# An Innovative Grey Approach for Group Multi-Criteria Decision Analysis Based on the Median of Ratings by Using Python

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

_{(λ)}of interval grey number$\otimes x$is defined as

## 3. The Newly Proposed Approach

#### Example

_{l}= 2.00 and the mean value of a number greater or equal to the median (3, 5, 3, 3, 4, 5) is x

_{u}= 3.83.

## 4. A Numerical Illustration

_{1}, Structure and navigability—C

_{2}, Content—C

_{3}, Innovation—C

_{4}, Personalization—C

_{5}.

_{i}= (0.25, 0.24, 0.22, 0.20, 0.10), and the data from Table 7, are shown in Table 9.

## 5. Analysis and Discussion

_{i}= (0.2, 0.2, 0.2, 0.2, 0.2) is used in this evaluation.

_{4}is best ranked according to all methods, with all crisp methods gave the same order of ranking A

_{4,}A

_{1}, A

_{2}, A

_{5}, A

_{3}, while the proposed grey approach gave the following rankings order A

_{4,}A

_{1}, A

_{2}, A

_{3}, A

_{5}. However, from Table 20 it is observable that there are very small differences in overall performance between the second-placed, third-placed and fourth-placed alternatives, which is why it can be expected that different ranking orders of alternatives could be obtained by using another weighting vector.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**Difference between the mean value of the sequence of numbers and the value obtained by the proposed approach.

Sample | Mean | Median | x_{l} | x_{u} | x_{m} = (x_{u} − x_{l})/2 | d = abs(Mean − x_{m}) | d (%) |
---|---|---|---|---|---|---|---|

5 | 5.60 | 5.00 | 4.00 | 7.00 | 5.50 | 0.10 | 1.79 |

10 | 7.10 | 7.50 | 5.40 | 8.80 | 7.10 | 0.00 | 0.00 |

15 | 5.13 | 6.00 | 2.88 | 7.50 | 5.19 | 0.05 | 1.06 |

20 | 6.50 | 7.00 | 5.08 | 8.00 | 6.54 | 0.04 | 0.64 |

25 | 4.52 | 4.00 | 2.43 | 6.50 | 4.46 | 0.06 | 1.23 |

50 | 5.20 | 5.00 | 3.07 | 7.14 | 5.11 | 0.09 | 1.81 |

100 | 5.01 | 5.00 | 2.93 | 7.09 | 5.01 | 0.00 | 0.02 |

150 | 5.16 | 5.00 | 3.05 | 6.91 | 4.98 | 0.18 | 3.45 |

**Table 2.**Ratings of alternative A

_{1}in relation to the evaluation criteria obtained from 10 respondents.

A_{1} | I | II | III | IV | V | VI | VII | VIII | IX | X |
---|---|---|---|---|---|---|---|---|---|---|

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

C_{2} | 3 | 3 | 4 | 5 | 3 | 4 | 4 | 4 | 2 | 4 |

C_{3} | 3 | 3 | 4 | 3 | 3 | 4 | 4 | 5 | 5 | 4 |

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

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

**Table 3.**Ratings of alternative A

_{2}in relation to the evaluation criteria obtained from 10 respondents.

A_{2} | I | II | III | IV | V | VI | VII | VIII | IX | X |
---|---|---|---|---|---|---|---|---|---|---|

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

C_{2} | 5 | 5 | 4 | 5 | 4 | 5 | 5 | 4 | 3 | 4 |

C_{3} | 2 | 4 | 4 | 4 | 3 | 4 | 5 | 3 | 3 | 3 |

C_{4} | 4 | 4 | 5 | 4 | 2 | 2 | 5 | 3 | 5 | 3 |

C_{5} | 4 | 4 | 5 | 3 | 4 | 3 | 4 | 3 | 2 | 4 |

**Table 4.**Ratings of alternative A

_{3}in relation to the evaluation criteria obtained from 10 respondents.

A_{3} | I | II | III | IV | V | VI | VII | VIII | IX | X |
---|---|---|---|---|---|---|---|---|---|---|

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

C_{2} | 3 | 5 | 5 | 4 | 2 | 2 | 3 | 4 | 4 | 4 |

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

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

C_{5} | 3 | 5 | 5 | 3 | 4 | 4 | 3 | 4 | 4 | 4 |

**Table 5.**Ratings of alternative A

_{4}in relation to the evaluation criteria obtained from 10 respondents.

