# An Extended Picture Fuzzy VIKOR Approach for Sustainable Supplier Management and Its Application in the Beef Industry

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Preliminaries

#### 3.1. Picture Fuzzy Sets

**Definition**

**1.**

**Definition**

**2.**

- (1)
- ${\tilde{p}}_{1}\oplus {\tilde{p}}_{2}=({\mu}_{1}+{\mu}_{2}-{\mu}_{1}{\mu}_{2},{\eta}_{1}{\eta}_{2},{v}_{1}{v}_{2});$
- (2)
- ${\tilde{p}}_{1}\otimes {\tilde{p}}_{2}=\left({\mu}_{1}{\mu}_{2},{\eta}_{1}+{\eta}_{2}-{\eta}_{1}{\eta}_{2},{v}_{1}+{v}_{2}-{v}_{1}{v}_{2}\right);$
- (3)
- $\lambda {\tilde{p}}_{1}=\left(1-{\left(1-{\mu}_{1}\right)}^{\lambda},{\left({\eta}_{1}\right)}^{\lambda},{\left({v}_{1}\right)}^{\lambda}\right),\text{}\lambda 0;$
- (4)
- ${{\tilde{p}}_{1}}^{\lambda}=\left({\left({\mu}_{1}\right)}^{\lambda},1-{\left(1-{\eta}_{1}\right)}^{\lambda},1-{\left(1-{v}_{1}\right)}^{\lambda}\right),\text{}\lambda 0.$

**Definition**

**3.**

**Definition**

**4.**

- (1)
- if $S\left({\tilde{p}}_{1}\right)>S\left({\tilde{p}}_{2}\right)$, then ${\tilde{p}}_{1}>{\tilde{p}}_{2}$;
- (2)
- if $S\left({\tilde{p}}_{1}\right)=S\left({\tilde{p}}_{2}\right)\text{}and\text{}H\left({\tilde{p}}_{1}\right)H\left({\tilde{p}}_{2}\right)$, then ${\tilde{p}}_{1}>{\tilde{p}}_{2}$;
- (3)
- if $S\left({\tilde{p}}_{1}\right)=S\left({\tilde{p}}_{2}\right)\text{}and\text{}H\left({\tilde{p}}_{1}\right)=H\left({\tilde{p}}_{2}\right)$, then ${\tilde{p}}_{1}={\tilde{p}}_{2}$.

**Definition**

**5.**

**Definition**

**6.**

#### 3.2. The OWAD Operator

**Definition**

**7.**

## 4. Picture Fuzzy Distance Operators

#### 4.1. The Picture Fuzzy-Ordered Weighted Standardized Distance (PFOWSD) Operator

**Definition**

**8.**

_{j}, $i=1,2,\dots ,m.$

**Example**

**1.**

#### 4.2. The Picture Fuzzy Euclidean-Ordered Weighted Standardized Distance (PFEOWSD) Operator

**Definition**

**9.**

_{j}, $i=1,2,\dots ,m.$

**Example**

**2.**

#### 4.3. The Hybrid Picture Fuzzy-Ordered Weighted Standardized Distance (HPFOWSD) Operator

**Definition**

**10.**

_{j}with ${w}_{j}\in \left[0,1\right]$ and ${\mathrm{\Sigma}}_{j=1}^{n}{w}_{j}=1$, n is the balancing coefficient, and ${\tilde{r}}_{j}^{*}$ and ${\tilde{r}}_{j}^{-}$ denote the idea value and the nadir value of the jth criterion, respectively. ${f}_{ij}$ is the picture fuzzy value of ith alternative regarding C

_{j}, $i=1,2,\dots ,m.$

**Example**

**3.**

#### 4.4. The Hybrid Picture Fuzzy Euclidean-Ordered Weighted Standardized Distance (HPFEOWSD) Operator

**Definition**

**11.**

_{j}with ${w}_{j}\in \left[0,1\right]$ and ${\mathrm{\Sigma}}_{j=1}^{n}{w}_{j}=1$, n is the balancing coefficient, and ${\tilde{r}}_{j}^{*}$ and ${\tilde{r}}_{j}^{-}$ are the ideal value and the nadir value of the jth criterion, respectively. ${\tilde{r}}_{ij}$ is the picture fuzzy value of ith alternative regarding C

_{j}, $i=1,2,\dots ,m.$

**Example**

**4.**

## 5. The Proposed Approach for Sustainable Supplier Selection

**Step 1.**Obtain the picture fuzzy evaluation matrix ${\tilde{R}}^{k}={\left({\tilde{r}}_{ij}^{k}\right)}_{m\times n}$ of each decision-maker.

