All steel manufactured by Steelco must mart the following re

All steel manufactured by Steelco must mart the following requirements 3.2-3.5% carbon; 18-2.5% silicon; 0.9-1.2% nickel; tensile strength of at least 44,000 pounds per square inch (psi). Steelco manufactures steel by combining two alloys. The cost and properties of each alloy are given in the following table. Assume that the tensile strength of a mixture of the two alloys can be determined by averaging that of the alloys that are mixed together. for example, a one-ton mixture that is 40% alloy 1 and 60% alloy 2 has a tensile strength of 0.4(42,000) + 0.6(50.0001). Formulate an LP to determine how to minimize the cost of producing a ton of steel.

Solution

Let the contribution of alloy be x

Then the contribution of other alloy be (1-x)

Constraints:

Percent Carbon = x * 3 + (1-x) * 4 = 4 - x

Percent Silicon = x * 2 + (1-x) * 2.5 = 2.5 - 0.5x

Percent Nickel = x + (1-x) * 1.5 = 1.5 - 0.5x

Tensile Strength = x * 42000 + (1-x) * 50000 = 50000 - 8000x

We need to maximize x such that all the constraints are satisfied because x is the percentage of alloy 1 which is cheaper in price

For carbon, xmax = 0.8 in order to satisfy carbon constraints

For silicon, xmax = 1 in order to satisfy silicon constraints

For nickel, xmax = 1 in order to satisfy nickel constraints

For tensile strength, xmax = 1 in order to satisfy tensile strength constraints

Hence the optimum value to satisfy all the constraints is x=0.8, which we are getting from copper constraint

Steel must be 80% of alloy 1 and 20% of alloy 2

Price of 1 ton of steel = 0.80 * 190 + 0.20 * 200

=> 192$