Set up initial Simplex tableau

A question is this type if and only if it asks to represent or display a given linear programming problem as an initial Simplex tableau, without performing any iterations.

2 questions · Moderate -0.3

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Edexcel D1 2001 January Q7
20 marks Moderate -0.3
A tailor makes two types of garment, A and B. He has available 70 m² of cotton fabric and 90 m² of woollen fabric. Garment A requires 1 m² of cotton fabric and 3 m² of woollen fabric. Garment B requires 2 m² of each fabric. The tailor makes \(x\) garments of type A and \(y\) garments of type B.
  1. Explain why this can be modelled by the inequalities $$x + 2y \leq 70,$$ $$3x + 2y \leq 90,$$ $$x \geq 0, y \geq 0.$$ [2 marks]
The tailor sells type A for £30 and type B for £40. All garments made are sold. The tailor wishes to maximise his total income.
  1. Set up an initial Simplex tableau for this problem. [3 marks]
  2. Solve the problem using the Simplex algorithm. [8 marks]
Figure 4 shows a graphical representation of the feasible region for this problem. \includegraphics{figure_4}
  1. Obtain the coordinates of the points A, C and D. [4 marks]
  2. Relate each stage of the Simplex algorithm to the corresponding point in Fig. 4. [3 marks]
Edexcel D1 2005 June Q7
15 marks Moderate -0.3
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
  1. Explain the meaning of the variables \(r\), \(s\) and \(t\) in the context of this question. [2]
  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. [5]
  3. State the value of every variable as given by tableau \(T\). [3]
  4. Write down the profit equation given by tableau \(T\). [2]
  5. 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,
  1. identify clearly the location of the next pivotal element. [2]
(Total 15 marks)