# Learning Path Optimization Based on Multi-Attribute Matching and Variable Length Continuous Representation

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Learning Path Personalization

#### 2.2. Evolutionary Algorithms for Learning Path Generation

#### 2.3. Adaptive Differential Evolution for Continuous Problems

## 3. Problem Formulation

#### 3.1. Learner Attributes

#### 3.2. Material Attributes

#### 3.3. Decision Variables

#### 3.4. Cost Function

#### 3.4.1. Learner Ability and Material Difficulty

#### 3.4.2. Learning Objectives and Covered Concepts of Material

#### 3.4.3. Learning Style and Material Type

#### 3.4.4. Material and Its Prerequisites

#### 3.4.5. The Required Learning Time and Expected Learning Time

#### 3.4.6. Weight Selection of the Sub-Cost Functions

## 4. Algorithm

## 5. Evaluation

- (1)
- Hardware environment: 3.4 GHz Intel Core i7 processor with 32 GB memories;
- (2)
- Software environment: MATLAB R2018a.

#### 5.1. Convergence Analysis

#### 5.2. Influence of the Number of Materials

#### 5.3. Scalability Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Symmetrical attributes of learners and materials. The learning path is a sequence of learning materials selected by an evolutionary algorithm to optimize the cost function that describes the symmetrical attributes’ matching degree.

**Figure 2.**The average fitness convergence curves of BPSO, PSO, DE, and LSHADE-cnEpSin. The error bars of the LSHADE-cnEpSin are drawn in every 1000 function evaluations.

**Figure 3.**Comparison of the average cost function values obtained by BPSO, PSO, DE, and LSHADE-cnEpSin with a different number of materials.

**Figure 6.**Comparison of the averaged computation time consumed by BPSO, PSO, DE, and LSHADE-cnEpSin with a different number of materials.

Parameter Class | Parameter Symbols | Parameters Settings |
---|---|---|

Parameters regarding learners | ${A}_{k}$, $1\le k\le 100$ | The ability of the learner is randomly divided into five levels: (0.2, 0.4, 0.6, 0.8, 1.0) |

$L{C}_{k}$, $1\le k\le 100$ | The learning targets are randomly assigned to 100 learners, respectively. | |

$L{S}_{k}$, $1\le k\le 100$ | The normalized learning styles are randomly assigned to 100 learners, respectively. | |

$L{T}_{k}$, $1\le k\le 100$ | The expected learning time (including upper and lower bound) is randomly assigned to 100 learners. | |

Parameters regarding materials | ${D}_{n}$, $1\le n\le 10,000$ | Five difficulty levels, i.e., (0.2, 0.4, 0.6, 0.8, 1.0), are randomly assigned to 10,000 materials, respectively. |

$M{C}_{n}$, $1\le n\le 10,000$ | The covered concepts are randomly assigned to 10,000 materials, respectively. | |

$M{S}_{n}$, $1\le n\le 10,000$ | The supported learning styles are randomly assigned to 10,000 materials, respectively. | |

$M{T}_{n}$, $1\le n\le 10,000$ | The required time is randomly assigned to 10,000 materials, respectively. | |

$M{P}_{n}$, $1\le n\le 10,000$ | The prerequisites are randomly assigned to 10,000 materials, respectively. | |

Parameters regarding LSHADE-cnEpSin | $4\le NP\le 18\left(NU+1\right)$ | Initial population size is determined by the length of the encoded vector. The minimum population size is set to 4 to ensure the mutation is applicable. The population size is linearly reduced depending on the number of generations. |

${F}_{i}^{g}$ | The scaling factor is adaptively updated according to [14]. | |

$C{R}_{i}^{g}$ | Crossover probability is adaptively updated according to [14]. | |

$F{E}_{\mathrm{max}}=20,000$ | The maximum number of function evaluations. | |

Parameters regarding BPSO | $NP=50$ | Population size is set according to [46]. |

