G.; G-Bustelo, B.C.P.; Lovelle, J.M.C. An approach to improve the accuracy of probabilistic classifiers fordecision support systems in sentiment analysis. Appl. Soft Comput.
**2018**, 67, 822–833. [Google Scholar] [CrossRef] - Taibi, A.; Atmani, B. Combining fuzzy AHP with GIS and decision rules for industrial site selection. Int. J. Interact. Multimedia Artif. Intell.
**2017**, 4, 60–69. [Google Scholar] [CrossRef] - Lee, A.H.I.; Kang, H.Y.; Hsu, C.F.; Hung, H.C. A green supplier selection model for high-tech industry. Expert Syst. Appl.
**2009**, 36, 7917–7927. [Google Scholar] [CrossRef] - Chen, H.M.W.; Chou, S.Y.; Luu, Q.D.; Yu, H.K. A fuzzy MCDM approach for green supplier selection from the economic and environmental aspects. Math. Probl. Eng.
**2016**. [Google Scholar] [CrossRef] - Yazdani, M. An integrated MCDM approach to green supplier selection. Int. J. Ind. Eng. Comput.
**2014**, 5, 443–458. [Google Scholar] [CrossRef] - Tsui, C.W.; Wen, U.P. A hybrid multiple criteria group decision-making approach for green supplier selection in the TFT-LCD industry. Math. Probl. Eng.
**2014**. [Google Scholar] [CrossRef] - Dobos, I.; Vörösmarty, G. Green supplier selection and evaluation using DEA-type composite indicators. Int. J. Prod. Econ.
**2014**, 157, 273–278. [Google Scholar] [CrossRef] - Hashemi, S.H.; Karimi, A.; Tavana, M. An integrated green supplier selection approach with analytic network process and improved grey relational analysis. Int. J. Prod. Econ.
**2015**, 159, 178–191. [Google Scholar] [CrossRef] - Kuo, R.J.; Wang, Y.C.; Tien, F.C. Integration of artificial neural network and MADA methods for green supplier selection. J. Clean. Prod.
**2010**, 18, 1161–1170. [Google Scholar] [CrossRef] - Kuo, T.C.; Hsu, C.W.; Li, J.Y. Developing a green supplier selection model by using the DANP with VIKOR. Sustainability
**2015**, 7, 1661–1689. [Google Scholar] [CrossRef] - Sang, X.Z.; Liu, X.W. An interval type-2 fuzzy sets-based TODIM method and its application to green supplier selection. J. Oper. Res. Soc.
**2016**, 67, 722–734. [Google Scholar] [CrossRef] - Govindan, K.; Kadziński, M.; Sivakumar, R. Application of a novel PROMETHEE-based method for construction of a group compromise ranking to prioritization of green suppliers in food supply chain. Omega
**2016**, 71, 129–145. [Google Scholar] [CrossRef] - Quan, M.Y.; Wang, Z.L.; Liu, H.C.; Shi, H. A hybrid MCDM approach for large group green supplier selection with uncertain linguistic information. IEEE Access
**2018**, 6, 50372–50383. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Bali, O.; Kose, E.; Gumus, S. Green supplier selection based on IFS and GRA. Grey Syst.
**2013**, 3, 158–176. [Google Scholar] [CrossRef] - Yager, R.R. Pythagorean membership grades in multicriteria decision making. IEEE Trans. Fuzzy Syst.
**2014**, 22, 958–965. [Google Scholar] [CrossRef] - Li, L.; Zhang, R.T.; Wang, J.; Shang, X.P.; Bai, K.Y. A novel approach to multi-attribute group decision-making with q-rung picture linguistic information. Symmetry
**2018**, 10, 172. [Google Scholar] [CrossRef] - Cường, B.C. Picture fuzzy sets. J. Comput. Sci. Cybern.
**2014**, 30, 409. [Google Scholar] [CrossRef] - Ullah, K.; Mahmood, T.; Jan, N. Similarity measures for T-spherical fuzzy sets with applications in pattern recognition. Symmetry
**2018**, 10, 193. [Google Scholar] [CrossRef] - Herrera-Viedma, E.; Herrera, F.; Chiclana, F. A consensus model for multiperson decision making with different preference structures. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum.
**2002**, 32, 394–402. [Google Scholar] [CrossRef] [Green Version] - Herrera-Viedma, E.; Alonso, S.; Chiclana, F.; Herrera, F. A consensus model for group decision making with incomplete fuzzy preference relations. IEEE Trans. Fuzzy Syst.
**2007**, 15, 863–877. [Google Scholar] [CrossRef] - Chu, J.F.; Liu, X.W.; Wang, Y.M.; Chin, K.S. A group decision making model considering both the additive consistency and group consensus of intuitionistic fuzzy preference relations. Comput. Ind. Eng.
**2016**, 101, 227–242. [Google Scholar] [CrossRef] - Wu, J.; Chiclana, F.; Herrera-Viedma, E. Trust based consensus model for social network in an incomplete linguistic information context. Appl. Soft Comput.
**2015**, 35, 827–839. [Google Scholar] [CrossRef] [Green Version] - Wu, Z.B.; Xu, J.P. Possibility distribution-based approach for MAGDM with hesitant fuzzy linguistic information. IEEE Trans. Cybern.
