Optimization and Brand
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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: 4 4 Formula for a case of Brand Y = 4n+6f+15c
Flavor Additive: 12 6
Color Additive: 6 15
Objective Function:
Max 40X + 30Y = total profit contribution
Constraint 1: Units of nutrient C used < units of nutrient C available
Units of nutrient C used = 4X + 4Y
30 units of nutrient C are available so the mathematical equation of constraint 1 is
4X + 4Y < 30
Constraint 2: Units of flavor additive used < units of flavor additive available
Units of flavor additive used = 12X + 6Y
72 units of flavor additive are available so the mathematical equation of constraint 2 is
12X + 6Y < 72
Constraint 3: Units of color additive used < units of color additive available
Units of color additive used = 6X + 15Y90 units of color additive are available so the mathematical equation of constraint 3 is
6X + 15Y < 90
A nonnegativey constraint must be added because Company A can't produce negative batches of Brand X or Brand Y.
X > 0 and Y > 0
Mathematical Model for Company A:
Max 40X + 30Y A.
Subject to (s.t.) Mathematical Equations for "Graph 1":
4X + 4Y < 30 Nutrient C 4X + 4Y = 30; when x = 0, y = 7.5; when x = 7.5, y = 0
12X + 6Y < 72 Flavor additive 12X + 6Y = 72; when x = 0, y = 12; when x = 6, y = 0
6X + 15Y < 90 Color additive
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: 4 4 Formula for a case of Brand Y = 4n+6f+15c
Flavor Additive: 12 6
Color Additive: 6 15
Objective Function:
Max 40X + 30Y = total profit contribution
Constraint 1: Units of nutrient C used < units of nutrient C available
Units of nutrient C used = 4X + 4Y
30 units of nutrient C are available so the mathematical equation of constraint 1 is
4X + 4Y < 30
Constraint 2: Units of flavor additive used < units of flavor additive available
Units of flavor additive used = 12X + 6Y
72 units of flavor additive are available so the mathematical equation of constraint 2 is
12X + 6Y < 72
Constraint 3: Units of color additive used < units of color additive available
Units of color additive used = 6X + 15Y90 units of color additive are available so the mathematical equation of constraint 3 is
6X + 15Y < 90
A nonnegativey constraint must be added because Company A can't produce negative batches of Brand X or Brand Y.
X > 0 and Y > 0
Mathematical Model for Company A:
Max 40X + 30Y A.
Subject to (s.t.) Mathematical Equations for "Graph 1":
4X + 4Y < 30 Nutrient C 4X + 4Y = 30; when x = 0, y = 7.5; when x = 7.5, y = 0
12X + 6Y < 72 Flavor additive 12X + 6Y = 72; when x = 0, y = 12; when x = 6, y = 0
6X + 15Y < 90 Color additive