7. Polly has a bird food stall at the local market. Each week she makes and sells three types of packs \(A , B\) and \(C\).
Pack \(A\) contains 4 kg of bird seed, 2 suet blocks and 1 kg of peanuts.
Pack \(B\) contains 5 kg of bird seed, 1 suet block and 2 kg of peanuts.
Pack \(C\) contains 10 kg of bird seed, 4 suet blocks and 3 kg of peanuts.
Each week Polly has 140 kg of bird seed, 60 suet blocks and 60 kg of peanuts available for the packs.
The profit made on each pack of \(A , B\) and \(C\) sold is \(\pounds 3.50 , \pounds 3.50\) and \(\pounds 6.50\) respectively. Polly sells every pack on her stall and wishes to maximise her profit, \(P\) pence.
Let \(x , y\) and \(z\) be the numbers of packs \(A , B\) and \(C\) sold each week.
An initial Simplex tableau for the above situation is
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 4 | 5 | 10 | 1 | 0 | 0 | 140 |
| \(s\) | 2 | 1 | 4 | 0 | 1 | 0 | 60 |
| \(t\) | 1 | 2 | 3 | 0 | 0 | 1 | 60 |
| \(P\) | - 350 | - 350 | - 650 | 0 | 0 | 0 | 0 |
- Explain the meaning of the variables \(r , s\) and \(t\) in the context of this question.
(2) - Perform one complete iteration of the Simplex algorithm to form a new tableau \(T\). Take the most negative number in the profit row to indicate the pivotal column.
- State the value of every variable as given by tableau \(T\).
- Write down the profit equation given by tableau \(T\).
- Use your profit equation to explain why tableau \(T\) is not optimal.
Taking the most negative number in the profit row to indicate the pivotal column,
- identify clearly the location of the next pivotal element.
(2)
(Total 15 marks)