**Step 2.**Determine the collective picture fuzzy evaluation matrix $\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{m\times n}$ by using the PFWA operator.

**Step 3.**Find the best ${\tilde{r}}_{j}^{*}$ and the worst ${\tilde{r}}_{j}^{-}$ values of all criteria ratings using Equations (14) and (15), for j = 1, 2, …, n.

**Step 4.**Obtain the values S

_{i}and R

_{i}, i = 1, 2, …, m, by employing the below equations:

**Step 5.**Obtain the values Q

_{i}, i = 1, 2, …, m, by the following equation:

**Step 6.**Based on the ascending orders of the values S, R, and Q, all the alternatives can be acquired. As a result, three ranking lists can be yielded.

**Step 7.**This step is to determine a compromise solution (A

^{(1)}), which is best ranked by the measure Q (minimum) and, at the same time, the following two conditions should be satisfied:

**C1.**- Acceptable advantage: $Q\left({A}^{(2)}\right)-Q\left({A}^{(1)}\right)\ge 1/\left(m-1\right),$ where ${A}^{(2)}$ ranks the second according to the values of Q.
**C2.**- Acceptable stability in decision making: The alternative A
^{(1)}should be the first alternative by S or/and R. The compromise solution is stable in a sustainable supplier selection process, which could be “voting by majority rule” (v > 0.5), or “by consensus” $v\approx 0.5$, or “with veto” (v < 0.5).

- Alternatives A
^{(1)}and A^{(2)}if only C2 is not fulfilled or - Alternatives A
^{(1)}, A^{(2)}, …, A^{(M)}if C1 is not fulfilled; A^{(M)}is determined by the relation $Q\left({A}^{\left(M\right)}\right)-Q\left({A}^{\left(1\right)}\right)<1/\left(m-1\right)$ for maximum M.

## 6. Case Study

#### 6.1. Background Description

_{1}), age of cattle (C

_{2}), diet fed to cattle (C

_{3}), average weight (C

_{4}), traceability (C

_{5}), carbon footprint (C

_{6}), and price (C

_{7}). An expert group consisting of three decision makers (DM

_{1}, DM

_{2}, and DM

_{3}) was established to conduct the performance rating of each supplier. The decision makers’ weights were assumed as ${\lambda}_{1}=0.3,{\lambda}_{2}=0.4,$ and ${\lambda}_{3}=0.3$ since they had different levels of technical knowledge and expertise. According to the linguistic terms defined in Table 1, the evaluation result of the three decision makers for the ten suppliers is listed in Table 2.

#### 6.2. Implementation Results

**Step 1.**The linguistic evaluation data of the decision makers are transformed into PFNs according to Table 1. For example, for DM

_{1}, the obtained picture fuzzy evaluation matrix ${\tilde{R}}^{1}={\left({\tilde{r}}_{ij}^{1}\right)}_{10\times 7}$ is demonstrated as in Table 3.

**Step 2.**The collective picture fuzzy evaluation matrix $\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{m\times n}$ of the suppliers is obtained by Equation (11). Table 4 displays the aggregated picture fuzzy evaluations of the alternatives.

**Step 3.**Based on the matrix $\tilde{R}$ and Equations (12) and (13), the best ${\tilde{r}}_{j}^{*}$ and the worst ${\tilde{r}}_{j}^{-}$ values for the seven criteria are determined as listed in Table 4.

**Steps 4 and 5.**Now we need to determine the values S

_{i}, R

_{i}, and Q

_{i}for the ten alternative suppliers with Equations (14)–(16).

_{i}index. In the case study, we considered the PFNHSD, the PFWHSD, the PFOWSD, the PFHOWSD, the PFNESD, the PFWESD, the PFEOWSD, and the HPFEOWSD operators. The following weighting vectors were selected: $\omega =\left(0.15,0.15,0.10,0.10,0.15,0.15,0.20\right)$ and $w=\left(0.170,0.103,0.129,0.116,0.140,0.171,0.171\right)$. The results of these calculations are shown in Table 5 and Table 6.

**Steps 6 and 7.**All the alternative suppliers are ranked in ascending order based on the values of S

_{i}, R

_{i}, and Q

_{i}. The lowest value of Q

_{i}in each method is the optimal result. As a result, the ranking results are displayed in Table 7.

