Optimizing Coffee Blends with Linear Programming

[rez123]

Homework Statement

A coffee firm sells "Premium blend" and "Economy blend" coffee. Both are blended
from three basic grades of coffee, A, B and C:
Premium blend = 50% A + 40% B + 10% C
Economy blend = 10% A + 40% B + 50% C
The market prices are $1130/tonne for Premium and $750/tonne for Economy. One
week the firm is given the option of buying up to 100 tonnes of grade A at $1000/tonne,
120 tonnes of grade B at $500/tonne and 100 tonnes of grade C at $300/tonne. Calculate
the profit per tonne of each blend of coffee. How much of each blend should the firm
produce to maximize its profit and what is the maximum profit?

Homework Equations

i know it is a linear programming problem

The Attempt at a Solution

Since i think it is a linear programming problem, i tried to solve it by using a Standard "table". so i got:

so let x1=be the number of tonnes of coffee A
x2= be the number of tonnes of coffee B
x3= be the number of tonnes of coffee C
then, Z= x1 + 0.5 x2 + 0.3 x3

And Z is subject to:
0.5 x1 + 0.4 x2 + 0.1 x3 >= NOT SURE HERE
0.1 x1 + 0.4 x2 + 0.5 x3 >= NOT SURE HERE
i am not sure how to continue, any help would be appreciated!

 

[Mark44]

Homework Statement

A coffee firm sells "Premium blend" and "Economy blend" coffee. Both are blended
from three basic grades of coffee, A, B and C:
Premium blend = 50% A + 40% B + 10% C
Economy blend = 10% A + 40% B + 50% C
The market prices are $1130/tonne for Premium and $750/tonne for Economy. One
week the firm is given the option of buying up to 100 tonnes of grade A at $1000/tonne,
120 tonnes of grade B at $500/tonne and 100 tonnes of grade C at $300/tonne. Calculate
the profit per tonne of each blend of coffee. How much of each blend should the firm
produce to maximize its profit and what is the maximum profit?

Homework Equations

i know it is a linear programming problem

The Attempt at a Solution

Since i think it is a linear programming problem, i tried to solve it by using a Standard "table". so i got:

so let x1=be the number of tonnes of coffee A
x2= be the number of tonnes of coffee B
x3= be the number of tonnes of coffee C
then, Z= x1 + 0.5 x2 + 0.3 x3

The goal is to maximize profit, for which I recommend using the letter P instead of Z.

They sell two grades of coffee, premium and economy. How much does it cost the company to make a tonne of premium coffee? How much to make a tonne of economy coffee? What is their profit on a tonne of premium coffee? economy coffee?

And Z is subject to:
0.5 x1 + 0.4 x2 + 0.1 x3 >= NOT SURE HERE
0.1 x1 + 0.4 x2 + 0.5 x3 >= NOT SURE HERE
i am not sure how to continue, any help would be appreciated!

 

[Ray Vickson]

Homework Statement

A coffee firm sells "Premium blend" and "Economy blend" coffee. Both are blended
from three basic grades of coffee, A, B and C:
Premium blend = 50% A + 40% B + 10% C
Economy blend = 10% A + 40% B + 50% C
The market prices are $1130/tonne for Premium and $750/tonne for Economy. One
week the firm is given the option of buying up to 100 tonnes of grade A at $1000/tonne,
120 tonnes of grade B at $500/tonne and 100 tonnes of grade C at $300/tonne. Calculate
the profit per tonne of each blend of coffee. How much of each blend should the firm
produce to maximize its profit and what is the maximum profit?

Homework Equations

i know it is a linear programming problem

The Attempt at a Solution

Since i think it is a linear programming problem, i tried to solve it by using a Standard "table". so i got:

so let x1=be the number of tonnes of coffee A
x2= be the number of tonnes of coffee B
x3= be the number of tonnes of coffee C
then, Z= x1 + 0.5 x2 + 0.3 x3

And Z is subject to:
0.5 x1 + 0.4 x2 + 0.1 x3 >= NOT SURE HERE
0.1 x1 + 0.4 x2 + 0.5 x3 >= NOT SURE HERE
i am not sure how to continue, any help would be appreciated!

Back when I was teaching this stuff, I always said to students: to figure out what the "right" variables are in a problem, ask yourself: if you are the manager, what do you need to know in order to ensure everybody can go and do their jobs?

You say you want to choose tonnes of A, B and C as variables. Will this do the job? Suppose I tell you to buy 4 T of A, 4 T of B and 5T of C. Do you now know everything you need to know? NO! You do not. What will you DO with those 4T of A? How will you use the 4T of B?, etc. Knowing the amounts of A, B and B does not help you, so those are not the "right" variables. Suppose, instead, you know the number of T of premium blend and economy blend to make. Is that enough? Now the answer is YES, because now you can figure out how much of A, B and C you will need, what are the costs and what will be the profit. So, those variables are appropriate.

RGV

 

[rez123]

Thank you this has really helped and i appreciate it!