6. A linear programming problem in \(x , y\) and \(z\) is described as follows. Maximise \(\quad P = 2 x + 2 y - z\)
subject to \(\quad 3 x + y + 2 z \leqslant 30\) $$\begin{aligned} x - y + z & \geqslant 8
4 y + 2 z & \geqslant 15
x , y , z & \geqslant 0 \end{aligned}$$

1. Explain why the Simplex algorithm cannot be used to solve this linear programming problem.
2. Set up the initial tableau for solving this linear programming problem using the big-M method. After a first iteration of the big-M method, the tableau is

   b.v.	\(x\)	\(y\)	\(z\)	\(s _ { 1 }\)	\(S _ { 2 }\)	\(S _ { 3 }\)	\(a _ { 1 }\)	\(a _ { 2 }\)	Value
   \(s _ { 1 }\)	3	0	1.5	1	0	0.25	0	-0.25	26.25
   \(a _ { 1 }\)	1	0	1.5	0	-1	-0.25	1	0.25	11.75
   \(y\)	0	1	0.5	0	0	-0.25	0	0.25	3.75
   \(P\)	\(- ( 2 + M )\)	0	2-1.5M	0	M	\(- 0.5 + 0.25 M\)	0	\(0.5 + 0.75 M\)	7.5-11.75M

3. State the value of each variable after the first iteration.
4. Explain why the solution given by the first iteration is not feasible. Taking the most negative entry in the profit row to indicate the pivot column,
5. obtain the most efficient pivot for a second iteration. You must give reasons for your answer.