# An Interval-Valued Intuitionistic Fuzzy Model Based on Extended VIKOR and MARCOS for Sustainable Supplier Selection in Organ Transplantation Networks for Healthcare Devices

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## Abstract

**:**

## 1. Introduction

- Developing an IVIF-similarity measure to compute weights of DMs. The similarity measure method is conducted based on the distance from the ideal decision matrix under IVIF conditions that considers membership and non-membership degrees to control vagueness and uncertain conditions.
- Extending an IVIF-Shannon entropy method to obtain the weights of the criteria. This method calculates the criteria weights with respecting entropy measures under IVIF situations.
- Developing an IVIF-E-VIKOR method for ranking the alternatives. In the IVIF-E-VIKOR method, a new indicator is presented for evaluating the alternatives under IVIF conditions by considering membership and non-membership degrees closer to the outcomes of real-world problems.
- Extending an IVIF-MARCOS approach to rank the alternatives. This method obtains the alternative ranking based on the utility function that is computed based on the utility degree by respecting membership and non-membership values to handle uncertain situations.
- Proposing a new hybrid ranking model to concurrently appraise alternatives based on two IVIF-E-VIKOR and IVIF-MARCOS approaches.

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

## 3. Proposed Soft Computing Model

**Step 1.**Constructing the decision matrix $\left({N}_{E}\right)$ based on the experts’ opinions $\left(E\right)$ from Equation (20).

**Step 2.**Calculating the DMs weights with an IVIF-similarity measure.

**Step 3.**Calculating the criteria weights with the Shannon entropy approach.

**Step 4.**Calculating the ranking of alternatives with proposed IVIF-integrating model.

**Step 5.**Obtaining the alternatives weights with IVIF-MARCOS method.

**Step 6.**Computing alternative final ranking based on a new integration method with Equation (40).

## 4. Illustrative Example

_{E}= DM

_{1}, DM

_{2}, DM

_{3}) are utilized to evaluate 17 criteria (Cr

_{j}= Cr

_{1}, Cr

_{2}, …, Cr

_{n}) that are related to three principal criteria, i.e., economic, social, and environmental. The criteria list is determined in Table 2 that is selected based on the related literature [55,56,57,58,59,60].

_{i}= A

_{1}, A

_{2}, A

_{3}, A

_{4}) are described in order to be evaluated and selected in organ transplantation networks. The first alternative (A

_{1}) is related to the foreign companies of medical items that supply the equipment and medical devices. The second alternative (A

_{2}) is regarded as the laboratory and surgery supplier. Moreover, the third alternative (A

_{3}) is a distributor of the surgery equipment. Finally, the fourth alternative (A

_{4}) is related to a supplier of surgery instruments.

#### 4.1. Computational Results

_{e}) is computed from Equation (23), of which the final values are determined in Table 5.

#### 4.2. Comparative Analysis

#### 4.3. Sensitivity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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References | Method Features | |||||
---|---|---|---|---|---|---|

Applying Linguistic Terms | IVIF | DM Weights | Criteria Weights | Hybrid/New Ranking Method | Group Decision Making | |

