# Optimization and Brand Essay

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Answers are hi-lighted yellow.
Company A's nationally advertised brand is Brand X. Contribution to profit with Brand X is \$40 per case.
Company A's re-proportioned formula is sold under a private label Brand Y. Contribution to profit with Brand Y is \$30 per case.
Company A's objective is to maximize the total contribution to profit.
Three constraints limit the number of cases of Brand X and Brand Y that can be produced.
Constraint 1: The available units of nutrient C (n) is 30.
Constraint 2: The available units of flavor additive (f) is 72.
Constraint 3: The available units of color additive (c) is 90.
Material units per case of Brand X and Brand Y:
Product
Brand X Brand Y Formula for a case of Brand X = 4n+12f+6c
Nutrient C:
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The other two extreme points must be solved using both line equations simultaneously: Of the five extreme points, one point will provide the maximum profit contribution.
Solving for the extreme point where the color and nutrient lines intersect subtract the two equations:
1) 6X + 15Y = 90 Multiply this equation by 2 to eliminate the X variable
4X + 4Y = 30 Multiply this equation by 3 to eliminate the X variable
2) 12X + 30Y = 180
12X + 12Y = 90 Subtract equation two from equation one 18Y = 90 Y = 5 Substituting Y = 5 back into the first equation
6X + 15(5) = 90 6X + 75 = 90 6X = 15 X = 2.5 Extreme point = (2.5,5)
Solving for the extreme point where the flavor and nutrient lines intersect subtract the two equatitions:
1) 12X + 6Y = 72 4X + 4Y = 30 Multiply this equation by 3 to eliminate the x variable
2) 12X + 6Y = 72 Subtract equation two from equation one
12x + 12y = 90 -6Y = -18 Y = 3 Substituting Y = 3 back into the first equation
12X + 6(3) = 72
12X + 18 = 72 12X = 54 X = 4.5 Extreme point = (4.5,3)
B. Determine the total contribution to profit if the company produces a combination of cases of Brand X and Brand Y that lies on the purple objective function (profit line) as it is plotted on "Graph 1".
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