Remove glossary
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@ -52,8 +52,6 @@
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\newpage
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\newpage
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\input{tex/apx/peak_results}
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\input{tex/apx/peak_results}
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\newpage
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\newpage
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\input{tex/apx/glossary}
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\newpage
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\bibliographystyle{static/elsarticle-harv}
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\bibliographystyle{static/elsarticle-harv}
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\bibliography{tex/references}
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\bibliography{tex/references}
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@ -11,8 +11,7 @@ A common feature of these platforms is that they do not operate kitchens but
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related processes in simple smartphone apps, and managing the delivery via
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related processes in simple smartphone apps, and managing the delivery via
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a fleet of either employees or crowd-sourced sub-contractors.
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a fleet of either employees or crowd-sourced sub-contractors.
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Various kinds of urban delivery platforms
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Various kinds of urban delivery platforms (UDP)
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(\gls{udp}; \ref{glossary} provides a glossary with all abbreviations)
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have received attention in recent scholarly publications.
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have received attention in recent scholarly publications.
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\cite{hou2018} look into heuristics to simultaneously optimize courier
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\cite{hou2018} look into heuristics to simultaneously optimize courier
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scheduling and routing in general, while \cite{masmoudi2018} do so
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scheduling and routing in general, while \cite{masmoudi2018} do so
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@ -20,8 +19,7 @@ Various kinds of urban delivery platforms
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the effect of different fulfillment strategies in the context of urban
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the effect of different fulfillment strategies in the context of urban
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meal delivery.
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meal delivery.
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\cite{ehmke2018} and \cite{alcaraz2019} focus their research on the routing
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\cite{ehmke2018} and \cite{alcaraz2019} focus their research on the routing
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aspect, which is commonly modeled as a so-called vehicle routing problem
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aspect, which is commonly modeled as a so-called vehicle routing problem (VRP).
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(\gls{vrp}).
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Not covered in the recent literature is research focusing on the demand
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Not covered in the recent literature is research focusing on the demand
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forecasting problem a UDP faces.
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forecasting problem a UDP faces.
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@ -69,7 +67,7 @@ In this paper, we develop a rigorous methodology as to how to build and
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We implement such a system with a broad set of commonly used forecasting
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We implement such a system with a broad set of commonly used forecasting
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methods.
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methods.
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We not only apply established (i.e., "classical") time series methods but also
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We not only apply established (i.e., "classical") time series methods but also
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machine learning (\gls{ml}) models that have gained traction in recent
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machine learning (ML) models that have gained traction in recent
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years due to advancements in computing power and availability of larger
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years due to advancements in computing power and availability of larger
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amounts of data.
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amounts of data.
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In that regard, the classical methods serve as benchmarks for the ML methods.
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In that regard, the classical methods serve as benchmarks for the ML methods.
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@ -100,4 +98,4 @@ The subsequent Section \ref{lit} reviews the literature on the forecasting
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Section \ref{mod} introduces our forecasting system, and Section \ref{stu}
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Section \ref{mod} introduces our forecasting system, and Section \ref{stu}
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discusses the results obtained in the empirical study.
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discusses the results obtained in the empirical study.
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Lastly, Section \ref{con} summarizes our findings and concludes
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Lastly, Section \ref{con} summarizes our findings and concludes
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with an outlook on further research opportunities.
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with an outlook on further research opportunities.
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@ -40,8 +40,7 @@ Their main advantages stem from the fact that the models calibrate themselves
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\cite{cleveland1990} introduce a seasonal and trend decomposition using a
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\cite{cleveland1990} introduce a seasonal and trend decomposition using a
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repeated locally weighted regression - the so-called Loess procedure - to
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repeated locally weighted regression - the so-called Loess procedure - to
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smoothen the trend and seasonal components, which can be viewed as a
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smoothen the trend and seasonal components, which can be viewed as a
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generalization of the methods above and is denoted by the acronym
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generalization of the methods above and is denoted by the acronym STL.
