## 2 Proposed Method
### MLP-Based Weighted Ensemble Method
#### 2.2.1 Construction of Training Datasets  
Our proposed method is to predict the parcel volume during special periods of sharp changes in logistics services. The proposed method constructs two MLP models based on the generated internal and external features. These models are learned from datasets prepared differently from the initial mail center datasets. The first predictive model (external features-based prediction model shown in Figure 2) is to predict the total parcel volume during special periods. Therefore, the first training dataset is produced by grouping daily information into summary rows belonging to the holiday. The Training Dataset-1 consists of the external features based on the large volume mailing and contract customer data, and the target variable is estimated by the sum of parcel volume during a peak period of the particular holidays.  
Figure 1: Feature-set by internal and external factors.  
For the second training dataset, instead of using the initial daily dataset directly, we prepare an enriched training dataset named Training Dataset-2 by adding the 500 synthetic mailing data generated from the VAE model. The VAE model is excluded on the first training dataset, which is a summarized representation of the daily dataset because the compressed dataset is produced from the original dataset and may not be able to properly represent the original data distribution. Therefore, the VAE model is applied to the second training dataset construction.  
The VAE is a neural network introduced first by [15]. It is mainly used to generate synthetic data. For example, in [16], the VAE was used to generate synthetic electronic health records. The effectiveness of the VAE was confirmed by the comparison of LSTM models trained on the synthetic and actual datasets. The authors of [17] proposed a coronary heart disease risk prediction method based on neural networks. They improved the prediction performance by augmenting rare instances with the VAE-based synthetic data. In [18], the VAE was used for image data generation and evaluated on the MNIST dataset. The VAE-based approach outperformed other traditional data generation methods, such as Synthetic Minority Over-sampling Technique (SMOTE) and Adaptive Synthetic (ADASYN).  
Figure 3 presents an architecture of the VAE model used in this study. Generally, the architecture of the VAE involves encoder and decoder parts. The encoder part encodes input data as a latent distribution in a lower-dimensional space and learns to return the mean and variance for the normal distribution of data. Then, the random point sampled from that distribution is decoded, and an error between decoded data and the initial data is calculated to adjust model weights. To generate synthetic data from the trained VAE, first, a random point \(z\) is estimated by Equation (1):  
\[z=\mu+\sigma\times\varepsilon \tag{1}\]  
where \(\varepsilon\) is randomly sampled from the standard normal distribution, and \(\mu\) and \(\sigma\) are the mean and standard deviations of the latent distribution. The sampled point \(z\) is decoded to obtain new data.  
Figure 2: Overall architecture of the proposed method. Dashed lines indicate prediction process; solid lines indicate training procedure.  
Input and output layers of the VAE model in the proposed method consist of 30 nodes that represent input features for an internal factors-based model and its target variable. Each hidden layer with 15 nodes uses the ReLU activation function described in Equation (2):  
\[\text{ReLU}(x)=\left\{\begin{smallmatrix}0\text{ \emph{if} }x\leq 0\\ x\text{ \emph{if} }x>0\end{smallmatrix}=\max\{0,x\}\right. \tag{2}\]