Bottled Coffee Production Optimization

Bottled Coffee Production

• Two types of bottled coffee: cappuccinos (6 oz coffee, 2 oz milk, $0.40 profit) and cafés au lait (4 oz coffee, 4 oz milk, $0.50 profit).
• Daily available resources: 720 ounces of coffee and 400 ounces of milk.
• Minimum requirement: at least 80 drinks produced daily.
• Variables defined: x = cappuccino bottles, y = café au lait bottles.

Objective Function

• Profit function defined as P = 0.40x + 0.50y.

Constraints for Coffee Production

• Constraints include:
  • x + y ≥ 80
  • 3x + 2y ≤ 360
  • x + 2y ≤ 200
  • x ≥ 0
  • y ≥ 0.

Feasible Region

• Graph C represents the feasible region for the coffee production constraints.
• Vertices of the feasible region: (0, 100), (0, 80), (80, 60), (80, 0), (120, 0).

Objective Function Analysis

• Objective function F = 8x + 5y evaluated for minimum (400) and maximum (960).

Printing Company Constraints

• Supplier X charges $25/case, Supplier Y charges $20/case.
• Constraints include:
  • At least 45 cases ordered daily.
  • Maximum of 30 cases from Supplier X.
  • Supplier Y can be no more than twice Supplier X’s cases.
• Variables defined: x = cases from Supplier X, y = cases from Supplier Y.

Objective Function for Printing

• Objective function to minimize costs: D = 25x + 20y.

Minimum Values

• For the objective function P = 20x + 16y, the minimum value is 780, occurring at x = 15, y = 30.

Steps for Maximization

• To maximize the objective function P = 3x + 4y:
  • Graph the inequalities from constraints.
  • Identify intersection points that form a polygon.
  • Calculate the objective function at each point to find the maximum value.

Product A and B Production

• Product A requires at least 30 units; Product B needs at least 10 units.
• Max production limit of 80 units.
• Profit: $15 from Product A, $8 from Product B.
• Objective function for profit maximization: P = 15a + 8b.

Maximum Profit Calculation

• Feasible region vertices for products are: (70, 10), (30, 10), and (30, 50).
• Maximum profit calculated at 1130.

Feasible Region Vertices

• Provided vertices of the feasible region include (0,15), (10,15), and (20,5).

Minimum Objective Function Value

• Minimum value for the objective function C = 4x + 9y is calculated as 125.

Feasible Region Graphing

• Constraints presented: y ≥ 2x, x + y ≤ 14, y ≥ 1, 5x + y ≥ 14, x + y ≥ 9.
• Region A represents the graph of the feasible region.