# Fuzzy Divergence Measure Based on Technique for Order of Preference by Similarity to Ideal Solution Method for Staff Performance Appraisal

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Fuzzy Sets

**Definition 1**

**.**Let $\mathit{Z}=\{{z}_{1},{z}_{2},\cdots ,{z}_{n}\}$ be a finite universe of discourse and let $H\subset \mathit{Z}$. Then, H is fuzzy set defined as:

**Definition 2**

**.**A triplet $({m}_{1},{m}_{2},{m}_{3})$ depicted in Figure 1 can be used to define a triangular fuzzy number, $\tilde{m}$. The definition of the membership function ${\nu}_{\tilde{m}}\left(z\right)$ is:

**Definition 3**

**.**When k is a positive real number, the definitions that follow are applicable to the arithmetic operations of triangular fuzzy number, $\tilde{\mathrm{\Xi}}=({\xi}_{1},{\xi}_{2},{\xi}_{3})$ and $\tilde{H}=({\eta}_{1},{\eta}_{2},{\eta}_{3})$:

- 1.
- Addition: $\tilde{\mathrm{\Xi}}(+)\tilde{H}=({\xi}_{1}+{\eta}_{1},{\xi}_{2}+{\eta}_{2},{\xi}_{3}+{\eta}_{3})$.
- 2.
- Subtraction: $\tilde{\mathrm{\Xi}}(-)\tilde{H}=({\xi}_{1}-{\eta}_{3},{\xi}_{2}-{\eta}_{2},{\xi}_{3}-{\eta}_{1})$.
- 3.
- Multiplication: $\tilde{\mathrm{\Xi}}(\times )\tilde{H}=\left(\mathrm{min}({\xi}_{1}{\eta}_{1},{\xi}_{1}{\eta}_{3},{\xi}_{3}{\eta}_{1},{\xi}_{3}{\eta}_{3}),{\xi}_{2}{\eta}_{2},\mathrm{max}({\xi}_{1}{\eta}_{1},{\xi}_{1}{\eta}_{3},{\xi}_{3}{\eta}_{1},{\xi}_{3}{\eta}_{3})\right)$, $c(\times )\xi =(c\times {\xi}_{1},c\times {\xi}_{2},c\times {\xi}_{3})$.
- 4.
- Division: $\tilde{\mathrm{\Xi}}(\xf7)\tilde{H}=\left(\mathrm{min}\left({\displaystyle \frac{{\xi}_{1}}{{\eta}_{1}}},{\displaystyle \frac{{\xi}_{1}}{{\eta}_{3}}},{\displaystyle \frac{{\xi}_{3}}{{\eta}_{1}}},{\displaystyle \frac{{\xi}_{3}}{{\eta}_{3}}}\right),{\displaystyle \frac{{\xi}_{2}}{{\eta}_{2}}},\mathrm{max}\left({\displaystyle \frac{{\xi}_{1}}{{\eta}_{1}}},{\displaystyle \frac{{\xi}_{1}}{{\eta}_{3}}},{\displaystyle \frac{{\xi}_{3}}{{\eta}_{1}}},{\displaystyle \frac{{\xi}_{3}}{{\eta}_{3}}}\right)\right).$

#### 2.2. Existing Theories of Divergence Measures

## 3. Generalized Divergence Measure

**Example 1.**

**Theorem 1.**

**Proof.**

**Case 1:**

**Case 2:**

**Case 3:**

**Case 4:**

**Theorem 2.**

- 1.
- $J(P,Q)\ge 0$,
- 2.
- $J(P,Q)=0$ if $P=Q$,
- 3.
- $J(P,Q)=J(Q,P)$.

**Proof.**(1) and (2):

**Theorem 3.**

- 1.
- $J(P\cup Q,P\cap Q)=J(P,Q)$,
- 2.
- $J(P\cup Q,P)+J(P\cap Q,P)=J(P,Q)$,
- 3.
- $J(P\cup Q,R)\le J(P,R)+J(Q,R)$,
- 4.
- $J(P\cap Q,R)\le J(P,R)+J(Q,R)$.

