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% Title Page
\title{SCM2912 Project Scheduling - Assignment 2}
\author{Bradley van Ree, s1505881}

\begin{document}
\maketitle
\tableofcontents
\listoffigures
\listoftables

\chapter{Question 1}

\paragraph{}  The project director for a software company is currently organising a project to develop new software for a management information system.  She has broken the project into eighteen activities as in table \ref{Question1:Table1} on page \pageref{Question1:Table1} (all times in weeks).


\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{cl|c|ccc}
Label & Activity & Predecessor & Most Likely & Optimistic & Pessimistic \\ 
\hline
A & Prepare project plan & - & 9 & 4 & 13 \\
B & Evaluate market research & A & 3 & 1 & 4 \\ 
C & Set screen formats & B & 7 & 4 & 16 \\ 
D & Set report formats & B & 7 & 4 & 16 \\ 
E & Code graphics & C & 9 & 7 & 18 \\ 
F & Code interface module & E & 9 & 7 & 18 \\ 
G & Code optimisations & B & 12 & 4 & 18 \\ 
H & Code database manager & D & 9 & 3 & 22 \\ 
I & Write manuals & F, H & 13 & 9 & 18 \\ 
J & Develop test data & D & 12 & 9 & 21 \\ 
K & Write interfacing protocols & F & 12 & 4 & 18 \\ 
L & Alpha testing & G, J & 4 & 3 & 21 \\ 
M & Obtain platform versions & G & 9 & 7 & 25 \\ 
N & Beta testing & I, M & 12 & 7 & 22 \\ 
O & Prepare marketing plan & L & 4 & 3 & 7 \\ 
P & Train marketing staff & O & 3 & 3 & 3 \\ 
Q & Place product on-line & K, N, P & 9 & 4 & 21 \\ 
R & Evaluate market reaction & Q & 4 & 3 & 7
\end{tabular}
} % end footnotesize
\end{center}
\caption{Raw data for Question 1 from assignment sheet.}
\label{Question1:Table1}
\end{table}


\section{Part (i)}

\paragraph{}Find the expected value and variance for each task.

\subparagraph{}The expected values for each task are in table \ref{Question1:ExpectedDuration} found on page \pageref{Question1:ExpectedDuration}.

\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{l|c|lll|l|ll}
� & � & (M) & (A) & (B)  & ${\Bigl( \frac{(B-A)}{6} \Bigr)}^2$ & $\frac{(A+B+4M)}{6}$ \\
Label & Predecessor & Most Likely & Optimistic & Pessimistic & Variance & Expected Duration\\
\hline
A & - & 9 & 4 & 13  & 2.25 & 8.83 \\
B & A & 3 & 1 & 4  & 0.25 & 2.83 \\
C & B & 7 & 4 & 16  & 4 & 8 \\
D & B & 7 & 4 & 16  & 4 & 8 \\
E & C & 9 & 7 & 18 & 3.36 & 10.17 \\
F & E & 9 & 7 & 18  & 3.36 & 10.17 \\
G & B & 12 & 4 & 18 & 5.44 & 11.67 \\
H & D & 9 & 3 & 22  & 10.03 & 10.17 \\
I & F, H & 13 & 9 & 18 & 2.25 & 13.17 \\
J & D & 12 & 9 & 21 & 4 & 13 \\
K & F & 12 & 4 & 18  & 5.44 & 11.67 \\
L & G, J & 4 & 3 & 21 & 9 & 6.67 \\
M & G & 9 & 7 & 25 & 9 & 11.33 \\
N & I, M & 12 & 7 & 22 & 6.25 & 12.83 \\
O & L & 4 & 3 & 7  & 0.44 & 4.33 \\
P & O & 3 & 3 & 3 & 0 & 3 \\
Q & K, N, P & 9 & 4 & 21 & 8.03 & 10.17 \\
R & Q & 4 & 3 & 7 & 0.44 & 4.33
\end{tabular}
} % end footnotesize
\end{center}
\caption{Values for the excpected duration of each arc within the project.}
\label{Question1:ExpectedDuration}
\end{table}


\section{Part (ii)}

\paragraph{}Find the critical path and the makespan for this project using the expected duration for each activity.  Verify your answer using MS PROJECT.