A_{4} | I | II | III | IV | V | VI | VII | VIII | IX | X |
---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 4 | 4 | 4 | 5 | 4 | 4 | 4 | 4 | 5 | 4 |

C_{2} | 5 | 5 | 4 | 5 | 3 | 3 | 4 | 5 | 5 | 3 |

C_{3} | 5 | 5 | 4 | 4 | 3 | 3 | 4 | 5 | 5 | 5 |

C_{4} | 4 | 4 | 5 | 5 | 5 | 3 | 5 | 5 | 4 | 3 |

C_{5} | 4 | 4 | 5 | 3 | 4 | 4 | 3 | 5 | 4 | 4 |

**Table 6.**Ratings of alternative A

_{5}in relation to the evaluation criteria obtained from 10 respondents.

A_{4} | I | II | III | IV | V | VI | VII | VIII | IX | X |
---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 4 | 3 | 5 | 5 | 5 | 3 | 5 | 3 | 4 | 3 |

C_{2} | 4 | 4 | 4 | 5 | 4 | 3 | 5 | 4 | 4 | 4 |

C_{3} | 5 | 4 | 4 | 4 | 3 | 3 | 4 | 4 | 3 | 5 |

C_{4} | 4 | 4 | 5 | 3 | 2 | 4 | 3 | 5 | 3 | 3 |

C_{5} | 3 | 4 | 4 | 4 | 4 | 3 | 4 | 5 | 4 | 4 |

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

A_{1} | 2.90 | 3.60 | 3.80 | 2.70 | 2.20 |

A_{2} | 3.60 | 4.40 | 3.50 | 3.70 | 3.60 |

A_{3} | 1.90 | 3.60 | 2.60 | 1.80 | 3.90 |

A_{4} | 4.20 | 4.20 | 4.30 | 4.30 | 4.00 |

A_{5} | 4.00 | 4.10 | 3.90 | 3.60 | 3.90 |

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

A_{1} | 2.96 | 3.81 | 3.92 | 2.70 | 2.11 |

A_{2} | 3.79 | 4.40 | 3.50 | 3.82 | 3.81 |

A_{3} | 1.95 | 3.79 | 2.31 | 1.40 | 3.96 |

A_{4} | 4.10 | 4.20 | 4.30 | 4.30 | 4.00 |

A_{5} | 4.00 | 4.05 | 3.96 | 3.60 | 3.95 |

ARAS | WASPAS | CoCoSo | WISP | |||||
---|---|---|---|---|---|---|---|---|

Alternatives | S_{i} | Rank | S_{i} | Rank | S_{i} | Rank | S_{i} | Rank |

A_{1} | 0.73 | 4 | 0.73 | 4 | 1.80 | 4 | 0.87 | 4 |

A_{2} | 0.88 | 3 | 0.88 | 3 | 2.17 | 3 | 0.95 | 3 |

A_{3} | 0.61 | 5 | 0.60 | 5 | 1.49 | 5 | 0.81 | 5 |

A_{4} | 0.99 | 1 | 0.99 | 1 | 2.43 | 1 | 1.00 | 1 |

A_{5} | 0.92 | 2 | 0.92 | 2 | 2.25 | 2 | 0.96 | 2 |

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

A_{1} | [2.50, 3.43] | [3.44, 4.17] | [3.50, 4.33] | [1.60, 3.80] | [1.71, 2.50] |

A_{2} | [3.25, 4.33] | [3.80, 5.00] | [2.80, 4.20] | [3.14, 4.50] | [3.44, 4.17] |

A_{3} | [1.78, 2.13] | [3.25, 4.33] | [1.83, 2.78] | [1.00, 1.80] | [3.62, 4.29] |

A_{4} | [4.00, 4.20] | [3.40, 5.00] | [3.60, 5.00] | [3.60, 5.00] | [3.75, 4.25] |

A_{5} | [3.33, 4.67] | [3.88, 4.22] | [3.63, 4.29] | [2.80, 4.40] | [3.78, 4.13] |

ARAS | WASPAS | CoCoSo | WISP | |||||
---|---|---|---|---|---|---|---|---|

Alternatives | S_{i} | Rank | S_{i} | Rank | S_{i} | Rank | S_{i} | Rank |

A_{1} | 0.76 | 4 | 0.46 | 4 | 1.90 | 4 | 0.74 | 4 |

A_{2} | 0.91 | 3 | 0.55 | 3 | 2.29 | 3 | 0.89 | 3 |

