# Interrelated Product Optimization

• Mar 1st 2011, 10:56 AM
iCONFUSED
Interrelated Product Optimization
We were given this today in class and told to complete it. I am still confused by it.

" A company produces and sells three interrelated products w, x, and y. They need to determine the optimum weekly production of each, ie w units of W, x units of X, y units of Y. However, the price demand function for the each of the products are linked together as follows:

Price of W = 40 -0.2w +0.1x +0.2y
Price of X= 60 +0.1w -0.15x +0.05y
Price of Y= 60 +0.05w +0.05x - 0.25y

The total weekly cost of producing w,x, and y units of the three products is given by:

Total weekly cost = 5000 +10w +20x +10y

A) From the price demand functions for W,X, and Y find the weekly revenues from the sales of each product in terms of w,x and y.

B) Write down the total weekly revenue and the total weekly profit from the combined sales of W, X and Y

C) Write down all the relevant partial derivatives of total weekly profit.

D) Find the optimum mix of production of W,X and Y that maximises the weekly profit.
• Mar 2nd 2011, 04:31 AM
Volga
(a) Revenue=price*quantity
(b) Profit=revenue-cost
(c) find partial derivatives from (b)
(d) find quanitities w,x,y which maximise the weekly profit function (usual exercise to find a maximum of a function)
• Mar 2nd 2011, 07:11 AM
iCONFUSED
Quote:

Originally Posted by Volga
(a) Revenue=price*quantity
(b) Profit=revenue-cost
(c) find partial derivatives from (b)
(d) find quanitities w,x,y which maximise the weekly profit function (usual exercise to find a maximum of a function)

a) So: P(w) X w.... W = 40w -0.2w^(2) +0.1wx + 0.2wy and so on?

b)For w again. = 40w -0.2w^(2) +0.1wx + 0.2wy - (5000 +10w +20x +10y)
= 30w-0.2w^(2) +0.1wx + 0.2wy -5000 -20x-10y
and so on?

c)Then w again... = 30 - 0.4w +0.1x +0.2y - 0 -0 - 0
=30 -0.4w +0.1x +0.2y

d) What do you do for d? Simultaneous equations?

Not sure if that is right, any help is GREATLY appreciated:)
• Mar 2nd 2011, 03:07 PM
Volga
Quote:

Originally Posted by iCONFUSED
a) So: P(w) X w.... W = 40w -0.2w^(2) +0.1wx + 0.2wy and so on?

b)For w again. = 40w -0.2w^(2) +0.1wx + 0.2wy - (5000 +10w +20x +10y)
= 30w-0.2w^(2) +0.1wx + 0.2wy -5000 -20x-10y
and so on?

c)Then w again... = 30 - 0.4w +0.1x +0.2y - 0 -0 - 0
=30 -0.4w +0.1x +0.2y

d) What do you do for d? Simultaneous equations?

Not sure if that is right, any help is GREATLY appreciated:)

(a) correct

(b) note you only have one cost function for all three products together, so you cannot find separate profit functions for each product. therefore, add all revenues from (a) and deduct the total cost function.

(c) since you have only one profit function with three variables, you find three partial derivatives of that profit function (on w, x and y).

(d) did you study this in calculus? to maximise function, you find the critical point (where it's first derivative=0) and check if this point is mix, max or something else. here you have one function with three variables, so you deal with points where their partial derivatives =0.
• Mar 3rd 2011, 04:41 AM
iCONFUSED
Quote:

Originally Posted by Volga
(a) correct

(b) note you only have one cost function for all three products together, so you cannot find separate profit functions for each product. therefore, add all revenues from (a) and deduct the total cost function.

(c) since you have only one profit function with three variables, you find three partial derivatives of that profit function (on w, x and y).

(d) did you study this in calculus? to maximise function, you find the critical point (where it's first derivative=0) and check if this point is mix, max or something else. here you have one function with three variables, so you deal with points where their partial derivatives =0.

Thank you.
So,
P.d w.r.t w= 30 -0.4w +0.2x +0.25y
... x=40 +0.2w-0.3x+0.1y
... y=50 +0.25w +0.1x -0.5y

Then I am confused. Can you show me what you mean possibly?

Thanks for the help.
• Mar 3rd 2011, 05:01 AM
Volga
Quote:

Originally Posted by iCONFUSED
Price of W = 40 -0.2w +0.1x +0.2y
Price of X= 60 +0.1w -0.15x +0.05y
Price of Y= 60 +0.05w +0.05x - 0.25y

The total weekly cost of producing w,x, and y units of the three products is given by:

Total weekly cost = 5000 +10w +20x +10y

(b) Total profit=w(40 -0.2w +0.1x +0.2y)+x(60 +0.1w -0.15x +0.05y)+y(60 +0.05w +0.05x - 0.25y)-(5000 +10w +20x +10y)

- open the brackets and simplify to get the final profit function. Then move to (c)

(c) Find partial derivatives of the above function w.r.t. w, x and y.

Example: if y=30a+40b+50c, then partial derivative of y with respect to a is 30 - treat other two variables as constants, and derivative of a constant is zero. I don't want to do your homework for you, so I gave a separate example.