\documentclass[a4paper,12pt]{article}

\begin{document}

\parindent=0pt

QUESTION

\begin{description}

\item[(a)]
A project consists of activities $A,B,\ldots,J$ whose
prerequisites 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.

\begin{tabular}{ccccc}
\hline &&Normal&Crash&Cost of\\ Activiyt&Prerequisites&duration
(days)&duration (days)&reduction (\pounds)\\ \hline A&-&7&5&14\\
B&-&6&5&10\\ C&-&4&3&15\\ D&A&5&2&12\\ E&B&7&4&18\\ F&C&9&6&30\\
G&B,D&8&5&27\\ H&E.F&7&4&45\\ I&G&5&2&50\\ J&G,H&6&5&20\\ \hline
\end{tabular}


Assuming that all activities have normal durations, as given in
the table, write all of the earliest and latest event times on the
network, and hence deduce \textit{all} critical paths.

The table above also lists the crash duration of each activity and
the corresponding cost of reducing the activity duration from its
normal duration to its crash duration. It is possible to set the
duration of any activity to any value between its normal and crash
duration, which incurs a cost that is proportional to the
reduction in duration.

Analyze how the overall project duration can be reduced by 2 days
from its normal duration at minimum total cost.


\item[(b)]
A company produces a special type of flour for a bakery.  The
bakery orders either 1, 2 or 3 batches of flour every three
months. Due to the lengthy production process, the company must
produce the flour before the order arrives.

It costs \pounds$15\,000$ for the company to produce each batch of
flour, and the bakery buys the flour at a price of {\it
\$}$20\,000$ per batch. If the bakery orders more flour than the
company has produced, then the company fills the order by
purchasing a substitute from another firm at a cost of {\it
\$}$24\,000$ per batch. Since the flour deteriorates over time, if
the bakery orders less flour than the company has produced, then
the company reprocesses any excess batches, where each excess
batch is valued at \pounds$5\,000$.

Based on historical data, the bakery orders 1 batch with
probability 0.3, orders 2 batches with probability 0.5, and orders
3 batches with probability 0.2. Develop a decision tree to find
how many batches the company should produce every three months.
Also, find the maximum discount per batch that should be offered
to the bakery for specifying in advance exactly how many batches
it will order.

\end{description}

ANSWER

\begin{description}

\item[(a)]\ \\

\setlength{\unitlength}{1.2cm}

\begin{picture}(10,6)

\put(1,3){\circle{.3}} \put(1,3){\makebox(0,0){1}}
\put(3,5){\circle{.3}} \put(3,5){\makebox(0,0){2}}
\put(3,3){\circle{.3}} \put(3,3){\makebox(0,0){3}}
\put(3,1){\circle{.3}} \put(3,1){\makebox(0,0){4}}
\put(5,5){\circle{.3}} \put(5,5){\makebox(0,0){5}}
\put(5,3){\circle{.3}} \put(5,3){\makebox(0,0){6}}
\put(7,5){\circle{.3}} \put(7,5){\makebox(0,0){7}}
\put(7,3){\circle{.3}} \put(7,3){\makebox(0,0){8}}
\put(9,4){\circle{.3}} \put(9,4){\makebox(0,0){9}}

\put(1.2,3){\vector(1,1){1.8}}\put(1.2,3){\vector(1,0){1.6}}\put(1.2,3){\vector(1,-1){1.8}}
\put(3.2,5){\vector(1,0){1.6}}
\put(3.2,3){\vector(1,1){.4}}\put(4,3.8){\vector(1,1){.4}}\put(3.2,3){\vector(1,0){1.6}}
\put(3.2,1){\vector(1,1){1.8}} \put(5.2,5){\vector(1,0){1.6}}
\put(5.2,3){\vector(1,0){1.6}}
\put(7.2,5){\vector(2,-1){1.7}}\put(7,4.8){\vector(0,-1){.4}}\put(7,4.){\vector(0,-1){.4}}
\put(7.2,3){\vector(2,1){1.7}}

