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\begin{document}

\title{General Mathematics and CPS II}
\author{Exercise 19}
\date{April 23, 2014}
\maketitle

\begin{enumerate}
\item  Use the simplex method to solve the following linear programming
problem. 


Maximize
\[
  z = 2 \, x_1 + x_2 
\]
subject to 
\begin{gather*}
  3 \, x_1 + x_2 \leq 6 \,, \\
  x_1 - x_2 \leq 2 \,, \\
  x_2 \leq 3 \,, \\
  \x \geq 0 \,.
\end{gather*}

\item Suppose each of the following tableaus occurs in the course of
performing the simplex algorithm on a linear programming problem.
\begin{enumerate}
\item \newcolumntype{B}{%
  >{\columncolor[gray]{.8}[.5\tabcolsep]}c}
\begin{tabular}{cBcB|c}
  $x_1$ & $x_2$ & $x_3$ & $x_4$ \\
  \hline
  $-1$ & $1$ & $0$ & $0$ & $4$  \\
  $-2$ & $0$ & $-2$ & $1$ & $1$ \\
  \hline
  $5$ & $0$ & $3$ & $0$ & $5$
\end{tabular}
\item
\begin{tabular}{BcBc|c}
  $x_1$ & $x_2$ & $x_3$ & $x_4$ \\
  \hline
  $0$ & $-1$ & $1$ & $-1$ & $4$  \\
  $1$ & $1$ & $0$ & $0$ & $1$ \\
  \hline
  $0$ & $1$ & $0$ & $-2$ & $5$
\end{tabular}
\item
\begin{tabular}{ccBB|c}
  $x_1$ & $x_2$ & $x_3$ & $x_4$ \\
  \hline
  $2$ & $-1$ & $0$ & $1$ & $0$  \\
  $1$ & $1$ & $1$ & $0$ & $1$ \\
  \hline
  $2$ & $1$ & $0$ & $0$ & $10$
\end{tabular}
\item
\begin{tabular}{cBBc|c}
  $x_1$ & $x_2$ & $x_3$ & $x_4$ \\
  \hline
  $2$ & $1$ & $0$ & $1$ & $3$  \\
  $1$ & $0$ & $1$ & $4$ & $3$ \\
  \hline
  $2$ & $0$ & $0$ & $0$ & $8$
\end{tabular}
\end{enumerate}
\newpage
State, for each case, whether
\begin{itemize}
\item The problem has a finite solution;
\item The solution is unique;
\item The solution is degenerate (i.e., one of the basic variables is
zero).
\end{itemize}

\end{enumerate}
\end{document}