**Proof.**

## 4. Formulation of $\mathit{\alpha}$-Cut Technique in Divergence Measure

## 5. Fuzzy Divergence Measure Based on TOPSIS Method

**Step 1**:

**Step 2**:

**Step 3**:

**Step 4**:

**Step 5**:

- The interval decision matrix for the normalized values:$$\begin{array}{c}{C}_{1}{C}_{2}\cdots {C}_{n}\\ {C}_{11}{C}_{12}\cdots {C}_{1a}{C}_{21}{C}_{22}\cdots {C}_{2b}{C}_{n1}{C}_{n2}\cdots {C}_{np}\end{array}\phantom{\rule{0ex}{0ex}}{\tilde{R}}_{\alpha}=\begin{array}{c}{A}_{1}\\ {A}_{2}\\ \vdots \\ {A}_{m}\end{array}\left(\begin{array}{ccccccccccccc}\hfill {\left({\tilde{r}}_{111}\right)}_{\alpha}& {\left({\tilde{r}}_{112}\right)}_{\alpha}& \cdots \hfill & {\left({\tilde{r}}_{11a}\right)}_{\alpha}& {\left({\tilde{r}}_{121}\right)}_{\alpha}& {\left({\tilde{r}}_{122}\right)}_{\alpha}& \cdots & {\left({\tilde{r}}_{12b}\right)}_{\alpha}& \cdots & {\left({\tilde{r}}_{1n1}\right)}_{\alpha}& {\left({\tilde{r}}_{1n2}\right)}_{\alpha}& \cdots & {\left({\tilde{r}}_{1np}\right)}_{\alpha}\\ \hfill {\left({\tilde{r}}_{211}\right)}_{\alpha}& {\left({\tilde{r}}_{212}\right)}_{\alpha}& \cdots \hfill & {\left({\tilde{r}}_{21a}\right)}_{\alpha}& {\left({\tilde{r}}_{221}\right)}_{\alpha}& {\left({\tilde{r}}_{222}\right)}_{\alpha}& \cdots & {\left({\tilde{r}}_{22b}\right)}_{\alpha}& \cdots & {\left({\tilde{r}}_{2n1}\right)}_{\alpha}& {\left({\tilde{r}}_{2n2}\right)}_{\alpha}& \cdots & {\left({\tilde{r}}_{2np}\right)}_{\alpha}\\ \vdots & \vdots & \ddots \hfill & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ \hfill {\left({\tilde{r}}_{m11}\right)}_{\alpha}& {\left({\tilde{r}}_{m12}\right)}_{\alpha}& \cdots \hfill & {\left({\tilde{r}}_{m1a}\right)}_{\alpha}& {\left({\tilde{r}}_{m21}\right)}_{\alpha}& {\left({\tilde{r}}_{m22}\right)}_{\alpha}& \cdots & {\left({\tilde{r}}_{m2b}\right)}_{\alpha}& \cdots & {\left({\tilde{r}}_{mn1}\right)}_{\alpha}& {\left({\tilde{r}}_{mn2}\right)}_{\alpha}& \cdots & {\left({\tilde{r}}_{mnp}\right)}_{\alpha}\end{array}\right)$$
- The interval decision matrix for the PIS and NIS, respectively:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\tilde{Z}}_{\alpha}^{+}=\{{\left({\tilde{\zeta}}_{11}^{+}\right)}_{\alpha},{\left({\tilde{\zeta}}_{12}^{+}\right)}_{\alpha},\cdots ,{\left({\tilde{\zeta}}_{np}^{+}\right)}_{\alpha}\},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\tilde{Z}}_{\alpha}^{-}=\{{\left({\tilde{\zeta}}_{11}^{-}\right)}_{\alpha},{\left({\tilde{\zeta}}_{12}^{-}\right)}_{\alpha},\cdots ,{\left({\tilde{\zeta}}_{np}^{-}\right)}_{\alpha}\},\hfill \end{array}$$

**Step 6**:

**Step 6.1**:

**Step 6.2**:

**Step 6.3**:

**Step 6.4**:

**Step 6.5**:

**Step 7**:

**Step 8**:

**Step 9**:

## 6. Application of Staff Performance Appraisal

- (i)
- Case Study 1:Let $A=\{{A}_{1},{A}_{2},\cdots ,{A}_{25}\}$ represent the university’s selected candidates, who are evaluated using four primary criteria $C=\{{C}_{1},{C}_{2},{C}_{3},{C}_{4}\}$ and 13 sub-criteria $S=\{{C}_{11},{C}_{12},\cdots ,{C}_{41}\}.$
- (ii)
- Case Study 2:Let $A=\{{A}_{1},{A}_{2},\cdots ,{A}_{20}\}$ represent the university’s selected candidates, who are evaluated using four primary criteria $C=\{{C}_{1},{C}_{2}\}$ and eight sub-criteria $S=\{{C}_{11},{C}_{12},\cdots ,{C}_{22}\}.$

