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\title{Operations Research}
\author{Homework 9}
\date{Due via Moodle or Mailbox Wednesday, December 2, 2020}
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\begin{enumerate}

\item \emph{(From HL, Exercise 19.3-4.)}  The Blue Cab Company is the
primary taxi company in the city of Maintown. It uses gasoline at the
rate of 8\,500 gallons per month. Because this is such a major cost,
the company has made a special arrangement with the Amicable Petroleum
Company to purchase a huge quantity of gasoline at a reduced price of
\$1.05 per gallon every few months. The cost of arranging for each
order, including placing the gasoline into storage, is \$1\,000. The
cost of holding the gasoline in storage is estimated to be \$0.01 per
gallon per month.

Use the EOQ model to find the optimal order quantity.

\item An airplane manufacturer is contracted to produce a small number
of a particular type of airplane during the coming years.  The
manufacturer will need to decide each year whether to set up a
production run with a fixed set-up cost of EUR\,1\,000\,000 per run.
During each production run, the manufacturer can make at most 6
airplanes.  If an airplane is not delivered during the year it is
produced, it will incur a holding cost of EUR\,100\,000 per year.  The
number of airplanes required are $r_1=1$, $r_2=6$, $r_3=2$, and
$r_4=3$ during each of the years.

Which production schedule(s) minimize the total cost for setup and
storage? 

\item
\begin{enumerate}
\item
Use the graphical method to maximize
\begin{gather*}
  z = x_1 + 2 \, x_2
\end{gather*}
subject to
\begin{gather*}
  x_1^2 + x_2^2 \leq 1 \,, \\
  x_1, x_2 \geq 0 \,.
\end{gather*}

\item Write a Pyomo program to confirm your answer.  (Use the
\texttt{ipopt} solver instead of \texttt{glpk}.)
\end{enumerate}
\end{enumerate}

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