A2 October 2020 Q7

EdexcelCurrent spec17 marksLinear Programming

7. A maximisation linear programming problem in \(x\), \(y\) and \(z\) is to be solved using the two-stage simplex method.

The partially completed initial tableau is shown below.

Basic variable\(x\)\(y\)\(z\)\(s_1\)\(s_2\)\(s_3\)\(a_1\)\(a_2\)Value
\(s_1\)1231000045
\(a_1\)3200\(-1\)0109
\(a_2\)\(-1\)0400\(-1\)014
\(P\)\(-2\)\(-1\)\(-3\)000000
\(A\)         
(a) Using the information in the above tableau, formulate the linear programming problem. State the objective and list the constraints as inequalities. (4)
(b) Complete the bottom row of Table 1 in the answer book. You should make your method and working clear. (2)

[Table 1 in the answer book is a copy of the partially completed initial tableau above.]

The following tableau is obtained after two iterations of the first stage of the two-stage simplex method.

Basic variable\(x\)\(y\)\(z\)\(s_1\)\(s_2\)\(s_3\)\(a_1\)\(a_2\)Value
\(s_1\)0\(\frac{5}{6}\)01\(\frac{7}{12}\)\(\frac{3}{4}\)\(-\frac{7}{12}\)\(-\frac{3}{4}\)\(\frac{147}{4}\)
\(x\)1\(\frac{2}{3}\)00\(-\frac{1}{3}\)0\(\frac{1}{3}\)03
\(z\)0\(\frac{1}{6}\)10\(-\frac{1}{12}\)\(-\frac{1}{4}\)\(\frac{1}{12}\)\(\frac{1}{4}\)\(\frac{7}{4}\)
\(P\)0\(\frac{5}{6}\)00\(-\frac{11}{12}\)\(-\frac{3}{4}\)\(\frac{11}{12}\)\(\frac{3}{4}\)\(\frac{45}{4}\)
\(A\)000000110
(c)
(i) Explain how the above tableau shows that a basic feasible solution has been found for the original linear programming problem.
(ii) Write down the basic feasible solution for the second stage. (3)
(d) Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the second stage of the two-stage simplex method, to obtain a new tableau, \(T\). Make your method clear by stating the row operations you use. (5)
(e)
(i) Explain, using \(T\), whether or not an optimal solution to the original linear programming problem has been found.
(ii) Write down the value of the objective function.
(iii) State the values of the basic variables. (3)
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