MHY Inc is a small firm that produces two chemicalbased prod

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.

Solution

This is a typical product mix problem, having two products as Fuel1 and Fuel2 to be produced from same three types of raw materials to have the maximum profits.

Decision Variables: Let X be the quantity of Fuel1 and Y be the quantity of Fuel2 to be produced

Objective Function: Maximization of Profit is given as Maximize 40X + 30Y

Constraints: These are on account of requirements and availabilities of materials as follows:

      .4X + .5Y <= 20 (Material1 availability)

      .2Y <= 5 (Material2 availability)

   .6X + .3Y <= 21 (Material3 availability)

X >= 0   and Y >= 0

B. Graphically the optimal solution can be found by drawing the lines represented by the constraints and forming the feasible region bounded by the lines .4X + .5Y = 20 , .2Y = 5 and .6X + .3Y = 21

The extreme points are X=35, Y=0 and X=25, Y=20

The values at the above mentioned extreme points are 35*40 =1400 and 25*40 +20*30 =1600

Therefore the optimal solution is 25 tons of Fuel1 and 20tons of Fuel2 with Maximum profit of $1600