Re: Linear Programming Question

On Apr 13, 2:58 pm, quasi <q...@xxxxxxxx> wrote:
On 13 Apr 2007 13:23:20 -0700, "Protoman" <Protoman2...@xxxxxxxxx>
wrote:





On Apr 12, 4:21 pm, quasi <q...@xxxxxxxx> wrote:
On 12 Apr 2007 13:21:50 -0700, "Protoman" <Protoman2...@xxxxxxxxx>
wrote:

On Apr 12, 5:34 am, quasi <q...@xxxxxxxx> wrote:
On 11 Apr 2007 20:42:44 -0700, "Protoman" <Protoman2...@xxxxxxxxx>
wrote:

I have this LP problem:

You have $20K to invest in stocks, bonds, and the money market. The
stocks yield 20%, the bonds yield 10%, and the money market fund
yields 7.5%. You'll invest whatever you have left after investing in
the stocks and bonds in the MM fund. But your broker limits
investments in the fund to $10K. Fees of your broker is 20% of assets.
What is your max return?

x=stocks
y=bonds
20-x-y=MM fund

x>=0,y>=0, y>=-x+20,y>=-x+10,y<=x/5 Y=.20x+.10y+.075(20-x-y) //Y=.025x
+.125y+1.5

-x+10=x/5
-5x+50=x
-4x=-50
x=12.5
y=-2.5

We can throw this one out b/c y is negative.

-x+20=x/5
-5x+100=x
-4x=-100
x=25
y=-5

This one can also be thrown out, b/c y is also negative.

-x+10=0
x=10
y=0

(10,0)

-x+20=0
x=20
y=0

(20,0)

Y=.025(10)+.125(0)+1.5
Y=1.75

Y=.025(20)+.125(0)+1.5
Y=2

Your max return is $2,000 if you invest all of your $20,000 in stocks.

Is this correct?

Thanks!!!!

I'm not even looking at your analysis, but it seems to me that the
problem can be solved with just common sense.

Since stocks give the highest return, invest as much as possible in
stocks (i.e. 100% of available funds).

The fee is presumably deducted in advance. Then:

$20,000 [initial amount] - $4,000 [broker's fee]

leaves $16,000 to invest.

Investing $16,000 in stocks produces a profit of $3,200.

So your ending asset value is $19,200, and thus a loss of $800.

If the fee is deducted at the end, it comes out exactly the same:

Investing $20,000 in stocks produces a profit of $4,000.

So your ending asset value (before the fee is deducted) is $24,000.

$24,000 [ending asset value before fee] - $4,800 [broker's fee]

leaves $19,200 [ending asset value], and thus a loss of $800.

By the way, the fact that it came out the same either way is no
accident. In the first calculation, you are effectively multiplying
the starting amount by (1-1/5)*(1+1/5), whereas in the second,
you are multiplying by (1+1/5)*(1-1/5).

Bottom line: If you stay with this broker long enough, you will
eventually get to zero balance.

quasi- Hide quoted text -

- Show quoted text -

OK, what about this one? I think the problem w/ his is that I forgot
how to solve systems oflinearequations w/ more than two equations:

You have to buy filing cabinets for your office. Cabinet X takes up 10
square feet, holds 1000 cubic feet of stuff, and costs $150. Cabinet
Y takes up 8 square feet, holds 512 cubic feet of stuff and cost $100.
You have no more than $10,000, and 200 square feet. How many of each
should you buy to maximize volume?

OK, we know that you can't buy a negative amount of either, so x>=0
and y>=0.
Cost is 150x+100y<=10000
Square footage is 10x+8y<=200
Volume is V=1000x=512y

So the system is
V=1000x+512y, subject to:
x,y>=0
150x+100y<=10000
10x+8y<=200

Graph the feasible region in the xy-plane.

The feasible region is the region bounded by the 4 lines:

150x+100y=10000 [but reduce the equation first in the obvious way]
10x+8y=200 [also can be reduced first]
x=0
y=0

Identify the convex extreme points of the region.

The optimum value of the objective function can always be found at one
of the extreme points.

So simply evaluate the objective function at each extreme point to see
which one is optimum. Although an optimal extreme point is usually
unique, it might not be, so if not, take any one them.

quasi- Hide quoted text -

- Show quoted text -

My graphing calculator somehow can't handle this. How do I solve it w/
o graphing?

Graph the boundary equations approximately, and then use elementary
algebra to see where selected pairs of boundary lines cross. Thus, you
will have a picture of the basic shape of the region, and also, you'll
have the coordinates of the convex extreme points. It should then be
obvious which extreme point yields the optimal value for the objective
function, but if not, try them all.

quasi- Hide quoted text -

- Show quoted text -

No, I mean analytically, by messing around w/ equations and
variables.

And why do they call it linear "programming"? I know why it's linear --
there's no squared terms--, but why is it called "programming"?
Thanks!!!

.

Relevant Pages

  • Re: Linear Programming Question
    ... stocks yield 20%, the bonds yield 10%, and the money market fund ... the stocks and bonds in the MM fund. ... Y takes up 8 square feet, holds 512 cubic feet of stuff and cost $100. ... Identify the convex extreme points of the region. ...
    (sci.math)
  • Re: Linear Programming Question
    ... the stocks and bonds in the MM fund. ... Since stocks give the highest return, invest as much as possible in ... Y takes up 8 square feet, holds 512 cubic feet of stuff and cost $100. ...
    (sci.math)
  • Re: Linear Programming Question
    ... stocks yield 20%, the bonds yield 10%, and the money market fund ... You'll invest whatever you have left after investing in ... the stocks and bonds in the MM fund. ... Y takes up 8 square feet, holds 512 cubic feet of stuff and cost $100. ...
    (sci.math)
  • Re: Linear Programming Question
    ... the stocks and bonds in the MM fund. ... Since stocks give the highest return, invest as much as possible in ... Y takes up 8 square feet, holds 512 cubic feet of stuff and cost $100. ...
    (sci.math)
  • Re: Linear Programming Question
    ... You have $20K to invest in stocks, bonds, and the money market. ... stocks yield 20%, the bonds yield 10%, and the money market fund ... the stocks and bonds in the MM fund. ...
    (sci.math)