\subparagraph{}The ``hand made'' network diagram used to work out the critical path and the makespan for this project can be found in figure \ref{Question1:NetworkDiagram} on page \pageref{Question1:NetworkDiagram}.  The MS Project varification for my answer is found in figure \ref{Question1:MSProject} on page \pageref{Question1:MSProject}.

\begin{figure}[!htbp]
\begin{center}
% \includegraphics[angle = 90, scale = .8]{Q2i.eps}
\includegraphics[angle = 0, scale = .58]{Q1ii_OO.eps}
\caption{Question 1 Network Diagram}
\label{Question1:NetworkDiagram}
\end{center}
\end{figure}

\begin{figure}[!htbp]
\begin{center}
% \includegraphics[angle = 90, scale = .8]{Q2i.eps}
\includegraphics[angle = 0, scale = .8]{mspQ01.eps}
\caption{Question 1 MS Project output}
\label{Question1:MSProject}
\end{center}
\end{figure}

\begin{figure}[!htbp]
\begin{center}
% \includegraphics[angle = 90, scale = .8]{Q2i.eps}
\includegraphics[angle = 0, scale = .6]{mspQ01_summary.eps}
\caption{Question 1 MS Project summary output}
\label{Question1:MSProjectSummary}
\end{center}
\end{figure}

\subparagraph{}The critical path I found was $A \rightarrow B \rightarrow C \rightarrow E \rightarrow F \rightarrow I \rightarrow N \rightarrow Q \rightarrow R$.  Using the aforementioned critical path I have calculated the makespan of this project to be 80.5 weeks.  These figures line up with the results from MS Project.

\section{Part (iii)}

\paragraph{}What is the probability that the critical path does not exceed 72 weeks?

\subparagraph{}The probability that the critical path does not exceed 72 weeks can be calculated by using the formula $z=\frac{\overline{x}-\mu}{\hat{\sigma}}$.  Mathmatically put, we are stating the problem as found in equation \ref{Question1:problemstatement}

\begin{eqnarray}
&&\textrm{Probability the project completed by or before week 72}\\
&&P(\textrm{Project completed by or before week 72})\\
&&P(\overline{x} \leq 72) 
\label{Question1:problemstatement}
\end{eqnarray}

\subparagraph{}Re-arranged the prolem satement can be written as in the equation \ref{Question1:problemstatementwithZ}.

\begin{eqnarray}
P\left(z \leq \frac{72-\mu}{\hat{\sigma}}\right)
\label{Question1:problemstatementwithZ}
\end{eqnarray}

\subparagraph{}To begin with, we need to calculate the variance as depicted by ${\hat{\sigma}}^2$.  The variance can be calculated by gathering the sum of the varience of each arc along the critical path.  The result of this calculation can be found in equation \ref{Question1:varience}.

%\begin{displaymath}
%\begin{math}
%\begin{array}[!htbp]{cc}
\begin{eqnarray}
{\hat{\sigma}}^2 & = & {\hat{\sigma}_{10,20}}^2 + {\hat{\sigma}_{20,30}}^2 + {\hat{\sigma}_{30,40}}^2 + {\hat{\sigma}_{40,60}}^2  + {\hat{\sigma}_{60,70}}^2 + {\hat{\sigma}_{70,90}}^2 + {\hat{\sigma}_{90,110}}^2 + {\hat{\sigma}_{110,130}}^2 + {\hat{\sigma}_{130,150}}^2 \\
{\hat{\sigma}}^2 & = & {\hat{\sigma}_{A}}^2 + {\hat{\sigma}_{B}}^2 + {\hat{\sigma}_{C}}^2 + {\hat{\sigma}_{E}}^2  + {\hat{\sigma}_{F}}^2 + {\hat{\sigma}_{I}}^2 + {\hat{\sigma}_{N}}^2 + {\hat{\sigma}_{Q}}^2 + {\hat{\sigma}_{R}}^2 \\
{\hat{\sigma}}^2 & = & 2.25 + 0.25 + 4 + 3.36 + 3.36 + 2.25 + 6.25 + 8.03 + 0.44 \\
{\hat{\sigma}}^2 & = & 30.19
\label{Question1:varience}
\end{eqnarray}

