Expert: Paul Klarreich - 7/4/2009

Question
Hi Paul,

My name is Jessica and I live in Salt Lake City, UT. I go to an online school from which I can get no help. I am ready to graduate, but I have one assignment to pass before I can do this and I have been trying to get it right for 3 months now. I have submitted the problem many times, but always get it wrong.  I have included the graph that goes with the problem.

I have identified the equations for the three constraints plotted on my attached graph as:
4X +4Y ≤ 30    Nutrient C
12X + 6Y ≤ 72   Flavor    
6X + 15Y ≤ 90   Color
X,Y ≥ 0
*X = Brand X. Y = Brand Y.

Now what I need to do that I cannot determine how to do is:

1) determine the total contribution to profit that lies on the objective function (profit line) as it is plotted on the graph if the company produces a combination of cases of Brand X and Brand Y.

2) Determine how many cases each of Brand X and of Brand Y you recommend should be produced during each production period for optimum production if Company A wants to generate the greatest amount of profit.

Where I am really confused is that there seems to be a “function OF profit” and a “function TO profit”. I think I need to figure these both out. Could you PLEASE help me in any way possible? I am about to have to pay another $2,500 for an additional semester simply so I can have more time to solve this one last problem! Eeeek!

Regards,
Jessica


Answer
Questioner:   Jessica
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  Profit Function
Question:  Hi Paul,

My name is Jessica and I live in Salt Lake City, UT. I go to an online school from which I can get no help.

>> Do you pay them?  Then they should help you.

I am ready to graduate, but I have one assignment to pass before I can do this and I have been trying to get it right for 3 months now. I have submitted the problem many times, but always get it wrong.  I have included the graph that goes with the problem.

>> I don't see one.

I have identified the equations for the three constraints plotted on my attached graph as:
4X +4Y ≤ 30 Nutrient C
12X + 6Y ≤ 72 Flavor
6X + 15Y ≤ 90 Color
X,Y ≥ 0
*X = Brand X.
Y = Brand Y.

Now what I need to do that I cannot determine how to do is:

1) determine the total contribution to profit that lies on the objective function (profit line) as it is plotted on the graph if the company produces a combination of cases of Brand X and Brand Y.

2) Determine how many cases each of Brand X and of Brand Y you recommend should be produced during each production period for optimum production if Company A wants to generate the greatest amount of profit.

Where I am really confused is that there seems to be a “function OF profit” and a “function TO profit”. I think I need to figure these both out. Could you PLEASE help me in any way possible? I am about to have to pay another $2,500 for an additional semester simply so I can have more time to solve this one last problem! Eeeek!

Regards,
Jessica
............................................
You are missing the essential thing -- the functions to be max-ed or min-ed.  You will have a 'feasible region' from your inequalities:


4X +4Y ≤ 30  --  Nutrient C --- What does this mean?
12X + 6Y ≤ 72 Flavor  --  or  2x + y <= 12
6X + 15Y ≤ 90 Color  --  or 2x + 5y <= 30

Now you are expected to find the 'corners' of the F.R. and evaluate your functions at these, then pick the 'best'.

P.S. Don't say:
*X = Brand X.
Y = Brand Y.

That is meaningless.  Write:

x = number of cases of <whatever> made.

ALWAYS ALWAYS be specific about the number that your variable stands for.