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\title{General Mathematics and Computational Science II}
\author{Exercise 15}
\date{April 17, 2007}
\maketitle

\begin{enumerate}

\item The second problem on the April 12 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*}


\end{enumerate}
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