A new mine in Florida has been divided into three distinct pits according to the grade of the manganese, these being High (H), Medium (M) and Low (L). Because of the pit contours, it costs $ 1,000,000 to mine 200 tonnes, 250 tonnes and 500 tonnes of H, M, L grade respectively in Pit 1, and$ 900,000 to mine 250 tonnes, 150 tonnes and 300 tonnes of H, M, L grade respectively in Pit 2 and $ 1,500,000 to mine 400 tonnes, 400 tonnes and 300 tonnes of H, M, L grade respectively in Pit 3 respectively. The company has the processing capacity of 300,000, 250,000 and 500,000 tonnes of H, M and L grade respectively.

a. Formulate the above problem as a linear programming model.

b. Formulate a dual maximization problem. Do not solve.