[20] | * | |||||

[21] | * | * | * | |||

[23] | * | * | * | |||

[25] | * | * | * | * | * | * |

[30] | * | * | * | * | ||

[32] | * | * | * | |||

[33] | * | * | * | |||

[36] | * | * | * | |||

[38] | * | * | * | |||

[43] | * | * | * | * | * | |

This paper | * | * | * | * | * | * |

Segments | Criteria | Definition |
---|---|---|

Economic | C_{1} | Price |

C_{2} | Quality | |

C_{3} | Delivery on time | |

C_{4} | Contributions | |

C_{5} | Management | |

C_{6} | Reliability | |

Social | C_{7} | Credibility |

C_{8} | Safety | |

C_{9} | Information revelation | |

C_{10} | Employee benefits and rights | |

C_{11} | Security acts | |

C_{12} | Education | |

C_{13} | Policies | |

Environmental | C_{14} | Environmental suitability |

C_{15} | Management systems of environment | |

C_{16} | Pollution control | |

C_{17} | Consider the requirements of ISO |

**Table 3.**Linguistic IVIF values [61].

Linguistic Terms | IVIF Values |
---|---|

Absolutely low (AL) | ([0.1,0.25],[0.65,0.75]) |

Very low (VL) | ([0.15,0.30],[0.60,0.70]) |

Low (L) | ([0.20,0.35],[0.55,0.65]) |

Medium low (ML) | ([0.25,0.40],[0.50,0.60]) |

Equal (E) | ([0.45,0.55],[0.30,0.45]) |

Medium high (MH) | ([0.50,0.60],[0.25,0.40]) |

High (H) | ([0.55,0.65],[0.20,0.35]) |

Very high (VH) | ([0.60,0.70],[0.15,0.30]) |

Absolutely high (AH) | ([0.65,0.75],[0.10,0.25]) |

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

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

C_{1} | DM_{1} | E | MH | VH | AH |

DM_{2} | AH | E | AH | MH | |

DM_{3} | MH | ML | VH | E | |

C_{2} | DM_{1} | MH | MH | ML | MH |

DM_{2} | VH | VH | AH | H | |

DM_{3} | MH | MH | ML | ML | |

C_{3} | DM_{1} | ML | E | MH | E |

DM_{2} | MH | VH | MH | H | |

DM_{3} | MH | AH | MH | MH | |

C_{4} | DM_{1} | E | MH | MH | MH |

DM_{2} | AH | MH | AH | MH | |

DM_{3} | VH | MH | MH | H | |

C_{5} | DM_{1} | MH | VH | MH | VH |

DM_{2} | VH | AH | VH | AH | |

DM_{3} | AH | VH | MH | MH | |

C_{6} | DM_{1} | MH | ML | MH | ML |

DM_{2} | E | AH | ML | E | |

DM_{3} | ML | ML | H | ML | |

C_{7} | DM_{1} | MH | MH | MH | VH |

DM_{2} | VH | MH | E | AH | |

DM_{3} | MH | MH | MH | VH | |

C_{8} | DM_{1} | E | MH | VH | MH |

DM_{2} | VH | AH | VH | AH | |

DM_{3} | AH | MH | VH | MH | |

C_{9} | DM_{1} | MH | MH | VH | MH |

DM_{2} | MH | VH | AH | MH | |

DM_{3} | MH | MH | VH | MH | |

C_{10} | DM_{1} | VH | MH | ML | MH |

DM_{2} | AH | ML | AH | AH | |

DM_{3} | VH | H | ML | MH | |

C_{11} | DM_{1} | ML | L | E | MH |

DM_{2} | AH | MH | AH | VH | |

DM_{3} | ML | MH | MH | MH | |

C_{12} | DM_{1} | MH | MH | MH | MH |

DM_{2} | MH | H | VH | ML | |

DM_{3} | MH | VH | MH | H | |

C_{13} | DM_{1} | MH | AH | ML | MH |

DM_{2} | AH | E | MH | E | |

DM_{3} | MH | VH | MH | MH | |

C_{14} | DM_{1} | MH | AH | E | VH |

DM_{2} | VH | VH | AH | VH | |

DM_{3} | MH | H | VH | VH | |

C_{15} | DM_{1} | MH | E | MH | VH |

DM_{2} | ML | MH | VH | AH | |

DM_{3} | H | AH | AH | VH | |

C_{16} | DM_{1} | MH | H | MH | ML |

DM_{2} | E | E | E | AH | |

DM_{3} | MH | MH | E | H | |

C_{17} | DM_{1} | VH | MH | MH | MH |

DM_{2} | VH | H | VH | MH | |

DM_{3} | VH | MH | MH | MH |

DMs | $SM({N}_{E},{N}^{\ast})$ | W_{e} |
---|---|---|

DM_{1} | 0.630 | 0.337 |

DM_{2} | 0.617 | 0.330 |

DM_{3} | 0.623 | 0.333 |

DMs | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | C_{13} | C_{14} | C_{15} | C_{16} | C_{17} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

DM_{1} | 0.429 | 0.464 | 0.471 | 0.467 | 0.436 | 0.464 | 0.450 | 0.453 | 0.450 | 0.450 | 0.462 | 0.464 | 0.444 | 0.429 | 0.453 | 0.458 | 0.450 |