#### 6.1. Comparison with Existing Methods

#### 6.2. Sensitivity Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{14}$ | ${\mathit{C}}_{15}$ | ${\mathit{C}}_{21}$ | ${\mathit{C}}_{22}$ | ${\mathit{C}}_{23}$ | ${\mathit{C}}_{31}$ | ${\mathit{C}}_{32}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{34}$ | ${\mathit{C}}_{41}$ | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${A}_{1}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | |

${A}_{2}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | |

${A}_{3}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | |

${A}_{4}$ | ${E}_{1}$ | $\alpha $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | $\gamma $ |

${E}_{2}$ | $\alpha $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | $\gamma $ | |

${A}_{5}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | $\gamma $ | $\gamma $ | $\gamma $ | $\beta $ | $\gamma $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\gamma $ | $\gamma $ | $\beta $ | $\beta $ | $\gamma $ | $\gamma $ | $\gamma $ | $\gamma $ | $\beta $ | $\gamma $ | |

${A}_{6}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | |

${A}_{7}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | |

${A}_{8}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | $\beta $ | $\beta $ | $\gamma $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | $\beta $ | $\beta $ | $\gamma $ | |

${A}_{9}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\gamma $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\gamma $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | |

${A}_{10}$ | ${E}_{1}$ | $\beta $ | $\gamma $ | $\gamma $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\beta $ | $\gamma $ | $\gamma $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | |

${A}_{11}$ | ${E}_{1}$ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | |

${A}_{12}$ | ${E}_{1}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\gamma $ | |

${A}_{13}$ | ${E}_{1}$ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ |

${E}_{2}$ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | |

${A}_{14}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | |

${A}_{15}$ | ${E}_{1}$ | $\alpha $ | $\gamma $ | $\gamma $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ |

${E}_{2}$ | $\alpha $ | $\gamma $ | $\gamma $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | |

${A}_{16}$ | ${E}_{1}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\beta $ | |

${A}_{17}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | |

${A}_{18}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\gamma $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\gamma $ | |

${A}_{19}$ | ${E}_{1}$ | $\alpha $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ |

${E}_{2}$ | $\alpha $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | |

${A}_{20}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | |

${A}_{21}$ | ${E}_{1}$ | $\beta $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ |

${E}_{2}$ | $\beta $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | |

${A}_{22}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | |

${A}_{23}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\gamma $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | |

${A}_{24}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ |

${E}_{2}$ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | |

${A}_{25}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\beta $ |

${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{14}$ | ${\mathit{C}}_{15}$ | ${\mathit{C}}_{16}$ | ${\mathit{C}}_{21}$ | ${\mathit{C}}_{22}$ | ||
---|---|---|---|---|---|---|---|---|---|

${A}_{1}$ | ${E}_{1}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | |

${A}_{2}$ | ${E}_{1}$ | $\alpha $ | $\gamma $ | $\alpha $ | $\alpha $ | $\u03f5$ | $\gamma $ | $\alpha $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\gamma $ | $\alpha $ | $\alpha $ | $\u03f5$ | $\gamma $ | $\alpha $ | $\alpha $ | |

${A}_{3}$ | ${E}_{1}$ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | |

${A}_{4}$ | ${E}_{1}$ | $\alpha $ | $\gamma $ | $\alpha $ | $\alpha $ | $\gamma $ | $\alpha $ | $\beta $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\gamma $ | $\alpha $ | $\alpha $ | $\gamma $ | $\alpha $ | $\beta $ | $\alpha $ | |

${A}_{5}$ | ${E}_{1}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\beta $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\beta $ | $\alpha $ | |

${A}_{6}$ | ${E}_{1}$ | $\gamma $ | $\gamma $ | $\alpha $ | $\alpha $ | $\delta $ | $\u03f5$ | $\alpha $ | $\alpha $ |

${E}_{2}$ | $\gamma $ | $\gamma $ | $\alpha $ | $\alpha $ | $\delta $ | $\u03f5$ | $\alpha $ | $\alpha $ | |

${A}_{7}$ | ${E}_{1}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\gamma $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\gamma $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | |

${A}_{8}$ | ${E}_{1}$ | $\alpha $ | $\delta $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\beta $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\delta $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\beta $ | $\alpha $ | |

${A}_{9}$ | ${E}_{1}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\u03f5$ | $\gamma $ | $\alpha $ | $\alpha $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\u03f5$ | $\gamma $ | $\alpha $ | $\alpha $ | $\alpha $ | |

${A}_{10}$ | ${E}_{1}$ | $\beta $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ |

${E}_{2}$ | $\beta $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | |

${A}_{11}$ | ${E}_{1}$ | $\alpha $ | $\gamma $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\beta $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\gamma $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\beta $ | $\alpha $ | |

${A}_{12}$ | ${E}_{1}$ | $\alpha $ | $\u03f5$ | $\theta $ | $\alpha $ | $\beta $ | $\alpha $ | $\gamma $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\u03f5$ | $\theta $ | $\alpha $ | $\beta $ | $\alpha $ | $\gamma $ | $\alpha $ | |

${A}_{13}$ | ${E}_{1}$ | $\beta $ | $\delta $ | $\delta $ | $\gamma $ | $\beta $ | $\alpha $ | $\alpha $ | $\beta $ |

${E}_{2}$ | $\beta $ | $\delta $ | $\delta $ | $\gamma $ | $\beta $ | $\alpha $ | $\alpha $ | $\beta $ | |

${A}_{14}$ | ${E}_{1}$ | $\alpha $ | $\eta $ | $\alpha $ | $\eta $ | $\beta $ | $\alpha $ | $\beta $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\eta $ | $\alpha $ | $\eta $ | $\beta $ | $\alpha $ | $\beta $ | $\alpha $ | |

${A}_{15}$ | ${E}_{1}$ | $\alpha $ | $\gamma $ | $\alpha $ | $\alpha $ | $\gamma $ | $\alpha $ | $\alpha $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\gamma $ | $\alpha $ | $\alpha $ | $\gamma $ | $\alpha $ | $\alpha $ | $\alpha $ | |

${A}_{16}$ | ${E}_{1}$ | $\alpha $ | $\delta $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\delta $ | $\alpha $ | $\alpha $ | $\beta $ | $\alpha $ | $\alpha $ | $\alpha $ | |

${A}_{17}$ | ${E}_{1}$ | $\alpha $ | $\delta $ | $\alpha $ | $\gamma $ | $\zeta $ | $\alpha $ | $\alpha $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\delta $ | $\alpha $ | $\gamma $ | $\zeta $ | $\alpha $ | $\alpha $ | $\alpha $ | |

${A}_{18}$ | ${E}_{1}$ | $\alpha $ | $\delta $ | $\alpha $ | $\alpha $ | $\u03f5$ | $\alpha $ | $\beta $ | $\beta $ |

${E}_{2}$ | $\alpha $ | $\delta $ | $\alpha $ | $\alpha $ | $\u03f5$ | $\alpha $ | $\beta $ | $\beta $ | |

${A}_{19}$ | ${E}_{1}$ | $\beta $ | $\gamma $ | $\alpha $ | $\beta $ | $\beta $ | $\alpha $ | $\gamma $ | $\beta $ |

${E}_{2}$ | $\beta $ | $\gamma $ | $\alpha $ | $\beta $ | $\beta $ | $\alpha $ | $\gamma $ | $\beta $ | |

${A}_{20}$ | ${E}_{1}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\zeta $ | $\alpha $ | $\alpha $ | $\alpha $ |

${E}_{2}$ | $\alpha $ | $\alpha $ | $\alpha $ | $\alpha $ | $\zeta $ | $\alpha $ | $\alpha $ | $\alpha $ |

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**Figure 4.**Correlation coefficient value between original ranking and the ranking with decreasing criteria weight for Case Study 1.

**Figure 5.**Correlation coefficient value between original ranking and the ranking with increasing criteria weight for Case Study 1.

**Figure 6.**Correlation coefficient value between original ranking and the ranking with decreasing criteria weight for Case Study 2.

**Figure 7.**Correlation coefficient value between original ranking and the ranking with increasing criteria weight for Case Study 2.

