# A Novel Dynamic Multi-Criteria Decision Making Method Based on Generalized Dynamic Interval-Valued Neutrosophic Set

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary

#### 2.1. Multi-Criteria Decision-Making Model Based on History

#### 2.2. Dynamic Interval-Valued Neutrosophic Set and Hesitant Fuzzy Set

**Definition**

**1.**

**Example**

**1.**

**Definition**

**2.**

**Example**

**2.**

## 3. Generalized Dynamic Interval-Valued Neutrosophic Set

**Definition**

**3.**

**Example**

**3.**

**Definition**

**4.**

- (i)
- Addition$$\begin{array}{l}{\tilde{h}}_{1}\oplus {\tilde{h}}_{2}={\displaystyle {\cup}_{\forall {\gamma}_{1}\in {\tilde{h}}_{1};\forall {\gamma}_{2}\in {\tilde{h}}_{2}}\left\{{\gamma}_{1}\oplus {\gamma}_{2}\right\}}\\ =\left\{\langle \begin{array}{l}\left[{T}_{{\gamma}_{1}}^{L}\left(x\left(\tau \right)\right)+{T}_{{\gamma}_{2}}^{L}\left(x\left(\tau \right)\right)-{T}_{{\gamma}_{1}}^{L}\left(x\left(\tau \right)\right)\times {T}_{{\gamma}_{2}}^{L}\left(x\left(\tau \right)\right),{T}_{{\gamma}_{1}}^{U}\left(x\left(\tau \right)\right)+{T}_{{\gamma}_{2}}^{U}\left(x\left(\tau \right)\right)-{T}_{{\gamma}_{1}}^{U}\left(x\left(\tau \right)\right)\times {T}_{{\gamma}_{2}}^{U}\left(x\left(\tau \right)\right)\right],\\ \left[{I}_{{\gamma}_{1}}^{L}\left(x\left(\tau \right)\right)\times {I}_{{\gamma}_{2}}^{L}\left(x\left(\tau \right)\right),{I}_{{\gamma}_{1}}^{U}\left(x\left(\tau \right)\right)\times {I}_{{\gamma}_{2}}^{U}\left(x\left(\tau \right)\right)\right],\left[{F}_{{\gamma}_{1}}^{L}\left(x\left(\tau \right)\right)\times {F}_{{\gamma}_{2}}^{L}\left(x\left(\tau \right)\right),{F}_{{\gamma}_{1}}^{U}\left(x\left(\tau \right)\right)\times {F}_{{\gamma}_{2}}^{U}\left(x\left(\tau \right)\right)\right]\end{array}\rangle \right\}\end{array}$$
- (ii)
- Multiplication$$\begin{array}{l}{\tilde{h}}_{1}\otimes {\tilde{h}}_{2}={\displaystyle {\cup}_{\forall {\gamma}_{1}\in {\tilde{h}}_{1};\forall {\gamma}_{2}\in {\tilde{h}}_{2}}\left\{{\gamma}_{1}\otimes {\gamma}_{2}\right\}}\\ =\left\{\langle \begin{array}{l}\left[{T}_{{\gamma}_{1}}^{L}\left(x\left(\tau \right)\right)\times {T}_{{\gamma}_{2}}^{L}\left(x\left(\tau \right)\right),{T}_{{\gamma}_{1}}^{U}\left(x\left(\tau \right)\right)\times {T}_{{\gamma}_{2}}^{U}\left(x\left(\tau \right)\right)\right],\\ \left[{I}_{{\gamma}_{1}}^{L}\left(x\left(\tau \right)\right)+{I}_{{\gamma}_{2}}^{L}\left(x\left(\tau \right)\right)-{I}_{{\gamma}_{1}}^{L}\left(x\left(\tau \right)\right)\times {I}_{{\gamma}_{2}}^{L}\left(x\left(\tau \right)\right),{I}_{{\gamma}_{1}}^{U}\left(x\left(\tau \right)\right)+{I}_{{\gamma}_{2}}^{U}\left(x\left(\tau \right)\right)-{I}_{{\gamma}_{1}}^{U}\left(x\left(\tau \right)\right)\times {I}_{{\gamma}_{2}}^{U}\left(x\left(\tau \right)\right)\right],\\ \left[{F}_{{\gamma}_{1}}^{L}\left(x\left(\tau \right)\right)+{F}_{{\gamma}_{2}}^{L}\left(x\left(\tau \right)\right)-{F}_{{\gamma}_{1}}^{L}\left(x\left(\tau \right)\right)\times {F}_{{\gamma}_{2}}^{L}\left(x\left(\tau \right)\right),{F}_{{\gamma}_{1}}^{U}\left(x\left(\tau \right)\right)+{F}_{{\gamma}_{2}}^{U}\left(x\left(\tau \right)\right)-{F}_{{\gamma}_{1}}^{U}\left(x\left(\tau \right)\right)\times {F}_{{\gamma}_{2}}^{U}\left(x\left(\tau \right)\right)\right]\end{array}\rangle \right\}\end{array}$$
- (iii)
- Scalar Multiplication$$\begin{array}{l}\lambda \tilde{h}={\displaystyle {\cup}_{\forall \gamma \in \tilde{h}}\left\{\lambda \gamma \right\}}\\ ={\displaystyle {\cup}_{\forall \gamma \in \tilde{h}}\left\{\langle \begin{array}{l}\left[1-{\left(1-{T}^{L}\left(x\left(\tau \right)\right)\right)}^{\lambda},1-{\left(1-{T}^{U}\left(x\left(\tau \right)\right)\right)}^{\lambda}\right],\\ \left[{\left({I}^{L}\left(x\left(\tau \right)\right)\right)}^{\lambda},{\left({I}^{U}\left(x\left(\tau \right)\right)\right)}^{\lambda}\right],\left[{\left({F}^{L}\left(x\left(\tau \right)\right)\right)}^{\lambda},{\left({F}^{U}\left(x\left(\tau \right)\right)\right)}^{\lambda}\right]\end{array}\rangle \right\}}\end{array}$$
- (iv)
- Power$$\begin{array}{l}{\tilde{h}}^{\lambda}={\displaystyle {\cup}_{\forall \gamma \in \tilde{h}}\left\{{\gamma}^{\lambda}\right\}}\\ ={\displaystyle {\cup}_{\forall \gamma \in \tilde{h}}\left\{\langle \begin{array}{l}\left[{\left({T}^{L}\left(x\left(\tau \right)\right)\right)}^{\lambda},{\left({T}^{U}\left(x\left(\tau \right)\right)\right)}^{\lambda}\right],\left[1-{\left(1-{I}^{L}\left(x\left(\tau \right)\right)\right)}^{\lambda},1-{\left(1-{I}^{U}\left(x\left(\tau \right)\right)\right)}^{\lambda}\right],\\ \left[1-{\left(1-{F}^{L}\left(x\left(\tau \right)\right)\right)}^{\lambda},1-{\left(1-{F}^{U}\left(x\left(\tau \right)\right)\right)}^{\lambda}\right]\end{array}\rangle \right\}}\end{array}$$

