# Table Organization Optimization in Schools for Preserving the Social Distance during the COVID-19 Pandemic

^{*}

## Abstract

**:**

## Featured Application

**The obtainment of a methodology for maximizing the social distancing by increasing the distance among the school desks in the classrooms during the coronavirus pandemic through a Genetic Algorithm optimization.**

## Abstract

## 1. Introduction

## 2. Analysis and Complexity Studies on the Table Location Problem

## 3. Problem Definition and Scenario of Application

#### 3.1. Problem Definition

#### 3.2. Scenario of Application

## 4. Genetic Algorithm Optimization for the TLP

#### 4.1. Codification of the Individuals

#### 4.2. Evaluation of the Individuals

#### 4.3. Selection and Elitism

#### 4.4. Crossover and Mutation

#### 4.5. Stop Criteria

## 5. Results

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GA | Genetic Algorithm |

MP2 | Two-Points MultiPoint crossover |

MP3 | Three-Points MultiPoint crossover |

NLP | Node Location Problem |

R | Roulette selection |

SARS-CoV-2 | Severe Acute Respiratory Syndrome |

TLE | Table Location Environment |

TLP | Table Location Problem |

T2 | Tournament-2 selection |

T3 | Tournament-3 selection |

U | Uniform crossover |

WHO | World Health Organization |

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**Figure 1.**Picture of both classrooms from the Marist Brothers School San José, used for the scenario modeling and optimization in this paper.

**Figure 2.**Models for the two classrooms under study, Class A (

**a**) deployed an orthogonal mesh distribution while Class B (

**b**) displayed a triangular mesh table allocation.

**Figure 3.**Binary codification of the population in the Genetic Algorithm (GA) proposed for the Table Location Problem (TLP).

**Figure 5.**Table distributions obtained for each scenario. The application of each optimization is based on the configuration proposed in Table 4.

**Figure 6.**Evolution of the GA optimization for both scenarios and convergence to the final solution. Class B requires additional generations to obtain an adequate solution, due to the restrictive limitations of the scenario. The rapid increase in fitness value for both functions when surpassing the 0 axis is due to the change in the fitness evaluation technique.

**Table 1.**List of parameters measured from the studied classrooms, implemented into the scenario modeling and table distribution optimization.

Parameters | Values |
---|---|

Table Length | 69.5 cm |

Table Width | 49.5 cm |

Student Space Radius | 10 cm |

Distance from Student to Table | 20 cm |

**Table 2.**GA hyperparameters used for all simulations, the hyperparameter values were optimized and adjusted experimentally. Elitism and Mutation values are provisional and require further adjustment.

GA Hyperparameters | Class A | Class B |
---|---|---|

Stop criteria | 800 Generations | |

30% Population equal | ||

Number of Individuals | 200 | |

Elitism | 10% | |

Mutation | 10% | |

$\rho $ | 2 m | |

$\kappa $ | 500 m | |

Number of Tables | 16 | 21 |

TLP Points | 4096 | 16,384 |

Number of Possible Combinations | 6.09 × ${10}^{57}$ | 3.14 × ${10}^{88}$ |

**Table 3.**Comparison of the fitness value obtained by the multiple selection and crossover genetic operators previously proposed.

Crossover Operators | Tournament 2 | Tournament 3 | Roulette | |||
---|---|---|---|---|---|---|

Max | Mean | Max | Mean | Max | Mean | |

Single point—Class A | 1.648 | 1.553 | 1.602 | 1.264 | 1.515 | 1.169 |

Two-point—Class A | 1.634 | 1.229 | 1.572 | 1.237 | 1.563 | 1.158 |

Three-point—Class A | 1.652 | 1.57 | 1.646 | 1.579 | 1.638 | 1.545 |

Uniform—Class A | 1.622 | 1.089 | 1.669 | 0.739 | 1.506 | 0.11 |

Single point—Class B | −0.262 | −4.76 | 1.681 | −2.312 | −1.296 | −.611 |

Two-point—Class B | 1.676 | −0.628 | −0.876 | −2.017 | 1.675 | −1.264 |

Three-point—Class B | 1.677 | −1.046 | −0.482 | −3.102 | 1.645 | −2.04 |

Uniform—Class B | −0.657 | −2.961 | −0.866 | −4.204 | −1.64 | −4.59 |

**Table 4.**Adjustment of the most rewarding elitism and mutation values for each particular scenario, for the methodologies selected. Values were obtained experimentally.

Scenario | Methodology | Elitism | Mutation |
---|---|---|---|

Class A | T2-MP3 | 15% | 12.5% |

Class B | T2-MP2 | 7.5% | 7.5% |

**Table 5.**Comparison of the mean table separation achieved between the GA optimized distribution and the initial mesh models.

Scenario | GA Optimization | Original Distribution | Improvement |
---|---|---|---|

Class A | 1.79 m | 1.5 m | 19.33% |

Class B | 1.65 m | 1.5 m | 10% |

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**MDPI and ACS Style**

Ferrero-Guillén, R.; Díez-González, J.; Verde, P.; Álvarez, R.; Perez, H.
Table Organization Optimization in Schools for Preserving the Social Distance during the COVID-19 Pandemic. *Appl. Sci.* **2020**, *10*, 8392.
https://doi.org/10.3390/app10238392

**AMA Style**

Ferrero-Guillén R, Díez-González J, Verde P, Álvarez R, Perez H.
Table Organization Optimization in Schools for Preserving the Social Distance during the COVID-19 Pandemic. *Applied Sciences*. 2020; 10(23):8392.
https://doi.org/10.3390/app10238392

**Chicago/Turabian Style**

Ferrero-Guillén, Rubén, Javier Díez-González, Paula Verde, Rubén Álvarez, and Hilde Perez.
2020. "Table Organization Optimization in Schools for Preserving the Social Distance during the COVID-19 Pandemic" *Applied Sciences* 10, no. 23: 8392.
https://doi.org/10.3390/app10238392