# Fuzzy Analytic Hierarchy Process-Based Multi-Criteria Decision Making for Universities Ranking

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Fuzzy Set**. Let $X$ be the universe set of the set of objects and $x$ be the elements of the universe set $X$. Let y be the subset of $X$ which is used for the membership, and the characteristic function ${\mu}_{y}$ from $X$ to {0, 1} can be described as follows:

**Fuzzy Number. Triangular Fuzzy Number**. Let y be a fuzzy subset of the universe set. Fuzzy number y is a fuzzy subset of real numbers that has important characteristics:

- The membership function ${\mu}_{y}\left(x\right)$ is continuous from R to [0, 1].
- The membership function ${\mu}_{y}\left(x\right)$ is normal, that is, there exists the number ${x}_{0}$ so that ${\mu}_{y}\left({x}_{0}\right)=1$.
- If all of the level sets are convex in classical sense for a fuzzy set y, that means that this fuzzy set y is convex.

**Fuzzy Decision Making.**Fuzzy decision making is used to choose the best alternative among several ones in the presence of uncertainty. A set of alternatives ${A}_{1},{A}_{2},\dots ,{A}_{n}$depends on some criteria ${H}_{1},{H}_{2},\dots ,{H}_{m}$. So, the best alternative is one that fulfills all criteria [24,25].

**Fuzzy Preferences**. Fuzzy preferences are actually based on fuzzy logic and fuzzy sets. In MCDM, fuzzy or uncertain preferences can be written as fuzzy utilities or weighted sums. These fuzzy utilities and fuzzy weighted sums are fuzzy numbers. A fuzzy preference is a significant type for fuzzy binary relation, and is used to generate the degree of preference between two alternatives when there are certainty and uncertainty preferences.

## 3. Statement of the Problem

## 4. Methodology

- Step 1. Construct the fuzzy matrix $\tilde{C}$ and then decompose it into three matrices called ${C}_{l}$, ${C}_{m}$, ${C}_{u}$ [31].
- Fuzzy matrix is a matrix with entries as triangular fuzzy numbers. Such a matrix shows the pairwise comparisons of the criteria (mxm matrix) or pairwise comparison of the alternatives with respect to each other (nxn matrix).
- After constructing the fuzzy triangular matrix, it is divided into three matrices as ${C}_{l}$,${C}_{m}$,${C}_{u}$ which mean matrices of lower, medium and upper values of triangular fuzzy numbers-based entries, respectively.
- Step 2. The three matrices obtained in step 1 will be used in the next step to calculate the system of fuzzy linear homogeneous equations [31].

- Step 3. Calculate the eigenvalues ${\overline{\lambda}}_{l}$, ${\overline{\lambda}}_{m}$, ${\overline{\lambda}}_{u}$ of matrices ${\overline{C}}_{l}$, ${\overline{C}}_{m}$, ${\overline{C}}_{u}$ that were determined in step 2. After that, calculate ${\lambda}_{l}$, ${\lambda}_{m}$, ${\lambda}_{u}$ by using the following equations [31]:

- Step 4. Calculate the eigenvectors ${w}_{l}$, ${w}_{m}$, ${w}_{u}$ of matrices ${\overline{C}}_{l}$, ${\overline{C}}_{m}$, ${\overline{C}}_{u}$. Next, calculate ${\overline{w}}_{l}$, ${\overline{w}}_{m}$, ${\overline{w}}_{u}$ by using the following formulas [31]:

- Step 5. Calculate the consistency index ($CI$) and consistency ratio ($CR$) of the matrix ${C}_{m}$ by using the following formulas. $CR$ should be ≤ 0.1 to claim that the comparison matrix is consistent and $RI$ is the random index [31]. $RI$ is used for random consistency which depends on the size of the matrix. The values of random index are recommended by Saaty in [2]. For example, if $n=4$ then $RI=0.90$, if $n=5$ then $RI=1.12$ [31].

