Write down the objective function and the constraints for the given linear programming problem.
A local herb shop is producing two natural perfumes: Gentle Roseand Rich Gardenia. The owner, who has equipment that can make up to 3000 ounces of perfume, cannot afford to spend more than $9000. The Gentle Rose is 2 ounces large and costs $3 to make. The Rich Gardenia is 1.5 ounces big and costs $6 to make. Each bottle of Gentle Rose sold results in a $4 profit and each bottle of Rich Gardenia produces a $5 profit. The owner wnats to know the number of bottles of each perfume should be made for maximum profit.
x=10
To write the objective function and constraints for this linear programming problem, we need to define the decision variables, objective function, and constraints.
Decision variables:
Let x be the number of bottles of Gentle Rose perfume produced.
Let y be the number of bottles of Rich Gardenia perfume produced.
Objective function:
The objective is to maximize the total profit. The profit from selling each bottle of Gentle Rose is $4, and the profit from selling each bottle of Rich Gardenia is $5. Therefore, the objective function is:
Maximize Profit = 4x + 5y
Constraints:
1. The amount of perfume produced cannot exceed the available equipment capacity of 3000 ounces:
2x + 1.5y ≤ 3000
2. The total cost of producing the perfumes cannot exceed $9000:
3x + 6y ≤ 9000
3. The number of bottles produced cannot be negative:
x ≥ 0
y ≥ 0
So, the objective function is Maximize Profit = 4x + 5y
subject to the constraints:
2x + 1.5y ≤ 3000
3x + 6y ≤ 9000
x ≥ 0
y ≥ 0