\subparagraph{}Next up we need to find the standard deviation of the critical path.  This is as simple as getting the square root of the varience.  The result of this calculation can be found in equation \ref{Question1:stdev}.

\begin{eqnarray}
\hat{\sigma} & = & \sqrt{30.19} \\
\hat{\sigma} & = & 5.49
\label{Question1:stdev}
\end{eqnarray}


\subparagraph{}Due to the network diagram and other elements of the question we know the other values in the equation.  By definition the expected value, as denoted by $\overline{x}$, is the average time of the project.  The last element to find $z$ is the length of the project, as denoted by $\mu$, is the projects makespan.  The values are outlined in equations \ref{Question1:knownVal} and \ref{Question1:knownValues1}.

\begin{eqnarray}
\overline{x} & = & 72 \textrm{ weeks}
\label{Question1:knownVal}
\end{eqnarray}
\begin{eqnarray}
\mu & = & 80.5 \textrm{ weeks}
\label{Question1:knownValues1}
\end{eqnarray}

\subparagraph{}Now we know all the facts, the problem statement can be re-written with all the variables filled in.

\begin{eqnarray}
\textrm{As stated in equation \ref{Question1:problemstatementwithZ}}&&P\left(z \leq \frac{72-\mu}{\hat{\sigma}}\right)\\
\textrm{Substitute all known values} &&P\left(z \leq \frac{72-80.5}{5.49}\right)\\
\textrm{Process the values}&&P\left(z \leq \frac{-8.5}{5.49}\right)\\
\textrm{Process the values}&&P\left(z \leq -1.54826\right)
\label{Question1:problemsolution}
\end{eqnarray}

\subparagraph{}Now we use the $z$ lookup table and we end up with the result shown in \ref{Question1:zvalue}.

\begin{eqnarray}
&&P\left(0.06078\right)
\label{Question1:zvalue}
\end{eqnarray}

\subparagraph{}This probability equates to a 6.078\% chance that the project does not exceed 72 weeks.

\chapter{Question 2}

\paragraph{}Given the project as stated in table \ref{Question2:Table1} found on page \pageref{Question2:Table1}

\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{ll|ll|ll}
� & � & Normal & � & Crash & � \\
Label & Activity & Duration & Cost & Duration & Cost \\ 
\hline
A & (10,20) & 9 & \$4500 & 6 & \$9000 \\ 
B & (10,30) & 12 & \$11250 & 9 & \$18000 \\ 
C & (20,30) & 15 & \$4500 & 8 & \$11250 \\ 
D & (20,40) & 15 & \$4500 & 14 & \$15750 \\ 
E & (20,80) & 13 & \$6000 & 8 & \$13500 \\ 
F & (30,40) & 14 & \$4500 & 12 & \$9000 \\ 
G & (30,50) & 18 & \$2250 & 6 & \$6750 \\ 
H & (30,90) & 12 & \$9000 & 10 & \$13500 \\ 
I & (40,60) & 15 & \$1500 & 10 & \$3750 \\ 
J & (40,80) & 12 & \$2250 & 11 & \$3750 \\ 
K & (50,70) & 14 & \$13500 & 7 & \$18000 \\ 
L & (50,90) & 14 & \$12000 & 7 & \$16500 \\ 
M & (60,70) & 9 & \$2250 & 5 & \$4050 \\ 
N & (70,80) & 5 & \$4050 & 2 & \$5550 \\ 
O & (70,90) & 12 & \$6750 & 6 & \$11250 \\ 
P & (80,100) & 15 & \$4500 & 12 & \$9000 \\ 
Q & (90,110) & 14 & \$13500 & 9 & \$18000 \\ 
R & (100,120) & 15 & \$12000 & 7 & \$16500 \\ 
S & (110,120) & 14 & \$4500 & 9 & \$9000 \\ 
T & (110,130) & 13 & \$4500 & 9 & \$9000 \\ 
U & (120,130) & 9 & \$3000 & 5 & \$4800 \\ 
V & (130,140) & 7 & \$4050 & 3 & \$5550 \\
\end{tabular}
} % end footnotesize
\end{center}
\caption{Raw data for Question 2 from assignment sheet.}
\label{Question2:Table1}
\end{table}

\section{Part (i)}

\paragraph{}Find the critical path and the cost of the project under normal conditions.