- Step 6. Set the priority fuzzy matrices ${\overline{P}}_{l},{\overline{P}}_{m},{\overline{P}}_{u}$ that contain normalized eigenvectors ${\overline{w}}_{l},{\overline{w}}_{m},{\overline{w}}_{u}$ of the alternatives with respect to each criterion (use ${{\overline{w}}_{l}}^{T},{{\overline{w}}_{m}}^{T},{{\overline{w}}_{u}}^{T}$) [32].
- Step 7. Vectors of global priorities ${g}_{l},{g}_{m},{g}_{u}$ are calculated according to the following formulas (where ${\overline{w}}_{l},{\overline{w}}_{m},{\overline{w}}_{u}$ are the eigenvectors of criteria) [32]:

- Step 8. Calculate the expected value (fuzzy mean) and standard deviation (fuzzy spread) by using the following formulas [32]:

## 5. Numerical Example

#### 5.1. Fuzzy Pairwise Matrix of Criteria

- ${w}_{l}$= [0.5714 0.5714 0.5714 0.1429]
- ${w}_{m}$= [0.5714 0.5714 0.5714 0.1429]
- ${w}_{u}$= [0.5714 0.5714 0.5714 0.1429]

- ${\overline{w}}_{l}$ = [0.2923 0.2923 0.2923 0.0731]
- ${\overline{w}}_{m}$ = [0.3077 0.3077 0.3077 0.0769]
- ${\overline{w}}_{u}$ = [0.3230 0.3230 0.3230 0.0807]

#### 5.2. Fuzzy Pairwise Matrix of Teaching Criterion

- ${w}_{l}$ = [0.5711 0.5228 0.2614 0.2424 0.5228]
- ${w}_{m}$ = [0.5711 0.5228 0.2614 0.2424 0.5228]
- ${w}_{u}$= [0.5711 0.5228 0.2614 0.2424 0.5228]

- ${\overline{w}}_{l}$ = [0.2558 0.2342 0.1171 0.1085 0.2342]
- ${\overline{w}}_{m}$ = [0.2693 0.2465 0.1232 0.1143 0.2465]
- ${\overline{w}}_{u}$ = [0.2827 0.2588 0.1294 0.1200 0.2588]

#### 5.3. Fuzzy Pairwise Matrix of Research Criterion

- ${w}_{l}$= [0.6618 0.4657 0.3525 0.2457 0.4007]
- ${w}_{m}$= [0.6618 0.4657 0.3525 0.2457 0.4007]
- ${w}_{u}$= [0.6618 0.4657 0.3525 0.2457 0.4007]

- ${\overline{w}}_{l}$ = [0.2956 0.2080 0.1574 0.1097 0.1790]
- ${\overline{w}}_{m}$ = [0.3112 0.2190 0.1657 0.1155 0.1884]
- ${\overline{w}}_{u}$ = [0.3267 0.2299 0.1740 0.1213 0.1978]

#### 5.4. Fuzzy Pairwise Matrix of Citations Criterion

- ${w}_{l}$= [0.5005 0.5005 0.5005 0.4642 0.1821]
- ${w}_{m}$= [0.5005 0.5005 0.5005 0.4642 0.1821]
- ${w}_{u}$= [0.5005 0.5005 0.5005 0.4642 0.1821]

- ${\overline{w}}_{l}$ = [0.2213 0.2213 0.2213 0.2053 0.0805]
- ${\overline{w}}_{m}$ = [0.2330 0.2330 0.2330 0.2161 0.0847]
- ${\overline{w}}_{u}$ = [0.2446 0.2446 0.2446 0.2269 0.0890]

#### 5.5. Fuzzy Pairwise Matrix of International Outlook Criterion

- ${w}_{l}$= [0.4472 0.4472 0.4472 0.4472 0.4472]
- ${w}_{m}$= [0.4472 0.4472 0.4472 0.4472 0.4472]
- ${w}_{u}$ = [0.4472 0.4472 0.4472 0.4472 0.4472]

