% m211oh99.tex (new 03) - Shipping of Commodities
%   - partial revision of m211oh95.tex
%
\newcommand{\titlesize}{30pt} % for title font -usual: 30pt 
\input{m211oh.tex}
%
%\begin{titlepage}
\vspace*{-30pt}
\begin{center}\tcr{\rmH Shipping of Commodities} \end{center} 
%\vspace{10pt}
%\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}$\scr20\%$}}\hspace*{5pt} 
& \tcg{\fbox{\parbox{2cm}{\hspace*{12pt}\mc{R2}}\rule[-.75cm]{0cm}{2cm}}}
 \\
\tcr{\phantom{\scr10\%}}\hspace{-10pt}
\updownarrow\hspace*{15pt}\rule[-.75cm]{0cm}{2cm} 
& \hspace*{10pt}\searrow\nwarrow\hspace{-50pt} %\tcr{\scr10\%}
\swarrow\nearrow\hspace{-5pt}
& \hspace*{15pt}\updownarrow\hspace{-5pt} %\tcr{\scr10\%}
\rule[-.75cm]{0cm}{2cm} \\
\tcg{\fbox{\parbox{2cm}{\hspace*{12pt}\mc{R3}}\rule[-.75cm]{0cm}{2cm}}}   
& \longleftrightarrow\vspace{-15pt}
& \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) \\
All unmarked arrows: {\LARGE\mc{10\%}}
\end{minipage}

\vspace{20pt}

\item[\tcg{Today's distribution:}]  \hspace*{20pt}\newline
\iffalse
\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}
\fi
\begin{tabular}{cccl}
\mc{R1} & has & \mc{r_1} & of all the goods \\ 
\mc{R2} & has & \mc{r_2} & of all the goods \\ 
\mc{R3} & has & \mc{r_3} & of all the goods \\  
\mc{R4} & has & \mc{r_4} & of all the goods  
\end{tabular}

%\mc{\Ra \vec{v}_0 = (p_1,p_2,p_3,p_4)^t} initial distribution 

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

\item[\tcg{Note:}] This is a \tcr{Markov chain}
\mc{\vv{n+1} = A\vv{n}} with %\vspace{-28pt}
\tcb{$$
A = \frac{1}{10}\bmat{cccc}
 1 & 2 & 1 & 1 \\
 7 & 6 & 1 & 1 \\
 1 & 1 & 7 & 1\\
 1 & 1 & 1 & 7 \emat 
%= \bmat{cc} T & 0 \\ B & I \emat,
\quad\mbox{and}\quad\vv{0} = \bmat{c} r_1 \\ r_2 \\ r_3 \\ r_4 \emat.
$$}  

\vspace*{-5pt}

\item[\tcr{Qualitative Solution:}] \tcg{Perron's theorem} \mc{\Ra} there is 
a vector \mc{\vec{p} = (p_1,p_2,p_3,p_4)^t} such that

-- \mc{\vec{p}} represents the distribution of goods among the regions \tcr{in the long run:}
$$
\tcb{\lim_{n\ra\infty} \vec{v}_n = \lim_{n\ra\infty} A^n\vec{v}_0 = \vec{p}}
$$
-- \mc{\vec{p}} does \tcr{not} depend on the \tcg{initial distribution} \mc{\vec{v}_0} 

-- \mc{p_j >0}, for \mc{j=1,\ldots,4}, i.e.\ \tcr{no} region is \tcg{completely drained} in the long run.

\item[\tcr{Quantitative Solution:}] 
Find the \tcg{Perron vector} \mc{\vec{p}}:

By row reduction (or by inspection)
$$
\tcb{E_A(1) = Solsp(A-I) = \{c(6,16,11,11)^t: c\in\C\},}
$$
and hence the \tcg{Perron vector} \mc{\vec{p}} is given by
$$
\tcb{\vec{p} = \frac{1}{\Sigma_4\vec{x}}\vec{x} = \frac{1}{44}(6,16,11,11)^t
= (.14,.36,.25,.25)^t}
$$
where \mc{\vec{x} = (6,16,11,11)^t}. \tcr{Thus:}

\begin{tabular}{cccl}
\mc{R1} & has & \mc{14\%} & of all the goods \tcg{in the long run}\\ 
\mc{R2} & has & \mc{36\%} & of all the goods \tcg{in the long run} \\ 
\mc{R3} & has & \mc{25\%} & of all the goods \tcg{in the long run}\\  
\mc{R4} & has & \mc{25\%} & of all the goods \tcg{in the long run}
\end{tabular}
\end{description} 
%\end{titlepage}
\end{document}                          