% m211oh95.tex (prev.: *oh91.*) - Shipping of Commodities
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\begin{titlepage}
\vspace*{-30pt}
\begin{center}\tcr{\rmH Shipping of Commodities} \end{center} 
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%\LARGE           
\huge

\newcommand{\ueq}{\vec{u}_{\scr eq}}
\newcommand{\vu}[1]{\vec{u}_{\scr #1}}
\newcommand{\vv}[1]{\vec{v}_{\scr #1}}
\newcommand{\rank}{\mbox{rank}}
\newcommand{\mult}{\mbox{mult}}
\newcommand{\bmat}{\left(\begin{array}}
\newcommand{\emat}{\end{array}\right)}

\begin{description}
\item[\tcg{Situation:}] Four regions (or cities) \mc{R1, R2, R3, R4} ship (non-renewable) \tcr{commodities} (such as \tcg{antique paintings, rental cars}) among themselves according to the following diagram: \vspace{8pt}

{\Huge
\mc{\begin{array}{ccc}
\tcg{\fbox{\parbox{2cm}{\hspace*{12pt}\mc{R1}}\rule[-.75cm]{0cm}{2cm}}} & \hspace*{5pt}\parbox{1.75cm}{\hspace*{3pt}\tcr{$\scr70\%$}\vspace{-15pt}\\
$\longrightarrow$\vspace{-15pt}\\
$\longleftarrow$\vspace{-15pt}\\\tcr{\hspace*{3pt}$\scr30\%$}}\hspace*{5pt} 
& \tcg{\fbox{\parbox{2cm}{\hspace*{12pt}\mc{R2}}\rule[-.75cm]{0cm}{2cm}}} \\
\tcr{\scr10\%}\hspace{-5pt}\downarrow\hspace*{15pt}\rule[-.75cm]{0cm}{2cm} 
& \hspace*{10pt}\searrow\hspace{-5pt}\tcr{\scr10\%}
& \hspace*{15pt}\downarrow\hspace{-5pt}\tcr{\scr10\%}
\rule[-.75cm]{0cm}{2cm} \\
\tcg{\fbox{\parbox{2cm}{\hspace*{12pt}\mc{R3}}\rule[-.75cm]{0cm}{2cm}}}   
&  & \tcg{\fbox{\parbox{2cm}{\hspace*{12pt}\mc{R4}}\rule[-.75cm]{0cm}{2cm}}}   
\end{array}}}\hspace{10pt}
\begin{minipage}{9cm}
The figures represent the percentage of goods each region ships per week. \\ 
(Percentage: in terms of goods present)
\end{minipage}

\item[\tcg{Today's distribution:}] \hspace*{20pt}\newline
\begin{tabular}{cccc}
\mc{R1} & has & \mc{^{\$}300,000} & worth of goods \\ 
\mc{R2} & has & \mc{^{\$}200,000} & worth of goods \\  
\mc{R3} & has & \mc{^{\$}100,000} & worth of goods \\  
\mc{R4} & has & \mc{^{\$}100,000} & worth of goods  
\end{tabular}

\item[\tcg{Question:}] What happens \tcr{in the long run}? 

\item[\tcg{Analysis:}] This is a \tcr{discrete linear system}
\mc{\vv{n+1} = A\vv{n}} with\vspace{-28pt}
\tcb{$$
A = \frac{1}{10}\bmat{cccc}
 1 & 3 & 0 & 0 \\
 7 & 6 & 0 & 0 \\
 1 & 0 & 10 & 0\\
 1 & 1 & 0 & 10 \emat 
= \bmat{cc} T & 0 \\ B & I \emat,
\quad\vv{0} = \bmat{c} 3 \\ 2 \\ 1 \\ 1 \emat.
$$}  
\end{description} 
\end{titlepage}
\end{document}                          