# An Extended VIKOR Method Using Intuitionistic Fuzzy Sets and Combination Weights for Supplier Selection

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^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Individual Methods

#### 2.2. Hybrid Methods

## 3. Preliminaries

#### 3.1. Intuitionistic Fuzzy Set Theory

**Definition**

**1.**

**Definition**

**2.**

- (1)
- ${\alpha}_{1}+{\alpha}_{2}=\left({\mu}_{{\alpha}_{1}}+{\mu}_{{\alpha}_{2}}-{\mu}_{{\alpha}_{1}}{\mu}_{{\alpha}_{2}},{v}_{{\alpha}_{1}}{v}_{{\alpha}_{2}}\right),$
- (2)
- ${\alpha}_{1}\times {\alpha}_{2}=\left({\mu}_{{\alpha}_{1}}{\mu}_{{\alpha}_{2}},{v}_{{\alpha}_{1}}+{v}_{{\alpha}_{2}}-{v}_{{\alpha}_{1}}{v}_{{\alpha}_{2}}\right),$
- (3)
- $\lambda \alpha =\left(1-{\left(1-{\mu}_{\alpha}\right)}^{\lambda},{v}_{\alpha}^{\lambda}\right),\lambda >0,$
- (4)
- ${\alpha}^{\lambda}=\left({\mu}_{\alpha}^{\lambda},1-{\left(1-{v}_{\alpha}\right)}^{\lambda}\right),\lambda >0.$

**Definition**

**3.**

- (1)
- If $S\left({\alpha}_{1}\right)<S\left({\alpha}_{2}\right)$, then ${\alpha}_{1}<{\alpha}_{2}$;
- (2)
- If $S\left({\alpha}_{1}\right)=S\left({\alpha}_{2}\right)$, and
- If $H\left({\alpha}_{1}\right)<H\left({\alpha}_{2}\right)$, then ${\alpha}_{1}<{\alpha}_{2}$;
- If $H\left({\alpha}_{1}\right)=H\left({\alpha}_{2}\right)$, then ${\alpha}_{1}={\alpha}_{2}$.

**Definition**

**4.**

**Definition**

**5.**

#### 3.2. Objective Weighting Method

**Step 1:**Establish the intuitionistic fuzzy decision matrix.

_{i}(i = 1, 2,..., m) to be performed over n criteria C

_{j}(j = 1, 2,..., n). The intuitionistic fuzzy decision matrix R is constructed as:

**Step 2:**Calculate the IFE values.

_{j}for each criterion:

**Step 3:**Obtain the objective weights of criteria by:

## 4. The Proposed Supplier Selection Method

**Step 1:**Aggregate the decision-makers’ individual assessments.

_{k}on the assessment of A

_{i}with respect to C

_{j}. Then, the aggregated intuitionistic fuzzy ratings (${r}_{ij}$) of alternatives with regard to each criterion can be acquired through the SIFWA operator as:

**Step 2:**Compute the subjective weights of criteria.

_{j}is provided as ${w}_{j}^{k}=\left({\mu}_{j}^{k},{v}_{j}^{k}\right)$ by the decision-maker DM

_{k}. Then, the collective intuitionistic fuzzy weights (${w}_{j}$) of the criteria are computed using the SIFWA operator, as:

**Step 3:**Calculate the objective weights of the criteria.

**Step 4:**Determine the intuitionistic fuzzy positive ideal solution ${f}_{j}^{*}=\left({\mu}_{j}^{*},{v}_{j}^{*}\right)$ and the intuitionistic fuzzy negative ideal solution ${f}_{j}^{-}=\left({\mu}_{j}^{-},{v}_{j}^{-}\right)$ of all criteria ratings, j = 1, 2,..., n.

**Step 5:**Determine the normalized intuitionistic fuzzy differences ${\overline{d}}_{ij}$, i = 1, 2,..., m, j = 1, 2,..., n.

**Step 6:**Obtain the values S

_{i}and R

_{i}, i = 1, 2,..., m, by using the formulas:

**Step 7:**Determine the values Q

_{i}, i = 1, 2,..., m, with Equation (18).

**Step 8:**Rank the alternative suppliers according to the values of S, R, and Q in increasing order. The results are three ranking lists.

