# Design and Optimization of a Fuzzy Logic System for Academic Performance Prediction

^{*}

## Abstract

**:**

## 1. Introduction

#### Proposed Approach and Organization

## 2. Methodology

#### 2.1. Data Collection and Analysis

- $i=1,2,3\dots ,N$: represents the academic marks; the rows in Table 1.
- N: the number of academic marks in the academic period (number of rows in Table 1).
- $p=1,2,3,4$: is the academic period of the year.
- $a=1,2,3,\dots ,8$: the subject, depending on the grade (see Table 2).
- $g=1,2,3$: the respective grade (level); $g=1$ for 8th, $g=2$ for 9th, $g=3$ for 10th, and $g=4$ for 11th.

- Time interval: 4 years
- Number of students: 21
- Number of input-output pairs: 168
- Total marks used: 4800

#### 2.2. Optimization Algorithm Selection

- Evolutionary algorithms.
- Swarm algorithms.
- Simulated annealing.
- Scattered search.
- Neighborhood search.
- Iterative local search.
- Multi-agent systems.

#### 2.3. Fuzzy Logic System Design

#### 2.4. Takagi–Sugeno Fuzzy Systems

- Zero-order TS fuzzy model: function f is a constant value $f=c$.
- First-order TS fuzzy model: function f is a first-order polynomial $f=ax+by+c$.

- Rule Number 1: If x is ${\mu}_{{A}_{1}}$ and y is ${\mu}_{{B}_{1}}$, then ${f}_{1}={p}_{1}x+{q}_{1}y+{r}_{1}$.
- Rule Number 2: If x is ${\mu}_{{A}_{2}}$ and y is ${\mu}_{{B}_{2}}$, then ${f}_{2}={p}_{2}x+{q}_{2}y+{r}_{2}$.

#### 2.5. Prediction Models

#### 2.6. System Optimization

## 3. Result Analysis

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Grade 9th, $(\mathit{g}=2)$ | ||||
---|---|---|---|---|

Mark | Academic Period $\mathit{p}$ | |||

i | 1 | 2 | 3 | 4 |

1 | 63.50 | 57.00 | 74.00 | 67.50 |

2 | 83.50 | 82.00 | 70.50 | 86.25 |

3 | 75.00 | 81.25 | 75.75 | 77.75 |

4 | 79.25 | 90.25 | 74.00 | 76.25 |

5 | 82.75 | 86.00 | 82.75 | 80.25 |

6 | 70.25 | 81.00 | 74.00 | 72.25 |

${\overline{x}}_{p,a,g}$ (Mean) | 76.80 | 79.30 | 75.40 | 77.60 |

a | 8th and 9th | 10th | 11th |
---|---|---|---|

1 | Sciences | Chemistry | Chemistry |

2 | Algebra and Geometry | Physics | Physics |

3 | Social Sciences | Trigonometry | Calculus |

4 | Spanish | Political Sciences | Political Sciences |

5 | English | Spanish | Spanish |

6 | Business training | English | English |

7 | - | Statistics | Statistics |

8 | - | Business Training | Business Training |

Features | Grade (Input) | Result (Output) | ||
---|---|---|---|---|

(Inputs) | 8th | 9th | 10th | 11th |

Term 1 | 83.50 | 87.00 | 92.33 | 81.00 |

Term 2 | 82.00 | 76.75 | 89.00 | 80.00 |

Term 3 | 70.50 | 70.00 | 84.00 | 76.67 |

Term 4 | 86.25 | 79.50 | 89.00 | 94.00 |

Final Exam | 85.00 | 86.00 | 72.00 | - |

# Fails | 6 | 1 | 1 | - |

ID Subject | - | - | 1 | - |

- | 5 | 10 | 20 | 50 | 70 | |||||
---|---|---|---|---|---|---|---|---|---|---|

- | Rules | Rules | Rules | Rules | Rules | |||||

Number of | Best | Mean | Best | Mean | Best | Mean | Best | Mean | Best | Mean |

Generations | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ |

1 | 18.5 | 36.75 | 22.08 | 36.92 | 19.03 | 35.08 | 19.58 | 34.9 | 18.93 | 34.82 |

5 | 17.34 | 27.54 | 17.24 | 29.78 | 16.96 | 28.03 | 15.8 | 29.2 | 17.16 | 27.75 |

10 | 15.63 | 21.95 | 14.23 | 20.17 | 13.37 | 24.76 | 14.06 | 21.62 | 13.37 | 22.72 |

50 | 11.4 | 11.85 | 9.405 | 9.841 | 8.379 | 8.737 | 9.648 | 9.956 | 8.204 | 8.337 |

80 | 11.4 | 11.4 | 9.333 | 9.558 | 7.712 | 8.066 | 8.288 | 8.583 | 7.786 | 7.988 |