DM_{2} | 0.418 | 0.405 | 0.444 | 0.414 | 0.386 | 0.446 | 0.429 | 0.386 | 0.425 | 0.389 | 0.400 | 0.444 | 0.446 | 0.397 | 0.425 | 0.450 | 0.430 |

DM_{3} | 0.453 | 0.464 | 0.439 | 0.444 | 0.425 | 0.458 | 0.450 | 0.425 | 0.450 | 0.444 | 0.464 | 0.444 | 0.450 | 0.430 | 0.394 | 0.462 | 0.450 |

DMs | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | C_{13} | C_{14} | C_{15} | C_{16} | C_{17} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

DM_{1} | 0.046 | 0.050 | 0.051 | 0.050 | 0.047 | 0.050 | 0.048 | 0.049 | 0.048 | 0.048 | 0.050 | 0.050 | 0.048 | 0.046 | 0.049 | 0.049 | 0.048 |

DM_{2} | 0.042 | 0.041 | 0.045 | 0.042 | 0.039 | 0.045 | 0.043 | 0.039 | 0.043 | 0.039 | 0.041 | 0.045 | 0.045 | 0.040 | 0.043 | 0.046 | 0.044 |

DM_{3} | 0.048 | 0.049 | 0.046 | 0.047 | 0.045 | 0.048 | 0.048 | 0.045 | 0.048 | 0.047 | 0.049 | 0.047 | 0.048 | 0.045 | 0.042 | 0.049 | 0.048 |

Criteria | Weights |
---|---|

${C}_{1}$ | 0.045 |

${C}_{2}$ | 0.127 |

${C}_{3}$ | 0.047 |

${C}_{4}$ | 0.046 |

${C}_{5}$ | 0.044 |

${C}_{6}$ | 0.048 |

${C}_{7}$ | 0.046 |

${C}_{8}$ | 0.044 |

${C}_{9}$ | 0.046 |

${C}_{10}$ | 0.045 |

${C}_{11}$ | 0.046 |

${C}_{12}$ | 0.047 |

${C}_{13}$ | 0.047 |

${C}_{14}$ | 0.044 |

${C}_{15}$ | 0.044 |

${C}_{16}$ | 0.048 |

${C}_{17}$ | 0.046 |

Alternatives | ${S}_{i}$ | ${R}_{i}$ | ${\lambda}^{i}$ | Final Score Values |
---|---|---|---|---|

${A}_{1}$ | ([0.379,0.391], [0.462,0.455]) | ([0.047,0.047], [0.127,0.127]) | ([0.032,0.030], [1.000,0.991]) | 0.030 |

${A}_{2}$ | ([0.369,0.381], [0.458,0.458]) | ([0.046,0.046], [0.127,0.127]) | ([0.010,0.010], [0.987,1.000]) | 0.010 |

${A}_{3}$ | ([0.443,0.460], [0.406,0.399]) | ([0.127,0.127], [0.048,0.048]) | ([0.198,0.192], [0.828,0.824]) | 0.191 |

${A}_{4}$ | ([0.465,0.492], [0.375,0.370]) | ([0.096,0.112], [0.046,0.046]) | ([0.266,0.279], [0.749,0.752]) | 0.266 |

${S}^{+}$ | ${S}^{-}$ | ${R}^{+}$ | ${R}^{-}$ |
---|---|---|---|

([0.369,0.381], [0.375,0.370]) | ([0.465,0.492], [0.462,0.458]) | ([0.046,0.046], [0.046,0.046]) | ([0.127,0.127], [0.127,0.127]) |

Alternatives | ${U}_{i}^{+}$ | ${U}_{i}^{-}$ | $f\left({U}_{i}^{+}\right)$ | $f\left({U}_{i}^{-}\right)$ | $f\left({U}_{i}\right)$ |
---|---|---|---|---|---|

${A}_{1}$ | ([0.588,0.592], [0.226,0.363]) | ([2.222,2.237], [0.855,1.374]) | ([0.791,0.791], [0.791,0.791]) | ([0.209,0.209], [0.209,0.209]) | ([0.557,0.561], [0.214,0.344]) |