**Step 9:**Propose a compromise solution, the alternative (A(1)), which is the best ranked by the measure Q (minimum) if the following two conditions are satisfied:

**C1.**Acceptable advantage: $Q\left({A}^{(2)}\right)-Q\left({A}^{(1)}\right)\ge 1/\left(m-1\right),$ where ${A}^{(2)}$ is the alternative with second position in the ranking list by Q.

**C2.**Acceptable stability: The alternative A

^{(1)}must also be in the first place by S or/and R. This compromise solution is stable within a decision-making process, which could be: “voting by majority rule” (when v >0.5 is needed), or “by consensus” $v\approx 0.5$, or “with veto” (v <0.5).

- Alternatives A
^{(1)}and A^{(2)}if only condition**C2**is not satisfied; or - Alternatives A
^{(1)}, A^{(2)},..., A^{(M)}if condition**C1**is not satisfied; A^{(M)}is calculated by the equation $Q\left({A}^{\left(M\right)}\right)-Q\left({A}^{\left(1\right)}\right)<1/\left(m-1\right)$ for maximum M.

## 5. Illustrative Examples

#### 5.1. Supplier Selection for a General Hosptial

_{1}, A

_{2}, A

_{3}, and A

_{4}have been determined as alternatives for further assessment. To perform the evaluation, a group composed of four decision-makers DM

_{1}, DM

_{2}, DM

_{3}, and DM

_{4}has been established. Five evaluation criteria for the software are considered, which include Functionality (C

_{1}), Reliability (C

_{2}), Usability (C

_{3}), Maintainability (C

_{4}), and Price (C

_{5}).

_{1}= 0.15, λ

_{2}= 0.20, λ

_{3}= 0.30, and λ

_{4}= 0.3 owing to their different domain knowledge backgrounds and expertise.

**Step 1:**After quantifying the linguistic evaluations by corresponding IFNs, the collective intuitionistic fuzzy decision matrix can be created using the SIFWA operator as given in Equation (8). The results are shown in Table 5.

**Step 2:**The assessments of decision-makers on criteria weights are fused by Equation (9) as listed in the last row of Table 5. Then, using Equation (10), the normalized subjective weights of criteria are obtained as displayed in Table 6.

**Step 3:**Based on the objective weighting method, the IFE value of each criterion is obtained by Equation (6) and the objective criteria weights are calculated based on Equation (7). The results of these calculations are displayed in Table 7.

**Step 4:**Functionality, reliability, usability, and maintainability are benefit criteria, and price is a cost criterion. Hence, we can determine the intuitionistic fuzzy positive ideal solution and the intuitionistic fuzzy negative ideal solution of all criteria ratings as seen below:

**Step 5:**The normalized intuitionistic fuzzy differences are calculated by applying Equation (13) and outlined in Table 6.

**Step 6:**The values of S, R, and Q are calculated by Equations (16)–(18) for the four alternatives and summarized in Table 8.

**Step 7:**The rankings of the four alternatives by the S, R, and Q values in increasing order are presented in Table 9.

**Step 8:**Based on Table 9, the ranking of the four alternatives is ${A}_{2}\succ {A}_{1}\succ {A}_{3}\succ {A}_{4}$ in accordance with the values of Q. Thus, A

_{2}is the most suitable company among the alternatives to provide the required software for this hospital.

#### 5.2. Supplier Selection for a Car Manufacturer

_{1}, A

_{2}, …, A

_{5}, are identified for the analysis. For assessing the suppliers, five decision-makers, i.e., DM

_{1}, DM

_{2}, …, DM

_{5}, from different departments are invited. The following criteria have been considered in the supplier evaluation and selection: Quality (C

_{1}), Reliability (C

_{2}), Functionality (C

_{3}), Customer satisfaction (C

_{4}), and Cost (C

_{5}). By using the seven-member linguistic term set in Table 1, the assessments of the alternative suppliers given by the decision-makers are shown in Table 10. Similarly, the decision-makers are asked to use another seven-member linguistic term set (Table 2) to rate the importance weights against individual criteria. The linguistic assessments regarding the criteria weights are given in Table 11. Note that λ

_{1}= λ

_{2}= ... =λ

_{5}= 0.2 in this case, since the same weights are allocated to the five decision-makers.

_{2}is the most suitable supplier. The above supplier selection problem was also solved by the fuzzy VIKOR [18] and the fuzzy TOPSIS [21] methods. The ranking results of the candidate suppliers as derived via the application of these methods and the proposed IFH-VIKOR method are shown in Figure 1.