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\gls{stl}.
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In contrast to the X11, X13, and SEATS methods, the STL supports seasonalities
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In contrast to the X11, X13, and SEATS methods, the STL supports seasonalities
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of any lag $k$ that must, however, be determined with additional
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of any lag $k$ that must, however, be determined with additional
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statistical tests or set with out-of-band knowledge by the forecaster
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statistical tests or set with out-of-band knowledge by the forecaster
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@ -4,8 +4,7 @@
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ML methods have been employed in all kinds of prediction tasks in recent
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ML methods have been employed in all kinds of prediction tasks in recent
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years.
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years.
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In this section, we restrict ourselves to the models that performed well in
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In this section, we restrict ourselves to the models that performed well in
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our study: Random Forest (\gls{rf}) and Support Vector Regression
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our study: Random Forest (RF) and Support Vector Regression (SVR).
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(\gls{svr}).
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RFs are in general well-suited for datasets without a priori knowledge about
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RFs are in general well-suited for datasets without a priori knowledge about
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the patterns, while SVR is known to perform well on time series data, as
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the patterns, while SVR is known to perform well on time series data, as
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shown by \cite{hansen2006} in general and \cite{bao2004} specifically for
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shown by \cite{hansen2006} in general and \cite{bao2004} specifically for
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@ -5,7 +5,7 @@ Because ML models are trained by minimizing a loss function $L$, the
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resulting value of $L$ underestimates the true error we see when
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resulting value of $L$ underestimates the true error we see when
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predicting into the actual future by design.
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predicting into the actual future by design.
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To counter that, one popular and model-agnostic approach is cross-validation
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To counter that, one popular and model-agnostic approach is cross-validation
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(\gls{cv}), as summarized, for example, by \cite{hastie2013}.
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(CV), as summarized, for example, by \cite{hastie2013}.
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CV is a resampling technique, which ranomdly splits the samples into a
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CV is a resampling technique, which ranomdly splits the samples into a
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training and a test set.
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training and a test set.
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Trained on the former, an ML model makes forecasts on the latter.
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Trained on the former, an ML model makes forecasts on the latter.
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@ -2,7 +2,7 @@
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\label{rf}
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\label{rf}
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\cite{breiman1984} introduce the classification and regression tree
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\cite{breiman1984} introduce the classification and regression tree
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(\gls{cart}) model that is built around the idea that a single binary
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(CART) model that is built around the idea that a single binary
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decision tree maps learned combinations of intervals of the feature
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decision tree maps learned combinations of intervals of the feature
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columns to a label.
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columns to a label.
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Thus, each sample in the training set is associated with one leaf node that
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Thus, each sample in the training set is associated with one leaf node that
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@ -2,7 +2,7 @@
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\label{svm}
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\label{svm}
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\cite{vapnik1963} and \cite{vapnik1964} introduce the so-called support vector
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\cite{vapnik1963} and \cite{vapnik1964} introduce the so-called support vector
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machine (\gls{svm}) model, and \cite{vapnik2013} summarizes the research
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machine (SVM) model, and \cite{vapnik2013} summarizes the research
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conducted since then.
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conducted since then.
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In its basic version, SVMs are linear classifiers, modeling a binary
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In its basic version, SVMs are linear classifiers, modeling a binary
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decision, that fit a hyperplane into the feature space of $\mat{X}$ to
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decision, that fit a hyperplane into the feature space of $\mat{X}$ to
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@ -41,7 +41,7 @@ These numerical instabilities occurred so often in our studies that we argue
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against using such measures.
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against using such measures.
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\item \textbf{Scaled Errors}:
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\item \textbf{Scaled Errors}:
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\cite{hyndman2006} contribute this category and introduce the mean absolute
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\cite{hyndman2006} contribute this category and introduce the mean absolute
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scaled error (\gls{mase}).
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scaled error (MASE).
