## 3 The Proposed Framework
### Triple Exponential Smoothing  
Exponential smoothing methods offer precise forecasts by relying on historical data and by assigning greater importance to recent observations. Specifically, time-series data can be modeled using three components: trend or long-term variation, seasonal components, and irregular or unpredictable components. These components can be combined using either additive or multiplicative procedures. In the additive model, the series demonstrated consistent cyclic variations, regardless of their level. However, the multiplicative model assumes that the amplitude of seasonal fluctuations changes as the series trend varies. It is important to note that a multiplicative HW model can be transformed into an additive HW model by applying a Box-Cox transformation. This statistical technique is used to modify a dataset so that the transformed data have a more suitable distribution for analysis. The additive method predicts the values of time series \(y_{t}\) using the following procedure:  
\[\hat{y}_{t}=L_{t-1}+T_{t-1}+S_{t-s}. \tag{1}\]  
At each iteration of index \(t\), the algorithm updates \(L_{t}\), \(T_{t}\), and \(S_{t}\) using three equations, each depending on a smoothing parameter with values in the interval \([0,1]\), that is, \(\alpha\), \(\beta\), and \(\gamma\), respectively. The parameter \(\alpha\) in the level equation is responsible for smoothing. A lower value of \(\alpha\) places greater emphasis on the historical data, whereas a higher value assigns more weight to recent observations. Parameter \(\beta\) is used to determine the trend estimation and is adjusted based on its proximity to 0 and 1. A value close to zero emphasizes the trend, whereas a value close to one assigns more weight to the observations. The parameter \(\gamma\) is responsible for controlling the smoothness of the seasonal component and affects the model's sensitivity to fluctuations in the series. As the value of \(\gamma\) increases, predictions become more sensitive. Algorithm 1 describes the additive HW (AHW) method used in the study. The algorithm requires three inputs: seasonal parameter \(s\), forecast time width \(h\), and historical time series of \(n\) values, denoted as \(\{y_{t}\}_{t=1}^{n}\). The output of the algorithm is \(\hat{y}_{n+h|n}\), that is, the forecast at the final instant \(t=n+h\). Therefore, the algorithm produces forecasts based on historical data, estimated trends, and seasonal patterns. ```
1:\(s\), \(h\) and \(\{y_{t}\}_{t=1}^{n}\)
2:\(\hat{y}_{n+h|n}\)
3:Set the initial values of \(L_{t<s}=0,L_{s}=\frac{1}{s}\sum_{t=1}^{s}y_{t}\)
4:Set the initial values of \(T_{t<s}=0,T_{s}=\frac{1}{s^{2}}\sum_{t=1}^{s}\left(y_{t+s}-y_{t}\right)\)
5:Set the initial values of \(S_{-s+2}=(y_{1}-L_{s}),\dots,S_{0}=(y_{s-1}-L_{s}),S_{1}=-L_{s}\)
6:for\(t=s+1\); \(t\leq n-1\); \(t=t+1\)do
7:\(L_{t}=\alpha(y_{t}-S_{t-s})+(1-\alpha)(L_{t-1}+T_{t-1})\)
8:\(T_{t}=\beta(L_{t}-L_{t-1})+(1-\beta)T_{t-1}\)
9:\(S_{t}=\gamma(y_{t}-L_{t})+(1-\gamma)S_{t-s}\)
10:\(\hat{y}_{t}=L_{t-1}+T_{t-1}+S_{t-s}\)
11:endfor
12:Select \(\texttt{error}=\text{MASE}\) as a function of \((\alpha,\beta,\gamma)\)
13:Obtain \((\alpha,\beta,\gamma)\) that minimize error by implementing the GA
14:Calculate \(L_{n},T_{n},S_{n-s+1},\dots,S_{n-1},S_{n}\)
15:Calculate \(\hat{y}_{n+h|n}=L_{n}+HT_{n}+S_{n-s+1+h\bmod s}\)
16:return\(\hat{y}_{n+h|n}\)
```  
**Algorithm 1** Additive Holt-Winters (AHW)  
In our implementation, we employed mean absolute scale error (MASE) as an accuracy measure for the AHW algorithm. This measure is preferred because it is not scale-dependent, does not result in undefined or infinite values, and provides accurate results. The MASE is calculated as \(\text{MASE}=\frac{\text{MAE}}{Q}\), where \(\text{MAE}=\frac{1}{n}\Sigma_{t=1}^{n}|y_{t}-\hat{y}_{t}|\), is the mean absolute error (MAE), \(y_{t}\) and \(\hat{y}_{t}\) are the actual and forecast values in period \(t\), respectively, and \(Q=\frac{1}{n-p}\Sigma_{t=1}^{n}|y_{t}-y_{t-p}|\) is the scaling factor, where \(p\) is the sampling frequency per day.