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QUESTION

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\item[(a)]
 A project consists of activities $A,B,\ldots,K$ whose prerequisites
and durations are given in the table below.  Draw a network,
suitable for analysis by the critical path method, to represent
the project.   You should avoid using dummy activities, where
possible. For each event, write the earliest and latest event
times on the network, and hence deduce all critical paths.

\begin{center}
\begin{tabular}{ccc}
\hline Activity&Prerequisite&Duration (days)\\ \hline A&-&8\\
B&-&5\\ C&A&7\\D&A&6\\ E&B&9\\f&c&9\\ G&C&4\\ H&D&7\\ I&D,E&5\\
J&G,H&2\\ K&G,H,I&3\\ \hline
\end{tabular}
\end{center}


 It is possible to
transfer resources from activity $B$ to either activity $A$ or
activity $C$, so that the duration of $B$ is increased and the
duration of $A$ or $C$ is decreased. Analyse whether such a
transfer can reduce the overall project duration.


\item[(b)]
A financial analyst has \pounds$1\,000\,000$ to invest. The entire
amount must first be invested for one year in either in stocks or
in bonds (but not both).  At the end of the year, the entire
amount that results from the investment in the first year must be
reinvested either in stocks or in bonds (but not both) for one
further year.  The aim is to maximize the return at the end of the
second year.


The annual return (in addition to the amount invested) depends on
the economy, as shown in the following table.

\begin{center}
\begin{tabular}{lcc}
\hline Economy&Stocks&Bonds\\ \hline Growth&20\%&5\%\\
Recession&$-10\%$&10\%\\ \hline
\end{tabular}
\end{center}

The probabilities of growth and recession in the first year are
0.7 and 0.3, respectively.  If growth occurs in the first year,
these probabilities remain the same for the second year. However,
if a recession occurs in the first year, these probabilities
change to 0.4 and 0.6, respectively.


Develop a decision tree to find an optimal strategy for investing
the money.

\end{description}


ANSWER


\begin{description}

\item[(a)]\ \\

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Critical paths A - C - F,\ A - D - H - K

Transferring resources to activity C does not help since C is only
on one of the paths. However, transferring resources to A reduces
the lengths of all critical paths. The total float of B is 2, so
that provided B does not increase by 2 days the overall project
duration will become smaller.

\item[(b)]\ \\

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From the tree, the best policy is to buy stocks in the first year.
If there is growth, buy stocks in the second year, if there is
recession, buy bonds in the second year.

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