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It is defined as the MAE from the actual forecasting method on the test day
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It is defined as the MAE from the actual forecasting method on the test day
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(i.e., "out-of-sample") divided by the MAE from the (seasonal) na\"{i}ve
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(i.e., "out-of-sample") divided by the MAE from the (seasonal) na\"{i}ve
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method on the entire training set (i.e., "in-sample").
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method on the entire training set (i.e., "in-sample").
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@ -84,4 +84,4 @@ We conjecture that percentage error measures may be usable for UDPs facing a
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higher overall demand with no intra-day down-times in between but have to
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higher overall demand with no intra-day down-times in between but have to
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leave that to a future study.
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leave that to a future study.
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Yet, even with high and steady demand, divide-by-zero errors are likely to
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Yet, even with high and steady demand, divide-by-zero errors are likely to
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occur.
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occur.
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@ -15,21 +15,21 @@ As the models in this family do not include the test day's demand data in
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The models in this family are as follows; we use prefixes, such as "h" here,
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The models in this family are as follows; we use prefixes, such as "h" here,
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when methods are applied in other families as well:
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when methods are applied in other families as well:
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\begin{enumerate}
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\begin{enumerate}
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\item \textit{\gls{naive}}:
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\item \textit{naive}:
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Observation from the same time step one week prior
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Observation from the same time step one week prior
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\item \textit{\gls{trivial}}:
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\item \textit{trivial}:
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Predict $0$ for all time steps
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Predict $0$ for all time steps
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\item \textit{\gls{hcroston}}:
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\item \textit{hcroston}:
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Intermittent demand method introduced by \cite{croston1972}
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Intermittent demand method introduced by \cite{croston1972}
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\item \textit{\gls{hholt}},
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\item \textit{hholt},
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\textit{\gls{hhwinters}},
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\textit{hhwinters},
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\textit{\gls{hses}},
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\textit{hses},
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\textit{\gls{hsma}}, and
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\textit{hsma}, and
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\textit{\gls{htheta}}:
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\textit{htheta}:
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Exponential smoothing without calibration
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Exponential smoothing without calibration
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\item \textit{\gls{hets}}:
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\item \textit{hets}:
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ETS calibrated as described by \cite{hyndman2008b}
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ETS calibrated as described by \cite{hyndman2008b}
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\item \textit{\gls{harima}}:
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\item \textit{harima}:
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ARIMA calibrated as described by \cite{hyndman2008a}
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ARIMA calibrated as described by \cite{hyndman2008a}
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\end{enumerate}
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\end{enumerate}
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\textit{naive} and \textit{trivial} provide an absolute benchmark for the
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\textit{naive} and \textit{trivial} provide an absolute benchmark for the
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@ -16,17 +16,17 @@ By decomposing the raw time series, all long-term patterns are assumed to be
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a potential trend and auto-correlations.
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a potential trend and auto-correlations.
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The models in this family are:
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The models in this family are:
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\begin{enumerate}
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\begin{enumerate}
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\item \textit{\gls{fnaive}},
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\item \textit{fnaive},
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\textit{\gls{pnaive}}:
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\textit{pnaive}:
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Sum of STL's trend and seasonal components' na\"{i}ve forecasts
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Sum of STL's trend and seasonal components' na\"{i}ve forecasts
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\item \textit{\gls{vholt}},
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\item \textit{vholt},
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\textit{\gls{vses}}, and
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\textit{vses}, and
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\textit{\gls{vtheta}}:
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\textit{vtheta}:
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Exponential smoothing without calibration and seasonal
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Exponential smoothing without calibration and seasonal
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fit
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fit
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\item \textit{\gls{vets}}:
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\item \textit{vets}:
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ETS calibrated as described by \cite{hyndman2008b}
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ETS calibrated as described by \cite{hyndman2008b}
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\item \textit{\gls{varima}}:
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\item \textit{varima}:
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ARIMA calibrated as described by \cite{hyndman2008a}
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ARIMA calibrated as described by \cite{hyndman2008a}
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\end{enumerate}
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\end{enumerate}
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As mentioned in Sub-section \ref{unified_cv}, we include the sum of the
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As mentioned in Sub-section \ref{unified_cv}, we include the sum of the
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@ -8,13 +8,13 @@ Instead of obtaining an $H$-step-ahead forecast, we retrain a model after
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every time step and only predict one step.
