P3 is a small firm that produces a variety of chemical products. In a particular

Question

P3 is a small firm that produces a variety of chemical products. In a particular production process, three raw materials (#1, #2, and #3) are blended to produce two products: a fuel addictive and a solvent. Each ton of fuel addictive is a mixture of 0.4 ton of material 1 and 0.6 ton of material 3. A ton of solvent is a mixture of 0.5 ton of material 1, 0.2 ton of material 2, and 0.3 ton of material 3. After deducting relevant costs, the gross profit margin is $40 for each ton of fuel addictive and $30 for each ton of solvent. P3's production is constrained by a limited availability of the three raw materials. For the current period, P3 has the following quantities on hand: Develop a linear program to determine the optimal production quantities for fuel addictive and solvent. Use x_1 and x_2 to denote the production quantities for fuel addictive and solvent, respectively.

Explanation / Answer

Solution:

The decision variables are:
x1 = tons of fuel additive to produce
x2 = tons of solvent to produce

Now need to define the objective function (in terms of the decision variables) and the constraints.

Here the objective is to maximize profits, so
Maximize
40x1 + 30x2

You are constrained by raw material availability.
0.4x1 + 0.5x2 <= 20 (material 1 constraint)
0.2x2 <= 5 (material 2 constraint)
0.6x1 + 0.3x2 <= 21 (material 3 constraint)
x1 >= 0, x2 >= 0 (assumed)

Therefore your LP is:

maximize
40x1 + 30x2
subject to
0.4x1 + 0.5x2 <= 20
0.2x2 <= 5
0.6x1 + 0.3x2 <= 21

The optimal solution is x1 = 25, x2 = 20 for a max profit of $1600.

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