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\begin{titlepage}

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{\bf
\begin{tabular}{lll}
MA 162 & \hspace{2.8in} & Russell Brown \\
Exam 2 & \hspace{2.8in} & 12 March, 1993
\end{tabular}}

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Answer all of the following questions.
Use the backs of the question papers for scratch paper.
Additional sheets are available if necessary.  No books
or notes may be used.  When answering these questions,
please be sure to 1) check answers when possible, 2)
clearly indicate your answer and the reasoning used to
arrive at that answer {\bf (unsupported answers will receive
NO credit)}, and 3) label all variables and equations.

\vfill

\begin{center}
\begin{tabular}{|r|c|r|}
\hline
Question & Score & Total \\
\hline \hline
1 && 20 \\ & & \\
\hline
2 && 20 \\ & & \\
\hline
3 && 20 \\ & & \\
\hline
4 && 20 \\ & & \\
\hline
5 && 20 \\ & & \\
\hline \hline
Total && 100 \\
\hline  
\end{tabular}
\end{center}

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\noindent
Name \underline{$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $}
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\noindent
Section \underline{$ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $}
\vspace{.1in}

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\end{titlepage}

\begin{enumerate}

\item
A company manufactures textbooks using labor, coverboard and paper.
The MA109 text requires 0.4 hours of labor, 1.5 square feet 
of coverboard and 150 sheets of
paper.  The MA123 text requires 0.5 hours of labor, 
2 square feet of coverboard and 125 sheets of paper.
The MA162 text requires 0.6 hours of labor, 2.2 square 
feet of coverboard and 200
sheets of paper.  The company receives a profit of \$5.00 on a 
MA109 book, \$4.50 on a MA123 book and
\$6.75 on a MA162 book.  The company has available a total
of 800 hours of labor, 1500 square feet of coverboard,
and 200,000 sheets of paper.

Write down a linear program whose solution gives the number
of each text they should produce to maximize their profit.
Be sure to define each variable and label the constraints.
Do not solve the linear program.

\newpage

\item
Consider the linear program

\begin{center}
\begin{tabular}{lll}
$ \min \: 2x + y $ && \\
subject to && $ x + y \geq 5 $ \\
&& $ x + 2y \geq 6 $ \\
&& $ x \geq 0 , ~~ y \geq 0 $. \end{tabular} 
\end{center}

a) Graph the feasible set and find the coordinates
of each corner point.
\vspace{3.5in}

b) Find the solution of the above linear program.
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c) What will the solution be if the objective function is
$ x + 3y $?
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d) What will the solution be if the objective function is $ -x $?
\vfill

\newpage

\item
Consider the tableau
\[ \begin{array}{c|crrccc}
\mbox{Row} & P & x & y & s_1 & s_2 & \mbox{RHS} \\
\hline 
1 & 1 & -2 & -1 & 0 & 0 & 0 \\
2 & 0 & 1 & 2 & 1 & 0 & 6 \\
3 & 0 & 2 & 1 & 0 & 1 & 6 
\end{array} \]

a)  List the basic and nonbasic variables.
\vspace{.7in}

b)  What are the values of the decision variables
when $ s_1 = 2 $ and $ s_2 = 4 $.  Is this point feasible?
Why or why not?
\vspace{.7in}

c)  Circle the pivot element in the above tableau and carry out
exactly one pivot.
\vspace{1.3in}

d)  For the tableau you found in part c), give the value
of the objective function and the augmented coordinates of the
associated corner point.
\vspace{1.3in}

e)  Is the tableau found in part c) optimal?  Why or why not?

\newpage

\item
Solve the following linear program by the simplex method.

\begin{center}
\begin{tabular}{lll}
$ \max \: x + 2y $ && \\
subject to && $ x + y \leq 6 $ \\
&& $ 2x + y \leq 10 $ \\
&& $ x \geq 0 , ~~ y \geq 0 $. \end{tabular}
\end{center}


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\newpage

\item
A company produces chairs, desks and tables.  The primary resources
they use are wood, assembly labor and finishing labor.  The Lindo printout
on the following page gives a linear program which 
maximizes their profit subject to constraints on labor and wood.
In the printout the variables $ C , T $ and $D$ stand for the number
of chairs, tables and desks produced.  Use this printout to answer
the following questions.

a)  How many tables, desks and chairs should they produce?
What will be the resulting profit?
\vspace{.9in}

b)  Suppose that they can hire additional labor.  Should they put
these workers in the assembly shop or the finishing shop?  Why?
\vspace{.9in}

c)  Suppose a shipment of 100 board feet of lumber is lost.
What will be the effect on their profit?
\vspace{.9in}

d)  Suppose that the profit on chairs decreases to \$10.
What will be their new total profit?
\vspace{.9in}

e)  Suppose that the profit on chairs remains at \$18
and the profit on tables increases to \$35.  What will their
new profit be?
\vspace{.9in}

\end{enumerate}
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