\subparagraph{}The network diagram to calculate the critical path can be found in figure \ref{Question2:NetworkDiagram} on page \pageref{Question2:NetworkDiagram}.

\subparagraph{}As demonstrated in the network diagram found in figure \ref{Question2:NetworkDiagram} on page \pageref{Question2:NetworkDiagram}, I have found the critical path to be $A \rightarrow C \rightarrow F \rightarrow I \rightarrow M \rightarrow O \rightarrow Q \rightarrow S \rightarrow U \rightarrow V$.

\begin{figure}[!htbp]
\begin{center}
% \includegraphics[angle = 90, scale = .8]{Q2i.eps}
\includegraphics[angle = 0, scale = .58]{Q2i.eps}
\caption{Question 2 Network Diagram}
\label{Question2:NetworkDiagram}
\end{center}
\end{figure}

\subparagraph{}The Excel calculation for cost under normal conditions can be found in table \ref{Question2:NormalCosts} on page \pageref{Question2:NormalCosts}

\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{ll|ll|ll}
� & � & Normal & � & Crash & � \\
Label & Activity & Duration & Cost & Duration & Cost \\ 
\hline
A & (10,20) & 9 & \$4500 & 6 & \$9000 \\ 
B & (10,30) & 12 & \$11250 & 9 & \$18000 \\ 
C & (20,30) & 15 & \$4500 & 8 & \$11250 \\ 
D & (20,40) & 15 & \$4500 & 14 & \$15750 \\ 
E & (20,80) & 13 & \$6000 & 8 & \$13500 \\ 
F & (30,40) & 14 & \$4500 & 12 & \$9000 \\ 
G & (30,50) & 18 & \$2250 & 6 & \$6750 \\ 
H & (30,90) & 12 & \$9000 & 10 & \$13500 \\ 
I & (40,60) & 15 & \$1500 & 10 & \$3750 \\ 
J & (40,80) & 12 & \$2250 & 11 & \$3750 \\ 
K & (50,70) & 14 & \$13500 & 7 & \$18000 \\ 
L & (50,90) & 14 & \$12000 & 7 & \$16500 \\ 
M & (60,70) & 9 & \$2250 & 5 & \$4050 \\ 
N & (70,80) & 5 & \$4050 & 2 & \$5550 \\ 
O & (70,90) & 12 & \$6750 & 6 & \$11250 \\ 
P & (80,100) & 15 & \$4500 & 12 & \$9000 \\ 
Q & (90,110) & 14 & \$13500 & 9 & \$18000 \\ 
R & (100,120) & 15 & \$12000 & 7 & \$16500 \\ 
S & (110,120) & 14 & \$4500 & 9 & \$9000 \\ 
T & (110,130) & 13 & \$4500 & 9 & \$9000 \\ 
U & (120,130) & 9 & \$3000 & 5 & \$4800 \\ 
V & (130,140) & 7 & \$4050 & 3 & \$5550 \\
Total & � & � & \$134,850 & � & � 
\end{tabular}
} %end footnotesize
\end{center}
\caption{Raw data for Question 2 from assignment sheet.}
\label{Question2:NormalCosts}
\end{table}

\subparagraph{}I have found that the total cost of the project under normal conditions is \$134,850.

\section{Part (ii)}

\paragraph{}Suppose the project must be completed in minimum time.  What is the least cost for the project?

\subparagraph{}The calculations for the costs of the project can be found in table \ref{Question2:CrashedCosts} on page \pageref{Question2:CrashedCosts}.  They are supported by the network diagram, being figure \ref{Question2:CrashedNetworkDiagram} found on page \pageref{Question2:CrashedNetworkDiagram}.