$c1=c2=2$ | Learning factors are set according to [46]. | |

Parameters regarding DE | $NP=18$ | Population size is set according to [37]. |

$CR=0.5026$ | Crossover probability is set according to [37]. | |

$F=0.6714$ | The scaling factor is set according to [37]. | |

Parameters regarding PSO | $NP=30$ | Population size is set according to [48]. |

$c1=c2=2$ | Learning factors are set according to [48]. | |

${w}_{\mathrm{min}}=0.4$ ${w}_{\mathrm{max}}=0.9$ | The upper and lower bounds of the inertia weight are set according to [48]. |

Algorithms | Success Rate | Avg. Mean | Avg. Min | Avg. Std |
---|---|---|---|---|

BPSO | 44% | 0.57 | 0.470972 | 0.0930082 |

PSO | 25% | 0.7 | 0.489049 | 0.122281 |

DE | 60% | 0.55 | 0.457864 | 0.0816821 |

LSHADE-cnEpSin | 80% | 0.53 | 0.448074 | 0.0584142 |

Ability Level | Learning Objectives | Learning Style | Expected Learning Time | ||
---|---|---|---|---|---|

Learner | 0.4 | (1, 5, 9, 10) | (2, 0, −1, 5) | (5.10, 8.24) | |

Selected Materials | Difficulty Level | Covered Concepts | Supported Learning Style | Prerequisite | Required Learning Time |

Material No. 1 | 0.6 | (5, 9) | (−6, −4, −3, −7) | NA | 1.92 |

Material No. 2 | 0.4 | 1 | (0, 5, −2, 6) | 9 (covered by material No. 1) | 1.43 |

Material No. 3 | 0.4 | 10 | (7, 4, −2, 2) | 1 (covered by material No. 2) | 1.99 |

Affinity Function Values | ${F}_{1}=0.067$ | ${F}_{2}=0$ | ${F}_{3}=0.267$ | ${F}_{4}=0$ | ${F}_{5}=0$ |

Remark | Average difficulty level: 0.47 | Total covered concepts: (1, 5, 9, 10) | Averaged supported learning style: (0.33, 1.67, −2.33, 0.33) | No prerequisite violation | Total required learning time: 5.34 |

Ability Level | Learning Objectives | Learning Style | Expected Learning Time | ||
---|---|---|---|---|---|

Learner | 0.4 | (1, 5, 9, 10) | (2, 0, −1, 5) | (5.10, 8.24] | |

Selected Materials | Difficulty Level | Covered Concepts | Supported Learning Style | Prerequisite | Required Learning Time |

Material No. 1 | 0.4 | 10 | (−1, −4, −3, 1) | NA | 1.20 |

Material No. 2 | 0.4 | 1 | (6, −4, 7, 7) | NA | 2.00 |

Material No. 3 | 0.4 | (5, 9) | (−3, −1, 0, 2) | [1; 10] (covered by materials No. 1 and 2) | 1.91 |

Affinity Function Values | ${F}_{1}=0$ | ${F}_{2}=0$ | ${F}_{3}=0.2278$ | ${F}_{4}=0$ | ${F}_{5}=0$ |

Remark | Average difficulty level: 0.4 | Total covered concepts: (1, 5, 9, 10) | Averaged supported learning style: (0.67, −3, 1.33, 3.33) | No prerequisite violation | Total required learning time: 5.11 |

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**MDPI and ACS Style**

Zhang, Y.-W.; Xiao, Q.; Song, Y.-L.; Chen, M.-M.
Learning Path Optimization Based on Multi-Attribute Matching and Variable Length Continuous Representation. *Symmetry* **2022**, *14*, 2360.
https://doi.org/10.3390/sym14112360

**AMA Style**

Zhang Y-W, Xiao Q, Song Y-L, Chen M-M.
Learning Path Optimization Based on Multi-Attribute Matching and Variable Length Continuous Representation. *Symmetry*. 2022; 14(11):2360.
https://doi.org/10.3390/sym14112360

**Chicago/Turabian Style**

Zhang, Yong-Wei, Qin Xiao, Ying-Lei Song, and Mi-Mi Chen.
2022. "Learning Path Optimization Based on Multi-Attribute Matching and Variable Length Continuous Representation" *Symmetry* 14, no. 11: 2360.
https://doi.org/10.3390/sym14112360