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\title{Operations Research}
\author{Homework 10}
\date{Due in class Wednesday, November 30, 2016}
\maketitle

\begin{enumerate}

\item \emph{(From HL, Exercise 13.3-5.)}
Consider the following problem, which is called a linear fractional
programming problem:
\begin{gather}
  \text{Maximize } 
  f(\textbf{x}) = \frac{10\,x_1 + 20 \, x_2 + 10}%
                       {3\,x_1 + 4\,x_2 + 20}
\intertext{subject to}
  x_1 + 3 \, x_2 \leq 50 \,, \\
  3 \, x_1 + 2 \, x_2 \leq 80 \,, \\
  x_1, x_2 \geq 0 \,.
\end{gather}

\begin{itemize}
\item Transform this problem into an equivalent linear programming
problem. 

\item Solve the linear programming problem using Pyomo.

\item Solve the original nonlinear programming problem using Pyomo.
Verify that the two solutions coincide.
\end{itemize}
Please submit printouts of your Ipython notebooks for (b) and (c).




\item \emph{(Variation of Homework 9, Problem 2.)}
The B. J. Jensen Company specializes in the production
of power saws and power drills for home use.  

B. J. Jensen, Jr., the current president of the company, is overseeing
the production plans being made for the month of January.  He has
managed to negotiate volume discounts with one of his suppliers
resulting in increased profits if more than $2\,000$ units of any one
of his products are produced.  He has obtained the following data.
\begin{center}
\begin{tabular}{|l|cc|cc|}
\hline
& \multicolumn{2}{c|}{\textbf{Maximum Monthly}$\vphantom{1^{1^1}}$} 
& \multicolumn{2}{c|}{\textbf{Profit per}$\vphantom{1^{1^1}}$} \\
& \multicolumn{2}{c|}{\textbf{Production}$\vphantom{1^{1^1}}$} 
& \multicolumn{2}{c|}{\textbf{Unit Produced}$\vphantom{1^{1^1}}$} \\
\hline
& \textbf{Regular Supply}$\vphantom{1^{1^1}}$ & 
  \textbf{Discounted Supply}$\vphantom{1^{1^1}}$ & 
  \textbf{Regular Supply}$\vphantom{1^{1^1}}$ & 
  \textbf{Discounted Supply}$\vphantom{1^{1^1}}$ \\
\hline
\textbf{Power saws}$\vphantom{1^{1^1}}$ & 2\,000 & 1\,000 & \$150 & \$180
  \\
\textbf{Power drills}$\vphantom{1^{1^1}}$ & 2\,000 & 3\,000 & \$100 & \$110
  \\
\hline
\end{tabular}
\end{center}

Two other factors will limit the production levels that can be
achieved. One is that the company's vendor for power supply units will
only be able to provide 7\,000 of these units for January. Each power
saw and each power drill requires one of these units. Second, the
vendor who supplies a key part for the gear assemblies will only be
able to provide 10\,000 for January. Each power saw requires two of
these parts and each power drill requires one.

Mr.\ Jensen now wants to determine how many power saws
and how many power drills to produce in January to maximize
the company's total profit.

Modify your Pyomo code from Homework 9 and solve this problem.  

\emph{Hint:} Note that the payoff function is now convex rather than
concave.  This means that you need introduce an additional binary
decision variable (in Pyomo: \texttt{within=Boolean}) which tracks
whether the discount threshold has been reached.



\end{enumerate}

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