**2016**, 46, 694–705. [Google Scholar] [CrossRef] [PubMed] - Dong, Y.C.; Chen, X.; Herrera, F. Minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making. Inf. Sci.
**2015**, 297, 95–117. [Google Scholar] [CrossRef] - Gong, Z.W.; Zhang, H.H.; Forrest, J.; Li, L.S.; Xu, X.X. Two consensus models based on the minimum cost and maximum return regarding either all individuals or one individual. Eur. J. Oper. Res.
**2015**, 240, 183–192. [Google Scholar] [CrossRef] - Gong, Z.W.; Xu, X.X.; Li, L.S.; Xu, C. Consensus modeling with nonlinear utility and cost constraints: A case study. Knowl.-Based Syst.
**2015**, 88, 210–222. [Google Scholar] [CrossRef] - Xu, G.L.; Wan, S.P.; Wang, F.; Dong, J.Y.; Zeng, Y.F. Mathematical programming methods for consistency and consensus in group decision making with intuitionistic fuzzy preference relations. Knowl.-Based Syst.
**2016**, 98, 30–43. [Google Scholar] [CrossRef] - Zhang, Z.M.; Pedrycz, W. Goal programming approaches to managing consistency and consensus for intuitionistic multiplicative preference relations in group decision-making. IEEE Trans. Fuzzy Syst.
**2018**. In press. [Google Scholar] [CrossRef] - Zhou, L.G.; Chen, H.Y.; Liu, J.P. Generalized power aggregation operators and their applications in group decision making. Comput. Ind. Eng.
**2012**, 62, 989–999. [Google Scholar] [CrossRef] - Xu, Z.S. Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators. Knowl.-Based Syst.
**2011**, 24, 749–760. [Google Scholar] [CrossRef] - Wan, S.P. Power average operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making. Appl. Math. Model.
**2013**, 37, 4112–4126. [Google Scholar] [CrossRef] - Liu, P.D.; Liu, Y. An approach to multiple attribute group decision making based on intuitionistic trapezoidal fuzzy power generalized aggregation operator. Int. J. Comput. Intell. Syst.
**2014**, 7, 291–304. [Google Scholar] [CrossRef] - He, Y.D.; Chen, H.Y.; Zhou, L.G.; Liu, J.P.; Tao, Z.F. Generalized interval-valued Atanassov’s intuitionistic fuzzy power operators and their application to group decision making. Int. J. Fuzzy Syst.
**2013**, 15, 401–411. [Google Scholar] - Zhang, X.; Liu, P.D.; Wang, Y.M. Multiple attribute group decision making methods based on intuitionistic fuzzy Frank power aggregation operators. J. Intell. Fuzzy Syst.
**2015**, 29, 2235–2246. [Google Scholar] [CrossRef] - Wei, G.W.; Lu, M. Pythagorean fuzzy power aggregation operators in multiple attribute decision making. Int. J. Intell. Syst.
**2018**, 33, 169–186. [Google Scholar] [CrossRef] - Song, M.X.; Jiang, W.; Xie, C.H.; Zhou, D.Y. A new interval numbers power average operator in multiple attribute decision making. Int. J. Intell. Syst.
**2017**, 32, 631–644. [Google Scholar] [CrossRef] - Liu, P.D.; Shi, L.L. The generalized hybrid weighted average operator based on interval neutrosophic hesitant set and its application to multiple attribute decision making. Neural Comput. Appl.
**2015**, 26, 457–471. [Google Scholar] [CrossRef] - Xu, Y.J.; Merigó, J.M.; Wang, H.M. Linguistic power aggregation operators and their application to multiple attribute group decision making. Appl. Math. Model.
**2012**, 36, 5427–5444. [Google Scholar] [CrossRef] - Wu, X.H.; Qian, J.; Peng, J.J.; Xue, C.C. A multi-criteria group decision-making method with possibility degree and power aggregation operators of single trapezoidal neutrosophic numbers. Symmetry
**2018**, 10, 590. [Google Scholar] [CrossRef] - Du, W.S. Minkowski-type distance measures for generalized orthopair fuzzy sets. Int. J. Intell. Syst.
**2018**, 33, 802–817. [Google Scholar] [CrossRef] - Xu, Z.S. A deviation-based approach to intuitionistic fuzzy multiple attribute group decision making. Group Decis. Negot.
**2010**, 19, 57–76. [Google Scholar] [CrossRef] - Karaşan, A.; Ilbahar, E.; Cebi, S.; Kahraman, C. A new risk assessment approach: Safety and critical effect analysis (SCEA) and its extension with Pythagorean fuzzy sets. Saf. Sci.
**2018**, 108, 173–187. [Google Scholar] [CrossRef] - Tosun, Ö.; Akyüz, G. A fuzzy TODIM approach for the supplier selection problem. Int. J. Comput. Intell. Syst.
**2015**, 8, 317–329. [Google Scholar] [CrossRef] - Kou, G.; Lu, Y.Q.; Peng, Y.; Shi, Y. Evaluation of classification algorithms using MCDM and rank correlation. Int. J. Inf. Technol. Decis. Mak.
**2012**, 11, 197–225. [Google Scholar] [CrossRef] - Xu, Z.S. Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst.
**2013**, 14, 108–116. [Google Scholar] [CrossRef]

