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{\Large \bf C\&O 370} \\
{\Large \bf HOMEWORK 3}
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Due: Monday, October 30

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\noindent
{\bf Reading} \\
\begin{tabular}{lll}
Chapter 8 & (pp. 310-326) & Network Models\\
Chvatal & (pp. 291-299) & Transshipment Problem/(pp. 367-378) Maximum Flow
Problem
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{\bf Problems}
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\begin{enumerate}
\item[3.1]
[25]\\
A mathematics class is partitioned into 20 teams of students.
Each team is required to produce an applied research project based upon one
of eight given
problems.
The instructor stipulates that at most three teams may work on the same
problem and that every
problem must be done by at least one of the teams.
Each team submits a ranking of the eight problems, giving a ``1" to the
most desirable, a ``2"
to the next most desirable and, finally, an ``8" to the least desirable.
Thus, large ranking numbers indicate {\em undesirability}.
\begin{itemize}
\item[(a)]
Develop a model for determining how the instructor should assign problems
to teams so as to
minimize the sum of the ranking numbers of the assigned problems.\\
Draw and label the appropriate network.
\item[(b)]
Describe how your model from part (a) would change if the objective was to
minimize the maximum
ranking number of an assigned problem.
\end{itemize}

\item[3.2]
[25]\\
On a given morning, 26 patients arrive at Heal-Fast Hospital.
Before being admitted for treatment, each patient is first classified into
one of 7 treatment
groups according to the type of treatment he/she requires.
Patients assigned to a given treatment group can be treated only in certain
designated
units from among the set $(a,b,c,d,e,f)$.
The number of available spaces in these units is 5,5,4,4,4,4, respectively.
The treatment group classification and designated units data are as follows:
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\begin{tabular}{c|c|c}
Number of & Treatment & Designated\\
Incoming Patients & Group & Units \\ \hline
5 & 1 & $c,f$\\
3 & 2 & $b,c$\\
2 & 3 & $a,e$\\
5 & 4 & $b,c,d$\\
2 & 5 & $a,d$\\
5 & 6 & $b,d,f$\\
4 & 7 & $a,c,e$
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\begin{itemize}
\item[(a)]
Develop a model for determining how the hospital should assign these patients to
allowable designated units so as to maximize the total number who can be
admitted for
treatment.
\item[(b)]
Solve your model from part (a), and verify the optimality of your solution.
\end{itemize}
\end{enumerate}

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\noindent
{\bf Case Study \#2} \newline
[50]\\
A manufacturer of nuclear reactor components has a monthly production
capacity of 100 components
in each of the months January, February and March and a demand of 70,90 and
90 components for
delivery at the end of January, February and March, respectively.
The plant is required to produce a minimum of 50 components in each of the
months of January
and February and may store in inventory a maximum of 30 components in each
of the months
February and March.

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\noindent
Unit production costs, in $(\$ \times 100)$/component, for January,
February and March are
4,7 and 4, respectively.
If February's demand is not met on time, then any shortage can be made up
from March's capacity
at a cost of $6(\$ \times 100)$/component.
Inventory-holding cost is $2(\$ \times 100)$/component in both February and
March.
In Feburary, there is a penalty cost of $2(\$ \times 100)$/component for
unused production capacity.

\vspace{3mm}
\noindent
You have been asked by the manufacturer to develop and solve a model for
determining
a production schedule which meets these demands and minimizes total cost.
Total cost is the sum of production, inventory-holding and penalty (for
unused production
capacity) costs.

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\noindent
The manufacturer would also like to know how to solve similar problems
using GAMS.
A carefully labeled network might help the manufacturer understand your model.
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