Linguistics Terms | Fuzzy Numbers |
---|---|

Excellent ($\alpha $) | $(9,10,10)$ |

Medium Excellent ($\beta $) | $(8,9,10)$ |

Very Good ($\gamma $) | $(7,8,9)$ |

Good ($\delta $) | $(6,7,8)$ |

Medium Good ($\u03f5$) | $(5,6,7)$ |

Fair ($\zeta $) | $(4,5,6)$ |

Medium Fair ($\eta $) | $(3,4,5)$ |

Poor ($\theta $) | $(2,3,4)$ |

Very Poor ($\iota $) | $(1,2,3)$ |

Medium Terrible ($\kappa $) | $(0,1,2)$ |

Terrible ($\lambda $) | $(0,0,1)$ |

Main Criteria | Weight |
---|---|

${C}_{1}$ | 0.50 |

${C}_{2}$ | 0.25 |

${C}_{3}$ | 0.20 |

${C}_{4}$ | 0.05 |

Main Criteria | Weight |
---|---|

${C}_{1}$ | 0.80 |

${C}_{2}$ | 0.20 |

Primary Criteria | Sub-Criteria | Weight, ${\mathit{w}}_{\mathit{jk}}$ |
---|---|---|

${C}_{11}$ | 0.2892 | |

${C}_{12}$ | 0.1776 | |

${C}_{1}$ | ${C}_{13}$ | 0.1818 |

${C}_{14}$ | 0.1742 | |

${C}_{15}$ | 0.1772 | |

${C}_{21}$ | 0.3437 | |

${C}_{2}$ | ${C}_{22}$ | 0.3317 |

${C}_{23}$ | 0.3246 | |

${C}_{3}$ | ${C}_{31}$ | 0.2619 |

${C}_{32}$ | 0.2529 | |

${C}_{33}$ | 0.2490 | |

${C}_{34}$ | 0.2362 | |

${C}_{4}$ | ${C}_{41}$ | 1.0000 |

Primary Criteria | Sub-Criteria | Weight, ${\mathit{w}}_{\mathit{jk}}$ |
---|---|---|

${C}_{1}$ | ${C}_{11}$ | 0.1989 |

${C}_{12}$ | 0.1181 | |

${C}_{13}$ | 0.2022 | |

${C}_{14}$ | 0.1722 | |

${C}_{15}$ | 0.1039 | |

${C}_{16}$ | 0.2048 | |

${C}_{2}$ | ${C}_{21}$ | 0.4637 |

${C}_{22}$ | 0.5363 |

**Table 6.**Ranking of alternatives corresponding to closeness coefficient for $\alpha =0.1$, $\alpha =0.5$, and $\alpha =\phantom{\rule{3.33333pt}{0ex}}0.9$ for Case Study 1.

Alternative | Ranking | ||
---|---|---|---|

$\mathit{\alpha}=\mathbf{0}.\mathbf{1}$ | $\mathit{\alpha}=\mathbf{0}.\mathbf{5}$ | $\mathit{\alpha}=\mathbf{0}.\mathbf{9}$ | |

${A}_{1}$ | 7 | 7 | 7 |

${A}_{2}$ | 6 | 6 | 6 |

${A}_{3}$ | 4 | 4 | 4 |

${A}_{4}$ | 7 | 7 | 7 |

${A}_{5}$ | 3 | 3 | 3 |

${A}_{6}$ | 5 | 5 | 5 |

${A}_{7}$ | 7 | 7 | 7 |

${A}_{8}$ | 13 | 13 | 13 |

${A}_{9}$ | 16 | 16 | 17 |

${A}_{10}$ | 20 | 20 | 20 |

${A}_{11}$ | 13 | 13 | 13 |

${A}_{12}$ | 22 | 22 | 22 |

${A}_{13}$ | 19 | 18 | 18 |

${A}_{14}$ | 23 | 23 | 23 |

${A}_{15}$ | 25 | 25 | 25 |

${A}_{16}$ | 17 | 17 | 16 |

${A}_{17}$ | 18 | 19 | 19 |

${A}_{18}$ | 12 | 12 | 12 |

${A}_{19}$ | 7 | 7 | 7 |

${A}_{20}$ | 2 | 2 | 2 |

${A}_{21}$ | 1 | 1 | 1 |

${A}_{22}$ | 7 | 7 | 7 |

${A}_{23}$ | 13 | 13 | 13 |

${A}_{24}$ | 24 | 24 | 24 |

${A}_{25}$ | 21 | 21 | 21 |

**Table 7.**Ranking of alternatives corresponding to closeness coefficient for $\alpha =0.1$, $\alpha =0.5$, and $\alpha =0.9$ for Case Study 2.

