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\title{Operations Research}
\author{Homework 3}
\date{Due in class Wednesday, February 24, 2016}
\maketitle

\noindent Reconsider Problem~1 from Homework Set~1: \emph{Minimize}
\[
  z = 8 \, x + 12 \, y
\]
\emph{subject to}
\begin{gather*}
  5 \, x + 2 \, y \geq 20 \,, \\
  4 \, x + 3 \, y \geq 24 \,, \\
  y \geq 2 \,, \\
  x, y \geq 0 \,.
\end{gather*}


\begin{enumerate}

\item Introduce slack variables to write this linear programming
problem in the standard form: \emph{Minimize}
\begin{gather*}
  z = \c^T \x \\
\intertext{\emph{subject to}} 
  A\x = \b \, \\ 
  \x \geq 0
\end{gather*}
where the coefficients $\b, \c$, and the decision variables $\x$ are
written as column vectors, and $A$ is a matrix of matching dimension.


\item Solve this problem using the simplex method \emph{on paper.}

\item On Homework Set~2, you solved this problem using Pyomo.  To
extract the values of slack variables, you can use the \verb+lslack()+
and \verb+uslack()+ attributes on each constraint.

Re-run your code, call \verb+lslack()+ and \verb+uslack()+ on each
constraint, and explain the numbers which are returned.  

There is no need to re-submit a printout of your program, your
homework will be graded on the explanation of the result.

\end{enumerate}

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