- ${\overline{w}}_{l}$ = [0.19 0.19 0.19 0.19 0.19]
- ${\overline{w}}_{m}$ = [0.2 0.2 0.2 0.2 0.2]
- ${\overline{w}}_{u}$ = [0.21 0.21 0.21 0.21 0.21]

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Criterion | Teaching | Research | Citations | International Outlook |
---|---|---|---|---|

Teaching | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (3.8,4,4.2) |

Research | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (3.8,4,4.2) |

Citations | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (3.8,4,4.2) |

International Outlook | (0.2375,0.25,0.2625) | (0.2375,0.25,0.2625) | (0.2375,0.25,0.2625) | (0.95,1,1.05) |

**Table 2.**Decomposing of fuzzy pairwise matrix of criteria into three matrices ${C}_{l},{C}_{m},{C}_{u}$.

$\mathbf{Matrix}{\mathit{C}}_{\mathit{l}}$ | $\mathbf{Matrix}{\mathit{C}}_{\mathit{m}}$ | $\mathbf{Matrix}{\mathit{C}}_{\mathit{u}}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

0.95 | 0.95 | 0.95 | 3.8 | 1 | 1 | 1 | 4 | 1.05 | 1.05 | 1.05 | 4.2 |

0.95 | 0.95 | 0.95 | 3.8 | 1 | 1 | 1 | 4 | 1.05 | 1.05 | 1.05 | 4.2 |

0.95 | 0.95 | 0.95 | 3.8 | 1 | 1 | 1 | 4 | 1.05 | 1.05 | 1.05 | 4.2 |

0.2375 | 0.2375 | 0.2375 | 0.95 | 0.25 | 0.25 | 0.25 | 1 | 0.2625 | 0.2625 | 0.2625 | 1.05 |

$\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{l}}$ | $\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{m}}$ | $\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{u}}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

2.9 | 2.9 | 2.9 | 11.6 | 6 | 6 | 6 | 24 | 3.1 | 3.1 | 3.1 | 12.4 |

2.9 | 2.9 | 2.9 | 11.6 | 6 | 6 | 6 | 24 | 3.1 | 3.1 | 3.1 | 12.4 |

2.9 | 2.9 | 2.9 | 11.6 | 6 | 6 | 6 | 24 | 3.1 | 3.1 | 3.1 | 12.4 |

0.725 | 0.725 | 0.725 | 2.9 | 1.5 | 1.5 | 1.5 | 6 | 0.775 | 0.775 | 0.775 | 3.1 |

Teaching | A | B | C | D | E |
---|---|---|---|---|---|

A | (0.95,1,1.05) | (0.95,1,1.05) | (1.9,2,2.1) | (2.85,3,3.15) | (0.95,1,1.05) |

B | (0.95,1,1.05) | (0.95,1,1.05) | (1.9,2,2.1) | (1.9,2,2.1) | (0.95,1,1.05) |

C | (0.475,0.5,0.525) | (0.475,0.5,0.525) | (0.95,1,1.05) | (0.95,1,1.05) | (0.475,0.5,0.525) |

D | (0.31635,0.333,0.34965) | (0.475,0.5,0.525) | (0.95,1,1.05) | (0.95,1,1.05) | (0.475,0.5,0.525) |

E | (0.95,1,1.05) | (0.95,1,1.05) | (1.9,2,2.1) | (1.9,2,2.1) | (0.95,1,1.05) |

**Table 5.**Decomposing of fuzzy pairwise matrix of teaching criterion into three matrices ${C}_{l},{C}_{m},{C}_{u}$.

$\mathbf{Matrix}{\mathit{C}}_{\mathit{l}}$ | $\mathbf{Matrix}{\mathit{C}}_{\mathit{m}}$ | $\mathbf{Matrix}{\mathit{C}}_{\mathit{u}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.95 | 0.95 | 1.9 | 2.85 | 0.95 | 1 | 1 | 2 | 3 | 1 | 1.05 | 1.05 | 2.1 | 3.15 | 1.05 |

0.95 | 0.95 | 1.9 | 1.9 | 0.95 | 1 | 1 | 2 | 2 | 1 | 1.05 | 1.05 | 2.1 | 2.1 | 1.05 |