_{10}because it had the lowest distance to the ideal alternative. For other circumstances, we observed that the optimal choice was A

_{6}. However, according to the related rules in Section 5, both A

_{10}and A

_{6}were compromise solutions since only the acceptability condition C1 was valid. Besides, by using the dominance theory [54], an aggregated ranking of the alternatives can be acquired based on the considered picture fuzzy distance operators (i.e., ${A}_{10}\sim {A}_{6}\succ {A}_{8}\succ {A}_{3}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}\sim {A}_{2}\succ {A}_{7}\succ {A}_{9}$). Therefore, in the given application, the focal abattoir and processor company can select A

_{10}or A

_{6}as the best beef cattle supplier for procurement.

#### 6.3. Comparative Analysis

_{10}was rated first among the four methods. Especially, the top two and the last two suppliers determined by the proposed model agreed with the ones using the fuzzy TOPSIS and the IF-VIKOR methods. This confirms the efficiency of the proposed PFOWD-VIKOR method.

_{8}ranked third and was better than A

_{4}when using the proposed approach. However, the third highest ranking supplier was A

_{4}and A

_{8}had a lower priority when the fuzzy TOPSIS and the IF-VIKOR methods were used. According to the IF-GRA method, A

_{3}was ranked behind A

_{5}. In reality, the former was more important, and, thus, the result of the proposed approach suggested that A

_{3}had a higher priority in comparison with A

_{5}. This was also validated by the fuzzy TOPSIS and the IF-VIKOR methods. Besides, when balancing all the criteria the two suppliers A

_{1}and A

_{2}should be given the same priority for selection, as derived by the proposed approach. But the other three methods produced the opposite results: A

_{1}was better than A

_{2}by the fuzzy TOPSIS and the IF-GRS, and A

_{2}ranked higher than A

_{1}using the IF-VIKOR method. Thus, a more reasonable ranking can be achieved by the use of the PFOWD-VIKOR algorithm.

#### 6.4. Managerial Implications

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Linguistic Terms | Picture Fuzzy Numbers (PFNs) |
---|---|

Very Poor (VP) | <0.10, 0.00, 0.85> |

Poor (P) | <0.25, 0.05, 0.60> |

Moderately Poor (MP) | <0.30, 0.00, 0.60> |

Fair (F) | <0.50, 0.10, 0.40> |

Moderately Good (MG) | <0.60, 0.00, 0.30> |

Good (G) | <0.75, 0.05, 0.10> |

Very good (VG) | <0.90, 0.00, 0.05> |

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

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | A_{6} | A_{7} | A_{8} | A_{9} | A_{10} | ||

C_{1} | DM_{1} | VG | F | P | MG | F | G | MP | P | P | MG |

DM_{2} | VG | MG | MG | MG | MG | G | F | VG | MG | MG | |

DM_{3} | VG | F | G | MG | VG | G | G | VG | F | G | |

C_{2} | DM_{1} | MP | G | P | F | VG | F | MP | G | F | MP |

DM_{2} | MP | G | MP | F | VG | MP | MP | MG | MP | F | |

DM_{3} | MP | G | F | F | VG | P | MP | F | P | MG | |

C_{3} | DM_{1} | VG | F | MG | VG | VG | G | MP | F | MP | MG |

DM_{2} | VG | F | G | G | G | G | F | MG | F | MG | |

DM_{3} | VG | F | VG | MG | G | G | MG | G | MG | MG | |

C_{4} | DM_{1} | G | MG | MG | P | P | G | P | P | MG | VG |

DM_{2} | G | G | MG | P | P | G | MP | MP | G | VG | |

DM_{3} | G | VG | MG | P | MP | G | F | F | VG | VG | |

C_{5} | DM_{1} | G | VG | G | VG | VP | VG | P | P | MG | VG |

DM_{2} | VG | VG | MG | G | P | VG | P | P | MG | VG | |

DM_{3} | VG | VG | F | VG | MP | VG | MP | P | MG | VG | |

C_{6} | DM_{1} | MP | G | MG | VG | G | F | MP | VG | F | VG |

DM_{2} | F | MG | MG | G | G | MG | F | G | F | VG | |

DM_{3} | MG | F | MG | VG | G | G | MG | MG | F | VG | |

C_{7} | DM_{1} | F | F | P | MP | F | F | F | F | F | P |

DM_{2} | F | F | MP | F | F | MP | MP | MP | F | F | |

DM_{3} | F | MP | F | MP | P | F | F | F | F | F |

Alternatives | Criteria | ||||||
---|---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | |

