# Time Series Forecasting Using a Two-Level Multi-Objective Genetic Algorithm: A Case Study of Maintenance Cost Data for Tunnel Fans

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Collection

#### 2.2. The ARIMA Model

- ${y}_{t}$: the actual data over time;
- $\mu $: the mean value of the time series data;
- $p$: the number of autoregressive cut-off lags;
- $d$: the number of differences calculated with the equation $\Delta {y}_{t}={y}_{t}-{y}_{t-1};$
- $q$: the number of cut-off lags of the moving average process;
- $\sigma $: autoregressive coefficients (AR);
- $\theta $: moving average coefficients (MA);
- $t$: time $\left\{1,\dots ,k\right\};$
- $\u03f5$: the white noise of the time series data.

#### 2.3. Two-Level System of Multi-Objective Genetic Algorithms

#### 2.3.1. Level 1: Multi-Objective GA Based on the ARIMA Model

- ${y}_{t}$: the actual data over time;
- $\alpha $: constant estimated value of the time series data;
- $p$: the hypothesis is either $p$ = 1 or $p$ < 1;
- $t$: time $\left\{1,\dots ,k\right\}$;
- $\epsilon $: the white noise of the time series data.

- $\mu $: the mean value of the time series data;
- $p$: the number of autoregressive cut-off lags;
- $d$: the number of differences calculated with the equation $\Delta {y}_{t}={y}_{t}-{y}_{t-1}$;
- $q$: the number of cut-off lags of the moving average process;
- $\sigma $: autoregressive coefficients (AR);
- $\theta $: moving average coefficients (MA);
- $t$: time $\left\{1,\dots ,k\right\}$;
- $\u03f5$: the white noise of the time series data.

- $m$: months {1, 2, 3,…, $m$}.

#### 2.3.2. Level 2: Multi-Objective GA for Measuring the Forecasting Accuracy

- $t$: time $\left\{1,\dots ,k\right\}$;
- ${Y}_{t}$: the actual data over time;
- ${F}_{t}$: the forecasted data over time.

#### 2.4. Multi-Objective Genetic Algorithms (GAs) Based on the Dynamic Regression Model

- $C$: constant value calculated with the normal equation, where ${X}^{T}XA={X}^{T}b$;
- ${b}_{1}$ and ${b}_{2}$: calculated with the normal equation, where ${X}^{T}XA={X}^{T}b$;
- ${Y}_{t}$: related to ${Y}_{t-1}$ and ${Y}_{t-2}$;
- $WN$: white noise.

#### 2.5. Models Evaluation Method

## 3. Results and Discussion

#### 3.1. Results of the ARIMA Model

#### 3.2. Results of the Two-Level System of Multi-Objective GAs

#### 3.2.1. Results for Level 1: Multi-Objective GA Based on the ARIMA Model

#### 3.2.2. Results of Level 2: Multi-Objective GA for Measuring the Forecasting Accuracy

#### 3.3. Results of the Multi-Objective Genetic Algorithms (GAs) Based on the Dynamic Regression Model

#### 3.4. Results of a Comparison of the Methods

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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DV1 | DV2 | |
---|---|---|

ARIMA model | 0.2374 | 1.2169 |

Multi-objective GA based on the ARIMA model | 0.1192 | 0.3869 |

Multi-objective GA based on the dynamic regression model | 1.2630 | 6.2324 |

DV1 | DV2 | |
---|---|---|

ARIMA model | 0.4171 | 1.0438 |

Multi-objective GA based on the ARIMA model | 0.3555 | 0.8477 |

Multi-objective GA based on the dynamic regression model | 0.5615 | 3.9543 |

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**MDPI and ACS Style**

Al-Douri, Y.K.; Hamodi, H.; Lundberg, J.
Time Series Forecasting Using a Two-Level Multi-Objective Genetic Algorithm: A Case Study of Maintenance Cost Data for Tunnel Fans. *Algorithms* **2018**, *11*, 123.
https://doi.org/10.3390/a11080123

**AMA Style**

Al-Douri YK, Hamodi H, Lundberg J.
Time Series Forecasting Using a Two-Level Multi-Objective Genetic Algorithm: A Case Study of Maintenance Cost Data for Tunnel Fans. *Algorithms*. 2018; 11(8):123.
https://doi.org/10.3390/a11080123

**Chicago/Turabian Style**

Al-Douri, Yamur K., Hussan Hamodi, and Jan Lundberg.
2018. "Time Series Forecasting Using a Two-Level Multi-Objective Genetic Algorithm: A Case Study of Maintenance Cost Data for Tunnel Fans" *Algorithms* 11, no. 8: 123.
https://doi.org/10.3390/a11080123