4 Lucy is making party bags which she will sell to raise money for charity. She has three colours of party bag: red, yellow and blue. The bags contain balloons, sweets and toys. Lucy has a stock of 40 balloons, 80 sweets and 30 toys. The table shows how many balloons, sweets and toys are needed for one party bag of each colour.
| Colour of party bag | Balloons | Sweets | Toys |
| Red | 5 | 3 | 5 |
| Yellow | 4 | 7 | 2 |
| Blue | 6 | 6 | 3 |
Lucy will raise \(\pounds 1\) for each bag that she sells, irrespective of its colour. She wants to calculate how many bags of each colour she should make to maximise the total amount raised for charity.
Lucy has started to model the problem as an LP formulation.
$$\begin{array} { l l }
\text { Maximise } & P = x + y + z ,
\text { subject to } & 3 x + 7 y + 6 z \leqslant 80 .
\end{array}$$
- What does the variable \(x\) represent in Lucy's formulation?
- Explain why the constraint \(3 x + 7 y + 6 z \leqslant 80\) must hold and write down another two similar constraints.
- What other constraints and restrictions apply to the values of \(x , y\) and \(z\) ?
- What assumption is needed for the objective to be valid?
- Represent the problem as an initial Simplex tableau. Do not carry out any iterations yet.
- Perform one iteration of the Simplex algorithm, choosing a pivot from the \(\boldsymbol { x }\) column. Explain how the choice of pivot row was made and show how each row was calculated.
- Write down the values of \(x , y\) and \(z\) from the first iteration of the Simplex algorithm and hence find the number of bags of each colour that Lucy should make according to this non-optimal tableau.
In the optimal solution Lucy makes 10 bags.
- Without carrying out further iterations of the Simplex algorithm, find a solution in which Lucy should make 10 bags.