Question

A chemical manufacturer uses chemicals 1 and 2 to produce two drugs. Drug 1 must be at least 70% chemical 1, and drug 2 must be at least 60%
chemical 2. Up to 50,000 ounces of drug 1 can be sold at $30 per ounce; up to 60,000 ounces of drug 2 can be sold at $25 per ounce. Up to 45,000 ounces of chemi- cal 1 can be purchased at $15 per ounce, and up to 55,000 ounces of chemical 2 can be purchased at 
$18 per ounce. Formulate this problem as a linear programming model

Answer 1

Define variables:

A = ounces of drug A to be produced

B = ounces of drug B to be produced

C1 = ounces of chemical 1 purchased (& used)

C2 = ounces of chemical 2 purchased (& used)

X1A = ounces of chemical 1 used to produce drug A

X2A = ounces of chemical 2 used to produce drug A

X1B = ounces of chemical 1 used to produce drug B

X2B = ounces of chemical 2 used to produce drug B

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Objective:

Maximize 30A + 25B - 15C1 - 18C2

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Drug A is composed entirely of chemicals 1 & 2: A = X1A + X2A

Drug B is composed entirely of chemicals 1 & 2: B = X1B + X2B

Usage of chemical 1 is limited by the amount purchased: X1A + X1B < = C1

Usage of chemical 2 is limited by the amount purchased: X2A + X2B < = C2

Chemical 1 must be at least 70% of drug A: X1A > = 0.7A

Chemical 2 must be at least 60% of drug B: X2B > = 0.6B

A maximum of 50000 ounces of drug A can be produced: A <=50000

A maximum of 60000 ounces of drug B can be produced: B <=60000

A maximum of 45000 ounces of chemical 1 can be purchased: C1 <= 45000

A maximum of 55000 ounces of chemical 2 can be purchased: C2 <= 55000

All variables are restricted to be nonnegative

MAX:30A + 25B - 15C1 - 18C2

SUBJECT TO

A - X1A - X2A = 0

B - X1B - X2B = 0

- C1 + X1A + X1B <= 0

- C2 + X2A + X2B <= 0

- 0.7 A + X1A >= 0

- 0.6 B + X2B >= 0