0.475 | 0.475 | 0.95 | 0.95 | 0.475 | 0.5 | 0.5 | 1 | 1 | 0.5 | 0.525 | 0.525 | 1.05 | 1.05 | 0.525 |

0.31635 | 0.475 | 0.95 | 0.95 | 0.475 | 0.333 | 0.5 | 1 | 1 | 0.5 | 0.34965 | 0.525 | 1.05 | 1.05 | 0.525 |

0.95 | 0.95 | 1.9 | 1.9 | 0.95 | 1 | 1 | 2 | 2 | 1 | 1.05 | 1.05 | 2.1 | 2.1 | 1.05 |

$\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{l}}$ | $\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{m}}$ | $\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{u}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2.9 | 2.9 | 5.8 | 8.7 | 2.9 | 6 | 6 | 12 | 18 | 6 | 3.1 | 3.1 | 6.2 | 9.3 | 3.1 |

2.9 | 2.9 | 5.8 | 5.8 | 2.9 | 6 | 6 | 12 | 12 | 6 | 3.1 | 3.1 | 6.2 | 6.2 | 3.1 |

1.45 | 1.45 | 2.9 | 2.9 | 1.45 | 3 | 3 | 6 | 6 | 3 | 1.55 | 1.55 | 3.1 | 3.1 | 1.55 |

0.9657 | 1.45 | 2.9 | 2.9 | 1.45 | 1.998 | 3 | 6 | 6 | 3 | 1.0323 | 1.55 | 3.1 | 3.1 | 1.55 |

2.9 | 2.9 | 5.8 | 5.8 | 2.9 | 6 | 6 | 12 | 12 | 6 | 3.1 | 3.1 | 6.2 | 6.2 | 3.1 |

Research | A | B | C | D | E |
---|---|---|---|---|---|

A | (0.95,1,1.05) | (0.95,1,1.05) | (1.9,2,2.1) | (2.85,3,3.15) | (1.9,2,2.1) |

B | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (1.9,2,2.1) | (0.95,1,1.05) |

C | (0.475,0.5,0.525) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) |

D | (0.31635,0.333,0.34965) | (0.475,0.5,0.525) | (0.95,1,1.05) | (0.95,1,1.05) | (0.475,0.5,0.525) |

E | (0.475,0.5,0.525) | (0.95,1,1.05) | (0.95,1,1.05) | (1.9,2,2.1) | (0.95,1,1.05) |

**Table 8.**Decomposing of fuzzy pairwise matrix of research criterion into three matrices ${C}_{l},{C}_{m},{C}_{u}$.

$\mathbf{Matrix}{\mathit{C}}_{\mathit{l}}$ | $\mathbf{Matrix}{\mathit{C}}_{\mathit{m}}$ | $\mathbf{Matrix}{\mathit{C}}_{\mathit{u}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.95 | 0.95 | 1.9 | 2.85 | 1.9 | 1 | 1 | 2 | 3 | 2 | 1.05 | 1.05 | 2.1 | 3.15 | 2.1 |

0.95 | 0.95 | 0.95 | 1.9 | 0.95 | 1 | 1 | 1 | 2 | 1 | 1.05 | 1.05 | 1.05 | 2.1 | 1.05 |

0.475 | 0.95 | 0.95 | 0.95 | 0.95 | 0.5 | 1 | 1 | 1 | 1 | 0.525 | 1.05 | 1.05 | 1.05 | 1.05 |

0.31635 | 0.475 | 0.95 | 0.95 | 0.475 | 0.333 | 0.5 | 1 | 1 | 0.5 | 0.34965 | 0.525 | 1.05 | 1.05 | 0.525 |

0.475 | 0.95 | 0.95 | 1.9 | 0.95 | 0.5 | 1 | 1 | 2 | 1 | 0.525 | 1.05 | 1.05 | 2.1 | 1.05 |

