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\title{Operations Research}
\author{Homework 10}
\date{Due in class Tuesday, December 5, 2017}
\maketitle

\begin{enumerate}

\item \emph{(From HL, Exercise 15.3-4.)}
Consider the decision analysis problem with the following payoff table
(in units of thousands of dollars) under risk-neutral valuation:
\begin{center}
\begin{tabular}{|c|ccc|}
\hline
& \multicolumn{3}{c|}{\textbf{State of Nature}$\vphantom{1^{1^1}}$} \\
\textbf{Alternative}$\vphantom{1^{1^1}}$ & 
  $S_1$ & $S_2$ & $S_3$ \\
\hline
$A_1\vphantom{1}^{\vphantom{\tfrac11}}$ & $-100$ & $10$ & $100$ \\
$A_2$ & $-10$ & $20$ & $50$ \\
$A_3$ & $10$ & $10$ & $60$ \\
\hline
Prior Probability$\vphantom{1}^{\vphantom{\tfrac11}}$ 
& $0.2$ & $0.3$ & $0.5$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}
\item Which alternative should be chosen?  What is the resulting
expected payoff?
\item You are offered the opportunity to obtain information which will
tell you with certainty whether the first state of nature $S_1$ will
occur.  What is the maximum amount you should pay for the information?
Assuming you will obtain the information, how should it be used to
choose an alternative?  What is the resulting expected payoff
(excluding the payment)?

\item You are offered the opportunity to obtain information which
asserts the true state of nature with an accuracy of $60\%$ and
wrongly identifies one of the other possible states of nature in
$\%20$ of cases each.  What is the maximum amount you should pay for
the information?  Assuming you will obtain the information, how should
it be used to choose an alternative?  What is the resulting expected
payoff (excluding the payment)?
\end{enumerate}

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

\end{enumerate}

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