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{\large\bf C\&O 350 \\ 
Assignment 1 
}
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{\large  Due: Thurs. Jan. 19 (\S 1),
          Fri. Jan. 20 (\S 2,3 )
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(The details should be worked out by hand. They can be verified on the
computer.)
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\item
In the following two modelling problems, you do {\em not} need to solve the models.
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\item
A small firm specializes in making five types of spare automobile parts.
Each part is first cast from iron in the casting shop and then sent to
the finishing shop where holes are drilled, surfaces are turned, and
edges are ground. The required worker-hours (per 100 units) for each of
the parts of the two shops are shown below:

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\begin{tabular}{|c|ccccc|} \hline
Part & 1 & 2 & 3 & 4 & 5 \\
\hline
Casting & 2 & 1 & 3 & 3 & 1 \\
Finishing & 3 & 2 & 2 & 1 & 1 \\
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The profits from the parts are \$30, \$20, \$40, \$25, and \$10 (per 100
units), respectively. The capacities of the casting and finishing shops
over the next month are 700 and 1000 worker-hours, respectively.
Formulate the problem of determining the quanitities of each spare part
to be made during the month so as to maximize profit.

\item
There are three types of machines that can be used to make a chemical.
The chemical can be manufactured at four different purity levels. 


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\begin{tabular}{|c|c|cccc|} \hline
 &  Capacity: & \multicolumn{4}{c}{Production Costs per Day} \\
Machine Type  &  Production per Day & 
         \multicolumn{4}{c}{If Used to Manufacture Purity} \\
\hline
&   & 1 & 2 & 3 & 4 \\
\hline
1 & 50 tons & \$900 & 1000 & 1250 & 1500 \\
2 & 40 tons & 600 & 750 & 1000 & 1050 \\
3 & 25 tons & 600 & 700 & 800 & 900 \\
\hline
\hline
\multicolumn{2}{|c}{Monthly demand for chemical at} &&&&  \\
\multicolumn{2}{|c}{purity level (tons)} & 150 & 300 & 90 & 175  \\
\hline
\multicolumn{2}{|c}{Penalty per ton short} & 40 & 60 & 75 & 90  \\
\hline
\hline
\end{tabular}
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The chemical is used by the company internally; if the company does not
manufacture enough of it, it can be bought at a price, which is called
the penalty for shortage. Assume that there are 30 production days per
month at the company. Formulate the problem of minimizing the overall
cost as an LP.


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\item
Chvatal, Problem 2.1 b. page 26.
\item
Solve Chvatal, Problem 2.1 c. page 26 geometrically (do not use the simplex
method). In addition,\\
(i) draw the feasible set in $\Re^2;$ \\
(ii) find all the extreme points of the feasible set and, for each of
these extreme points, write down the corresponding dictionary.

\item
Chvatal, Problem 2.2  page 26.


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