$\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{l}}$ | $\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{m}}$ | $\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{u}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2.9 | 2.9 | 5.8 | 8.7 | 5.8 | 6 | 6 | 12 | 18 | 12 | 3.1 | 3.1 | 6.2 | 9.3 | 6.2 |

2.9 | 2.9 | 2.9 | 5.8 | 2.9 | 6 | 6 | 6 | 12 | 6 | 3.1 | 3.1 | 3.1 | 6.2 | 3.1 |

1.45 | 2.9 | 2.9 | 2.9 | 2.9 | 3 | 6 | 6 | 6 | 6 | 1.55 | 3.1 | 3.1 | 3.1 | 3.1 |

0.9657 | 1.45 | 2.9 | 2.9 | 1.45 | 1.998 | 3 | 6 | 6 | 3 | 1.0323 | 1.55 | 3.1 | 3.1 | 1.55 |

1.45 | 2.9 | 2.9 | 5.8 | 2.9 | 3 | 6 | 6 | 12 | 6 | 1.55 | 3.1 | 3.1 | 6.2 | 3.1 |

Citations | A | B | C | D | E |
---|---|---|---|---|---|

A | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (2.85,3,3.15) |

B | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (2.85,3,3.15) |

C | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (2.85,3,3.15) |

D | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (1.9,2,2.1) |

E | (0.31635,0.333,0.34965) | (0.31635,0.333,0.34965) | (0.31635,0.333,0.34965) | (0.475,0.5,0.525) | (0.95,1,1.05) |

**Table 11.**Decomposing of fuzzy pairwise matrix of citations criterion into three matrices ${C}_{l},{C}_{m},{C}_{u}$.

$\mathbf{Matrix}{\mathit{C}}_{\mathit{l}}$ | $\mathbf{Matrix}{\mathit{C}}_{\mathit{m}}$ | $\mathbf{Matrix}{\mathit{C}}_{\mathit{u}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.95 | 0.95 | 0.95 | 0.95 | 2.85 | 1 | 1 | 1 | 1 | 3 | 1.05 | 1.05 | 1.05 | 1.05 | 3.15 |

0.95 | 0.95 | 0.95 | 0.95 | 2.85 | 1 | 1 | 1 | 1 | 3 | 1.05 | 1.05 | 1.05 | 1.05 | 3.15 |

0.95 | 0.95 | 0.95 | 0.95 | 2.85 | 1 | 1 | 1 | 1 | 3 | 1.05 | 1.05 | 1.05 | 1.05 | 3.15 |

0.95 | 0.95 | 0.95 | 0.95 | 1.9 | 1 | 1 | 1 | 1 | 2 | 1.05 | 1.05 | 1.05 | 1.05 | 2.1 |

0.31635 | 0.31635 | 0.31635 | 0.475 | 0.95 | 0.333 | 0.333 | 0.333 | 0.5 | 1 | 0.34965 | 0.34965 | 0.34965 | 0.525 | 1.05 |

**Table 12.**Matrices ${\overline{C}}_{l},{\overline{C}}_{m},{\overline{C}}_{u}$ of citations criterion.

$\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{l}}$ | $\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{m}}$ | $\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{u}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2.9 | 2.9 | 2.9 | 2.9 | 8.7 | 6 | 6 | 6 | 6 | 18 | 3.1 | 3.1 | 3.1 | 3.1 | 9.3 |

2.9 | 2.9 | 2.9 | 2.9 | 8.7 | 6 | 6 | 6 | 6 | 18 | 3.1 | 3.1 | 3.1 | 3.1 | 9.3 |

2.9 | 2.9 | 2.9 | 2.9 | 8.7 | 6 | 6 | 6 | 6 | 18 | 3.1 | 3.1 | 3.1 | 3.1 | 9.3 |

2.9 | 2.9 | 2.9 | 2.9 | 5.8 | 6 | 6 | 6 | 6 | 12 | 3.1 | 3.1 | 3.1 | 3.1 | 6.2 |

0.9657 | 0.9657 | 0.9657 | 1.45 | 2.9 | 1.998 | 1.998 | 1.998 | 3 | 6 | 1.0323 | 1.0323 | 1.0323 | 1.55 | 3.1 |

Inter. Outlook | A | B | C | D | E |
---|---|---|---|---|---|

A | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) |

B | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) |

C | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) |

D | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) |

E | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) | (0.95,1,1.05) |

**Table 14.**Decomposing of fuzzy pairwise matrix of international outlook criterion into three matrices ${C}_{l},{C}_{m},{C}_{u}$.