\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{ll|ll|ll|l|l|l|l}
� & � & Normal & � & Crash & � & � & � & � & � \\
Label & Activity & Duration & Cost & Duration & Cost & Crash On Offer & Slope & Crash Used & Cost\\ 
\hline
A & (10,20) & 9 & \$4500 & 6 & \$9000 & 3 & \$1,500.00 & 3 & \$4,500.00\\ 
B & (10,30) & 12 & \$11250 & 9 & \$18000 & 3 & \$2,250.00 & 0 & \$0.00\\ 
C & (20,30) & 15 & \$4500 & 8 & \$11250 & 7 & \$964.29 & 7 & \$6,750.00\\ 
D & (20,40) & 15 & \$4500 & 14 & \$15750 & 1 & \$11,250.00 & 0 & \$0.00\\ 
E & (20,80) & 13 & \$6000 & 8 & \$13500 & 5 & \$1,500.00 & 0 & \$0.00\\ 
F & (30,40) & 14 & \$4500 & 12 & \$9000 & 2 & \$2,250.00 & 2 & \$4,500.00\\ 
G & (30,50) & 18 & \$2250 & 6 & \$6750 & 12 & \$375.00 & 5 & \$1,875.00\\ 
H & (30,90) & 12 & \$9000 & 10 & \$13500 & 2 & \$2,250.00 & 0 & \$0.00\\ 
I & (40,60) & 15 & \$1500 & 10 & \$3750 & 5 & \$450.00 & 5 & \$2,250.00\\ 
J & (40,80) & 12 & \$2250 & 11 & \$3750 & 1 & \$1,500.00 & 0 & \$0.00\\ 
K & (50,70) & 14 & \$13500 & 7 & \$18000 & 7 & \$642.86 & 0 & \$0.00\\ 
L & (50,90) & 14 & \$12000 & 7 & \$16500 & 7 & \$642.86 & 0 & \$0.00\\ 
M & (60,70) & 9 & \$2250 & 5 & \$4050 & 4 & \$450.00 & 4 & \$1,800.00\\ 
N & (70,80) & 5 & \$4050 & 2 & \$5550 & 3 & \$500.00 & 3 & \$1,500.00\\ 
O & (70,90) & 12 & \$6750 & 6 & \$11250 & 6 & \$750.00 & 6 & \$4,500.00\\ 
P & (80,100) & 15 & \$4500 & 12 & \$9000 & 3 & \$1,500.00 & 0 & \$0.00\\ 
Q & (90,110) & 14 & \$13500 & 9 & \$18000 & 5 & \$900.00 & 5 & \$4,500.00\\ 
R & (100,120) & 15 & \$12000 & 7 & \$16500 & 8 & \$562.50 & 8 & \$4,500.00\\ 
S & (110,120) & 14 & \$4500 & 9 & \$9000 & 5 & \$900.00 & 5 & \$4,500.00\\ 
T & (110,130) & 13 & \$4500 & 9 & \$9000 & 4 & \$1,125.00 & 0 & \$0.00\\ 
U & (120,130) & 9 & \$3000 & 5 & \$4800 & 4 & \$450.00 & 4 & \$1,800.00\\ 
V & (130,140) & 7 & \$4050 & 3 & \$5550 & 4 & \$375.00 & 4 & \$1,500.00\\ 
Totals & � & � &  \$134,850 & � & � &� & � & � & \$44,475
\end{tabular}
} % end footnotesize text
\end{center}
\caption{Calculation and results for crash costs}
\label{Question2:CrashedCosts}
\end{table}

\begin{figure}[!htbp]
\begin{center}
% \includegraphics[angle = 90, scale = .8]{Q2i.eps}
\includegraphics[angle = 0, scale = .55]{Q2CrashedNetworkDiagram.eps}
\caption{Question 2 Crashed Network Diagram}
\label{Question2:CrashedNetworkDiagram}
\end{center}
\end{figure}

\subparagraph{}I have found that the total extra cost under crash conditions is \$44,475.  This brings the total cost of the project  which has been forced to run in minimum time to \$179,325.

\section{Part (iii)}

\paragraph{}Verify your answers to parts (i) and (ii) using MS EXCEL.

\subparagraph{}MS Excel was used to produce tables \ref{Question2:CrashedCosts} and \ref{Question2:NormalCosts}.

\chapter{Question 3}

\paragraph{}This question was not attempted.