A_{1} | <0.90,0.00,0.05> | <0.30,0.00,0.60> | <0.90,0.00,0.05> | <0.75,0.05,0.10> | <0.75,0.05,0.10> | <0.30,0.00,0.60> | <0.50,0.10,0.40> |

A_{2} | <0.25,0.05,0.60> | <0.75,0.05,0.10> | <0.50,0.10,0.40> | <0.25,0.05,0.60> | <0.25,0.05,0.60> | <0.90,0.00,0.05> | <0.50,0.10,0.40> |

A_{3} | <0.25,0.05,0.60> | <0.25,0.05,0.60> | <0.60,0.00,0.30> | <0.60,0.00,0.30> | <0.75,0.05,0.10> | <0.60,0.00,0.30> | <0.25,0.05,0.60> |

A_{4} | <0.50,0.10,0.40> | <0.75,0.05,0.10> | <0.50,0.10,0.40> | <0.60,0.00,0.30> | <0.90,0.00,0.05> | <0.75,0.05,0.10> | <0.50,0.10,0.40> |

A_{5} | <0.50,0.10,0.40> | <0.90,0.00,0.05> | <0.90,0.00,0.05> | <0.25,0.05,0.60> | <0.10,0.00,0.85> | <0.75,0.05,0.10> | <0.50,0.10,0.40> |

A_{6} | <0.75,0.05,0.10> | <0.50,0.10,0.40> | <0.75,0.05,0.10> | <0.75,0.05,0.10> | <0.90,0.00,0.05> | <0.50,0.10,0.40> | <0.50,0.10,0.40> |

A_{7} | <0.25,0.05,0.60> | <0.50,0.10,0.40> | <0.30,0.00,0.60> | <0.60,0.00,0.30> | <0.60,0.00,0.30> | <0.50,0.10,0.40> | <0.50,0.10,0.40> |

A_{8} | <0.90,0.00,0.05> | <0.50,0.10,0.40> | <0.90,0.00,0.05> | <0.25,0.05,0.60> | <0.90,0.00,0.05> | <0.90,0.00,0.05> | <0.30,0.00,0.60> |

A_{9} | <0.30,0.00,0.60> | <0.30,0.00,0.60> | <0.30,0.00,0.60> | <0.25,0.05,0.60> | <0.25,0.05,0.60> | <0.30,0.00,0.60> | <0.50,0.10,0.40> |

A_{10} | <0.60,0.00,0.30> | <0.30,0.00,0.60> | <0.60,0.00,0.30> | <0.90,0.00,0.05> | <0.90,0.00,0.05> | <0.90,0.00,0.05> | <0.25,0.05,0.60> |

Alternatives | Criteria | ||||||
---|---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | |

A_{1} | <0.9,0,0.05> | <0.3,0,0.6> | <0.9,0,0.05> | <0.75,0.05,0.1> | <0.868,0,0.062> | <0.483,0,0.414> | <0.5,0.1,0.4> |

A_{2} | <0.543,0,0.357> | <0.75,0.05,0.1> | <0.5,0.1,0.4> | <0.781,0,0.113> | <0.9,0,0.05> | <0.629,0,0.235> | <0.447,0,0.452> |

A_{3} | <0.581,0,0.266> | <0.354,0,0.531> | <0.781,0,0.113> | <0.6,0,0.3> | <0.629,0,0.235> | <0.6,0,0.3> | <0.354,0,0.531> |

A_{4} | <0.6,0,0.3> | <0.5,0.1,0.4> | <0.781,0,0.113> | <0.25,0.05,0.6> | <0.9,0,0.05> | <0.9,0,0.05> | <0.388,0,0.51> |

A_{5} | <0.718,0,0.191> | <0.9,0,0.05> | <0.81,0,0.081> | <0.265,0,0.6> | <0.224,0,0.666> | <0.75,0.05,0.1> | <0.435,0.081,0.452> |

A_{6} | <0.75,0.05,0.1> | <0.354,0,0.531> | <0.75,0.05,0.1> | <0.75,0.05,0.1> | <0.9,0,0.05> | <0.629,0,0.235> | <0.428,0,0.47> |

A_{7} | <0.551,0,0.298> | <0.3,0,0.6> | <0.483,0,0.414> | <0.354,0,0.531> | <0.265,0,0.6> | <0.483,0,0.414> | <0.428,0,0.47> |

