Optimization given only variables

swellbor

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I have a problem in Calculus that I am having trouble with because it deals only with variables, and I am having a difficult time interpreting all of the data

Our forward-thinking company, Tiddlywinks & Sons Manufacturing Corporation, is developing two new products that may revolutionize the world of office products. We are getting ready for production and will be hiring at least six new workers to start after Thanksgiving. It is my responsibility to come up with a production plan that will lead to the highest profit possible.
I am not allowed to divulge any details about our new products, (everything is top secret as we don’t want our competition to know what we’re up to). Let me call our new items product X and product Y, and I need to determine how many units of each product to produce. We expect to realize a profit of a dollars per unit on product X and b dollars per unit with product Y (as determined by our marketing department).
The actual number of units that we will be able to produce is, however, limited by the number of hours that the workers have available, which we call person hours. Once we’ve made our new hires, we will have N total person hours that we can devote to the production of products X and Y. (Again, I apologize, but I’m not allowed even to tell you what the number N is.) Each unit of X requires c person hours to produce. For the second product, however, economies of scale are more pronounced, so that the number of person hours required to produce y units of Y is proportional to y^(1/2) , with a constant of proportionality d.

In order for our company to be successful (and for me to have a chance at a promotion), it is imperative that we determine the best production strategy for these two products. When I present my suggested production plan, I will also need to justify to Mr. Tiddlywinks why this will be the best possible plan. I have heard through the lunchroom gossip that Mr. Tiddlywinks is expecting that the highest profit will result if we produce twice as many of product Y as we do of product X (because of the economies of scale mentioned above.) I, however, think the situation will be much more complicated, and our plan will depend on all the constants that I’ve described. But I don’t know just how to work it all out, and so I’m hoping that you can help me. I depend on you to explain everything to me clearly. Since we are gearing up for production soon, I would appreciate an answer as soon as possible.
 
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I have a problem in Calculus that I am having trouble with because it deals only with variables, and I am having a difficult time interpreting all of the data

Our forward-thinking company, Tiddlywinks & Sons Manufacturing Corporation, is developing two new products that may revolutionize the world of office products. We are getting ready for production and will be hiring at least six new workers to start after Thanksgiving. It is my responsibility to come up with a production plan that will lead to the highest profit possible.
I am not allowed to divulge any details about our new products, (everything is top secret as we don’t want our competition to know what we’re up to). Let me call our new items product X and product Y, and I need to determine how many units of each product to produce. We expect to realize a profit of a dollars per unit on product X and b dollars per unit with product Y (as determined by our marketing department).
The actual number of units that we will be able to produce is, however, limited by the number of hours that the workers have available, which we call person hours. Once we’ve made our new hires, we will have N total person hours that we can devote to the production of products X and Y. (Again, I apologize, but I’m not allowed even to tell you what the number N is.) Each unit of X requires c person hours to produce. For the second product, however, economies of scale are more pronounced, so that the number of person hours required to produce y units of Y is proportional to y^(1/2) , with a constant of proportionality d.

In order for our company to be successful (and for me to have a chance at a promotion), it is imperative that we determine the best production strategy for these two products. When I present my suggested production plan, I will also need to justify to Mr. Tiddlywinks why this will be the best possible plan. I have heard through the lunchroom gossip that Mr. Tiddlywinks is expecting that the highest profit will result if we produce twice as many of product Y as we do of product X (because of the economies of scale mentioned above.) I, however, think the situation will be much more complicated, and our plan will depend on all the constants that I’ve described. But I don’t know just how to work it all out, and so I’m hoping that you can help me. I depend on you to explain everything to me clearly. Since we are gearing up for production soon, I would appreciate an answer as soon as possible.

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I'm also working on this problem

I'm in a college calculus class.

x=total number of product X
y=total number of product Y

What I have is that the xa+yb=P(profit)

N(total hours)=cx+dy^(1/2)

How do I go on from here?

Also if the total number of workers (6) relevant to my equation?
 
I'm in a college calculus class.

x=total number of product X
y=total number of product Y

What I have is that the xa+yb=P(profit)

N(total hours)=cx+dy^(1/2)

How do I go on from here?

Also if the total number of workers (6) relevant to my equation?
Are you working on the same exercise as is in the original posting? Or is this a separate question? Thank you! ;)
 
I am working on the same problem as well. As lordhelpme stated, we are trying to maximize profit with the constraint being the number of hours that the workers have available.

Profit is represented by:

P = ax+by

Number of hours that the workers have available:

N = cx + dy
1/2

I then solved this equation for x so I could make a substitution in the profit equation:

x = (N - dy1/2)/c


So the profit equation now reads:

P = a((N - dy1/2)/c) + by

How do I differentiate this, set it equal to zero, and solve for y (finding critical points)?
 
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I'm in a college calculus class.

x=total number of product X
y=total number of product Y

What I have is that the (1)xa+yb=P(profit)

(2)N(total hours)=cx+dy^(1/2)

How do I go on from here?

Also if the total number of workers (6) relevant to my equation?
First, the last question: The total number of workers should somehow be implicit in the total number of hours they work.

To continue with what you have, from (2) above we have
x = (N-d y1/2)/c
[Note we are assuming c is not zero]. Substituting that into (1) we have
P(y) = a (N-d y1/2)/c + b y = b y - e y1/2 + f
where
e = \(\displaystyle \frac{a\, d}{c}\)
and
f = \(\displaystyle \frac{a\, N}{c}\)

Now find the y value at the maximum of P(y).
 
First, the last question: The total number of workers should somehow be implicit in the total number of hours they work.

To continue with what you have, from (2) above we have
x = (N-d y1/2)/c
[Note we are assuming c is not zero]. Substituting that into (1) we have
P(y) = a (N-d y1/2)/c + b y = b y - e y1/2 + f
where
e = \(\displaystyle \frac{a\, d}{c}\)
and
f = \(\displaystyle \frac{a\, N}{c}\)

Now find the y value at the maximum of P(y).


I took the derivative of P(y) and got a maximum value for y=(e/2b)^2
Is that correct?

And then am I suppose to do the same for the x value?
 
I took the derivative of P(y) and got a maximum value for y=(e/2b)^2
Is that correct?

And then am I suppose to do the same for the x value?
Yes,
y = [e/(2b)]2
is where the first derivative is zero. But is it a maximum or a minimum? Anyway, one you find y use the equation
(A) x = (N-d y1/2)/c
to get x. Then use the equation
P = a x + b y
to find the profit.

EDIT: Note that both x and y should be positive so that, for example, if equation (A) above were to give an answer less that zero, then, because of the boundaries on x and y you might possibly set x to its minimum (x=0) and see what that gives for y.
 
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I got the equation [(N/c)/(a(d^2)/(2b(c^2))](a)+(b)[(ab)/(2bc)]^2= Profit

profit = ((Na)/c)/[((a^2)(d^2))/(4b(c^2))]

is this correct?
 
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