**Definition**

**5.**

**Example**

**4.**

**Definition**

**6.**

**Theorem**

**1.**

**Proof.**

**Definition**

**7.**

**Theorem**

**2.**

**Proof.**

**Definition**

**8.**

**Theorem**

**3.**

**Proof.**

## 4. Dynamic TOPSIS Method

^{th}, $r=\left\{1,2,\dots ,s\right\}$. For a decision-maker ${D}_{q};q=1,\dots ,{h}_{r},$ the evaluation of an alternative ${A}_{m};m=1,\dots ,{v}_{r},$ on a criteria ${C}_{p};p=1,\dots ,{n}_{r},$ in time sequence $\tau =\left\{{\tau}_{1},{\tau}_{2},\dots ,{\tau}_{{k}_{r}}\right\}$ is represented by the Neutrosophic decision matrix ${\Re}^{q}\left({t}_{r}\right)={\left({\xi}_{mp}^{q}\left(\tau \right)\right)}_{{v}_{r}\times {n}_{r}};l=1,2,\dots ,{k}_{r}.$ where

**Step 1. Calculate aggregate ratings at period r**

^{th}.**Step 2. Calculate importance weight aggregation at period r**

^{th}.**Step 3. Evaluation for aggregate ratings of alternatives with history data.**

**Step 4. Evaluation for importance weight aggregation of criteria with history data.**

**Step 5. Calculate the average weighted ratings at period r**

^{th}.**Step 6. Determination of**${A}_{r}^{+},{A}_{r}^{-}$

**and**${d}_{r}^{+},{d}_{r}^{-}$

**at period r**

^{th}.**Step 7. Determination the closeness coefficient.**

**Step 8. Rank the alternatives.**

## 5. Applications

#### 5.1. ASK Model for Ranking Students

#### 5.2. Application Model

#### 5.3. Comparison with the Related Methods

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Language Labels | Values |
---|---|

Very Poor | ([0.1, 0.26], [0.4, 0.5], [0.63, 0.76]) |

Poor | ([0.26, 0.38], [0.47, 0.6], [0.51, 0.6]) |

Medium | ([0.38, 0.5], [0.4, 0.61], [0.44, 0.55]) |

Good | ([0.5, 0.65], [0.36, 0.5], [0.31, 0.48]) |

Very Good | ([0.65, 0.8], [0.1, 0.2], [0.12, 0.2]) |

Language Labels | Values |
---|---|

Unimportant | ([0.1, 0.19], [0.32, 0.47], [0.64, 0.8]) |

Slightly Important | ([0.2, 0.38], [0.46, 0.62], [0.36, 0.55]) |

Important | ([0.45, 0.63], [0.41, 0.53], [0.2, 0.42]) |

Very Important | ([0.66, 0.8], [0.3, 0.39], [0.22, 0.32]) |

Absolutely Important | ([0.8, 0.94], [0.18, 0.29], [0.1, 0.2]) |

Criteria | Students | ||
---|---|---|---|

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | |

${C}_{1}$ | ([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) | ([0.43, 0.577], [0.021, 0.053], [0.017, 0.048]) |

${C}_{2}$ | ([0.488, 0.632], [0.005, 0.025], [0.008, 0.021]) | ([0.419,0.578], [0.011,0.037], [0.011,0.026]) | ([0.419,0.578], [0.011,0.037], [0.011,0.026]) |

${C}_{3}$ | ([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.423, 0.556], [0.02, 0.066], [0.02, 0.051]) |

${C}_{4}$ | ([0.423, 0.556], [0.02, 0.066], [0.02, 0.051]) | ([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.388, 0.523], [0.023, 0.065], [0.024, 0.056]) |

${C}_{5}$ | ([0.523, 0.673], [0.005, 0.021], [0.005, 0.018]) | ([0.423, 0.556], [0.02, 0.066], [0.02, 0.051]) | ([0.43, 0.577], [0.021, 0.053], [0.017, 0.048]) |

${C}_{6}$ | ([0.43, 0.577], [0.021, 0.053], [0.017, 0.048]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) | ([0.342, 0.463], [0.026, 0.081], [0.034, 0.065]) |

${C}_{7}$ | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) | ([0.388, 0.523], [0.023, 0.065], [0.024, 0.056]) | ([0.342, 0.463], [0.026, 0.081], [0.034, 0.065]) |

${C}_{8}$ | ([0.26, 0.38], [0.036, 0.078], [0.046, 0.078]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) |

${C}_{9}$ | ([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.523, 0.673], [0.005, 0.021], [0.005, 0.018]) | ([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) |

${C}_{10}$ | ([0.5, 0.65], [0.016, 0.044], [0.01, 0.038]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) | ([0.43, 0.577], [0.021, 0.053], [0.017, 0.048]) |