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every time step and only predict one step.
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The remainder is as in the previous sub-section, and the models are:
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The remainder is as in the previous sub-section, and the models are:
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\begin{enumerate}
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\begin{enumerate}
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\item \textit{\gls{rtholt}},
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\item \textit{rtholt},
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\textit{\gls{rtses}}, and
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\textit{rtses}, and
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\textit{\gls{rttheta}}:
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\textit{rttheta}:
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Exponential smoothing without calibration and seasonal fit
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Exponential smoothing without calibration and seasonal fit
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\item \textit{\gls{rtets}}:
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\item \textit{rtets}:
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ETS calibrated as described by \cite{hyndman2008b}
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ETS calibrated as described by \cite{hyndman2008b}
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\item \textit{\gls{rtarima}}:
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\item \textit{rtarima}:
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ARIMA calibrated as described by \cite{hyndman2008a}
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ARIMA calibrated as described by \cite{hyndman2008a}
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\end{enumerate}
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\end{enumerate}
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Retraining \textit{fnaive} and \textit{pnaive} did not increase accuracy, and
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Retraining \textit{fnaive} and \textit{pnaive} did not increase accuracy, and
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@ -10,7 +10,7 @@ Based on the seasonally-adjusted time series $a_t$, we employ the feature
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The ML models are trained once before a test day starts.
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The ML models are trained once before a test day starts.
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For training, the matrix and vector are populated such that $y_T$ is set to
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For training, the matrix and vector are populated such that $y_T$ is set to
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the last time step of the day before the forecasts, $a_T$.
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the last time step of the day before the forecasts, $a_T$.
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As the splitting during CV is done with whole days, the \gls{ml} models are
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As the splitting during CV is done with whole days, the ML models are
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trained with training sets consisting of samples from all times of a day
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trained with training sets consisting of samples from all times of a day
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in an equal manner.
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in an equal manner.
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Thus, the ML models learn to predict each time of the day.
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Thus, the ML models learn to predict each time of the day.
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As a result, real-time data are included.
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As a result, real-time data are included.
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The models in this family are:
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The models in this family are:
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\begin{enumerate}
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\begin{enumerate}
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\item \textit{\gls{vrfr}}: RF trained on the matrix as described
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\item \textit{vrfr}: RF trained on the matrix as described
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\item \textit{\gls{vsvr}}: SVR trained on the matrix as described
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\item \textit{vsvr}: SVR trained on the matrix as described
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\end{enumerate}
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\end{enumerate}
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We tried other ML models such as gradient boosting machines but found
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We tried other ML models such as gradient boosting machines but found
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only RFs and SVRs to perform well in our study.
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only RFs and SVRs to perform well in our study.
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\label{overall_results}
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\label{overall_results}
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Table \ref{t:results} summarizes the overall best-performing models grouped by
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Table \ref{t:results} summarizes the overall best-performing models grouped by
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training horizon and a pixel's average daily demand (\gls{add}) for a
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training horizon and a pixel's average daily demand (ADD) for a
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pixel size of $1~\text{km}^2$ and 60-minute time steps.
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pixel size of $1~\text{km}^2$ and 60-minute time steps.
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Each combination of pixel and test day counts as one case, and the total
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Each combination of pixel and test day counts as one case, and the total
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number of cases is denoted as $n$.
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number of cases is denoted as $n$.
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@ -19,11 +19,11 @@ Thus, we suggest conducting more detailed analyses on how to incorporate model
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Future research should also integrate our forecasting system into a predictive
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Future research should also integrate our forecasting system into a predictive
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routing application and evaluate its business impact.