\begin{small}
\put(1,3.5){\makebox(0,0){(0,0)}}
\put(3,5.5){\makebox(0,0){(7,7)}}
\put(3,2.5){\makebox(0,0){(6,6)}} \put(3,.5){\makebox(0,0){(4,4)}}
\put(5,5.5){\makebox(0,0){(10,12,12)}}
\put(5,3.5){\makebox(0,0){(13,13)}}
\put(7,5.5){\makebox(0,0){(18,20,20)}}
\put(7,2.5){\makebox(0,0){(20,20,18)}}
\put(9.5,3.5){\makebox(0,0){(26,26,24)}}

\put(1.2,4){A 7} \put(2,3.2){B 6} \put(1.2,1.5){C 4}
\put(3.5,5.2){D 5} \put(3.9,3.2){E 7} \put(4,1.5){F 9}
\put(5.7,5.2){G 8} \put(5.7,3.2){H 7} \put(8,4.7){I 5} \put(8,3){J
6}

\end{small}

\end{picture}

Critical paths are A - D - G - I, B - E - H - J, C - F - H- J

Possibilities to reduce duration are

Reduce J: unit cost 20

Reduce D,H: unit cost 4+15=19

Reduce D,E,F: unit cost 4+6+10=20

Reduce D,H by 2 day to 3 and 5 days.

Since the project duration becomes 24, this is the desired
solution.

\item[(b)]\ \\

\setlength{\unitlength}{1.5cm}

\begin{picture}(8,10)

\put(6,9){5000} \put(6,8){1000} \put(6,7){$-3000$}
\put(6,6){$-5000$} \put(6,5){10000} \put(6,4){6000}
\put(6,3){$-15000$} \put(6,2){0} \put(6,1){15000}

\put(4,8){\circle{.3}} \put(4,5){\circle{.3}}
\put(4,2){\circle{.3}}

\put(.8,4.8){\framebox(.3,.3)}

\put(1.2,5){\line(1,1){2.8}} \put(1.2,5){\line(1,0){2.6}}
\put(1.2,5){\line(1,-1){2.8}}

\put(4.2,8){\line(2,1){1.8}}\put(4.2,8){\line(1,0){1.6}}\put(4.2,8){\line(2,-1){1.8}}
\put(4.2,5){\line(2,1){1.8}}\put(4.2,5){\line(1,0){1.6}}\put(4.2,5){\line(2,-1){1.8}}
\put(4.2,2){\line(2,1){1.8}}\put(4.2,2){\line(1,0){1.6}}\put(4.2,2){\line(2,-1){1.8}}

\put(.5,5.5){4700} \put(3.5,8.5){1400} \put(3.5,5.5){4700}
\put(3.5,2.5){$-1500$}

\put(5,9){1 0.3} \put(5,8.2){2 0.5} \put(5,7){3 0.2} \put(5,6){1
0.3} \put(5,5.2){2 0.5} \put(5,4){3 0.2} \put(5,3){1 0.3}
\put(5,2.2){2 0.5} \put(5,1){3 0.2}

\put(1.5,8){Production} \put(4.5,9.5){Order}

\put(2,6.2){1} \put(2,5.2){2} \put(2,4){3}

\put(1.6,5.7){\line(0,-1){.5}}\put(1.6,4.8){\line(0,-1){.5}}

\end{picture}

The optimal decision is to produce 2 batches each time.

If the bakery specifies in advance its order, then the profits are
$5000-x,\ 10000-2x,\ 15000-3x$ with probabilities 0.3, 0.5, 0.2
where $x$ is the discount for each batch.

The expected profit is

$$(5000-x)0.3+(10000-2x)0.5+(15000-3x)0.2=9500-1.9x$$

The break even discount is given by

$$9500-1.9x=4700,\ x=2526.32$$

\end{description}






\end{document}
