5. A garden centre makes hanging baskets to sell to its customers. Three types of hanging basket are made, Sunshine, Drama and Peaceful. The plants used are categorised as Impact, Flowering or Trailing.
Each Sunshine basket contains 2 Impact plants, 4 Flowering plants and 3 Trailing plants. Each Drama basket contains 3 Impact plants, 2 Flowering plants and 4 Trailing plants. Each Peaceful basket contains 1 Impact plant, 3 Flowering plants and 2 Trailing plants.
The garden centre can use at most 80 Impact plants, at most 140 Flowering plants and at most 96 Trailing plants each day.
The profit on Sunshine, Drama and Peaceful baskets are \(\pounds 12 , \pounds 20\) and \(\pounds 16\) respectively. The garden centre wishes to maximise its profit.
Let \(x , y\) and \(z\) be the number of Sunshine, Drama and Peaceful baskets respectively, produced each day.
- Formulate this situation as a linear programming problem, giving your constraints as inequalities.
- State the further restriction that applies to the values of \(x , y\) and \(z\) in this context.
The Simplex algorithm is used to solve this problem. After one iteration, the tableau is
| b.v. | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | \(- \frac { 1 } { 4 }\) | 0 | \(- \frac { 1 } { 2 }\) | 1 | 0 | \(- \frac { 3 } { 4 }\) | 8 |
| \(s\) | \(\frac { 5 } { 2 }\) | 0 | 2 | 0 | 1 | \(- \frac { 1 } { 2 }\) | 92 |
| \(y\) | \(\frac { 3 } { 4 }\) | 1 | \(\frac { 1 } { 2 }\) | 0 | 0 | \(\frac { 1 } { 4 }\) | 24 |
| \(P\) | 3 | 0 | -6 | 0 | 0 | 5 | 480 |
- State the variable that was increased in the first iteration. Justify your answer.
- Determine how many plants in total are being used after only one iteration of the Simplex algorithm.
- Explain why for a second iteration of the Simplex algorithm the 2 in the \(z\) column is the pivot value.
After a second iteration, the tableau is
| b.v. | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | \(\frac { 3 } { 8 }\) | 0 | 0 | 1 | \(\frac { 1 } { 4 }\) | \(- \frac { 7 } { 8 }\) | 31 |
| \(s\) | \(\frac { 5 } { 4 }\) | 0 | 1 | 0 | \(\frac { 1 } { 2 }\) | \(- \frac { 1 } { 4 }\) | 46 |
| \(y\) | \(\frac { 1 } { 8 }\) | 1 | 0 | 0 | \(- \frac { 1 } { 4 }\) | \(\frac { 3 } { 8 }\) | 1 |
| \(P\) | \(\frac { 21 } { 2 }\) | 0 | 0 | 0 | 3 | \(\frac { 7 } { 2 }\) | 756 |
- Use algebra to explain why this tableau is optimal.
- State the optimal number of each type of basket that should be made.
The manager of the garden centre is able to increase the number of Impact plants available each day from 80 to 100 . She wants to know if this would increase her profit.
- Use your final tableau to determine the effect of this increase. (You should not carry out any further calculations.)