How many pounds of each blend should the shop prepare each day to maximize profit?

Question

Pete's Coffee sells two blends of coffee beans, Morning Blend (MB) and South American Blend (SMB). Morning Blend is 1/3 Mexican beans and 2/3 Colombian beans, while South American Blend is 2/3 Mexican beans and 1/3 Colombian beans. The profit for Morning Blend is $3 per pound, while the profit for South American Blend is $2.50 per pound. Each day the shop can obtain up to 100 pounds of Mexican beans and up to 80 pounds of Colombian beans. How many pounds of each blend should the shop prepare each day to maximize profit?

Answer 1

For this linear programming problem,

Let x = number of pounds of Morning Blend (MB).

Let y = number of pounds of South American Blend (SMB).

 

Now, let's write down our profit equation:

 

profit = 3x + 2.5y

 

Now setting up our formulation table,

 

          Mexican beans      Colombian beans

MB:            1/3                        2/3

SMB:          2/3                        1/3

 

the constraint equations are:

x ≥ 0

y ≥ 0

1/3x + 2/3y ≤ 100 lb.

2/3x + 1/3y ≤ 80 lb.

 

Now to graph these equations, we solve for y in those equations that have both x and y in them to get:

 

x ≥ 0
y ≥ 0

y ≤ 150 - 1/2x

y ≤ 240 - 2x

 

x = 0 is the same line as the y-axis.

y = 0 is the same line as the x-axis.

 

The area of the graph that is bounded by the x-axis, y-axis, and both inequality lines is called the region of feasibility, and the answer to our optimization problem lies on one of the corner points (vertices) of this region.

 

the four corner points produced by this graph are:

(0,0), (0,150), (120,0), and (60,120).

 

Now, we just have to see which point of these four produces the biggest value and we have found our answer!

 

Recall that at the start of this problem we set

x = number of pounds of Morning Blend (MB), and

y = number of pounds of South American Blend (SMB).

 

So, checking our four points:

 

(0, 0) means 0 pounds of MB plus 0 pounds of SMB, which means the profit is 0*3 + 0*2.5 = $0

 

(0, 150) yields 0 pounds of MB plus 150 pounds of SMB, so the profit is 0*3 + 150*2.5 = $375

 

(120, 0) yields 120 pounds of MB plus 0 pounds of SMB, so the profit is 120*3 + 0*2.5 = $360

 

(60, 120) yields 60 pounds of MB and 120 pounds of SMB, so the profit is 60*3 + 120*2.5 = $180 + $300 = $480

 

Therefore, based on the given constraints, the shop should produce 60 pounds of Morning Blend (MB), and 120 pounds of South American Blend (SMB) to maximize profit.

Answer 2

m is mexican beans -- 3.00 / lb profit -- max 80 pounds /day

c is colombian beans -- 2.50 / lb profit -- max 100 pounds / day

 

MB::  m + 2c = 3.00 

SMB:: 2m + c = 2.50

 

divide smb by 2

    2m/2 + c/2 = 2.50/2

== m + c/2 = 1.25

 

subtract this from MB

3/2 c = 1.75

c = 1.167

 

substitute c in either equation and m = 66.7 (these are profit / pound of each bean m,c)

since c has maximum profit / pound, maximize this via MB

2:1 ratio with max of 100 pounds c gives 50 pounds of m