Alternative | Ranking | ||
---|---|---|---|

$\mathit{\alpha}=\mathbf{0}.\mathbf{1}$ | $\mathit{\alpha}=\mathbf{0}.\mathbf{5}$ | $\mathit{\alpha}=\mathbf{0}.\mathbf{9}$ | |

${A}_{1}$ | 1 | 1 | 1 |

${A}_{2}$ | 14 | 14 | 14 |

${A}_{3}$ | 3 | 3 | 3 |

${A}_{4}$ | 8 | 8 | 8 |

${A}_{5}$ | 2 | 2 | 2 |

${A}_{6}$ | 18 | 17 | 17 |

${A}_{7}$ | 6 | 6 | 6 |

${A}_{8}$ | 10 | 10 | 10 |

${A}_{9}$ | 15 | 15 | 15 |

${A}_{10}$ | 4 | 4 | 4 |

${A}_{11}$ | 5 | 5 | 5 |

${A}_{12}$ | 20 | 20 | 20 |

${A}_{13}$ | 17 | 18 | 18 |

${A}_{14}$ | 19 | 19 | 19 |

${A}_{15}$ | 7 | 7 | 7 |

${A}_{16}$ | 9 | 9 | 9 |

${A}_{17}$ | 16 | 16 | 16 |

${A}_{18}$ | 13 | 13 | 13 |

${A}_{19}$ | 11 | 11 | 11 |

${A}_{20}$ | 12 | 12 | 12 |

**Table 8.**Rankings of the alternatives in descending order for $\alpha =0.1,0.5$, and $0.9$ for Case Study 1.

$\mathit{\alpha}$ Value | Ranking |
---|---|

0.1 | ${A}_{21}\succ {A}_{20}\succ {A}_{5}\succ {A}_{3}\succ {A}_{6}\succ {A}_{2}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{18}\succ {A}_{8}\succ {A}_{8}\succ {A}_{8}\succ {A}_{9}\succ {A}_{16}\succ {A}_{17}\succ {A}_{13}\succ {A}_{10}\succ {A}_{25}\succ {A}_{12}\succ {A}_{14}\succ {A}_{24}\succ {A}_{15}$ |

0.5 | ${A}_{21}\succ {A}_{20}\succ {A}_{5}\succ {A}_{3}\succ {A}_{6}\succ {A}_{2}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{18}\succ {A}_{8}\succ {A}_{8}\succ {A}_{8}\succ {A}_{9}\succ {A}_{16}\succ {A}_{13}\succ {A}_{17}\succ {A}_{10}\succ {A}_{25}\succ {A}_{12}\succ {A}_{14}\succ {A}_{24}\succ {A}_{15}$ |

0.9 | ${A}_{21}\succ {A}_{20}\succ {A}_{5}\succ {A}_{3}\succ {A}_{6}\succ {A}_{2}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{18}\succ {A}_{8}\succ {A}_{8}\succ {A}_{8}\succ {A}_{16}\succ {A}_{9}\succ {A}_{13}\succ {A}_{17}\succ {A}_{10}\succ {A}_{25}\succ {A}_{12}\succ {A}_{14}\succ {A}_{24}\succ {A}_{15}$ |

**Table 9.**Rankings of the alternatives in descending order for $\alpha =0.1,0.5$, and $0.9$ for Case Study 2.

$\mathit{\alpha}$ Value | Ranking |
---|---|

0.1 | ${A}_{1}\succ {A}_{5}\succ {A}_{3}\succ {A}_{10}\succ {A}_{11}\succ {A}_{7}\succ {A}_{15}\succ {A}_{4}\succ {A}_{16}\succ {A}_{8}\succ {A}_{19}\succ {A}_{20}\succ {A}_{18}\succ {A}_{2}\succ {A}_{9}\succ {A}_{17}\succ {A}_{13}\succ {A}_{6}\succ {A}_{14}\succ {A}_{12}$ |

0.5 | ${A}_{1}\succ {A}_{5}\succ {A}_{3}\succ {A}_{10}\succ {A}_{11}\succ {A}_{7}\succ {A}_{15}\succ {A}_{4}\succ {A}_{16}\succ {A}_{8}\succ {A}_{19}\succ {A}_{20}\succ {A}_{18}\succ {A}_{2}\succ {A}_{9}\succ {A}_{17}\succ {A}_{6}\succ {A}_{13}\succ {A}_{14}\succ {A}_{12}$ |