_{2}, using the proposed approach and the fuzzy VIKOR method. Nevertheless, according to the fuzzy TOPSIS method, A

_{4}has a higher priority as compared to A

_{2}, and is the best option for the considered supplier selection case. The ranking orders of the other alternatives (A

_{1}, A

_{3}, A

_{5},) obtained by the proposed IFH-VIKOR method are different from those produced by the fuzzy VIKOR and the fuzzy TOPSIS approaches. The main reasons that brought about the inconsistencies are as follows: (1) fuzzy set theory is used by the two compared methods to handle the ambiguity information that arises in the supplier selection process. However, there is no means to incorporate the hesitation or uncertainty in the fuzzy set. In contrast, the theory of IFSs adopted in this study is helpful for addressing the uncertainty of supplier evaluation and for quantifying the ambiguous nature of subjective assessments in a convenient way; (2) only subjective weights of criteria are taken into account in the fuzzy VIKOR and the fuzzy TOPSIS methods. In the proposed IFH-VIKOR approach, both subjective and objective criteria weights are considered in the prioritization of alternative suppliers, which makes the method here proposed more realistic and more flexible; and (3) the ranking lists determined by using the proposed model and the fuzzy TOPSIS method are greatly different. This is mainly because the aggregation approaches employed in the two approaches are dissimilar. The IFH-VIKOR approach is based on an aggregating function which represents the distance from the ideal solution. The fuzzy TOPSIS method, in contrast, is based on the idea that the optimum alternative should have the shortest distance from the positive ideal solution and the farthest from the negative ideal solution.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Linguistic Terms | IFNs |
---|---|

Very Poor (VP) | (0.10, 0.90) |

Poor (P) | (0.20, 0.65) |

Moderately Poor (MP) | (0.35, 0.55) |

Fair (F) | (0.50, 0.50) |

Moderately Good (MG) | (0.65, 0.25) |

Good (G) | (0.80, 0.05) |

Very Good (VG) | (0.90, 0.10) |

Linguistic Terms | IFNs |
---|---|

Very low (VL) | (0.15, 0.80) |

Low (L) | (0.25, 0.65) |

Medium Low (ML) | (0.40, 0.50) |

Medium (M) | (0.50, 0.50) |

Medium High (MH) | (0.60, 0.30) |

High (H) | (0.75, 0.15) |

Very High (VH) | (0.90, 0.05) |

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

A_{1} | A_{2} | A_{3} | A_{4} | ||

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

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

DM_{3} | F | MG | F | F | |

DM_{4} | F | G | MG | F | |

C_{2} | DM_{1} | F | MG | MG | MG |

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

DM_{3} | MG | MG | MG | MG | |

DM_{4} | MG | MG | G | F | |

C_{3} | DM_{1} | F | F | F | MG |

DM_{2} | F | MG | F | F | |

DM_{3} | MP | MG | MP | MG | |

DM_{4} | F | F | MP | F | |

C_{4} | DM_{1} | MG | VG | MG | F |

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

DM_{3} | MG | G | G | MG | |

DM_{4} | F | VG | MG | MP | |

C_{5} | DM_{1} | G | MG | G | MG |

DM_{2} | G | G | MG | MG | |

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

DM_{4} | MG | MG | G | F |

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

DM_{1} | DM_{2} | DM_{3} | DM_{4} | |

C_{1} | H | MH | MH | H |

C_{2} | H | H | H | VH |

C_{3} | VH | H | VH | VH |

C_{4} | M | M | MH | MH |

C_{5} | ML | M | ML | M |

Alternatives | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
---|---|---|---|---|---|

A_{1} | (0.446,0.518) | (0.629,0.282) | (0.454,0.515) | (0.569,0.379) | (0.708,0.149) |

A_{2} | (0.761,0.084) | (0.650,0.250) | (0.577,0.366) | (0.857,0.071) | (0.732,0.117) |

A_{3} | (0.577,0.366) | (0.739,0.108) | (0.401,0.533) | (0.762,0.133) | (0.814,0.087) |

A_{4} | (0.500,0.500) | (0.569,0.379) | (0.569,0.379) | (0.492,0.436) | (0.554,0.405) |

${w}_{j}$ | (0.680,0.216) | (0.815,0.104) | (0.878,0.063) | (0.566,0.366) | (0.455,0.500) |

Alternatives | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
---|---|---|---|---|---|

