D1 June 2005 Q7

EdexcelOld spec15 marksLinear Programming

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 £3.50, £3.50 and £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
\(r\)4510100140
\(s\)21401060
\(t\)12300160
\(P\)\(-350\)\(-350\)\(-650\)0000
(a) Explain the meaning of the variables \(r\), \(s\) and \(t\) in the context of this question. (2)
(b) 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. (5)
(c) State the value of every variable as given by tableau \(T\). (3)
(d) Write down the profit equation given by tableau \(T\). (2)
(e) Use your profit equation to explain why tableau \(T\) is not optimal. (1)

Taking the most negative number in the profit row to indicate the pivotal column,

(f) identify clearly the location of the next pivotal element. (2)
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