90 | 11.4 | 11.4 | 9.322 | 9.641 | 7.51 | 7.793 | 7.93 | 8.26 | 7.732 | 7.824 |

100 | 11.4 | 11.4 | 9.314 | 9.569 | 7.487 | 7.725 | 7.772 | 7.955 | 7.526 | 7.677 |

150 | - | - | 9.3 | 9.74 | 7.375 | 8.27 | 7.339 | 7.48 | 7.065 | 7.224 |

200 | - | - | 9.296 | 9.561 | 7.339 | 7.698 | 7.225 | 7.278 | 7 | 7.054 |

- | 5 | 10 | 20 | 50 | 70 | |||||
---|---|---|---|---|---|---|---|---|---|---|

- | Rules | Rules | Rules | Rules | Rules | |||||

Number of | Best | Mean | Best | Mean | Best | Mean | Best | Mean | Best | Mean |

Iterations | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ |

1 | 21.53 | 40.44 | 19.29 | 37.04 | 17.49 | 35.37 | 17.1 | 33.68 | 17.96 | 33.44 |

5 | 17.28 | 35.27 | 17.53 | 33.97 | 16.74 | 30.75 | 17.1 | 32.26 | 16.04 | 29.42 |

10 | 17.28 | 35.95 | 16.25 | 32.34 | 16.38 | 30.05 | 16.19 | 30.6 | 16.04 | 27.09 |

50 | 13.05 | 20.38 | 11.74 | 18.6 | 10.48 | 19.18 | 11.63 | 24.65 | 8.565 | 16.24 |

80 | 13.03 | 14.84 | 11.67 | 17.38 | 8.467 | 13.49 | 10.59 | 16.82 | 7.688 | 12.24 |

90 | 13.03 | 15.14 | 10.92 | 15.26 | 8.43 | 14.68 | 10.7 | 16.82 | 7.492 | 11.2 |

100 | 13.03 | 15.04 | 9.707 | 19.84 | 8.215 | 12.09 | 10.33 | 16.24 | 7.373 | 9.977 |

150 | 13 | 14 | 9.348 | 11.74 | 7.893 | 9.314 | 9.906 | 18.63 | 7.25 | 10.83 |

160 | 13 | 13.85 | 9.337 | 10.79 | 7.813 | 12.55 | 9.827 | 13.79 | 7.237 | 8.013 |

190 | - | - | 9.175 | 9.936 | 7.724 | 9.467 | 9.634 | 13.98 | 7.198 | 7.802 |

200 | - | - | 9.174 | 10.14 | 7.72 | 12.97 | 9.599 | 13.39 | 7.196 | 8.773 |

- | 5 | 10 | 20 | 50 | 70 | |||||
---|---|---|---|---|---|---|---|---|---|---|

- | Rules | Rules | Rules | Rules | Rules | |||||

Number of | Best | Mean | Best | Mean | Best | Mean | Best | Mean | Best | Mean |

Iterations | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ | $\mathit{f}\left(\mathit{X}\right)$ |

1 | 24.38 | 29.37 | 33.29 | 33.29 | 27.82 | 29.22 | 26.5 | 26.5 | 41.45 | 41.44 |

5 | 24.38 | 46.64 | 21.69 | 26.53 | 21.64 | 35.06 | 26.5 | 42.88 | 19.99 | 22.56 |

10 | 24.38 | 36.78 | 21.69 | 22.68 | 21.64 | 30.23 | 26.5 | 36.12 | 19.99 | 24.06 |

50 | 19.38 | 25.2867 | 16.07 | 18.82 | 17.94 | 29.59 | 22.66 | 24.96 | 19.44 | 19.44 |

100 | 19.38 | 35.73 | 16.07 | 20.76 | 16.03 | 33.02 | 20.39 | 30.93 | 18.95 | 20.6 |

150 | 17.75 | 36.28 | 16.07 | 32.04 | 16.03 | 29.66 | 20.39 | 41.83 | 18.95 | 34.86 |

200 | 17.75 | 33.26 | 16.07 | 25.61 | 16.03 | 33.25 | 20.39 | 33.68 | 18.04 | 18.09 |

- | 5 | 10 | 20 | 50 | 70 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Number of | Rules | Rules | Rules | Rules | Rules | |||||

Iterations | $\mathit{f}\left(\mathit{X}\right)$ | Size Grid | $\mathit{f}\left(\mathit{X}\right)$ | Size Grid | $\mathit{f}\left(\mathit{X}\right)$ | Size Grid | $\mathit{f}\left(\mathit{X}\right)$ | Size Grid | $\mathit{f}\left(\mathit{X}\right)$ | Size Grid |

1 | 59.45 | 2 | 59.45 | 2 | 59.45 | 2 | 59.45 | 2 | 59.45 | 2 |

5 | 35.21 | 2 | 35.21 | 2 | 35.21 | 2 | 35.21 | 2 | 35.21 | 2 |

10 | 16.77 | 1 | 16.77 | 1 | 16.77 | 1 | 16.77 | 1 | 16.77 | 1 |

15 | 16.77 | 0.125 | 16.77 | 0.125 | 16.77 | 0.125 | 16.77 | 0.125 | 16.77 | 0.125 |