A_{1} | 1.000 | 0.642 | 0.822 | 0.830 | 0.760 |

A_{2} | 0.000 | 0.524 | 0.000 | 0.000 | 0.859 |

A_{3} | 0.639 | 0.000 | 1.000 | 0.238 | 1.000 |

A_{4} | 0.933 | 1.000 | 0.069 | 1.000 | 0.000 |

${w}_{j}^{S}$ | 0.207 | 0.242 | 0.255 | 0.166 | 0.130 |

Weights | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
---|---|---|---|---|---|

${E}_{j}^{}$ | 0.816 | 0.742 | 0.929 | 0.688 | 0.604 |

${w}_{j}^{O}$ | 0.151 | 0.211 | 0.058 | 0.256 | 0.324 |

Indexes | A_{1} | A_{2} | A_{3} | A_{4} |
---|---|---|---|---|

S | 0.801 | 0.314 | 0.548 | 0.616 |

R | 0.179 | 0.195 | 0.227 | 0.227 |

Q | 0.500 | 0.166 | 0.740 | 0.810 |

Indexes | A_{1} | A_{2} | A_{3} | A_{4} |
---|---|---|---|---|

By S | 4 | 1 | 2 | 3 |

By R | 1 | 2 | 3 | 3 |

By Q | 2 | 1 | 3 | 4 |

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

DM_{1} | DM_{2} | DM_{3} | DM_{4} | DM_{5} | ||

A_{1} | C_{1} | MG | F | G | MG | VG |

C_{2} | F | G | MG | F | G | |

C_{3} | F | G | G | G | F | |

C_{4} | F | G | G | G | G | |

C_{5} | G | MG | F | VG | MG | |

A_{2} | C_{1} | VG | VG | G | G | G |

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

C_{3} | G | VG | MG | VG | VG | |

C_{4} | MG | G | MG | G | VG | |

C_{5} | F | VG | F | MP | VG | |

A_{3} | C_{1} | G | MG | MG | MG | G |

C_{2} | VG | MG | MG | MG | MG | |

C_{3} | G | MP | MG | MP | G | |

C_{4} | VG | G | MG | VG | VG | |

C_{5} | F | G | G | MP | MP | |

A_{4} | C_{1} | G | MP | F | F | MP |

C_{2} | G | G | VG | G | VG | |

C_{3} | VG | VG | VG | G | G | |

C_{4} | VG | G | VG | VG | VG | |

C_{5} | VG | MG | G | G | G | |

A_{5} | C_{1} | G | G | VG | VG | G |

C_{2} | MG | VG | MG | VG | MG | |

C_{3} | MG | VG | MG | G | VG | |

C_{4} | G | G | F | MG | MG | |

C_{5} | G | G | MG | VG | MG |

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

DM_{1} | DM_{2} | DM_{3} | DM_{4} | DM_{5} | |

C_{1} | H | H | M | H | H |

C_{2} | VH | VH | VH | H | H |

C_{3} | H | H | MH | H | MH |

C_{4} | M | VH | H | H | H |

C_{5} | VH | H | VH | H | H |

Indexes | A_{1} | A_{2} | A_{3} | A_{4} | A_{5} |
---|---|---|---|---|---|

S | 0.643 | 0.326 | 0.433 | 0.375 | 0.556 |

R | 0.202 | 0.119 | 0.199 | 0.192 | 0.224 |

Q | 0.895 | 0.000 | 0.551 | 0.424 | 0.862 |

© 2017 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**

Zhao, J.; You, X.-Y.; Liu, H.-C.; Wu, S.-M.
An Extended VIKOR Method Using Intuitionistic Fuzzy Sets and Combination Weights for Supplier Selection. *Symmetry* **2017**, *9*, 169.
https://doi.org/10.3390/sym9090169

**AMA Style**

Zhao J, You X-Y, Liu H-C, Wu S-M.
An Extended VIKOR Method Using Intuitionistic Fuzzy Sets and Combination Weights for Supplier Selection. *Symmetry*. 2017; 9(9):169.
https://doi.org/10.3390/sym9090169

**Chicago/Turabian Style**

Zhao, Jiansen, Xiao-Yue You, Hu-Chen Liu, and Song-Man Wu.
2017. "An Extended VIKOR Method Using Intuitionistic Fuzzy Sets and Combination Weights for Supplier Selection" *Symmetry* 9, no. 9: 169.
https://doi.org/10.3390/sym9090169