${C}_{11}$ | ([0.463, 0.606], [0.018, 0.054], [0.014, 0.044]) | ([0.302, 0.423], [0.03, 0.079], [0.04, 0.071]) | ([0.38, 0.5], [0.022, 0.082], [0.029, 0.059]) |

Criteria | Importance Aggregated Weights |
---|---|

${C}_{1}$ | ([0.963, 0.996], [0.022, 0.06], [0.004, 0.027]) |

${C}_{2}$ | ([0.908, 0.968], [0.041, 0.094], [0.017, 0.056]) |

${C}_{3}$ | ([0.758, 0.89], [0.077, 0.174], [0.014, 0.097]) |

${C}_{4}$ | ([0.648, 0.816], [0.087, 0.204], [0.026, 0.127]) |

${C}_{5}$ | ([0.604, 0.794], [0.06, 0.154], [0.046, 0.185]) |

${C}_{6}$ | ([0.963, 0.992], [0.022, 0.06], [0.004, 0.027]) |

${C}_{7}$ | ([0.834, 0.925], [0.069, 0.149], [0.008, 0.074]) |

${C}_{8}$ | ([0.758, 0.89], [0.077, 0.174], [0.014, 0.097]) |

${C}_{9}$ | ([0.758, 0.89], [0.077, 0.174], [0.014, 0.097]) |

${C}_{10}$ | ([0.936, 0.975], [0.037, 0.081], [0.01, 0.043]) |

${C}_{11}$ | ([0.897, 0.959], [0.05, 0.11], [0.009, 0.056]) |

Students | Weighted Ratings |
---|---|

${A}_{1}$ | ([0.368, 0.409], [0.069, 0.168], [0.03, 0.114]) |

${A}_{2}$ | ([0.34, 0.382], [0.071, 0.181], [0.035, 0.12]) |

${A}_{3}$ | ([0.338, 0.377], [0.072, 0.178], [0.035, 0.121]) |

**Table 6.**The distance of each student from ${A}_{{t}_{1}}^{+}$ and ${A}_{{t}_{1}}^{-}$ at period ${t}_{1}$.

Students | ${\mathit{d}}_{{\mathit{t}}_{1}}^{+}$ | ${\mathit{d}}_{{\mathit{t}}_{1}}^{-}$ |
---|---|---|

${A}_{1}$ | 0.364193 | 0.773329 |

${A}_{2}$ | 0.380989 | 0.763987 |

${A}_{3}$ | 0.382736 | 0.763579 |

Students | Closeness Coefficients | Ranking Order |
---|---|---|

${A}_{1}$ | 0.679837 | 1 |

${A}_{2}$ | 0.667251 | 2 |

${A}_{3}$ | 0.666116 | 3 |

Criteria | Students | |||
---|---|---|---|---|

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |

${C}_{1}$ | ([0.699, 0.83], [0.001, 0.005], [0, 0.002]) | ([0.566, 0.75], [0.001, 0.009], [0.001, 0.003]) | ([0.637, 0.759], [0.001, 0.007], [0.001, 0.003]) | ([0.5, 0.6], [0.022, 0.046], [0.009, 0.022]) |

${C}_{2}$ | ([0.707, 0.852], [0.001, 0.007], [0, 0.002]) | ([0.686, 0.834], [0.001, 0.008], [0, 0.003]) | ([0.72, 0.862], [0.001, 0.006], [0, 0.002]) | ([0.498, 0.6], [0.023, 0.049], [0.009, 0.023]) |

${C}_{3}$ | ([0.709, 0.848], [0.003, 0.016], [0, 0.005]) | ([0.643, 0.783], [0.003, 0.018], [0, 0.006]) | ([0.603, 0.767], [0.003, 0.019], [0.001, 0.006]) | ([0.56, 0.669], [0.008, 0.029], [0.004, 0.016]) |

${C}_{4}$ | ([0.598, 0.766], [0.004, 0.022], [0.001, 0.007]) | ([0.639, 0.782], [0.004, 0.021], [0.001, 0.007]) | ([0.634, 0.793], [0.004, 0.02], [0.001, 0.008]) | ([0.506, 0.643], [0.009, 0.034], [0.006, 0.021]) |

