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\begin{document}


\title{Response to Reviewer \# 2}

\date{ }
\maketitle



\section*{}\label{intro}
%
%We thank you for your comments and suggestions. We have carefully
%considered all major and minor points you raised. In particular:\\




\noindent

We thank the reviewer for comments and suggestions.
%We have carefully
%considered all major and minor points you raised. In particular:\\
%
All the required modifications %he/she suggested
have been carefully addressed. In particular:



\be

\ii Herewith enclosed the explanation of how we calculate the weight of the graph

\textit{The whole graph is designed for a single production plant, and there are as many graphs as production plants. The goal is to find the production plan with minimal reduced cost
amongst all possible production plans of a single production plant. Hopefully, this "t" was not the source of confusion, but it seems better to point it out, just in case.}

\textit{The notations we used in this part of the paper might not be clear enough. On a production plan, there are as many $r$ variables as the number of production campaigns, and each arc
$a_k$ is associated with a single campaign $k$. Variables $x$ are the remaining fuel at the end of an arc; for most arcs, they are not variables at all, because this quantity is fixed once for all when the graph is built; the exceptions are terminal arcs, i.e. arcs that end on the end of the timeline, for which variable $x$ determines the $x_T$ quantity of fuel wich participates in the objective function of the main problem. Variables $e$ are the quantities of energy produced during the time spanned by the arc; as two arcs in a same path are
associated with disjoint time spans, these also are independent variables.}

\textit{Taking your example, and if we take :
\begin{itemize}
\item $d$ as an arc associated with campaign $k$
\item $d$ spanning from time $t_1$ to $T$ (i.e. it is a terminal arc)
\end{itemize}}

\textit{Then, if we suppose that $(b,c)$ is a path in the graph (considering
that $b$ and $c$ are OUTAGE arcs):}

\textit{\begin{itemize}
\item $b$ has to be associated with campaign $k-1$
\item $b$ has to span from some time $t_0$ to $t_1$
\end{itemize}}

\textit{We should note the example like this:}

\textit{\begin{itemize}
\item[-] $ \bar{c}_b (r_{k_1,1}, e_{t_0,t_1,1}) = -5 $
\item[-] $ \bar{c}_b (r_{k_1,2}, e_{t_0,t_1,2}) = -2 $
\item[-] $ \bar{c}_d (r_{k,1}, e_{t_1,T,1}, x_{T_1}) = -3 $
\item[-] $ \bar{c}_d (r_{k,2}, e_{t_1,T,2}, x_{T_2}) = -5 $
\end{itemize}}


\textit{where
\begin{itemize}
\item $r_{k_1,1}$ and $r_{k_1,2}$ (resp. $r_{k,1}$ and $r_{k,2}$) are the
 possible refuel quantities at the beginning of campaign $k_1$ (resp. $k$)
\item $e_{t_0,t_1,1}$ and $e_{t_0,t_1,2}$ (resp. $e_{t_1,T,1}$ and
 $e_{t_1,T,2}$ are the possible quantities of energy produced during
 the time period $[t_0,t_1]$ (resp. $[t_1, T]$)
\item $x_{T_1}$ and $x_{T_2}$ are the possible quantities of fuel remaining
 at the end of the timeline.
\end{itemize}}

\textit{By construction, two arcs that can participate in a same path have costs that depend on distinct variables, because they are associated with distinct campaigns (for $r$) and different timespans (for $e$). Then, there is no problem in mixing the first solution of $b$ with the second of $c$ in the example.}


\ii The section Conclusion has been added as suggested by several reviewers


\ee


\noindent
{\bf Minor comments:}

\smallskip

We thank the referee for the long list of precious and punctual minor comments that he/she made on the paper. All these comments have been taken into account. Thanks.



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