9 Janet and Samantha play a zero-sum game.
The game is represented by the following pay-off matrix for Janet.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Samantha}
| \multirow{5}{*}{Janet} | Strategy | \(\mathbf { S } _ { \mathbf { 1 } }\) | \(\mathbf { S } _ { \mathbf { 2 } }\) | \(\mathbf { S } _ { \mathbf { 3 } }\) |
| \(\mathbf { J } _ { \mathbf { 1 } }\) | 2 | 7 | 6 |
| \(\mathbf { J } _ { \mathbf { 2 } }\) | 5 | 5 | 1 |
| \(\mathbf { J } _ { \mathbf { 3 } }\) | 4 | 3 | 8 |
| \(\mathbf { J } _ { \mathbf { 4 } }\) | 1 | 6 | 4 |
\end{table}
\(\mathbf { 9 }\) (a) Explain why Janet should never play strategy \(\mathbf { J } _ { \mathbf { 4 } }\)
9 (b) Janet wants to maximise her winnings from the game.
She defines the following variables.
\(p _ { 1 } =\) the probability of Janet playing strategy \(\mathbf { J } _ { \mathbf { 1 } }\)
\(p _ { 2 } =\) the probability of Janet playing strategy \(\mathbf { J } _ { 2 }\)
\(p _ { 3 } =\) the probability of Janet playing strategy \(\mathbf { J } _ { \mathbf { 3 } }\)
\(v =\) the value of the game for Janet
Janet then formulates her situation as the following linear programming problem.
$$\begin{array} { l l }
\text { Maximise } & P = v
\text { subject to } & 2 p _ { 1 } + 5 p _ { 2 } + 4 p _ { 3 } \geq v
& 7 p _ { 1 } + 5 p _ { 2 } + 3 p _ { 3 } \geq v
& 6 p _ { 1 } + p _ { 2 } + 8 p _ { 3 } \geq v
\text { and } & p _ { 1 } + p _ { 2 } + p _ { 3 } \leq 1
& p _ { 1 } , p _ { 2 } , p _ { 3 } \geq 0
\end{array}$$
9 (b) (i) Complete the initial Simplex tableau for Janet's situation in the grid below.
Find the probability of Janet playing strategy \(\mathbf { J } _ { \mathbf { 3 } }\) when she is playing to maximise her winnings from the game.
\includegraphics[max width=\textwidth, alt={}, center]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-17_2491_1755_173_123}
A project is undertaken by Higton Engineering Ltd.
The project is broken down into 11 separate activities \(A , B , \ldots , K\)
Figure 3 below shows a completed activity network for the project, along with the earliest start time, duration, latest finish time and the number of workers required for each activity. All times and durations are given in days.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-18_930_1714_724_148}
\end{figure}