${C}_{5}$ | ([0.721, 0.866], [0.002, 0.012], [0.001, 0.015]) | ([0.651, 0.823], [0.002, 0.013], [0.002, 0.016]) | ([0.616, 0.765], [0.002, 0.014], [0.002, 0.017]) | ([0.461, 0.604], [0.013, 0.042], [0.009, 0.035]) |

${C}_{6}$ | ([0.685, 0.81], [0.001, 0.005], [0, 0.002]) | ([0.623, 0.803], [0.001, 0.007], [0, 0.002]) | ([0.546, 0.72], [0.001, 0.009], [0.001, 0.004]) | ([0.3, 0.5], [0.022, 0.08], [0.022, 0.044]) |

${C}_{7}$ | ([0.62, 0.802], [0.002, 0.013], [0, 0.004]) | ([0.618, 0.769], [0.002, 0.013], [0.001, 0.005]) | ([0.543, 0.72], [0.002, 0.015], [0.001, 0.006]) | ([0.438, 0.569], [0.024, 0.061], [0.012, 0.03]) |

${C}_{8}$ | ([0.491, 0.648], [0.005, 0.025], [0.002, 0.013]) | ([0.686, 0.862], [0.004, 0.02], [0, 0.006]) | ([0.499, 0.709], [0.005, 0.025], [0.001, 0.009]) | ([0.43, 0.567], [0.026, 0.071], [0.012, 0.033]) |

${C}_{9}$ | ([0.702, 0.847], [0.004, 0.021], [0, 0.007]) | ([0.761, 0.891], [0.004, 0.019], [0, 0.006]) | ([0.682, 0.828], [0.004, 0.022], [0, 0.007]) | ([0.488, 0.598], [0.026, 0.062], [0.009, 0.027]) |

${C}_{10}$ | ([0.687, 0.8], [0.002, 0.01], [0, 0.003]) | ([0.663, 0.836], [0.001, 0.008], [0, 0.003]) | ([0.718, 0.842], [0.001, 0.008], [0, 0.003]) | ([0.534, 0.636], [0.012, 0.032], [0.006, 0.018]) |

${C}_{11}$ | ([0.608, 0.751], [0.001, 0.009], [0.001, 0.003]) | ([0.557, 0.722], [0.001, 0.01], [0.001, 0.006]) | ([0.565, 0.75], [0.001, 0.011], [0.001, 0.004]) | ([0.499, 0.6], [0.023, 0.048], [0.009, 0.023]) |

${C}_{12}$ | ([0.36, 0.533], [0.043, 0.12], [0.021, 0.06]) | ([0.4, 0.516], [0.049, 0.11], [0.023, 0.065]) | ([0.463, 0.606], [0.033, 0.089], [0.012, 0.047]) | ([0.258, 0.439], [0.049, 0.133], [0.037, 0.087]) |

${C}_{13}$ | ([0.229, 0.373], [0.05, 0.119], [0.055, 0.108]) | ([0.229, 0.373], [0.05, 0.119], [0.055, 0.108]) | ([0.43, 0.568], [0.038, 0.095], [0.017, 0.047]) | ([0.43, 0.568], [0.038, 0.095], [0.017, 0.047]) |

${C}_{14}$ | ([0.284, 0.408], [0.083, 0.167], [0.046, 0.123]) | ([0.284, 0.408], [0.083, 0.167], [0.046, 0.123]) | ([0.269, 0.486], [0.071, 0.179], [0.03, 0.098]) | ([0.431, 0.592], [0.061, 0.137], [0.017, 0.076]) |

