\documentclass[a4paper,11pt]{article}
\newenvironment{example}{\subsection{Example}}{\\\textbf{End of example}}
\author{Ariel Cal\'o}
\title{Some math background for supply-chain}
\begin{document}
\maketitle
\begin{abstract}
to be done
\end{abstract}

\newpage
\tableofcontents
\newpage

\section{Notations and Conventions}
Trought all the paper we assume that we have two series of pairs:
$O=(o{i},d{i})_{i=0}^n$ (orders) and
$S=(q{i},d{i})_{i=0}^n$ (stock movements)
each pair up to $n-1$ is composed of a quantity and a date and represents a
 log in our inventory archive. the last pair has $0$ for quantity by convention and the date is the last day for the time interval under consideration. 
\\
For simplicity we assume that we have only one article in stock, so if we have
the following logs for the period 1-10 january:
\\
\begin{tabular}{|r|l|}
1 Jan & 32 items in stock.\\
3 Jan & Ordered 100 items.\\
4 Jan & 20 items sold.\\
6 Jan & 100 items arrived.\\
6 Jan & 15 items sold.\\
\end{tabular}\\
then we have the following series:
\[ O= \{ (0,\textrm{1-jan}),(100,\textrm{3-jan}),(-100,\textrm{6-jan}),(0,10-jan)\} \] and
\[ S= \{ (32,\textrm{1-jan}),(-20,\textrm{4-jan}),(100,\textrm{6-jan}),(-15,\textrm{6-jan}),(0,10-jan) \} \]
we also denote with $t$ the time interval.

\section{Base statistics}

\subsection{Quantity in stock}
The \textbf{Quantity in stock} is denoted by $Q$ and is given by:
\begin{equation}
Q = \sum_{i=0}^n q_{i}
\end{equation}
\begin{example}
\begin{displaymath}
Q=32-20+100-15+0=97
\end{displaymath} 
\end{example}

\subsection{Total sales}
The \textbf{Total sales} is denoted by $TS$ and is given by:
\begin{equation}
TS = \sum_{i=0}^n IsSale( q_{i})\qquad 
\textrm{where} \qquad IsSale( q_{i}) = 
\left\{ \begin{array}{lcl}
-q_{i} & \textrm{if} & q_{i} < 0\\
0  & \textrm{if} & q_{i} \ge 0
\end{array} \right.
\end{equation}
\begin{example}
\begin{displaymath}
Q=0+20+0+15+0=35
\end{displaymath} 
\end{example}

\subsection{Average stock quantity}
The \textbf{Average stock quantity} is denoted by $\overline{S}$ and is given by:
\begin{equation}
\overline{S} =\frac{ \sum_{i=0}^{n-1}\left[ \left( \sum_{k=0}^i q_{k} \right) (d_{i+1} - d_{i})\right]}{t}
\end{equation}
\begin{example}

\begin{tabular}{|c|c|c|c|}
\hline
$i$ & $ \sum_{k=0}^i q_{k}$ & $d_{i+1} - d_{i}$ & $ \left( \sum_{k=0}^i q_{k} \right) (d_{i+1} - d_{i})$ \\
\hline
0 & 32 & 2 & 64 \\
1 & 12 & 2 & 24 \\
2 & 112 & 0 & 0 \\
3 & 97 & 1 & 97 \\
\hline
Total & - & - & 185 \\
\hline
\end{tabular}

\begin{displaymath}
\overline{S} =\frac{185}{9} = 20.\overline{5}
\end{displaymath} 
\end{example}

\subsection{Average supply time}
The \textbf{Average supply time} also called \textbf{Average lead time} is the average number of days occurring between the order issue and the receiving,
is denoted by $\overline{L}$

\subsection{Average daily sale}
The \textbf{Average daily sale} is denoted by $\overline{D}$
 and is given by:
\begin{equation}
\overline{D} = \frac{TS}{t}
\end{equation}

\subsection{Rotation Index}
The \textbf{Rotation Index} is denoted by $I_{rot}$
 and is given by:
\begin{equation}
I_{rot} =  \frac{\overline{S}}{TS}
\end{equation}

\section{Reorder Point and Safety Stock}

\subsection{Theoretical order point}
The \textbf{Theoretical order point} is denoted by $P_{theortical}$
 and is given by:
\begin{equation}
P_{theortical} = \overline{D}\quad  \overline{L}
\end{equation}

\subsection{Safety Stock}
The \textbf{Safety Stock} is denoted by $S_{safety}$
The classical definition is:
\begin{equation} \label{eq:ss1}
S_{safety} = \frac{P_{theortical}}{2}
\end{equation}
a more general general and flexible (altought requires human judgement and intervention) is:
\begin{equation}\label{eq:ss2}
S_{safety} =  \overline{D} x K
\end{equation}
where $K$ is a constant denoting the number of supply days yo hold in reserve and is defined by the inventory/sales  respnsible.
Note that when $K = \frac{ \overline{L} }{2}$ then (\ref{eq:ss2}) is equivalent to  (\ref{eq:ss1}).
\\
Jan Schreibfeder has proposed an alternative definition,
 details can be found at <www.effectiveinventory.com$\backslash$article29.html>

\subsection{Order point}
The \textbf{Order point} is denoted by $P$
 and is given by:
\begin{equation}
P = P_{theortical} + S_{safety}
\end{equation}

\section{Economic Order Quantity (EOQ)}
\begin{equation}
EOQ=\sqrt{\frac{2 x TS x cost of placing an order} {cost of inventory per item}}
\end{equation}


\section{Evalution uf the inventory}

\section{}
\end{document}