A_{3} | 0.59 | 5 | 0.36 | 5 | 1.47 | 5 | 0.62 | 5 |

A_{4} | 0.99 | 1 | 0.60 | 1 | 2.48 | 1 | 0.99 | 1 |

A_{5} | 0.92 | 2 | 0.56 | 2 | 2.32 | 2 | 0.92 | 2 |

WS | WP | WASPAS | CoCoSo | WISP | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Alternatives | S_{i} | Rank | P_{i} | Rank | Q_{i} | Rank | K_{i} | Rank | S_{i} | Rank |

A_{1} | 0.83 | 5 | 0.75 | 5 | 0.75 | 5 | 1.77 | 5 | 0.59 | 5 |

A_{2} | 1.00 | 1 | 0.91 | 1 | 0.91 | 1 | 2.16 | 1 | 1.00 | 1 |

A_{3} | 0.98 | 2 | 0.87 | 2 | 0.88 | 2 | 2.09 | 2 | 0.88 | 2 |

A_{4} | 0.84 | 4 | 0.75 | 4 | 0.76 | 4 | 1.79 | 4 | 0.61 | 4 |

A_{5} | 0.89 | 3 | 0.80 | 3 | 0.81 | 3 | 1.91 | 3 | 0.71 | 3 |

**Table 13.**Ranking of alternatives on the basis of 10 virtual respondents and the proposed grey approach.

WS | WP | WASPAS | CoCoSo | WISP | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Alternatives | S_{i} | Rank | P_{i} | Rank | Q_{i} | Rank | K_{i} | Rank | S_{i} | Rank |

A_{1} | 0.82 | 5 | 0.74 | 5 | 0.75 | 5 | 1.78 | 5 | 0.58 | 5 |

A_{2} | 1.00 | 1 | 0.92 | 1 | 0.92 | 1 | 2.18 | 1 | 1.00 | 1 |

A_{3} | 0.95 | 2 | 0.84 | 2 | 0.86 | 2 | 2.04 | 2 | 0.80 | 2 |

A_{4} | 0.83 | 4 | 0.75 | 4 | 0.75 | 4 | 1.79 | 4 | 0.59 | 4 |

A_{5} | 0.88 | 3 | 0.80 | 3 | 0.81 | 3 | 1.91 | 3 | 0.69 | 3 |

WS | WP | WASPAS | CoCoSo | WISP | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Alternatives | S_{i} | Rank | P_{i} | Rank | S_{i} | S_{i} | K_{i} | Rank | S_{i} | Rank |

A_{1} | 0.97 | 3 | 0.94 | 3 | 0.94 | 3 | 1.90 | 3 | 0.91 | 3 |

A_{2} | 0.92 | 4 | 0.89 | 4 | 0.89 | 4 | 1.80 | 4 | 0.78 | 4 |

A_{3} | 1.00 | 1 | 0.97 | 1 | 0.97 | 1 | 1.97 | 1 | 1.00 | 1 |

A_{4} | 0.91 | 5 | 0.88 | 5 | 0.89 | 5 | 1.79 | 5 | 0.77 | 5 |

A_{5} | 0.97 | 2 | 0.94 | 2 | 0.95 | 2 | 1.91 | 2 | 0.91 | 2 |

**Table 15.**Ranking of alternatives on the basis of 50 virtual respondents and the proposed grey approach.

WS | WP | WASPAS | CoCoSo | WISP | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Alternatives | S_{i} | Rank | P_{i} | Rank | Q_{i} | Rank | K_{i} | Rank | S_{i} | Rank |

A_{1} | 0.98 | 2 | 0.94 | 2 | 0.94 | 2 | 1.90 | 2 | 0.94 | 2 |

A_{2} | 0.92 | 5 | 0.89 | 5 | 0.89 | 5 | 1.79 | 5 | 0.80 | 5 |

A_{3} | 1.00 | 1 | 0.96 | 1 | 0.96 | 1 | 1.94 | 1 | 1.00 | 1 |

A_{4} | 0.93 | 4 | 0.89 | 4 | 0.89 | 4 | 1.81 | 4 | 0.82 | 4 |

A_{5} | 0.97 | 3 | 0.93 | 3 | 0.93 | 3 | 1.87 | 3 | 0.90 | 3 |

Crisp | Grey Approach | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Alternatives | WS | WP | WASPAS | CoCoSo | WISP | WS | WP | WASPAS | CoCoSo | WISP |