Criteria | Importance Aggregated Weights |
---|---|

${C}_{1}$ | ([0.999, 1], [0, 0.003], [0, 0.001]) |

${C}_{2}$ | ([0.997, 1], [0.001, 0.006], [0, 0.002]) |

${C}_{3}$ | ([0.985, 0.998], [0.003, 0.014], [0, 0.004]) |

${C}_{4}$ | ([0.978, 0.997], [0.003, 0.016], [0, 0.005]) |

${C}_{5}$ | ([0.959, 0.993], [0.002, 0.011], [0.001, 0.015]) |

${C}_{6}$ | ([0.999, 1], [0, 0.003], [0, 0.001]) |

${C}_{7}$ | ([0.993, 0.999], [0.002, 0.009], [0, 0.002]) |

${C}_{8}$ | ([0.975, 0.997], [0.004, 0.019], [0, 0.005]) |

${C}_{9}$ | ([0.975, 0.997], [0.004, 0.019], [0, 0.005]) |

${C}_{10}$ | ([0.996, 1], [0.001, 0.006], [0, 0.002]) |

${C}_{11}$ | ([0.998, 1], [0.001, 0.005], [0, 0.001]) |

${C}_{12}$ | ([0.963, 0.996], [0.022, 0.06], [0.004, 0.027]) |

${C}_{13}$ | ([0.977, 0.998], [0.016, 0.044], [0.005, 0.02]) |

${C}_{14}$ | ([0.897, 0.973], [0.05, 0.11], [0.009, 0.056]) |

Students | Weighted Ratings |
---|---|

${A}_{1}$ | ([0.605, 0.76], [0.004, 0.02], [0.001, 0.009]) |

${A}_{2}$ | ([0.594, 0.761], [0.004, 0.02], [0.001, 0.009]) |

${A}_{3}$ | ([0.581, 0.744], [0.004, 0.021], [0.001, 0.009]) |

${A}_{4}$ | ([0.458, 0.588], [0.022, 0.058], [0.011, 0.031]) |

**Table 11.**The distance of each student from ${A}_{{t}_{2}}^{+}$ and ${A}_{{t}_{2}}^{-}$ at period ${t}_{2}$.

Students | ${\mathit{d}}_{{\mathit{t}}_{2}}^{+}$ | ${\mathit{d}}_{{\mathit{t}}_{2}}^{-}$ |
---|---|---|

${A}_{1}$ | 0.188874 | 0.901553 |

${A}_{2}$ | 0.192392 | 0.900405 |

${A}_{3}$ | 0.200641 | 0.896588 |

${A}_{4}$ | 0.279475 | 0.848118 |

Students | Closeness Coefficients | Ranking Order |
---|---|---|

${A}_{1}$ | 0.826789 | 1 |

${A}_{2}$ | 0.823945 | 2 |

${A}_{3}$ | 0.817138 | 3 |

${A}_{4}$ | 0.752149 | 4 |

Criteria | Students | ||||
---|---|---|---|---|---|

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | |

${C}_{1}$ | ([0.794, 0.9], [0, 0], [0, 0]) | ([0.51, 0.75], [0, 0.006], [0, 0.002]) | ([0.764, 0.893], [0, 0], [0, 0]) | ([0.711, 0.822], [0, 0.003], [0, 0.001]) | ([0.441, 0.569], [0.022, 0.053], [0.012, 0.027]) |

${C}_{2}$ | ([0.871, 0.951], [0, 0], [0, 0]) | ([0.675, 0.818], [0, 0.002], [0, 0.001]) | ([0.881, 0.96], [0, 0], [0, 0]) | ([0.788, 0.891], [0, 0.001], [0, 0]) | ([0.441, 0.569], [0.022, 0.053], [0.012, 0.027]) |

${C}_{3}$ | ([0.829, 0.918], [0, 0.001], [0, 0]) | ([0.608, 0.785], [0.001, 0.005], [0, 0.001]) | ([0.728, 0.884], [0, 0.001], [0, 0]) | ([0.817, 0.91], [0, 0.001], [0, 0]) | ([0.569, 0.67], [0.005, 0.016], [0.004, 0.012]) |

${C}_{4}$ | ([0.711, 0.875], [0, 0.001], [0, 0]) | ([0.608, 0.785], [0.001, 0.005], [0, 0.001]) | ([0.816, 0.922], [0, 0.001], [0, 0]) | ([0.663, 0.795], [0, 0.002], [0, 0.001]) | ([0.569, 0.67], [0.005, 0.016], [0.004, 0.012]) |

