Math problem

[ChaoZ]

Doing a linear programming problem and need to plot the points for this:
8x1+8x2+2x3<= 11980

If I start by setting x1=0, what I do with x2 and x3? I feel so stupid for even having to ask...

 

[DarkThinker]

Originally posted by: ChaoZ
Doing a linear programming problem and need to plot the points for this:
8x1+8x2+2x3<= 11980

If I start by setting x1=0, what I do with x2 and x3? I feel so stupid for even having to ask...

You can't get an answer it will become 8x2 +2x3<=11980

You need a system of two equations at least.

And yes you are stupid

Plus you are just assuming x1=0, to be honest, you can't assume it's anything you like, you need a system of 3 equations to solve for x1, x2 and x3






😛
DarkThinker

 

[Heisenberg]

Yeah, you can't solve the system with just one equation. You could however solve for x1, and then plot the surface representing the solutions for x2 and x3.

 

[ChaoZ]

Can you point me to how to solve this? At least the name of what this type of equation is called.

 

[DarkThinker]

It's a one equation of 3 unkowns, and it's impossible to solve!

Am I missing something, what does the quesiton say?

DarkThinker

 

[ChaoZ]

So basically there's 3 products, I need to figure out how much to produce of each to maximize profits.

Material A costs $4 per pound. Product Q, R W each uses material A, with 2, 2, 1/2 pounds respectively. Need to start off with an equation and start plotting.

 

[DarkThinker]

I mean from a logical point of view product W uses the least amount of material A, but I have no idea for how much the prodcuts are selling for, so how on earth can I determine the profit if there is no sale price?

 

[homercles337]

If its three unknowns why is it in this form:

8x1+8x2+2x3<= 11980

rather than this form:

8x1+8y2+2z3<= 11980

???

Is x == * or is x == variable?

 

[Heller]

5

its always 5.

 

[TuxDave]

The solution is a volume under a plane. You could set X1=X2=0 and find X3. X1=X3=0 and find X2 and blah blah...

Now you have three points, connect them to make a plane. The solution is the volume of space beneath that.

 

[ChaoZ]

Ah, gotta type out the whole thing then...

So there's a budget of $11,980 for the 3 products combined. Material A costs $4 per pound, same goes for Material B. Labor costs $8/hour.

Product Q, R W each uses material A, with 2, 2, 1/2 pounds respectively. Material B uses 1, 2 pounds for Q, W respectively. Labor required for products Q, R, W are 5, 4, 2 hours respectively.

Product Q, R, W sell for $122, $115, $76 respectively per unit. Thought I'd figure the rest out by myself but oh well. It never specifically asked for plots, just how maximize each product.

 

[DarkThinker]

Originally posted by: ChaoZ
Ah, gotta type out the whole thing then...

So there's a budget of $11,980 for the 3 products combined. Material A costs $4 per pound, same goes for Material B. Labor costs $8/hour.

Product Q, R W each uses material A, with 2, 2, 1/2 pounds respectively. Material B uses 1, 2 pounds for Q, W respectively. Labor required for products Q, R, W are 5, 4, 2 hours respectively.

Product Q, R, W sell for $122, $115, $76 respectively per unit. Thought I'd figure the rest out by myself but oh well. It never specifically asked for plots, just how maximize each product.

Wow, how was I supposed to figure all this out with the initial info you provided me withy, oh and BTW you ,made a typo up there right?

 

[ChaoZ]

^ Yea, it was a typo. Edited.

 

[silverpig]

The equation you want is

(cost to make Q) + (cost to make R) + (cost to make W) <= 11980

cost to make Q = (2 lbs of A)*($4/lb) + (1 lb of B)*($4/lb) + (5 hrs labour)*($8/hr) = $52 per unit of Q
cost to make R = (2 lbs of A)*($4/lb) + (4 hrs labour)*($8/hr) = $40 per unit of R
cost to make W = (0.5 lbs of A)*($4/lb) + (2 lbs of B)*($4/lb) + (2 hrs labour)*($8/hr) = $26 per unit of W

profit(Q) = $122 - $52 = $60
profit(R) = $115 - $40 = $75
profit(W) = $76 - $26 = $50

So you want to maximize the profit while keeping the cost to $11890. Let q, r, w be the number of units of product Q, R, and W respectively.

Production cost = 52q + 40r + 26w
profit = 60q + 75r + 50w

Now, I think this is just a simple matter of looking at the cost/profit ratio per unit. For every dollar you spend on a W, you get almost two back. The others aren't quite as good.

11890/26 = 457
457*50 = $22850 profit if you make all W

 

[TuxDave]

Originally posted by: silverpig
The equation you want is

(cost to make Q) + (cost to make R) + (cost to make W) <= 11980

cost to make Q = (2 lbs of A)*($4/lb) + (1 lb of B)*($4/lb) + (5 hrs labour)*($8/hr) = $52 per unit of Q
cost to make R = (2 lbs of A)*($4/lb) + (4 hrs labour)*($8/hr) = $40 per unit of R
cost to make W = (0.5 lbs of A)*($4/lb) + (2 lbs of B)*($4/lb) + (2 hrs labour)*($8/hr) = $26 per unit of W

profit(Q) = $122 - $52 = $60
profit(R) = $115 - $40 = $75
profit(W) = $76 - $26 = $50

So you want to maximize the profit while keeping the cost to $11890. Let q, r, w be the number of units of product Q, R, and W respectively.