\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{ll|ll}
Label & Activity & Duration & Number of workers \\ 
\hline
A & (10,20) & 7 & 5 \\ 
B & (10,30) & 5 & 5 \\ 
C & (20,30) & 6 & 3 \\ 
D & (20,40) & 3 & 6 \\ 
E & (30,60) & 9 & 2 \\ 
F & (30,50) & 5 & 4 \\ 
G & (40,60) & 12 & 6 \\ 
H & (40,80) & 7 & 2 \\ 
I & (50,90) & 8 & 6 \\ 
J & (60.70) & 6 & 3 \\ 
K & (60,90) & 5 & 5 \\ 
L & (70,80) & 6 & 5 \\ 
M & (70,110) & 16 & 6 \\ 
N & (80,100) & 10 & 2 \\ 
O& (90,110) & 6 & 3 \\ 
P & (100,120) & 12 & 4 \\ 
Q & (110,130) & 10 & 6 \\ 
R & (120,140) & 9 & 6 \\ 
S & (130,140) & 7 & 7
\end{tabular}
} % end footnotesize
\end{center}
\caption{Raw data for Question 3 from assignment sheet.}
\label{Question3:Table1}
\end{table}

\section{Part (a)}

\paragraph{}Determine the minimum number of workers required for the project given the following requirements for each task of the project.

\section{Part (b)}

\paragraph{}Suppose we now consider the following hourly costs for the workers as follows, how does this alter your answer to part (a)?  Explain your reasoning and give the MS Project output for this part.

\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{l|l|l|l|l|l}
Worker 1 & Worker 2 & Worker 3 & Worker 4 & Worker 5 & Worker 6 \\ 
\hline
\$75 & \$90 & \$90 & \$110 & \$150 & \$120
\end{tabular}
} %end footnotesize
\end{center}
\caption{Hourly cost allocation for workers}
\label{Question3:WorkerCost}
\end{table}

\paragraph{}All additional workers cost \$150 per hour.  There are no more than 15 workers available for this project.

%\begin{figure}[!htbp]
%\begin{center}
%% \includegraphics[angle = 90, scale = .8]{Q2i.eps}
%\includegraphics[angle = 0, scale = .4]{placeholder.eps}
%\label{Question3:MSProjectOutput}
%\caption{Question 3 MS Project output for the worker allocation}
%\end{center}
%\end{figure}

%\subparagraph{}

\chapter{Question 4}

\paragraph{}A manufacturer of complex electronic equipment has just received a sizeable contract and plans to subcontract seven parts of this project.  He has available seven contractors A, B, C, D, E, F and G who can each produce one part.  Once a contractor completes a part there is a set-up period where the equipment is prepared for the next contractor.

\section{Part (i)}
\paragraph{}Given the following set-up-times between contractors, find the sequence that minimises the total set-up costs.

\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{l|l|lllllll}
 & � & � & To &  & � & � & � & � \\ 
\hline
� & � & A & B & C & D & E & F & G \\ 
\hline
� & A & $\infty$ & 2300 & 2130 & 2360 & 2370 & 2480 & 2240 \\ 
� & B & 2220 & $\infty$ & 2190 & 2240 & 2030 & 2460 & 2140 \\ 
From & C & 2240 & 2540 & $\infty$ & 2360 & 2240 & 2510 & 2370 \\ 
� & D & 2240 & 2600 & 2260 & $\infty$ & 2220 & 2440 & 2160 \\ 
� & E & 2300 & 2480 & 2120 & 2300 & $\infty$ & 2450 & 2240 \\ 
� & F & 2630 & 2500 & 2180 & 2380 & 2250 & $\infty$ & 2020 \\ 
� & G & 2480 & 2630 & 2300 & 2270 & 2360 & 2480 & $\infty$
\end{tabular}
} % end footnotesize
\end{center}
\caption{Setup cost allocations}
\label{Question4:CostTable}
\end{table}

\subparagraph{}The sequence that minimizes the total setup costs (or tour as it's known) is $A \rightarrow B \rightarrow E \rightarrow C \rightarrow F \rightarrow G \rightarrow D$.  It was worked out by hand as shown in the appendix (it's long and won't fit here.

\section{Part (ii)}
\paragraph{}Find the optimal cycle order (and associated costs) in which the products should be produced.  Verify your answer using MS EXCEL.

\subparagraph{}The cycle order as found by excel can be seen in figure \ref{Question4:Workings_B}.

\begin{figure}[!htbp]
\begin{center}
% \includegraphics[angle = 90, scale = .8]{Q2i.eps}
%\includegraphics[angle = 0, scale = 0.6]{scm2912Q4_smaller.eps}
\includegraphics[angle = 0, scale = 0.6]{scm2912Q4_1.eps}
\caption{Question 4 Workings for discovering the set-up costs using MS Excel}
\label{Question4:Workings_B}
\end{center}
\end{figure}

\chapter{Question 5}

\paragraph{}Orders have been received for 1000 units of parent item X with the following product structure tree and on hand inventory.