**Figure 1.**Geometric space range of the intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PFS), and q-rung orthopair fuzzy set (q-ROFS).

**Figure 2.**The flowchart of the proposed approach. Q-ROF: q-rung orthopair fuzzy. Q-ROFPWA: q-rung orthopair fuzzy power weighted average. Q-ROF-TODIM: q-rung orthopair fuzzy TOmada de Decisao Interativa e Multicritevio

**Figure 5.**Ranking results of intuitionistic fuzzy (IF)-technique for order performance by similarity to ideal solution (TOPSIS) method with different weights of criteria.

**Figure 6.**Ranking results of fuzzy TOmada de Decisao Interativa e Multicritevio (TODIM) method with different weights of criteria.

Linguistic Terms | Corresponding q-ROFNs |
---|---|

Extremely High (EH) | (0.95,0.15) |

Very High (VH) | (0.85,0.25) |

High (H) | (0.75,0.35) |

Medium High (MH) | (0.65,0.45) |

Medium (M) | (0.55,0.55) |

Medium Low (ML) | (0.45,0.65) |

Low (L) | (0.35,0.75) |

Very Low (VL) | (0.25,0.85) |

Extremely Low (EL) | (0.15,0.95) |

Decision Makers | Alternatives | ${\mathit{C}}_{\mathbf{1}}$ | ${\mathit{C}}_{\mathbf{2}}$ | ${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | ${\mathit{C}}_{\mathbf{5}}$ | ${\mathit{C}}_{\mathbf{6}}$ |
---|---|---|---|---|---|---|---|

${D}_{1}$ | ${A}_{1}$ | EH | H | L | EL | H | M |

${A}_{2}$ | MH | L | H | H | M | L | |

${A}_{3}$ | L | H | M | L | EH | H | |

${A}_{4}$ | L | EL | H | MH | MH | H | |

${D}_{2}$ | ${A}_{1}$ | VH | VL | M | L | H | ML |

${A}_{2}$ | H | MH | MH | MH | M | ML | |

${A}_{3}$ | L | VH | M | L | EL | H | |

${A}_{4}$ | ML | H | H | MH | MH | H | |

${D}_{3}$ | ${A}_{1}$ | ML | MH | ML | VL | VH | M |

${A}_{2}$ | VH | L | H | MH | MH | M | |

${A}_{3}$ | ML | MH | MH | ML | EL | MH | |

${A}_{4}$ | M | VH | EH | M | M | EL |

$\mathit{I}\mathit{R}$ | Individual Evaluation Information | Collective Evaluation Information | Direction Rules |
---|---|---|---|

(3,(1,1)) | ML, (0.45,0.65) | (0.4750,0.7136) | ML→M |

(2,(1,2)) | VL, (0.25,0.85) | (0.6345,0.5116) | VL→L |

(1,(3,5)) | EH, (0.95,0.15) | (0.7823,0.5135) | EH→VH |

(1,(4,2)) | EL, (0.15,0.95) | (0.7332,0.4364) | EL→VL |

(3,(4,6)) | EL, (0.15,0.95) | (0.6745,0.4882) | EL→VL |

Decision Makers | Alternatives | ${\mathit{C}}_{\mathbf{1}}$ | ${\mathit{C}}_{\mathbf{2}}$ | ${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | ${\mathit{C}}_{\mathbf{5}}$ | ${\mathit{C}}_{\mathbf{6}}$ |
---|---|---|---|---|---|---|---|

