Question

Bullco blends silicon and nitrogen to produce two types of fertilizers. Fertilizer 1 must be at least 40% nitrogen and sells for $70 per pound. Fertilizer 2 must be at least 70% silicon and sells for $40 per pound. Bullco can purchase up to 80 pounds of nitrogen at $15 per pound and up to 100 pounds of silicon at $10 per pound. Assuming all fertilizer products can be sold, formulate and solve an LP to help Bullco maximize profits.

Answer 1

Answer:

Maximize

z = 70(Xs1 + Xn1 ) + 40(Xs2 + Xn2 ) - 10 (Xs1 + Xs2 ) - 15(Xn1 + Xn2 )  

Subject to the constraints  

Xs1 + Xs2 ≤ 100

Xn1 + Xn2 ≤ 80

Xn1 ≥ 0.4 ( Xs1 + Xn1 )

Xs2 ≥ 0.7 ( Xs2 + Xn2 )

All Variables ≥ 0

Step-by-step explanation:

Firstly lets consider Xs1 and Xs2 to be the number of pounds of silicon used in fertilizer1 and fertilizer2 respectively

Also let Xn1 and Xn2 be the number of pounds of nitrogen used in fertilizer1 and fertilizer2 respectively

We know that the objective is to maximize the profits of Bullco.

z = [(Selling price of fertilizer1) (Amount of silicon and nitrogen used to produce fertilizer1) + (Selling price of fertilizer2) (Amount of silicon and nitrogen used to produce fertilizer2) - (Cost of silicon) (Amount of silicon used to produce fertilizer I and 2) - (Cost of nitrogen) (Amount of nitrogen used to produce fertilizer I and 2)]

so

z = 70(Xs1 + Xn1 ) + 40(Xs2 + Xn2 ) - 10 (Xs1 + Xs2 ) - 15(Xn1 + Xn2 )  

Now

Constraint 1;  At most, 100 lb of silicon can be purchased

Amount of silicon used to produce fertilizer 1 and 2 ≤ 100

Xs1 + Xs2 ≤ 100

Constraint 2; At most, 80 lb of nitrogen can be purchased

Amount of nitrogen used to produce fertilizer 1 and 2 ≤ 80

Xn1 + Xn2 ≤ 80

Constraint 3; Fertilizer 1 must be at least 40% of nitrogen

Amount of nitrogen used to produce fertilizer 1 ≥ 40% (fertilizer 1)

Xn1 ≥ 0.4 ( Xs1 + Xn1 )

Constraint 4; Fertilizer 2 must be at least 70% of silicon

Amount of silicon used to produce fertilizer 2  ≥ 70% (fertilizer 2)

Xs2 ≥ 0.7 ( Xs2 + Xn2 )  

so the formulization of the given linear program is,  

Maximize

z = 70(Xs1 + Xn1 ) + 40(Xs2 + Xn2 ) - 10 (Xs1 + Xs2 ) - 15(Xn1 + Xn2 )  

Subject to the constraints  

Xs1 + Xs2 ≤ 100

Xn1 + Xn2 ≤ 80

Xn1 ≥ 0.4 ( Xs1 + Xn1 )

Xs2 ≥ 0.7 ( Xs2 + Xn2 )

All Variables ≥ 0