4. The manager of a factory is planning the production schedule for the next three weeks for a range of cabinets. The following constraints apply to the production schedule.

• The total number of cabinets produced in week 3 cannot be fewer than the total number produced in weeks 1 and 2
• At most twice as many cabinets must be produced in week 3 as in week 2
• The number of cabinets produced in weeks 2 and 3 must, in total, be at most 125

The production cost for each cabinet produced in weeks 1,2 and 3 is \(\pounds 250 , \pounds 275\) and \(\pounds 200\) respectively.
The factory manager decides to formulate a linear programming problem to find a production schedule that minimises the total cost of production. The objective is to minimise \(250 x + 275 y + 200 z\)

1. Explain what the variables \(x , y\) and \(z\) represent.
2. Write down the constraints of the linear programming problem in terms of \(x , y\) and \(z\). Due to demand, exactly 150 cabinets must be produced during these three weeks. This reduces the constraints to $$\begin{gathered} x + y \leqslant 75
   x + 3 y \geqslant 150
   x \geqslant 25
   y \geqslant 0 \end{gathered}$$ which are shown in Diagram 1 in the answer book.
   Given that the manager does not want any cabinets left unfinished at the end of a week,
3. 1. use a graphical approach to solve the linear programming problem and hence determine the production schedule which minimises the cost of production. You should make your method and working clear.
   2. Find the minimum total cost of the production schedule.