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

A_{2} | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 |

A_{3} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

A_{4} | 5 | 5 | 5 | 5 | 5 | 4 | 4 | 4 | 4 | 4 |

A_{5} | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 |

Crisp | Grey Approach | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Alternatives | WS | WP | WASPAS | CoCoSo | WISP | WS | WP | WASPAS | CoCoSo | WISP |

A_{1} | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

A_{2} | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 2 |

A_{3} | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 |

A_{4} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

A_{5} | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |

WS | WP | WASPAS | CoCoSo | WISP | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Alternatives | S_{i} | Rank | P_{i} | Rank | S_{i} | S_{i} | K_{i} | Rank | S_{i} | Rank |

A_{1} | 0.993 | 2 | 0.960 | 2 | 0.961 | 2 | 1.840 | 2 | 0.977 | 2 |

A_{2} | 0.975 | 3 | 0.944 | 3 | 0.944 | 3 | 1.808 | 3 | 0.927 | 3 |

A_{3} | 0.971 | 5 | 0.938 | 5 | 0.939 | 5 | 1.799 | 5 | 0.914 | 5 |

A_{4} | 1.000 | 1 | 0.968 | 1 | 0.968 | 1 | 1.854 | 1 | 1.000 | 1 |

A_{5} | 0.973 | 4 | 0.942 | 4 | 0.942 | 4 | 1.805 | 4 | 0.923 | 4 |

WS | WP | WASPAS | CoCoSo | WISP | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Alternatives | S_{i} | Rank | P_{i} | Rank | Q_{i} | Rank | K_{i} | Rank | S_{i} | Rank |

A_{1} | 0.988 | 2 | 0.954 | 2 | 0.955 | 2 | 1.806 | 2 | 0.965 | 2 |

A_{2} | 0.988 | 3 | 0.953 | 3 | 0.954 | 3 | 1.805 | 3 | 0.963 | 3 |

A_{3} | 0.987 | 4 | 0.952 | 4 | 0.953 | 4 | 1.804 | 4 | 0.960 | 4 |

A_{4} | 1.000 | 1 | 0.966 | 1 | 0.966 | 1 | 1.828 | 1 | 1.000 | 1 |

A_{5} | 0.985 | 5 | 0.952 | 5 | 0.952 | 5 | 1.801 | 5 | 0.958 | 5 |

Crisp | Grey Approach | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Alternatives | WS | WP | WASPAS | CoCoSo | WISP | WS | WP | WASPAS | CoCoSo | WISP |

A_{1} | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

A_{2} | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

A_{3} | 5 | 5 | 5 | 5 | 5 | 4 | 4 | 4 | 4 | 4 |

A_{4} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

A_{5} | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 |

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Stanujkić, D.; Karabašević, D.; Popović, G.; Stanimirović, P.S.; Smarandache, F.; Saračević, M.; Ulutaş, A.; Katsikis, V.N. An Innovative Grey Approach for Group Multi-Criteria Decision Analysis Based on the Median of Ratings by Using Python. *Axioms* **2021**, *10*, 124.
https://doi.org/10.3390/axioms10020124

**AMA Style**

Stanujkić D, Karabašević D, Popović G, Stanimirović PS, Smarandache F, Saračević M, Ulutaş A, Katsikis VN. An Innovative Grey Approach for Group Multi-Criteria Decision Analysis Based on the Median of Ratings by Using Python. *Axioms*. 2021; 10(2):124.
https://doi.org/10.3390/axioms10020124

**Chicago/Turabian Style**

Stanujkić, Dragiša, Darjan Karabašević, Gabrijela Popović, Predrag S. Stanimirović, Florentin Smarandache, Muzafer Saračević, Alptekin Ulutaş, and Vasilios N. Katsikis. 2021. "An Innovative Grey Approach for Group Multi-Criteria Decision Analysis Based on the Median of Ratings by Using Python" *Axioms* 10, no. 2: 124.
https://doi.org/10.3390/axioms10020124