%3/4/92
%revised 10/12/92 included lindo output.
%Revised 1/19/94  Changed lindo problem.
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\begin{document}
\begin {flushleft}
MA162 \hfill
Review 2
\end{flushleft}
The following topics and skills will be examined in the second midterm test.
\begin{itemize}

\item Graphing the feasible set of a system of linear inequalities in
two variables.  Finding corner points
\item Formulating written descriptions of feasible sets as linear
inequalities. 
\item Solving linear programs by graphical, corner point and simplex
methods.
\item Formulate word problems as linear programs.
\item The augmented coordinate system. 
\item The simplex tableau: Basic and nonbasic variables, choosing the
pivot element, finding the augmented coordinates of a corner point and
determining if a tableau corresponds to the optimal feasible point.
\item Dual prices, righthand side ranging and changing coefficients in
objective function.
\end{itemize}
Below are some sample questions to help you prepare for the exam.
Be sure that you understand all of the topics from the above list, not
just the topics covered by the questions below!
In your answers on the exam, please be sure to 1) Check
answers when 
possible. 2) Clearly indicate your answer and the reasoning used to
arrive at that answer. 3)  Label all variables and equations.



\begin{enumerate}
\item
\begin {enumerate} \item Solve the following linear program by the
graphical method. 
$$
\begin {array}{rr}
\max &x+2y\\
\st & x+y\leq 4\\
 & 2x+y \leq 6\\
& x\geq 0, \ y \geq 0
\end{array}
$$
\item Check your answer by re-solving the problem by the corner point
method.
\end{enumerate}
\item The feasible set for the following system of linear inequalities
is sketched below.
\begin{eqnarray*}
x-y&\leq& 0
\\
\qquad \qquad \qquad 5x+6y& \leq &30 \\
y & \leq &4\\
x \geq 0,&  & y \geq 0 
\end{eqnarray*}

\psfig{file=r2.eps}

\begin{enumerate}
\item Rewrite the system of inequalities  by introducing slack variables.
\item Find the augmented coordinates of the points labelled A-E.
\end{enumerate}
\item You own a shipping company which transports two types of
packages, A and B.  Your plane can carry only 100 units of volume and
60 units of weight. 

You know that a package of type A has a volume of 5 units and a weight
of 1 unit, but will give a profit of \$3 a package. A package of type
B has a volume of 2 units and a weight of 2 units and gives you a
profit of \$2 per package. Customs will allow you  to transport no
more than 16 packages of type A per flight. 

How many of each type package should you transport to maximize your
profit?  What is the maximum profit?
\item  
\begin{enumerate}
\item Carry out one simplex pivot on the following tableau.
$$
\begin{array}{|rrrrr|r|}
\hline
P&x_1&x_2&s_1&s_2&RHS\\
\hline
1&-1& -2& 0& 0& 0\\
0& 1& 1& 1& 0& 4\\
0& 2& 4& 0& 1& 8\\
\hline
\end{array}
$$
After you have done the pivot, answer the following questions:
\begin{enumerate}
\item Are we done with the problem, that is have we reached an optimal
solution?
Why or why  not?
\item Find the associated corner point
\end{enumerate}
\item Carry out one pivot   so that the variable $x_2 $
becomes basic.  (Remark: There are two possible ways to do this.)
$$
\begin{array}{|rrrrrr|r|}
\hline
x_1& x_2 & x_3 & s_1 & s_2 & s_3 & RHS\\ \hline
1& -1 & 0& 0& 1& -2&4\\
0& -3 & 0& 1& 1& 0& 3\\
0& 0& 1& 0& -2 & 1& 2\\
\hline
\end{array}
$$
After you have pivoted, give  the basic variables and their values.
What are the nonbasic 
variables and their values?
\end{enumerate}

\item Use the attached LINDO printout to answer the following
questions.  

\begin{enumerate}
\item What is the optimal feasible point?  What is the maximum value
of the objective function?

\item  Compute the new value of the objective function if the
coefficient of $x$ is decreased by 1 unit.

\item Compute the new value of the objective function if the
coefficient of $x$ in the objective function is 8.
\item Compute the new value of the objective function if the
righthand side of constraint 2 is increased by 10 units.  (For this
question, assume that the coefficient of $x$ has been returned to its
original value, 4.)
\end{enumerate}

\begin{verbatim}

 MAX     4 X + 2 Y + 5 Z
 SUBJECT TO
        2)   X + 2 Y + 3 Z <=   30
        3)   3 X + Y + 2 Z <=   40
        4)   X + Y + Z <=   20
 END

 LP OPTIMUM FOUND AT STEP      2

        OBJECTIVE FUNCTION VALUE

        1)    70.0000000

  VARIABLE        VALUE          REDUCED COST
         X         8.571428          0.000000
         Y         0.000000          1.000000
         Z         7.142857          0.000000


       ROW   SLACK OR SURPLUS     DUAL PRICES
        2)         0.000000          1.000000
        3)         0.000000          1.000000
        4)         4.285714          0.000000

 NO. ITERATIONS=       2

 RANGES IN WHICH THE BASIS IS UNCHANGED:

                           OBJ COEFFICIENT RANGES
 VARIABLE         CURRENT        ALLOWABLE        ALLOWABLE
                   COEF          INCREASE         DECREASE
        X        4.000000         3.500000         2.333333
        Y        2.000000         1.000000         INFINITY
        Z        5.000000         6.999999         1.400000

                           RIGHTHAND SIDE RANGES
      ROW         CURRENT        ALLOWABLE        ALLOWABLE
                    RHS          INCREASE         DECREASE
        2       30.000000        30.000000        16.666664
        3       40.000000        15.000000        20.000000
        4       20.000000         INFINITY         4.285714


\end{verbatim}



\end{enumerate}
\end{document}