20 | 16.77 | 0.004 | 16.77 | 0.004 | 16.77 | 0.004 | 16.77 | 0.004 | 16.77 | 0.004 |

25 | 16.77 | 0.0001 | 16.77 | 0.0001 | 16.77 | 0.0001 | 16.77 | 0.0001 | 16.77 | 0.0001 |

30 | 16.77 | $3.80\times {10}^{-6}$ | 16.77 | $3.80\times {10}^{-6}$ | 16.77 | $3.80\times {10}^{-6}$ | 16.77 | $3.80\times {10}^{-6}$ | 16.77 | $3.80\times {10}^{-6}$ |

32 | 16.77 | $9.50\times {10}^{-7}$ | 16.77 | $9.50\times {10}^{-7}$ | 16.77 | $9.50\times {10}^{-7}$ | 16.77 | $9.50\times {10}^{-7}$ | 16.77 | $9.50\times {10}^{-7}$ |

**Table 8.**Error measurement considering different numbers of rules. PS, Pattern Search; SMAPE, Symmetric Mean Absolute Percentage Error.

5 Rules | ||||

Measure | GA | PSO | SA | PS |

MAD | 10.415 | 10.5289 | 14.1012 | 13.7737 |

RMSE | 13.1114 | 12.7891 | 16.8512 | 15.9708 |

SMAPE | 16.04% | 15.49% | 21.58% | 20.99% |

10 Rules | ||||

Measure | GA | PSO | SA | PS |

MAD | 6.8494 | 8.0779 | 13.2082 | 13.7737 |

RMSE | 9.0726 | 9.976 | 15.3469 | 15.9708 |

SMAPE | 10.16% | 12.40% | 20.19% | 20.99% |

20 Rules | ||||

Measure | GA | PSO | SA | PS |

MAD | 5.4282 | 5.9044 | 13.6077 | 13.7737 |

RMSE | 7.4833 | 8.1553 | 15.8417 | 15.9708 |

SMAPE | 8.32% | 9.13% | 21.30% | 20.99% |

50 Rules | ||||

Measure | GA | PSO | SA | PS |

MAD | 5.5452 | 7.6017 | 17.5366 | 13.7737 |

RMSE | 7.7689 | 9.7637 | 20.1783 | 15.9708 |

SMAPE | 8.69% | 11.81% | 28.31% | 20.99% |

70 Rules | ||||

Measure | GA | PSO | SA | PS |

MAD | 5.3774 | 5.4175 | 15.0029 | 13.7737 |

RMSE | 7.6123 | 7.6526 | 17.3534 | 15.9708 |

SMAPE | 8.38% | 8.38% | 22.70% | 20.99% |

- | Output | |||
---|---|---|---|---|

Measure | Mark 1 | Mark 2 | Mark 3 | Mark 4 |

MAD for 20 R | 4.8728 | 5.0576 | 7.6355 | 4.1469 |

MAD for 70 R | 5.2924 | 5.4486 | 7.1452 | 3.6234 |

RMSE for 20 R | 6.3535 | 6.673 | 10.7386 | 4.877 |

RMSE for 70 R | 6.6572 | 7.5007 | 10.44 | 4.7133 |

SMAPE for 20 R | 7.64% | 8.62% | 11.25% | 5.78% |

SMAPE for 70 R | 8.28% | 9.35% | 10.76% | 5.15% |

Rules | GA | PSO | SA | PS |
---|---|---|---|---|

5 | 30.08 | 23.89 | 4.21 | 6.48 |

10 | 81.40 | 43.46 | 10.98 | 23.47 |

20 | 190.78 | 89.34 | 38.93 | 98.86 |

50 | 610.77 | 273.19 | 288.67 | 828.35 |

70 | 1007.98 | 447.24 | 636.74 | 1945.20 |

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**MDPI and ACS Style**

Rojas, J.A.; Espitia, H.E.; Bejarano, L.A.
Design and Optimization of a Fuzzy Logic System for Academic Performance Prediction. *Symmetry* **2021**, *13*, 133.
https://doi.org/10.3390/sym13010133

**AMA Style**

Rojas JA, Espitia HE, Bejarano LA.
Design and Optimization of a Fuzzy Logic System for Academic Performance Prediction. *Symmetry*. 2021; 13(1):133.
https://doi.org/10.3390/sym13010133

**Chicago/Turabian Style**

Rojas, Juan A., Helbert E. Espitia, and Lilian A. Bejarano.
2021. "Design and Optimization of a Fuzzy Logic System for Academic Performance Prediction" *Symmetry* 13, no. 1: 133.
https://doi.org/10.3390/sym13010133