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routing application and evaluate its business impact.
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This embeds our research into the vast literature on the VRP.
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This embeds our research into the vast literature on the VRP.
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Initially introduced by \cite{dantzig1959}, \gls{vrp}s are concerned with
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Initially introduced by \cite{dantzig1959}, VRPs are concerned with
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finding optimal routes serving customers.
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finding optimal routes serving customers.
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We refer to \cite{toth2014} for a comprehensive overview.
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We refer to \cite{toth2014} for a comprehensive overview.
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The two variants relevant for the UDP case are the dynamic VRP and
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The two variants relevant for the UDP case are the dynamic VRP and
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the pickup and delivery problem (\gls{pdp}).
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the pickup and delivery problem (PDP).
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A VRP is dynamic if the data to solve a problem only becomes available
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A VRP is dynamic if the data to solve a problem only becomes available
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as the operations are underway.
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as the operations are underway.
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\cite{thomas2010}, \cite{pillac2013}, and \cite{psaraftis2016} describe how
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\cite{thomas2010}, \cite{pillac2013}, and \cite{psaraftis2016} describe how
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@ -39,7 +39,7 @@ Forecasts by our system extend this idea naturally as dummy nodes could be
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The concrete case of a meal delivering UDP is contained in a recent
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The concrete case of a meal delivering UDP is contained in a recent
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literature stream started by \cite{ulmer2017} and extended by
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literature stream started by \cite{ulmer2017} and extended by
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\cite{reyes2018} and \cite{yildiz2018}: They coin the term meal delivery
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\cite{reyes2018} and \cite{yildiz2018}: They coin the term meal delivery
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routing problem (\gls{mdrp}).
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routing problem (MDRP).
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The MDRP is a special case of the dynamic PDP where the defining
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The MDRP is a special case of the dynamic PDP where the defining
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characteristic is that once a vehicle is scheduled, a modification of the
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characteristic is that once a vehicle is scheduled, a modification of the
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route is inadmissible.
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route is inadmissible.
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@ -1,144 +0,0 @@
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\section{Glossary}
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\label{glossary}
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% Abbreviations for technical terms.
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\newglossaryentry{add}{
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name=ADD, description={Average Daily Demand}
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}
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\newglossaryentry{cart}{
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name=CART, description={Classification and Regression Trees}
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}
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\newglossaryentry{cv}{
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name=CV, description={Cross Validation}
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}
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\newglossaryentry{mase}{
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name=MASE, description={Mean Absolute Scaled Error}
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}
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\newglossaryentry{mdrp}{
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name=MDRP, description={Meal Delivery Routing Proplem}
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}
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\newglossaryentry{ml}{
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name=ML, description={Machine Learning}
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}
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\newglossaryentry{pdp}{
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name=PDP, description={Pickup and Delivery Problem}
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}
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\newglossaryentry{rf}{
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name=RF, description={Random Forest}
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}
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\newglossaryentry{stl}{
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name=STL, description={Seasonal and Trend Decomposition using Loess}
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}
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\newglossaryentry{svm}{
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name=SVM, description={Support Vector Machine}
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}
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\newglossaryentry{svr}{
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name=SVR, description={Support Vector Regression}
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}
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\newglossaryentry{udp}{
|
|
||||||
name=UDP, description={Urban Delivery Platform}
|
|
||||||
}
|
|
||||||
\newglossaryentry{vrp}{
|
|
||||||
name=VRP, description={Vehicle Routing Problem}
|
|
||||||
}
|
|
||||||
|
|
||||||
% Model names.
