Max/Min Problem- Need help forming equations

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TheNextOne

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Magna Inc. sells and manufactures two products- Product A and Product B. The cost to produce Product A and Product B are $36 and $45 respectively. 8,000 units of Product A are needed and 11,000 units of product B are needed. These products are produced using a machine and the machine hours required to manufacture each unit of Product A is 2.5 whereas it is 3 for Product B. In total, there are 41,000 machine hours that need to be used up. Netscape company offers to manufacture and sell the products to Magna for $33.75 (Product A) and $40.50 (Product B). How much of each product should Magna purchase and produce in order to minimize costs given that the 41,000 machine hours must be used up.

My Work:

I formed one equation which is 41,000 = 2.5x + 3y, where x is the number of Product A units to produce and y is the number of Product B units to produce. I need suggestions for what I should do next. Thank you.
 
When I first read this problem, it sounded like a typical linear programming question. But when I started to work it, I changed my mind. The requirement that 41,000 machine hours “must” be used set this problem apart from others of this type.

So here’s what I did. I agree with your equation: 41000 = 2.5x + 3y. Since 41000 hours must be used, any values of x and y that would be potential solutions to the problem must be solutions of this equation. Let’s rewrite the equation solving for y:

y = (-2.5x + 41000)/3

But x and y have constraints: x can be no larger than 8000, and y can be no larger than 11000. If we start substituting for x, we see that for values of x less than 3200, y is greater than 11000, so the domain of x has a lower limit of 3200 and an upper limit of 8000. Likewise, when x=8000, y has a value of 7000, so the range of this function is y=7000 to y=11000.

The way you have chosen your variable x and y, they represent the number of A and B items to be “machined”, respectively. If we choose to machine less than 8000 of the A items, we have to purchase the rest of them at 33.75 each. Therefore the cost of the total number of A items is:

costA = 36x + 33.75 * pA, where pA is the number of A item purchased, and x is the number of A items machined

Likewise, the cost of the total number of B items is:

costB = 45y + 40.50 pB, where pB is the number of B items purchased, and y is the number of B items machined

To get values for pA and pB:

pA = 8000 – x (this is from the relation: A items machined + A items purchased = 8000)

and

pB = 11000 – y (from: B items machined + B items purchased = 11000)

So the final cost equations are (after substituting for pA, pB, and y):

costA = 36x + 33.75(8000 – x)

and

costB = 45(-2.5x + 41000)/3 + 40.5[11000 – (-2.5x + 41000)/3]

The total cost is the sum of these. I didn’t take the time to simplify either of these equations – I just put them into my TI-83 and plotted the total cost from x=3200 to x = 8000. This is a linear function with negative slope and I found that the minimum total cost of $765,000 occurs when x = 8000.
Click to expand...
(I hope I didn’t make any typos while re-doing this problem from my scrap paper.) :)

Steve
 
Just out of curiousity, how would you do this problem is 41,000 was the upper limit but not a requirement to be filled?
 
Just out of curiousity, how would you do this problem is 41,000 was the upper limit but not a requirement to be filled?

------------

My quick answer is to use the techniques of linear programming. My long answer is that I have to think more about it. :) The perplexing factor for me is the purchasing vs. the machining aspect. I haven't done one like that before.

Steve
 
Linear programming

If i was to use linear programming,

Would I use:

2.5x + 3y < 41,000,
x < 8000
y < 11,000

or some other equations?
 
TheNextOne said:
Magna Inc. sells and manufactures two products- Product A and Product B. The cost to produce Product A and Product B are $36 and $45 respectively. 8,000 units of Product A are needed and 11,000 units of product B are needed. These products are produced using a machine and the machine hours required to manufacture each unit of Product A is 2.5 whereas it is 3 for Product B. In total, there are 41,000 machine hours that need to be used up.

Netscape company offers to manufacture and sell the products to Magna for $33.75 (Product A) and $40.50 (Product B). How much of each product should Magna purchase and produce in order to minimize costs given that the 41,000 machine hours must be used up.
Click to expand...

I'm simply UNABLE to SEE what's going on here!

What's the difference between "manufacture" and "produce"?

If Magna's costs are now $36 and $45 per units, and someone offers to "do the work"
such that the costs are reduced to $33.75 and $40.50, then it seems that Magna
should quit manufacturing and sell only.

Why should Magna care about machine hours (or anything else Netscape uses)?

Assume Magma sells A's at $50 each:
50 - 36 = 14 = profit now
50 - 33.75 = 16.25 = profit if Magna subcontracts production

What am I missing?
Can your problem be rewritten in a clearer fashion?
 
Hello, TheNextOne!

I think I know what the problem says . . .
\(\displaystyle \;\;\)but I still don't have a reasonable set-up.

Magna Inc. sells and manufactures two products- Product A and Product B.
The cost to produce Product A and Product B are $36 and $45 respectively.
8,000 units of Product A are needed and 11,000 units of product B are needed.

These products are produced by a machine.
2.5 machine hours are needed to produce one unit of A, and 3 hours for each B.
In total, there are 41,000 machine hours that need to be used up.

Netscape company offers to manufacture and sell the products to Magna
for $33.75 per A and $40.50 per B.

How much of each product should Magna purchase and produce
in order to minimize costs given that the 41,000 machine hours must be used up?
Click to expand...
If Magna tries to produce all the products itself, there are not enough machine hours.
8,000 A's at 2.5 hours each = 20,000 hours.
11,000 B's at 3 hours each = 33,000 hours.
\(\displaystyle \;\;\)The total (53,000) exceeds the maximum of 41,000 hours.

So, we assume that Magna will produce some of the A's and buy the rest from Netscape.
Similarly, Magna will produce some of the B's and buy the rest from Netscape.


Let \(\displaystyle x\) = number of A's that Magna will manufacture.
Then it will buy the other \(\displaystyle 8,000 - x\) units from Netscape.

Let \(\displaystyle y\) = number of B's that Magna will manufacture.
Then it will buy the other \(\displaystyle 11,000 - y\) units from Netscape.

The cost of making \(\displaystyle x\) units of A is: \(\displaystyle 36x\) dollars.
The cost of buying \(\displaystyle 8000-x\) units of A from Netscape is: \(\displaystyle 33.75(8,000-x)\) dollars.
\(\displaystyle \;\;\)The total cost of getting 8,000 A's is: \(\displaystyle \,36x\,+\,33.75(8,000-x)\) dollars.

The cost of making \(\displaystyle y\) units of B is: \(\displaystyle 45y\) dollars.
The cost of buying \(\displaystyle 11,000-y\) units of B from Netscape is: \(\displaystyle 40.50(11,000-y)\) dollars.
\(\displaystyle \;\;\)The total cost of getting 11,000 B's is: \(\displaystyle \,45y\,+\,40.40(11,000-y)\) dollars.

Hence, the total cost is: \(\displaystyle \,C\:=\:[36x\,+\,33.75(8,000-x)]\,+\,[45y\,+\,40.5(11,000-y)]\)

And we must minimize: \(\displaystyle \,C\;=\;715,500\,+\,2.25x\,+\,4.5y\)


The only constraint (other than \(\displaystyle x\,\geq\,0\) and \(\displaystyle y\,\geq\,0)\,\) is the number of machine hours available:

\(\displaystyle \;\;\;2.5x\,+\,3y\:\leq \:41,000\)


Is that it?
 
Yes, that is the only constraint I can get. How de we minimize a function given only one constraint?
 
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