A_{8} | <0.817,0,0.105> | <0.629,0,0.235> | <0.629,0,0.235> | <0.354,0,0.531> | <0.25,0.05,0.6> | <0.781,0,0.113> | <0.428,0,0.47> |

A_{9} | <0.484,0,0.403> | <0.354,0,0.531> | <0.483,0,0.414> | <0.781,0,0.113> | <0.6,0,0.3> | <0.5,0.1,0.4> | <0.5,0.1,0.4> |

A_{10} | <0.653,0,0.216> | <0.483,0,0.414> | <0.6,0,0.3> | <0.9,0,0.05> | <0.9,0,0.05> | <0.9,0,0.05> | <0.435,0.081,0.452> |

${\tilde{r}}_{j}^{*}$ | <0.9,0,0.05> | <0.9,0,0.05> | <0.9,0,0.05> | <0.9,0,0.05> | <0.9,0,0.05> | <0.781,0,0.113> | <0.354,0,0.531> |

${\tilde{r}}_{j}^{-}$ | <0.551,0,0.298> | <0.354,0,0.531> | <0.483,0,0.414> | <0.354,0,0.531> | <0.224,0,0.666> | <0.483,0,0.414> | <0.5,0.1,0.4> |

Distance Operators | Ranking Indexes | Alternatives | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | A_{6} | A_{7} | A_{8} | A_{9} | A_{10} | ||

Picture fuzzy normalized Hamming standardized distance (PFNHSD) | S | 0.489 | 0.497 | 0.539 | 0.527 | 0.532 | 0.41 | 0.905 | 0.521 | 0.877 | 0.464 |

R | 0.157 | 0.154 | 0.143 | 0.170 | 0.166 | 0.143 | 0.157 | 0.143 | 0.183 | 0.109 | |

Q | 0.401 | 0.390 | 0.357 | 0.528 | 0.506 | 0.227 | 0.822 | 0.339 | 0.971 | 0.054 | |

Picture fuzzy-weighted Hamming standardized distance (PFWHSD) | S | 0.494 | 0.518 | 0.521 | 0.497 | 0.533 | 0.392 | 0.882 | 0.478 | 0.913 | 0.472 |

R | 0.171 | 0.174 | 0.156 | 0.146 | 0.140 | 0.103 | 0.171 | 0.135 | 0.220 | 0.120 | |

Q | 0.389 | 0.425 | 0.349 | 0.285 | 0.294 | 0.000 | 0.762 | 0.218 | 1.000 | 0.151 | |

Picture fuzzy-ordered weighted standardized distance (PFOWSD) | S | 0.449 | 0.477 | 0.507 | 0.493 | 0.50 | 0.391 | 0.866 | 0.490 | 0.832 | 0.418 |

R | 0.165 | 0.162 | 0.150 | 0.179 | 0.174 | 0.15 | 0.165 | 0.150 | 0.193 | 0.115 | |

Q | 0.383 | 0.393 | 0.349 | 0.517 | 0.497 | 0.227 | 0.822 | 0.331 | 0.964 | 0.029 | |

Hybrid Picture Fuzzy-Ordered Weighted Standardized Distance (HPFOWSD) | S | 0.468 | 0.491 | 0.489 | 0.468 | 0.489 | 0.368 | 0.853 | 0.454 | 0.862 | 0.436 |

R | 0.180 | 0.183 | 0.163 | 0.153 | 0.147 | 0.108 | 0.180 | 0.141 | 0.231 | 0.126 | |

Q | 0.392 | 0.428 | 0.347 | 0.286 | 0.281 | 0.000 | 0.782 | 0.222 | 1.000 | 0.143 |

Distance Operators | Ranking Indexes | Alternatives | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | A_{6} | A_{7} | A_{8} | A_{9} | A_{10} | ||

Picture fuzzy-normalized Euclidean standardized distance (PFNESD) | S | 0.685 | 0.625 | 0.629 | 0.664 | 0.669 | 0.498 | 0.940 | 0.626 | 0.95 | 0.560 |

R | 0.415 | 0.408 | 0.378 | 0.450 | 0.439 | 0.378 | 0.415 | 0.378 | 0.485 | 0.307 | |

Q | 0.510 | 0.422 | 0.343 | 0.585 | 0.560 | 0.199 | 0.792 | 0.340 | 1.000 | 0.068 | |

Picture fuzzy-weighted Euclidean standardized distance (PFWESD) | S | 0.69 | 0.640 | 0.616 | 0.632 | 0.654 | 0.468 | 0.922 | 0.592 | 0.98 | 0.560 |