${C}_{5}$ | ([0.81, 0.912], [0, 0.001], [0, 0.001]) | ([0.635, 0.804], [0, 0.003], [0, 0.001]) | ([0.777, 0.889], [0, 0.001], [0, 0.001]) | ([0.751, 0.872], [0, 0.001], [0, 0.001]) | ([0.48, 0.608], [0.011, 0.032], [0.008, 0.021]) |

${C}_{6}$ | ([0.832, 0.923], [0, 0], [0, 0]) | ([0.608, 0.785], [0, 0.004], [0, 0.001]) | ([0.744, 0.902], [0, 0], [0, 0]) | ([0.482, 0.69], [0.001, 0.006], [0.001, 0.003]) | ([0.536, 0.637], [0.011, 0.026], [0.006, 0.016]) |

${C}_{7}$ | ([0.689, 0.86], [0, 0.001], [0, 0]) | ([0.591, 0.759], [0.001, 0.004], [0, 0.002]) | ([0.682, 0.86], [0, 0.001], [0, 0]) | ([0.586, 0.733], [0.001, 0.004], [0.001, 0.002]) | ([0.441, 0.569], [0.022, 0.053], [0.012, 0.027]) |

${C}_{8}$ | ([0.751, 0.898], [0, 0.001], [0, 0]) | ([0.662, 0.822], [0, 0.003], [0, 0.001]) | ([0.732, 0.89], [0, 0.001], [0, 0]) | ([0.699, 0.83], [0, 0.004], [0, 0.001]) | ([0.268, 0.441], [0.027, 0.079], [0.033, 0.062]) |

${C}_{9}$ | ([0.874, 0.95], [0, 0.002], [0, 0]) | ([0.749, 0.861], [0, 0.002], [0, 0.001]) | ([0.889, 0.963], [0, 0.002], [0, 0]) | ([0.743, 0.853], [0, 0.003], [0, 0.001]) | ([0.418, 0.578], [0.011, 0.039], [0.011, 0.026]) |

${C}_{10}$ | ([0.757, 0.891], [0, 0], [0, 0]) | ([0.636, 0.804], [0, 0.003], [0, 0.001]) | ([0.837, 0.926], [0, 0], [0, 0]) | ([0.712, 0.818], [0, 0.002], [0, 0.001]) | ([0.5, 0.6], [0.022, 0.044], [0.009, 0.022]) |

${C}_{11}$ | ([0.753, 0.88], [0, 0], [0, 0]) | ([0.521, 0.71], [0.001, 0.005], [0.001, 0.003]) | ([0.696, 0.875], [0, 0.001], [0, 0]) | ([0.651, 0.769], [0.001, 0.004], [0, 0.002]) | ([0.569, 0.67], [0.005, 0.015], [0.004, 0.012]) |

${C}_{12}$ | ([0.753, 0.884], [0.001, 0.007], [0, 0.002]) | ([0.53, 0.662], [0.002, 0.011], [0.001, 0.007]) | ([0.778, 0.903], [0.001, 0.007], [0, 0.002]) | ([0.544, 0.72], [0.002, 0.013], [0.001, 0.005]) | ([0.534, 0.636], [0.012, 0.032], [0.006, 0.018]) |

${C}_{13}$ | ([0.677, 0.845], [0, 0.002], [0, 0.001]) | ([0.338, 0.521], [0.001, 0.006], [0.003, 0.009]) | ([0.759, 0.881], [0, 0.002], [0, 0]) | ([0.699, 0.83], [0.001, 0.004], [0, 0.001]) | ([0.374, 0.536], [0.022, 0.065], [0.016, 0.035]) |

${C}_{14}$ | ([0.688, 0.837], [0.001, 0.005], [0, 0.001]) | ([0.407, 0.555], [0.002, 0.008], [0.002, 0.008]) | ([0.777, 0.916], [0.001, 0.005], [0, 0.001]) | ([0.699, 0.826], [0.001, 0.007], [0, 0.002]) | ([0.44, 0.569], [0.023, 0.057], [0.012, 0.029]) |

