LINEAR PROGRAMMING M1L6

• Shana works at a flower shop on the weekend.

• The owner told her she needs to sell at least 5 dozen roses, but no more than 20 dozen.

• Additionally, she needs to sell at least 1 dozen carnations, but no more than 10 dozen.

• Up to 24 dozen flowers arrive in each shipment.

THE SETUP

THE OBJECTIVE

If they make… • $25 for each dozen roses sold

and • $15 for each dozen carnations

sold, what combination of sales would bring in the most profit?

Identifying the VARIABLES… what do we need to know?

Since we are asked for the combination that would provide the most profit, we’re looking for the number of roses (by dozen) and number of carnations (by dozen) Shana should aim to sell to make the most money possible given her resources.

Let’s let x = number of roses (by dozen) and y = number of carnations (by dozen)

The number of roses should be at least 5, but no more than 20.

“At least” means you can go at or above that number but not below… ,

and “no more than” means you can be at that number or below but not above… .

When put together it is .

CONSTRAINT 1

5≤ 𝑥≤20

The number of carnations should be at least 1,

but no more than 10.“At least” means you can go at or above

that number but not below… ,and “no more than” means you can be at

that number or below but not above… .

When put together it is .

CONSTRAINT 2

1≤ 𝑦 ≤10

Up to 24 dozen flowers arrive in each shipment.

That means the total amount of roses & carnations (by dozen) together cannot go

over 24.However, it is possible for the total to =

24.

This would mean x + y 24

CONSTRAINT 3

x + y 24

Graph all the constraints together to find the feasible region (what is possible given our constraints)

The region shaded by all 3 constraints is feasible. We check the vertices (corner points) of the region because 1 of them will provide the most profit.

(5,10)

(14,10)

(20,4)

(20,1)

(5,1)

OBJECTIVE FUNCTION

Shana’s goal is to make the most profit. If she makes $25 for each dozen roses and $15 for each dozen carnations…

Profit = 25x + 15y

Now we just need to check each of the vertices of the feasible region by substituting them into the objective function. x y Profit = 25x +

15y5 1 25(5)+15(1)=14

05 10 25(5)+15(10)=2

7514 10 25(14)+15(10)=

50020 4 25(20)+15(4)=

56020 1 25(20)+15(1)=5

15

CONCLUSION

Shana should sell 20 dozen roses and 4 dozen carnations.

That combination will maximize the profit that is possible with the given constraints.