6 A company is planning its production of "MPowder" for the next three months.

• Over the next 3 months 20 tonnes must be produced.
• Production quantities must not be decreasing. The amount produced in month 2 cannot be less than the amount produced in month 1 , and the amount produced in month 3 cannot be less than the amount produced in month 2.
• No more than 12 tonnes can be produced in total in months 1 and 2.
• Production costs are \(\pounds 2000\) per tonne in month \(1 , \pounds 2200\) per tonne in month 2 and \(\pounds 2500\) per tonne in month 3.

The company planner starts to formulate an LP to find a production plan which minimises the cost of production: $$\begin{array} { l l } \text { Minimise } & 2000 x _ { 1 } + 2200 x _ { 2 } + 2500 x _ { 3 }
\text { subject to } & x _ { 1 } \geq 0 x _ { 2 } \geq 0 x _ { 3 } \geq 0
& x _ { 1 } + x _ { 2 } + x _ { 3 } = 20
& x _ { 1 } \leq x _ { 2 }
& \bullet \cdot \cdot \end{array}$$

1. Explain what the variables \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) represent, and write down two more constraints to complete the formulation.
2. Explain how the LP can be reformulated to: $$\begin{array} { l l } \text { Maximise } & 500 x _ { 1 } + 300 x _ { 2 }
   \text { subject to } & x _ { 1 } \geq 0 x _ { 2 } \geq 0
   & x _ { 1 } \leq x _ { 2 }
   & x _ { 1 } + 2 x _ { 2 } \leq 20
   & x _ { 1 } + x _ { 2 } \leq 12 \end{array}$$
3. Use a graphical approach to solve the LP in part (ii). Interpret your solution in terms of the company's production plan, and give the minimum cost.