\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{llllllll}
Level 0 & � & � & � & X & � & � & � \\ 
�  & � & � & � & � & � & � & � \\ 
�  & � & � & � & � & � & � & � \\ 
Level 1 & � & A(5) & � & � & � & B(3) & � \\ 
�  & � & � & � & � & � & � & � \\ 
�  & � & � & � & � & � & � & � \\ 
Level 2 & C(8) & � & $D_0(5)$ & � & $D_1(2)$ & � & E(6) \\ 
�  & � & � & � & � & � & � & � \\ 
�  & � & � & � & � & � & � & � \\ 
Level 3 & $G_0(15)$ & � & $F_0(12)$ & � & $F_1(3)$ & � & $G_1(1)$ \\ 
�  & � & � & � & � & � & � & � \\ 
�  & � & � & � & � & � & � & � \\ 
Level 4 & � & � & $H_0(3)$ & � & � & � & $H_1(2)$
\end{tabular}
} % end footnotesize
\end{center}
\caption{Heirarchy for materials}
\label{Quesiton5:MaterialHeirarchy}
\end{table}

\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{l|l|l|l|l|l|l|l|l}
Item & A & B & C & D & E & F & G & H \\ 
\hline
Inventory On Hand & 1500 & 2500 & 4000 & 6000 & 3000 & 10000 & 15000 & 15000
\end{tabular}
} %end footnotesize
\end{center}
\caption{Inventory On Hand}
\label{Quesiton5:Inventory}
\end{table}


\section{Part (i)}

\paragraph{}Calculate the gross and net requirements for the above Bill of Materials.

\subparagraph{}The gross and net requirements for the bill of materials can be seen in figure \ref{Question5:Workings_i}.

\begin{figure}[!htbp]
\begin{center}
% \includegraphics[angle = 90, scale = .8]{Q2i.eps}
\includegraphics[angle = 0, scale = 0.8]{Q5_component_numbers.eps}
\caption{Question 5 Workings for part (i)}
\label{Question5:Workings_i}
\end{center}
\end{figure}

\section{Part (ii)}

\paragraph{}Time phase the net requirements for the components, assuming the following lead times and scheduled receipts.

\subparagraph{}

\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{l|l|l|l|l|l|l|l|l}
Item & A & B & C & D & E & F & G & H \\ 
\hline
Lead Time (weeks) & 5 & 3 & 2 & 3 & 2 & 3 & 4 & 2
\end{tabular}
} % end footnotesize
\end{center}
\caption{Lead times for components}
\label{Quesiton5:LeadTime}
\end{table}

\begin{figure}[!htbp]
\begin{center}
% \includegraphics[angle = 90, scale = .8]{Q2i.eps}
\includegraphics[angle = 0, scale = 0.8]{Q5_TimePhase.eps}
\caption{Question 5 Time phase workings}
\label{Question5:TimePhase}
\end{center}
\end{figure}

\section{Part (iii)}

\paragraph{}Suppose additional items of component G are delivered, from another source, according to the following delivery plan, time phase the new net requirements for component G only.

\begin{table}[!htcb]
\begin{center}
{\footnotesize
% use packages: array
\begin{tabular}[htcb]{l|l|l|l|l|l|l|l|l|l|l}
Week & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 
\hline
Scheduled Receipts of G & - & 4000 & - & 5000 & - & 3000 & 2000 & - & 1000 & -
\end{tabular}
} % end footnotesize
\end{center}
\caption{Extra shipments for G}
\label{Quesiton5:ExtraG}
\end{table}

\begin{figure}[!htbp]
\begin{center}
% \includegraphics[angle = 90, scale = .8]{Q2i.eps}
\includegraphics[angle = 0, scale = 0.8]{Q5_TimePhaseG.eps}
\caption{Question 5 Time phase re-workings for the additional deliveries of G}
\label{Question5:TimePhase1}
\end{center}
\end{figure}

\subparagraph{}

\end{document}