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{\Large \bf C\&O 370} \\
{\Large \bf HOMEWORK 3}
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Due: Thursday Mar. 4.

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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}
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