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\title{General Mathematics and Computational Science I}
\author{Exercise 18}
\date{November 17, 2005}
\maketitle

\begin{enumerate}

\item The last problem on the November 15 exercises has the standard
form \emph{minimize}
$
  \zeta = -2 \, y_1 - y_2 + y_3 + 5
$
\emph{subject to}
\begin{gather*}
  y_1 + y_2 + s_1 = 4 \,, \\
  y_2 - y_3 + s_2 = 0 \,, \\
  -y_1 - y_3 + s_3 = 1 \,, \\
  y_i \geq 0 \text{ and } s_i \geq 0 \text{ for } i=1,2,3 \,.
\end{gather*}
Solve this linear programming problem using the simplex method.



\item Solve the following linear programming problem using the simplex
method.

Maximize
$
  z = 3 \, x_1 + 4 \, x_2
$
subject to 
\begin{gather*}
  2 \, x_1 + x_2 \leq 4 \,, \\
  3 \, x_1 + 2 \, x_2 \leq 8 \,, \\
  x_i \geq 0 \text{ for } i=1,2 \,.
\end{gather*}


\item (From Lial \emph{et al.})  Southwestern Oil supplies two
distributors in the Northwest from two outlets, $S_1$ and $S_2$,
respectively.  Distributor $D_1$ needs at least $3000$ barrels of oil,
and distributor $D_2$ needs at least $5000$ barrels.  The two outlets
can furnish up to $5000$ barrels of oil.  The costs per barrel to ship
the oil are given in the table
\begin{center}
\begin{tabular}{c|cc}
 & $D_1$ & $D_2$ \\
\hline
$S_1$ & \$30 & \$20 \\
$S_2$ & \$25 & \$22 
\end{tabular}
\end{center}
There is also a shipping tax per barrel as given in the table below.
Southwestern Oil is determined to spend no more than $\$40\,000$ on
shipping tax.
\begin{center}
\begin{tabular}{c|cc}
 & $D_1$ & $D_2$ \\
\hline
$S_1$ & \$2 & \$6 \\
$S_2$ & \$5 & \$4 
\end{tabular}
\end{center}
How should the oil be supplied to minimize shipping costs?
\end{enumerate}
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