0.9 | ${A}_{1}\succ {A}_{5}\succ {A}_{3}\succ {A}_{10}\succ {A}_{11}\succ {A}_{7}\succ {A}_{15}\succ {A}_{4}\succ {A}_{16}\succ {A}_{8}\succ {A}_{19}\succ {A}_{20}\succ {A}_{18}\succ {A}_{2}\succ {A}_{9}\succ {A}_{17}\succ {A}_{6}\succ {A}_{13}\succ {A}_{14}\succ {A}_{12}$ |

Methods | Principal Measure | Ranking Order |
---|---|---|

Fuzzy TOPSIS by Kabak et al. [57] | Fuzzy sets and Euclidean distance | ${A}_{21}\succ {A}_{20}\succ {A}_{3}\succ {A}_{6}\succ {A}_{5}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{16}\succ {A}_{2}\succ {A}_{18}\succ {A}_{9}\succ {A}_{13}\succ {A}_{8}\succ {A}_{8}\succ {A}_{8}\succ {A}_{17}\succ {A}_{10}\succ {A}_{25}\succ {A}_{12}\succ {A}_{14}\succ {A}_{24}\succ {A}_{15}$ |

Fuzzy Divergence Measures by Joshi and Kumar [39] | Intuitionistic fuzzy sets and divergence measure | ${A}_{21}\succ {A}_{20}\succ {A}_{5}\succ {A}_{3}\succ {A}_{6}\succ {A}_{2}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{18}\succ {A}_{8}\succ {A}_{8}\succ {A}_{8}\succ {A}_{9}\succ {A}_{16}\succ {A}_{17}\succ {A}_{13}\succ {A}_{10}\succ {A}_{25}\succ {A}_{12}\succ {A}_{14}\succ {A}_{24}\succ {A}_{15}$ |

Fuzzy Divergence Measures by Rani et al. [40] | Fuzzy sets and divergence measure | ${A}_{21}\succ {A}_{20}\succ {A}_{5}\succ {A}_{3}\succ {A}_{6}\succ {A}_{2}\succ {A}_{18}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{8}\succ {A}_{8}\succ {A}_{8}\succ {A}_{9}\succ {A}_{16}\succ {A}_{17}\succ {A}_{13}\succ {A}_{10}\succ {A}_{14}\succ {A}_{25}\succ {A}_{12}\succ {A}_{24}\succ {A}_{15}$ |

Proposed Method | Fuzzy sets, $\alpha $-cut, and divergence measure | ${A}_{21}\succ {A}_{20}\succ {A}_{5}\succ {A}_{3}\succ {A}_{6}\succ {A}_{2}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{1}\succ {A}_{18}\succ {A}_{8}\succ {A}_{8}\succ {A}_{8}\succ {A}_{9}\succ {A}_{16}\succ {A}_{13}\succ {A}_{17}\succ {A}_{10}\succ {A}_{25}\succ {A}_{12}\succ {A}_{14}\succ {A}_{24}\succ {A}_{15}$ |

Methods | Principal Measure | Ranking Order |
---|---|---|

Fuzzy TOPSIS by Kabak et al. [57] | Fuzzy sets and Euclidean distance | ${A}_{1}\succ {A}_{5}\succ {A}_{3}\succ {A}_{10}\succ {A}_{11}\succ {A}_{7}\succ {A}_{15}\succ {A}_{16}\succ {A}_{4}\succ {A}_{8}\succ {A}_{20}\succ {A}_{9}\succ {A}_{19}\succ {A}_{18}\succ {A}_{2}\succ {A}_{17}\succ {A}_{13}\succ {A}_{6}\succ {A}_{14}\succ {A}_{12}$ |

Fuzzy Divergence Measures by Joshi and Kumar [39] | Intuitionistic fuzzy sets and divergence measure | ${A}_{1}\succ {A}_{5}\succ {A}_{3}\succ {A}_{10}\succ {A}_{11}\succ {A}_{7}\succ {A}_{15}\succ {A}_{4}\succ {A}_{19}\succ {A}_{16}\succ {A}_{8}\succ {A}_{2}\succ {A}_{20}\succ {A}_{18}\succ {}_{}$ |