|
|
||||||
\newglossaryentry{naive}{
|
|
||||||
name=naive, description={(Seasonal) Na\"{i}ve Method}
|
|
||||||
}
|
|
||||||
\newglossaryentry{fnaive}{
|
|
||||||
name=fnaive, description={"Flexible" STL Decomposition,
|
|
||||||
with tuned ns parameter}
|
|
||||||
}
|
|
||||||
\newglossaryentry{pnaive}{
|
|
||||||
name=pnaive, description={"Periodic" STL Decomposition,
|
|
||||||
with ns parameter set to large number}
|
|
||||||
}
|
|
||||||
\newglossaryentry{trivial}{
|
|
||||||
name=trivial, description={Trivial Method}
|
|
||||||
}
|
|
||||||
\newglossaryentry{hcroston}{
|
|
||||||
name=hcroston, description={Croston's Method,
|
|
||||||
trained on horizontal time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{hholt}{
|
|
||||||
name=hholt, description={Holt's Linear Trend Method,
|
|
||||||
trained on horizontal time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{vholt}{
|
|
||||||
name=vholt, description={Holt's Linear Trend Method,
|
|
||||||
trained on vertical time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{rtholt}{
|
|
||||||
name=rtholt, description={Holt's Linear Trend Method,
|
|
||||||
(re)trained on vertical time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{hhwinters}{
|
|
||||||
name=hhwinters, description={Holt-Winter's Seasonal Method,
|
|
||||||
trained on horizontal time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{hses}{
|
|
||||||
name=hses, description={Simple Exponential Smoothing Method,
|
|
||||||
trained on horizontal time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{vses}{
|
|
||||||
name=vses, description={Simple Exponential Smoothing Method,
|
|
||||||
trained on vertical time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{rtses}{
|
|
||||||
name=rtses, description={Simple Exponential Smoothing Method,
|
|
||||||
(re)trained on vertical time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{hsma}{
|
|
||||||
name=hsma, description={Simple Moving Average Method,
|
|
||||||
trained on horizontal time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{htheta}{
|
|
||||||
name=htheta, description={Theta Method,
|
|
||||||
trained on horizontal time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{vtheta}{
|
|
||||||
name=vtheta, description={Theta Method,
|
|
||||||
trained on vertical time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{rttheta}{
|
|
||||||
name=rttheta, description={Theta Method,
|
|
||||||
(re)trained on vertical time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{hets}{
|
|
||||||
name=hets, description={ETS State Space Method,
|
|
||||||
trained on horizontal time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{vets}{
|
|
||||||
name=vets, description={ETS State Space Method,
|
|
||||||
trained on vertical time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{rtets}{
|
|
||||||
name=rtets, description={ETS State Space Method,
|
|
||||||
(re)trained on vertical time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{harima}{
|
|
||||||
name=harima, description={Autoregressive Integrated Moving Average
|
|
||||||
Method,
|
|
||||||
trained on horizontal time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{varima}{
|
|
||||||
name=varima, description={Autoregressive Integrated Moving Average
|
|
||||||
Method,
|
|
||||||
trained on vertical time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{rtarima}{
|
|
||||||
name=rtarima, description={Autoregressive Integrated Moving Average
|
|
||||||
Method,
|
|
||||||
(re)trained on vertical time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{vrfr}{
|
|
||||||
name=vrfr, description={Random Forest Regression Method,
|
|
||||||
trained on vertical time series}
|
|
||||||
}
|
|
||||||
\newglossaryentry{vsvr}{
|
|
||||||
name=vsvr, description={Support Vector Regression Method,
|
|
||||||
trained on vertical time series}
|
|
||||||
}
|
|
||||||
|
|
||||||
\printglossary[title=]
|
|
|
@ -1,9 +1,6 @@
|
||||||
% Use the document width more effectively.
|
% Use the document width more effectively.
|
||||||
\usepackage[margin=2.5cm]{geometry}
|
\usepackage[margin=2.5cm]{geometry}
|
||||||
|
|
||||||
\usepackage[acronym]{glossaries}
|
|
||||||
\makeglossaries
|
|
||||||
|
|
||||||
% Enable captions for figures and tables.
|
% Enable captions for figures and tables.
|
||||||
\usepackage{caption}
|
\usepackage{caption}
|
||||||
|
|
||||||
|
|
Loading…
Reference in a new issue