$\mathbf{Matrix}{\mathit{C}}_{\mathit{l}}$ | $\mathbf{Matrix}{\mathit{C}}_{\mathit{m}}$ | $\mathbf{Matrix}{\mathit{C}}_{\mathit{u}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 1 | 1 | 1 | 1 | 1 | 1.05 | 1.05 | 1.05 | 1.05 | 1.05 |

0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 1 | 1 | 1 | 1 | 1 | 1.05 | 1.05 | 1.05 | 1.05 | 1.05 |

0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 1 | 1 | 1 | 1 | 1 | 1.05 | 1.05 | 1.05 | 1.05 | 1.05 |

0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 1 | 1 | 1 | 1 | 1 | 1.05 | 1.05 | 1.05 | 1.05 | 1.05 |

0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 1 | 1 | 1 | 1 | 1 | 1.05 | 1.05 | 1.05 | 1.05 | 1.05 |

**Table 15.**Matrices ${\overline{C}}_{l},{\overline{C}}_{m},{\overline{C}}_{u}$ of international outlook criterion.

$\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{l}}$ | $\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{m}}$ | $\mathbf{Matrix}{\overline{\mathit{C}}}_{\mathit{u}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2.9 | 2.9 | 2.9 | 2.9 | 2.9 | 6 | 6 | 6 | 6 | 6 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 |

2.9 | 2.9 | 2.9 | 2.9 | 2.9 | 6 | 6 | 6 | 6 | 6 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 |

2.9 | 2.9 | 2.9 | 2.9 | 2.9 | 6 | 6 | 6 | 6 | 6 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 |

2.9 | 2.9 | 2.9 | 2.9 | 2.9 | 6 | 6 | 6 | 6 | 6 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 |

2.9 | 2.9 | 2.9 | 2.9 | 2.9 | 6 | 6 | 6 | 6 | 6 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 |

Alternative | Vector g _{l} | Vector g _{m} | Vector g _{u} | Exp. Val. g _{i,e} | Stand. Dev. (%) | CV_{i} |
---|---|---|---|---|---|---|

A | 0.2397 | 0.2657 | 0.2928 | 0.2660 | 0.8397 | 3.1568 |

B | 0.2078 | 0.2303 | 0.2538 | 0.2306 | 0.7274 | 3.1544 |

C | 0.1588 | 0.1760 | 0.1940 | 0.1762 | 0.5566 | 3.1589 |

D | 0.1377 | 0.1526 | 0.1682 | 0.1528 | 0.4823 | 3.1564 |

E | 0.1582 | 0.1753 | 0.1932 | 0.1755 | 0.5535 | 3.1538 |

© 2020 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**

Aliyev, R.; Temizkan, H.; Aliyev, R.
Fuzzy Analytic Hierarchy Process-Based Multi-Criteria Decision Making for Universities Ranking. *Symmetry* **2020**, *12*, 1351.
https://doi.org/10.3390/sym12081351

**AMA Style**

Aliyev R, Temizkan H, Aliyev R.
Fuzzy Analytic Hierarchy Process-Based Multi-Criteria Decision Making for Universities Ranking. *Symmetry*. 2020; 12(8):1351.
https://doi.org/10.3390/sym12081351

**Chicago/Turabian Style**

Aliyev, Rashad, Hasan Temizkan, and Rafig Aliyev.
2020. "Fuzzy Analytic Hierarchy Process-Based Multi-Criteria Decision Making for Universities Ranking" *Symmetry* 12, no. 8: 1351.
https://doi.org/10.3390/sym12081351