R | 0.414 | 0.422 | 0.377 | 0.405 | 0.396 | 0.321 | 0.414 | 0.360 | 0.531 | 0.292 | |

Q | 0.471 | 0.439 | 0.322 | 0.398 | 0.399 | 0.060 | 0.697 | 0.263 | 1.000 | 0.090 | |

picture fuzzy Euclidean-ordered weighted standardized distance (PFEOWSD) | S | 0.663 | 0.624 | 0.617 | 0.649 | 0.659 | 0.495 | 0.912 | 0.615 | 0.922 | 0.531 |

R | 0.426 | 0.418 | 0.387 | 0.461 | 0.45 | 0.387 | 0.426 | 0.387 | 0.497 | 0.296 | |

Q | 0.518 | 0.454 | 0.370 | 0.591 | 0.576 | 0.227 | 0.810 | 0.367 | 1.000 | 0.043 | |

Hybrid Picture Fuzzy Euclidean-Ordered Weighted Standardized Distance (HPFEOWSD) | S | 0.706 | 0.658 | 0.605 | 0.607 | 0.625 | 0.442 | 0.901 | 0.558 | 0.997 | 0.553 |

R | 0.464 | 0.471 | 0.422 | 0.396 | 0.380 | 0.279 | 0.464 | 0.365 | 0.595 | 0.326 | |

Q | 0.529 | 0.498 | 0.372 | 0.333 | 0.323 | 0.000 | 0.706 | 0.240 | 1.000 | 0.175 |

Distance Operators | Ranking | Distance Operators | Ranking |
---|---|---|---|

PFNHSD | ${A}_{10}\sim {A}_{6}\succ {A}_{8}\succ {A}_{3}\succ {A}_{2}\succ {A}_{1}\succ {A}_{5}\succ {A}_{4}\succ {A}_{7}\succ {A}_{9}$ | PFNESD | ${A}_{10}\sim {A}_{6}\succ {A}_{8}\succ {A}_{3}\succ {A}_{2}\succ {A}_{1}\succ {A}_{5}\succ {A}_{4}\succ {A}_{7}\succ {A}_{9}$ |

PFWHSD | ${A}_{6}\sim {A}_{10}\succ {A}_{8}\succ {A}_{4}\succ {A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{7}\succ {A}_{9}$ | PFWESD | ${A}_{6}\sim {A}_{10}\succ {A}_{8}\succ {A}_{3}\succ {A}_{4}\succ {A}_{5}\succ {A}_{2}\succ {A}_{1}\succ {A}_{7}\succ {A}_{9}$ |

PFOWSD | ${A}_{10}\sim {A}_{6}\succ {A}_{8}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{5}\succ {A}_{4}\succ {A}_{7}\succ {A}_{9}$ | PFEOWSD | ${A}_{10}\sim {A}_{6}\succ {A}_{8}\succ {A}_{3}\succ {A}_{2}\succ {A}_{1}\succ {A}_{5}\succ {A}_{4}\succ {A}_{7}\succ {A}_{9}$ |

HPFOWSD | ${A}_{6}\sim {A}_{10}\succ {A}_{8}\succ {A}_{5}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{7}\succ {A}_{9}$ | HPFEOWSD | ${A}_{6}\sim {A}_{10}\succ {A}_{8}\succ {A}_{5}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{7}\succ {A}_{9}$ |

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## Share and Cite

**MDPI and ACS Style**

Meksavang, P.; Shi, H.; Lin, S.-M.; Liu, H.-C.
An Extended Picture Fuzzy VIKOR Approach for Sustainable Supplier Management and Its Application in the Beef Industry. *Symmetry* **2019**, *11*, 468.
https://doi.org/10.3390/sym11040468

**AMA Style**

Meksavang P, Shi H, Lin S-M, Liu H-C.
An Extended Picture Fuzzy VIKOR Approach for Sustainable Supplier Management and Its Application in the Beef Industry. *Symmetry*. 2019; 11(4):468.
https://doi.org/10.3390/sym11040468

**Chicago/Turabian Style**

Meksavang, Phommaly, Hua Shi, Shu-Min Lin, and Hu-Chen Liu.
2019. "An Extended Picture Fuzzy VIKOR Approach for Sustainable Supplier Management and Its Application in the Beef Industry" *Symmetry* 11, no. 4: 468.
https://doi.org/10.3390/sym11040468