${A}_{2}$ | ([0.490,0.590], [0.229,0.366]) | ([1.852,2.228], [0.865,1.384]) | ([0.791,0.791], [0.791,0.791]) | ([0.209,0.209], [0.209,0.209]) | ([0.464,0.559], [0.217,0.347]) |

${A}_{3}$ | ([0.483,0.586], [0.242,0.375]) | ([1.824,2.215], [0.915,1.419]) | ([0.791,0.791], [0.791,0.791]) | ([0.209,0.209], [0.209,0.209]) | ([0.457,0.555], [0.229,0.356]) |

${A}_{4}$ | ([0.484,0.584], [0.235,0.371]) | ([1.824,2.209], [0.889,1.402]) | ([0.791,0.791], [0.791,0.791]) | ([0.209,0.209], [0.209,0.209]) | ([0.458,0.554], [0.223,0.351]) |

Alternatives | Final Score Values | Final Ranking Results |
---|---|---|

${A}_{1}$ | 0.340 | 3 |

${A}_{2}$ | 0.311 | 4 |

${A}_{3}$ | 0.397 | 2 |

${A}_{4}$ | 0.436 | 1 |

Alternatives | IVIF-VIKOR [62] | IVIF-VIKOR Ranking | IVIF-TOPSIS [63] | IVIF-TOPSIS Ranking | IVIF-Hybrid Proposed Model | Proposed Model Ranking |
---|---|---|---|---|---|---|

${A}_{1}$ | 0.052 | 3 | 0.522 | 3 | 0.340 | 3 |

${A}_{2}$ | 0.006 | 4 | 0.496 | 4 | 0.311 | 4 |

${A}_{3}$ | 0.359 | 2 | 0.524 | 2 | 0.397 | 2 |

${A}_{4}$ | 0.461 | 1 | 0.653 | 1 | 0.436 | 1 |

$\Gamma $ | $1-\Gamma $ | Ranking Results |
---|---|---|

0.100 | 0.900 | ${A}_{4}>{A}_{3}>{A}_{1}>{A}_{2}$ |

0.200 | 0.800 | ${A}_{4}>{A}_{3}>{A}_{1}>{A}_{2}$ |

0.300 | 0.700 | ${A}_{4}>{A}_{3}>{A}_{1}>{A}_{2}$ |

0.400 | 0.600 | ${A}_{4}>{A}_{3}>{A}_{1}>{A}_{2}$ |

0.500 | 0.500 | ${A}_{4}>{A}_{3}>{A}_{1}>{A}_{2}$ |

0.600 | 0.400 | ${A}_{4}>{A}_{3}>{A}_{1}>{A}_{2}$ |

0.700 | 0.300 | ${A}_{4}>{A}_{3}>{A}_{1}>{A}_{2}$ |

0.800 | 0.200 | ${A}_{4}>{A}_{1}>{A}_{3}>{A}_{2}$ |

0.900 | 0.100 | ${A}_{1}>{A}_{4}>{A}_{3}>{A}_{2}$ |

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## Share and Cite

**MDPI and ACS Style**

Salimian, S.; Mousavi, S.M.; Antucheviciene, J.
An Interval-Valued Intuitionistic Fuzzy Model Based on Extended VIKOR and MARCOS for Sustainable Supplier Selection in Organ Transplantation Networks for Healthcare Devices. *Sustainability* **2022**, *14*, 3795.
https://doi.org/10.3390/su14073795

**AMA Style**

Salimian S, Mousavi SM, Antucheviciene J.
An Interval-Valued Intuitionistic Fuzzy Model Based on Extended VIKOR and MARCOS for Sustainable Supplier Selection in Organ Transplantation Networks for Healthcare Devices. *Sustainability*. 2022; 14(7):3795.
https://doi.org/10.3390/su14073795

**Chicago/Turabian Style**

Salimian, Sina, Seyed Meysam Mousavi, and Jurgita Antucheviciene.
2022. "An Interval-Valued Intuitionistic Fuzzy Model Based on Extended VIKOR and MARCOS for Sustainable Supplier Selection in Organ Transplantation Networks for Healthcare Devices" *Sustainability* 14, no. 7: 3795.
https://doi.org/10.3390/su14073795