# Assessing a Multi-Objective Genetic Algorithm with a Simulated Environment for Energy-Saving of Air Conditioning Systems with User Preferences

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## Abstract

**:**

## 1. Introduction

## 2. Algorithm Design

#### 2.1. Problem Definition

#### 2.2. GA and NSGA II

**Population**. We use a vector of one hundred solutions for both algorithms, where each solution $\left({T}_{{o}_{i}}\in \mathbb{R}\right)$ is a possible temperature between 22 and 27 °C.

**Selection**. The algorithms use a binary tournament between random individuals of the population to select the parents for the crossover operator. The GA uses a binary tournament, comparing the solutions’ objective values, and the winners are selected for crossover. On the other hand, the NSGA II algorithm uses the rank of the Pareto front of each solution to identify the winner, which will be the solution with the lowest level on the Pareto front, meaning that it is less dominated than the other solution. However, suppose both solutions are in the same level of the Pareto front. In that case, we use the crowded comparison operator (${>}_{n}$), which identifies the solution with the largest crowding distance, which will be explained further in this section.

**Crossover**. As stated before, our GA and NSGA II have a static population of one hundred solutions; then, another one hundred solutions are produced as offspring, using the Simulated Binary Crossover operator [9]. This operator requires two parents ($p1$ and $p2$) to obtain two new individuals (${T}_{{o}_{i}}$ and ${T}_{{o}_{i+1}}$).

**Mutation**. As a mutation operator, we used the polynomial mutation operator [10], which is applied to fifty percent of the solutions generated in the crossover operator.

**Fast Non-Dominated Sort**. This procedure allocates the solutions of the pool of solutions ($P$) into Pareto fronts of non-dominated individuals; this allows elitism in selecting the parents for the offspring of NSGA II.

**Crowding Distance**. A particular solution’s crowding distance value is the average distance of its two neighboring solutions [11]. Figure 4 shows the calculation of the crowding distance of the solution i, which is an estimate of the size of the largest cuboid enclosing i without including any other solution. The algorithm used in this implementation is a normalized version of the original algorithm with the minimum ($MI{N}_{m}$) and maximum ($MA{X}_{m}$) values per objective ($m$) [12]; this algorithm is calculated for each generation for the current population. The algorithm requires a set of non-dominated individuals ($\mathrm{l}$) to calculate their distance.

## 3. Simulator

## 4. Experimental Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Example of user satisfaction gain (${G}_{us}$) and the energy-saving gain (${G}_{es}$) calculation.

**Figure 11.**Simulation of temperature according to the recommendation of the genetic algorithms for configuration A.

**Figure 12.**Simulation of temperature according to the recommendation of the genetic algorithms for configuration B.

**Figure 13.**Simulation of temperature according to the recommendation of the genetic algorithms for configuration C.

Configuration | ${\mathit{\alpha}}_{\mathit{u}\mathit{s}}$ | ${\mathit{\alpha}}_{\mathit{e}\mathit{s}}$ |
---|---|---|

A | 0.2 | 0.8 |

B | 0.5 | 0.5 |

C | 0.8 | 0.2 |

Configuration | NSGA II | GA |
---|---|---|

A | 698.10343 | 993.36546 |

B | 316.98898 | 1058.35429 |

C | 313.84696 | 313.88824 |

Configuration | NSGA II | GA | ||
---|---|---|---|---|

kWh | Cost | kWh | Cost | |

A | 83.74032 | $250.97 | 86.01629 | $257.79 |

B | 94.29362 | $282.60 | 93.56877 | $280.43 |

C | 89.98913 | $269.70 | 91.99613 | $275.71 |

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**MDPI and ACS Style**

García Ruiz, A.H.; Ibarra Martínez, S.; Castán Rocha, J.A.; Terán Villanueva, J.D.; Laria Menchaca, J.; Treviño Berrones, M.G.; Ponce Flores, M.P.; Santiago Pineda, A.A.
Assessing a Multi-Objective Genetic Algorithm with a Simulated Environment for Energy-Saving of Air Conditioning Systems with User Preferences. *Symmetry* **2021**, *13*, 344.
https://doi.org/10.3390/sym13020344

**AMA Style**

García Ruiz AH, Ibarra Martínez S, Castán Rocha JA, Terán Villanueva JD, Laria Menchaca J, Treviño Berrones MG, Ponce Flores MP, Santiago Pineda AA.
Assessing a Multi-Objective Genetic Algorithm with a Simulated Environment for Energy-Saving of Air Conditioning Systems with User Preferences. *Symmetry*. 2021; 13(2):344.
https://doi.org/10.3390/sym13020344

**Chicago/Turabian Style**

García Ruiz, Alejandro Humberto, Salvador Ibarra Martínez, José Antonio Castán Rocha, Jesús David Terán Villanueva, Julio Laria Menchaca, Mayra Guadalupe Treviño Berrones, Mirna Patricia Ponce Flores, and Aurelio Alejandro Santiago Pineda.
2021. "Assessing a Multi-Objective Genetic Algorithm with a Simulated Environment for Energy-Saving of Air Conditioning Systems with User Preferences" *Symmetry* 13, no. 2: 344.
https://doi.org/10.3390/sym13020344