\section{Tabular and Real-time Forecasts without Retraining}
\label{tabular_ml_models}

Regarding the structure of the feature matrix for the ML models in Sub-section
    \ref{ml_models}, we provide an alternative approach that works without
    the STL method.
Instead of decomposing a time series and arranging the resulting
    seasonally-adjusted time series $a_t$ into a matrix $\mat{X}$, one can
    create a matrix with two types of feature columns mapped to the raw
    observations in $\vec{y}$:
While the first group of columns takes all observations of the same time of
    day over a horizon of, for example, one week ($n_h=7$), the second group
    takes all observations covering a pre-defined time horizon, for example
    $3$ hours ($n_r=3$ for 60-minute time steps), preceding the time step to
    be fitted.
Thus, we exploit the two-dimensional structure of time tables as well, and
    conceptually model historical and recent demand.
The alternative feature matrix appears as follows where the first three
    columns are the historical and the last three the recent demand features:

$$
\vec{y}
=
\begin{pmatrix}
    y_T \\
    y_{T-1} \\
    \dots \\
    y_{1+n_hH}
\end{pmatrix}
~~~~~
\mat{X}
=
\begin{bmatrix}
    y_{T-H}              & y_{T-2H}       & \dots & y_{T-n_hH}
        & y_{T-1}        & y_{T-2}        & \dots & y_{T-n_r} \\
    y_{T-1-H}            & y_{T-1-2H}     & \dots & y_{T-1-n_hH}
        & y_{T-2}        & y_{T-3}        & \dots & y_{T-n_r-1} \\
    \dots                & \dots          & \dots & \dots
        & \dots          & \dots          & \dots & \dots \\
    y_{1+(n_h-1)H}       & y_{1+(n_h-2)H} & \dots & y_1
        & y^*_{1+n_hH-1} & y^*_{1+n_hH-2} & \dots & y^*_{1+n_hH-n_r}
\end{bmatrix}
$$
\

Being a detail, we note that the recent demand features lying on the end of
    the previous day are set to $0$, which is shown with the $^*$ notation
    above.
This alignment of the undecomposed order data $y_t$ ensures that the ML
    models learn the two seasonal patterns independently.
The parameters $n_h$ and $n_r$ must be adapted to the data, but we found the
    above values to work well.

As such matrices resemble time tables, we refer to them as tabular.
However, we found the ML models with vertical time series to outperform the
    tabular ML models, which is why we disregarded them in the study.
This tabular form could be beneficial for UDPs with a demand that exhibits
    a weaker seasonality such as a meal delivery platform.