Production cost = 52q + 40r + 26w
profit = 60q + 75r + 50w

Now, I think this is just a simple matter of looking at the cost/profit ratio per unit. For every dollar you spend on a W, you get almost two back. The others aren't quite as good.

11890/26 = 457
457*50 = $22850 profit if you make all W

Small typo that the budget is $11980 and not $11890. I find the best answer to be 459W and 1R

 

[DarkThinker]

If I am not mistaken :
--------------------------

Profits:
--------
Q= $122-$52=$70
R=$115-$40=$75
W=$76-$36=$40

so it's obvious product R yields the highest profit @ $75 a pop so it would make sense to produce as much as possible of product R, the max you can produce of R is:

11980 / 40 = 299.5 ===> 299 units
cost = 299 x 40 = 11960 < 11980

but then you can't producee anything else add to that that I haven't slept in a day and a half this makes me incapable of finishing, good luck.

DarkThinker

 

[ChaoZ]

Originally posted by: silverpig
The equation you want is

(cost to make Q) + (cost to make R) + (cost to make W) <= 11980

cost to make Q = (2 lbs of A)*($4/lb) + (1 lb of B)*($4/lb) + (5 hrs labour)*($8/hr) = $52 per unit of Q
cost to make R = (2 lbs of A)*($4/lb) + (4 hrs labour)*($8/hr) = $40 per unit of R
cost to make W = (0.5 lbs of A)*($4/lb) + (2 lbs of B)*($4/lb) + (2 hrs labour)*($8/hr) = $26 per unit of W

profit(Q) = $122 - $52 = $60
profit(R) = $115 - $40 = $75
profit(W) = $76 - $26 = $50

So you want to maximize the profit while keeping the cost to $11890. Let q, r, w be the number of units of product Q, R, and W respectively.

Production cost = 52q + 40r + 26w
profit = 60q + 75r + 50w

Now, I think this is just a simple matter of looking at the cost/profit ratio per unit. For every dollar you spend on a W, you get almost two back. The others aren't quite as good.

11890/26 = 457
457*50 = $22850 profit if you make all W

:beer: Thank You!

 

[DarkThinker]

silverpig, I think there is either a mistake in your profit calculations or mine??b

 

[TuxDave]

Originally posted by: DarkThinker
silverpig, I think there is either a mistake in your profit calculations or mine??b

His profit calculation is off but it doesn't matter since even with the right number the profit per cost is highest with W.

Profit(Q) = $70
Profit(R) = $75
Profit(W) = $50

Are the numbers that I used.

 

[DarkThinker]

Originally posted by: TuxDave

Originally posted by: DarkThinker
silverpig, I think there is either a mistake in your profit calculations or mine??b

His profit calculation is off but it doesn't matter since even with the right number the profit per cost is highest with W.

R yields a $75 profit, how can W be better?
And the budget is 11980 not the 11890.

 

[TuxDave]

Originally posted by: DarkThinker

Originally posted by: TuxDave

Originally posted by: DarkThinker
silverpig, I think there is either a mistake in your profit calculations or mine??b

His profit calculation is off but it doesn't matter since even with the right number the profit per cost is highest with W.

R yields a $75 profit, how can W be better?

Because you're limited by cost and not by the sheer number of units you can sell.

Imagine this case:

Product A cost $5 and you make a profit of $10 on each.
Product B cost $1 and you make a profit of $5 on each.

If you have $5 and you make 1 of A you make $10 but if you make 5 of B you make $25. So you can't go by the sheer profit number alone.

 

[DarkThinker]

Sorry my mistake, I for some reason ignored the budget limit for a step there and focused on the profit, ya your right, the cost is lowest for W so you can make more with the available budget.

DarkThinker

 

[oboeguy]

Wow you guys have never heard of linear programming? (obvious from the first replies) OP, what is the actual statement of the problem and what have you learned in class so far? Also is there a direct restriction on use of labor? (e.g. max number of labor hours allowed)

 

[ChaoZ]

Originally posted by: oboeguy
Wow you guys have never heard of linear programming? (obvious from the first replies) OP, what is the actual statement of the problem and what have you learned in class so far? Also is there a direct restriction on use of labor? (e.g. max number of labor hours allowed)

It's kinda of stupid cause this problem has nothing to do with what we learned in class so I wasn't sure how to go about it. It asks me a few questions

1) The optimal quantities of products and the necessary quantities of labor and materials
2) Same as #1, except there's a requirement of 85 units of R to be produced
3) Make equal amounts of the products. What amount of each will maximize contribution. How much less will total contribution be if this proposal is adopted.

 

[silverpig]

Originally posted by: oboeguy
Wow you guys have never heard of linear programming? (obvious from the first replies) OP, what is the actual statement of the problem and what have you learned in class so far? Also is there a direct restriction on use of labor? (e.g. max number of labor hours allowed)

Go ahead and code it up when you can solve it with a pencil in 10 minutes.


And yeah, I messed up with some typos. *shrug*