What are the constraints on this system

Question

A company produces 2 types of frames, an ATB frame and a race frame For the ATB frama you need 4kg of aluminum and 6kg of steel, for the race from you need 5kg of aluminum and 2kg of steel. They sell the ATB frames for 1960 euros a piece and the race frame for 1240 euros a piece. They only have a maximum of 70kg of aluminum a day and 72kg of steel a day. The company wants to maximize its profit. To do this they need to decide how many ATB frames and how many race framer they should make, given the amount of aluminum and steel they have.

What I have tried:

Let $x_1$ = number of ATB frames made in 1 day, and $x_2$ = number of race frames made in 1 day.

So before I tackle this problem using simplex, I need to know the constraints on this. But I have a hard time figuring out what they are. This is what I thought:

$$x_1 \leq 12$$

$$ x_2 \leq 14$$

Because that is the maximum they can produce I believe. But my problem is, this is all I have. I don't really know the $z$, I don't know any other constraints and I don't know what to do with the prices of these products, i.e. how to incorporate them into this mathematically.

Any help would be much appreciated.

They want to max profit...but you're only given what the revenue numbers are? You should get, for example, what steel and aluminum cost. — Ron Gordon, Apr 01 '13 at 13:22
@RonGordon The revenue given in these kind of problems is Gross Revenue per item , so we don't need to subtract the construction cost. — ABC, Apr 01 '13 at 13:27
@exploringnet Indeed I would think so too, otherwise they'd have given me more data. — Ylyk Coitus, Apr 01 '13 at 13:36
Please chose a correct answer so that it is marked as answered — ABC, Apr 04 '13 at 15:14

score 1 · Answer 1 · answered Apr 01 '13 at 13:21

1

The constraints are "They only have a maximum of 70kg of aluminum a day and 72kg of steel a day." Now translate that into (some number)*$x_1+$(some other number)$*x_2 \le 70$ and another similar constraint based on steel. The point is that the number of each kind of frame is reduced by the material used for the other kind.

answered Apr 01 '13 at 13:21

Ross Millikan

374,822

Would the first constraint be: $ 4x_1 + 5x_2 \leq 70$? – Ylyk Coitus Apr 01 '13 at 13:41
@YlykCoitus: exactly. You are showing that you don't use up more aluminum than you have. – Ross Millikan Apr 01 '13 at 13:53

score 1 · Accepted Answer · answered Apr 01 '13 at 13:23

1

Hint:

Profit: $1960\times x_1+1240\times x_2$ ie. to be maximum.

Maximum aluminium being used. $70Kg\ge 4Kg\times x_1+5Kg\times x_2$

Similarly for steel gives you another constraint which would be enough to solve problem.

answered Apr 01 '13 at 13:23

ABC

6,034

So you basically have the 2 constraints, and $z = 1960 x_1 + 1240x_2$? – Ylyk Coitus Apr 01 '13 at 13:43
Yes you got it. Also , there are two more important constraints ie. $x_1\ge 0$ and $x_2\ge 0$ which would bound the favourable reigon in your plot. – ABC Apr 01 '13 at 13:46
But aren't those constraints logically deduced from this question? As in you can't lose money in this problem? – Ylyk Coitus Apr 01 '13 at 13:49
Yes of-course they are and even more logically we can't create -ve number of frames or destroy frames that are not even constructed! lol – ABC Apr 01 '13 at 13:54
I said something extremely stupid - I derped. Every constraint can be deduced from the problem. I meant it would be kind of useless but now that I think about it, it's not. Thanks for this answer! One little question though: Let's say we live in a parallel universe and they want to minimize this. Would 0 be the minimum, since the origin is in our feasible area? – Ylyk Coitus Apr 01 '13 at 14:00

What are the constraints on this system

2 Answers2