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
