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\title{Operations Research}
\author{Homework 10}
\date{Due in class Friday, April 29, 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{(From HL, Exercise 15.8-5.)}
The B. J. Jensen Company specializes in the production
of power saws and power drills for home use. Sales are relatively
stable throughout the year except for a jump upward during the
Christmas season. Since the production work requires considerable
work and experience, the company maintains a stable employment
level and then uses overtime to increase production in November.
The workers also welcome this opportunity to earn extra money
for the holidays.

B. J. Jensen, Jr., the current president of the company, is overseeing
the production plans being made for the upcoming November. 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 Time}$\vphantom{1^{1^1}}$ & 
  \textbf{Overtime}$\vphantom{1^{1^1}}$ & 
  \textbf{Regular Time}$\vphantom{1^{1^1}}$ & 
  \textbf{Overtime}$\vphantom{1^{1^1}}$ \\
\hline
\textbf{Power saws}$\vphantom{1^{1^1}}$ & 3\,000 & 2\,000 & \$150 & \$50
  \\
\textbf{Power drills}$\vphantom{1^{1^1}}$ & 5\,000 & 3\,000 & \$100 & \$75
  \\
\hline
\end{tabular}
\end{center}

The values for maximum monthly production assume adequate supplies 
 of materials from the company's
vendors.

However, Mr.\ Jensen now has learned that, in addition to the
limited number of labor hours available, two other factors will limit
the production levels that can be achieved this November. One is
that the company's vendor for power supply units will only be able
to provide 10\,000 of these units for November (2\,000 more than
his usual monthly shipment). 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 15\,000 for
November (4\,000 more than for other months). 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 November to maximize
the company's total profit.

Set up a model and solve the model using Pyomo, e.g.\ by formulating
it as a separable programming problem and using a linear solver.

\end{enumerate}
\end{document}