Criteria | Importance Aggregated Weights |
---|---|

${C}_{1}$ | ([0.99999, 1], [0, 0.00009], [0, 0.00001]) |

${C}_{2}$ | ([0.99995, 1], [0.00001, 0.00019], [0, 0.00002]) |

${C}_{3}$ | ([0.99964, 1], [0.00005, 0.00062], [0, 0.00009]) |

${C}_{4}$ | ([0.99912, 0.99998], [0.00009, 0.00097], [0, 0.00018]) |

${C}_{5}$ | ([0.99776, 0.99995], [0.00004, 0.00077], [0.00001, 0.00053]) |

${C}_{6}$ | ([0.99999, 1], [0, 0.00006], [0, 0]) |

${C}_{7}$ | ([0.99985, 1], [0.00003, 0.00039], [0, 0.00005]) |

${C}_{8}$ | ([0.99907, 0.99999], [0.00009, 0.00114], [0, 0.00015]) |

${C}_{9}$ | ([0.99842, 0.99996], [0.00014, 0.00154], [0, 0.00024]) |

${C}_{10}$ | ([0.99991, 1], [0.00002, 0.00029], [0, 0.00004]) |

${C}_{11}$ | ([0.99997, 1], [0.00001, 0.00016], [0, 0.00001]) |

${C}_{12}$ | ([0.99615, 0.99988], [0.00112, 0.00657], [0.00004, 0.00152]) |

${C}_{13}$ | ([0.99969, 1], [0.00016, 0.00145], [0.00001, 0.00026]) |

${C}_{14}$ | ([0.99762, 0.99993], [0.00082, 0.00483], [0.00004, 0.00116]) |

Students | Weighted Ratings |
---|---|

${A}_{1}$ | ([0.78, 0.901], [0, 0.001], [0, 0]) |

${A}_{2}$ | ([0.589, 0.759], [0.001, 0.004], [0, 0.002]) |

${A}_{3}$ | ([0.785, 0.91], [0, 0.001], [0, 0]) |

${A}_{4}$ | ([0.693, 0.822], [0, 0.003], [0, 0.001]) |

${A}_{5}$ | ([0.476, 0.599], [0.014, 0.037], [0.009, 0.022]) |

**Table 16.**The distance of each student from ${A}_{{t}_{3}}^{+}$ and ${A}_{{t}_{3}}^{-}$ at period ${t}_{3}$.

Students | ${\mathit{d}}_{{\mathit{t}}_{3}}^{+}$ | ${\mathit{d}}_{{\mathit{t}}_{3}}^{-}$ |
---|---|---|

${A}_{1}$ | 0.37844 | 0.776416 |

${A}_{2}$ | 0.352522 | 0.752181 |

${A}_{3}$ | 0.381797 | 0.777005 |

${A}_{4}$ | 0.358066 | 0.764391 |

${A}_{5}$ | 0.325366 | 0.738391 |

Students | Closeness Coefficients | Ranking Order |
---|---|---|

${A}_{1}$ | 0.672305 | 4 |

${A}_{2}$ | 0.680890 | 3 |

${A}_{3}$ | 0.670525 | 5 |

${A}_{4}$ | 0.680998 | 2 |

${A}_{5}$ | 0.694135 | 1 |

Time Period | The Method in [14] | The Proposed Method |
---|---|---|

${t}_{1}$ | ${A}_{1}\succ {A}_{2}\succ {A}_{3}$ | ${A}_{1}\succ {A}_{2}\succ {A}_{3}$ |

${t}_{2}$ | ${A}_{4}\succ {A}_{2}\succ {A}_{3}\succ {A}_{1}$ | ${A}_{1}\succ {A}_{2}\succ {A}_{3}\succ {A}_{4}$ |

${t}_{3}$ | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}$ | ${A}_{5}\succ {A}_{4}\succ {A}_{2}\succ {A}_{1}\succ {A}_{3}$ |

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