MHY Inc. is a small firm that produces two chemical-based products: fuel 1 and fuel 2 using three raw materials. The three raw materials will be blended to make the tow products as follows. The proportions of the materials used for each ton of the two fuels are shown below along with the available amounts of the materials. (For example, a ton of fuel 1 is a mixture of 0.4 ton of Material 1 and 0.6 ton of material 3). The profit contributions are $40 for each ton of fuel 1 and $30 for each ton of fuel 2.

  Fuel 1    Fuel 2 Amount Available for Production

Material 1 0.4 0.5 20 tons

Material 2 0.2 5 tons

Material 3 0.6 0.3 21 tons

a. Define the variables are write up a linear programming model.

b. Graphically solve the model to find optimal solutions.