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\begin{document}
\begin{center}\bf Math 311\\Exam 2 - Fall 2000\\Prof. Leonid Faybusovich
\end{center}
\vskip .70 in
\noindent1.  Consider the matrix game with a payoff matrix.

\

$$\left[\begin{array}{ccc}

7&6&8\\
2&4&3\\
1&-1&8

 \end{array}\right]$$

\indent a)  Find the saddle point.\hfill (10)

\

\indent b)  Determine optimal strategies for the row and the column player and the value of the game. \hfill
(10)

\

\

\noindent 2.  Consider the game with the pay-off matrix. 

\

$$\left[\begin{array}{ccc}

3&5&2\\
2&1&3\\
2&3&1

 \end{array}\right]$$

\indent a)  Show that the game has a dominated row.\hfill(10)

\

\indent b)  Solve the reduced game using the graphic method.  \hfill (50)

\

\

\noindent3.  Solve the following optimization problem:  \hfill (20)

\

$$x_1 + 3x_2 + 2x_3 \rightarrow \rm{max}$$
$$x_1 + 2x_2 + x_3 \leq 8$$
$$x_i - \rm{nonnegative\  integers}$$

\

4.  Solve the following assignment problem:\hfill(50)


$$\begin{tabular}{|c|c|c|c|} 
\hline
1&4&6&3\\
\hline
9&7&10&9\\
\hline
4&5&11&7\\
\hline
8&7&8&5\\
\hline
\end{tabular}$$

\end{document}


