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\begin{document}

\title{Operations Research}
\author{Homework 3}
\date{Due in class Wednesday, September 28, 2016}
\maketitle

\emph{Note: All Pyomo homework is due Thursday, September 29, in class.}

\begin{enumerate}

\item 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*}

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 Problem 1 using the simplex method \emph{on paper.}


\item (From HL, problem 3.4-15.)  A cargo plane has three compartments
for storing cargo: front, center, and back.  These compartments have
capacity limits on both \emph{weight} and \emph{space}, as summarized
below:

\begin{center}
\begin{tabular}{lcc}
\hline
  \textbf{Compartment}$\vphantom{1^{1^1}}$ & \textbf{Weight Capacity} [t] & \textbf{Space
                                                        Capacity}
                                                        [ft$^3$] \\
\hline
Front$^{\vphantom{\tfrac11}}$ & 12 & 7\,000 \\
Center & 18 & 9\,000 \\
Back & 10 & 5\,000 \\
\hline
\end{tabular}
\end{center}
Furthermore, the weight of the cargo in the respective compartments
must be the same proportion of that compartment's weight capacity to
maintain the balance of the airplane.

The following four cargoes have been offered for shipment on an
upcoming flight as space is available:
\begin{center}
\begin{tabular}[lcc]{lccc}
  \hline
  \textbf{Cargo}$\vphantom{1^{1^1}}$ & \textbf{Weight} [t] & \textbf{Volume} [ft$^3$/t] &
                                                                      \textbf{Profit}
                                                                      [\$/t]
  \\
\hline
1$^{\vphantom{\tfrac11}}$ & 20 & 500 & 320 \\
2 & 16 & 700 & 400 \\
3 & 25 & 600 & 360 \\
4 & 13 & 400 & 290 \\
\hline
\end{tabular}
\end{center}
Any portion of these cargoes can be accepted.  The objective is to
determine how much (if any) of each cargo should be accepted and how
to distribute each among the compartments to maximize the total profit
for the flight.
\begin{enumerate}
\item Formulate a linear programming model for this problem.
(Submission in mathematical notation!)
\item Solve this model using Pyomo to find one of its multiple optimal
solutions. (Submission as printout of Ipython notebook.)
\end{enumerate}


\end{enumerate}

\end{document}