${D}_{1}$ | ${A}_{1}$ | (0.15,0.95) | (0.75,0.35) | (0.75,0.35) | (0.15,0.95) | (0.75,0.35) | (0.55,0.55) |

${A}_{2}$ | (0.45,0.65) | (0.35,0.75) | (0.35,0.75) | (0.75,0.35) | (0.55,0.55) | (0.35,0.75) | |

${A}_{3}$ | (0.75,0.35) | (0.75,0.35) | (0.55,0.55) | (0.35,0.75) | (0.85,0.25) | (0.75,0.35) | |

${A}_{4}$ | (0.75,0.35) | (0.45,0.65) | (0.35,0.75) | (0.65,0.45) | (0.65,0.45) | (0.75,0.35) | |

${D}_{2}$ | ${A}_{1}$ | (0.25,0.85) | (0.45,0.65) | (0.55,0.55) | (0.35,0.75) | (0.75,0.35) | (0.45,0.65) |

${A}_{2}$ | (0.35,0.75) | (0.65,0.45) | (0.45,0.65) | (0.65,0.45) | (0.55,0.55) | (0.45,0.65) | |

${A}_{3}$ | (0.75,0.35) | (0.85,0.25) | (0.55,0.55) | (0.35,0.75) | (0.15,0.95) | (0.75,0.35) | |

${A}_{4}$ | (0.65,0.45) | (0.75,0.35) | (0.35,0.75) | (0.65,0.45) | (0.65,0.45) | (0.75,0.35) | |

${D}_{3}$ | ${A}_{1}$ | (0.45,0.65) | (0.65,0.45) | (0.65,0.45) | (0.25,0.85) | (0.85,0.25) | (0.55,0.55) |

${A}_{2}$ | (0.25,0.85) | (0.35,0.75) | (0.35,0.75) | (0.65,0.45) | (0.65,0.45) | (0.55,0.55) | |

${A}_{3}$ | (0.65,0.45) | (0.65,0.45) | (0.45,0.65) | (0.45,0.65) | (0.15,0.95) | (0.65,0.45) | |

${A}_{4}$ | (0.55,0.55) | (0.85,0.25) | (0.15,0.95) | (0.55,0.55) | (0.55,0.55) | (0.45,0.65) |

Alternatives | ${\mathit{C}}_{\mathbf{1}}$ | ${\mathit{C}}_{\mathbf{2}}$ | ${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | ${\mathit{C}}_{\mathbf{5}}$ | ${\mathit{C}}_{\mathbf{6}}$ |
---|---|---|---|---|---|---|

${A}_{1}$ | (0.3331,0.8082) | (0.6512,0.4677) | (0.6660,0.4425) | (0.2747,0.8461) | (0.7904,0.3133) | (0.5221,0.5811) |

${A}_{2}$ | (0.3692,0.7456) | (0.4985,0.6341) | (0.3893,0.7155) | (0.6888,0.4143) | (0.5883,0.5149) | (0.4665,0.6449) |

${A}_{3}$ | (0.7225,0.3801) | (0.7690,0.3396) | (0.5221,0.5811) | (0.3893,0.7155) | (0.6271,0.6421) | (0.7225,0.3801) |

${A}_{4}$ | (0.6660,0.4425) | (0.7415,0.3869) | (0.3116,0.8097) | (0.6220,0.4807) | (0.6220,0.4807) | (0.6926,0.4264) |

Examples | ${\mathit{C}}_{\mathbf{1}}$ | ${\mathit{C}}_{\mathbf{2}}$ | ${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | ${\mathit{C}}_{\mathbf{5}}$ | ${\mathit{C}}_{\mathbf{6}}$ |
---|---|---|---|---|---|---|

Example 0 | 0.191 | 0.125 | 0.140 | 0.156 | 0.230 | 0.158 |

Example 1 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |

Example 2 | 0.750 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 |

Example 3 | 0.050 | 0.750 | 0.050 | 0.050 | 0.050 | 0.050 |

Example 4 | 0.050 | 0.050 | 0.750 | 0.050 | 0.050 | 0.050 |

Example 5 | 0.050 | 0.050 | 0.050 | 0.750 | 0.050 | 0.050 |

Example 6 | 0.050 | 0.050 | 0.050 | 0.050 | 0.750 | 0.050 |

Example 7 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.750 |

Methods | Average of Spearman’s Rank Correlation Coefficients |
---|---|

The proposed approach | 0.9429 |

IF-TOPSIS method | 0.9429 |

Fuzzy TODIM method | 0.9143 |

© 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**

Wang, R.; Li, Y.
A Novel Approach for Green Supplier Selection under a q-Rung Orthopair Fuzzy Environment. *Symmetry* **2018**, *10*, 687.
https://doi.org/10.3390/sym10120687

**AMA Style**

Wang R, Li Y.
A Novel Approach for Green Supplier Selection under a q-Rung Orthopair Fuzzy Environment. *Symmetry*. 2018; 10(12):687.
https://doi.org/10.3390/sym10120687

**Chicago/Turabian Style**

Wang, Rui, and Yanlai Li.
2018. "A Novel Approach for Green Supplier Selection under a q-Rung Orthopair Fuzzy Environment" *Symmetry* 10, no. 12: 687.
https://doi.org/10.3390/sym10120687