7. An engineer makes three components \(X , Y\) and \(Z\). Relevant details are as follows: Component \(X\) requires 6 minutes turning, 3 minutes machining and 1 minute finishing. Component \(Y\) requires 15 minutes turning, 3 minutes machining and 4 minutes finishing. Component \(Z\) requires 12 minutes turning, 1 minute machining and 4 minutes finishing. The engineer gets access to 185 minutes turning, 30 minutes machining and 60 minutes finishing each day. The profits from selling components \(X , Y\) and \(Z\) are \(\pounds 40 , \pounds 90\) and \(\pounds 60\) respectively and the engineer wishes to maximise the profit from her work each day. Let the number of components \(X , Y\) and \(Z\) the engineer makes each day be \(x , y\) and \(z\) respectively.

1. Write down the 3 inequalities that apply in addition to \(x \geq 0 , y \geq 0\) and \(z \geq 0\).
2. Explain why it is not appropriate to use a graphical method to solve the problem. It is decided to use the simplex algorithm to solve the problem.
3. Show that a possible initial tableau is: Workings:
4. Workings:

   \(E\)	\(\bullet\)	\(\bullet\)	\(O\)
   \(F\)	\(\bullet\)	\(\bullet\)	\(D\)
   \(G\)	\(\bullet\)	\(\bullet\)	\(C\)
   \(H\)	\(\bullet\)	\(\bullet\)	\(A\)
   \(I\)	\(\bullet\)	\(\bullet\)	\(S\)

   Alternative matching:

   \(E\)	\(\bullet\)	\(\bullet\)	\(O\)
   \(F\)	\(\bullet\)	\(\bullet\)	\(D\)
   \(G\)	\(\bullet\)	\(\bullet\)	\(C\)
   \(H\)	\(\bullet\)	\(\bullet\)	\(A\)
   \(I\)	\(\bullet\)	\(\bullet\)	\(S\)