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Ok I have to take finite mathematics for my criminal justice degree (why I have no idea-in ten years in law enforcement I have never used anything close to this.) Anyway I am struggling with setting up of word problems. In the section titled "The Simplex Method: Nonstandard Problems the following problem is given:

Natsano has at most $50,000 to invest in the common stocks of two companies. He estimates that an investment in company A will yield a return of 10%, whereas an investment in company B, which he feels is a riskier investment, will yeild a return of 20%. If he decides that his investment in the stock of company A is to exceed his investment in the stock of company B by at least $20,000, determine how much he should invest in the stock of each company in order to maximize the return on his investment.

I am so lost on even how to begin this. If there is anyone that can help me with this problem I would be so thankful.

thanks
 

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What the what???
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fireplug said:
Natsano has at most $50,000 to invest in the common stocks of two companies. He estimates that an investment in company A will yield a return of 10%, whereas an investment in company B, which he feels is a riskier investment, will yeild a return of 20%. If he decides that his investment in the stock of company A is to exceed his investment in the stock of company B by at least $20,000, determine how much he should invest in the stock of each company in order to maximize the return on his investment.

I am so lost on even how to begin this. If there is anyone that can help me with this problem I would be so thankful.

thanks

Focus on this part. The rest is a smoke-screen.

A has to be 20,000 more than B
He wants to invest as much as possible in B to maximize his return

A=35,000
B=15,000
 

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Steaming piles of opinion
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Opus51569 said:
Focus on this part. The rest is a smoke-screen.

A has to be 20,000 more than B
He wants to invest as much as possible in B to maximize his return

A=35,000
B=15,000
Correct answer - but remember this is schoolwork, not real life. How you get there is more important.

Gotta admit, I haven't touched linear programming in 25 years, but I remember hating it. So, sorry, but I will only empathize from here on out.

EXCEPT:

If you need a math guru, you need Wolfram Alpha:

http://www.wolframalpha.com/

Type this into their search box:

maximize .1A+.2B where A=20000+B and A+B = 50000

goofyspookycoolsmartcomputerstuff. Them wolfram folks are scary.
 

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Yeah, the rate of investment has nothing to do with the question, concentrate on amounts invested. Think of it like this:

Total investment = 50,000
Call investment A = X
Investment B must excede investment A by 20,000. Both and A and B together must total 50,000.

So,
A = X
B = X + 20,000
Total principle = 50,000.

X + (X + 20,000) = 50,000
2X = 50,000 - 20,000
2X = 30,000
X = 15,000

If X = 15,000 then that is one of the totals.

50,000 - 15,000 = 35,000

There is a difference of 20,000 between 15,000 and 35,000 so this answer is correct.

Also, it might help to think of each investment in terms of the other. This eliminates the need for multiple variables which always get confusing.
 

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What the what???
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Creakyknees said:
I'd recommend diversifying a bit more.
Yeah, my first thought was wondering how close to retirement Natsano might be :)
 

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Cowboy up
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This is linear programming. The Simplex Method is a general procedure for solving linear programs.

Problems of this form maximize or minimize something by selecting values for decision variables, and those decision variables are subject to a set of constraints.

These problems will have:

1. objective function (that is maximized or minimized)
2. decision variables (what values need to be decided)
3. constraints (what constraints do those decision variables have).

So when you read the problem ask what is the objective? What do you want to minimize or maximize?

How is that objective function determined and what are the decision variables?

What are the constraints on those decision variables?


In this problem you want to maximize the amount of money you get from your investment.
They tell you the way that money (return on investment) is made depends on the levels of A and B. You need to decide these levels, so A and B are your decision variables.

Money = (0.1 * A) + (0.2 * B)

And they tell you what constraints are placed on those decision variables.

This problem is set up as

Objective function

Maximize Z = (0.1 * A) + (0.2 * B)

Constraints

A + B <= 50,000 (you have "at most" $50,000, hence the less than or equal to sign)

A - B >= 20,000 ("at least" $20,000, hence the greater than or equal to sign)

A>=0 B>=0 (A and B have to be positive values)


Since you have just two decision variables you can solve this with a 2 dimensional graph on paper. If you have 3 or more you can use the simplex method to solve it.

Graphical method solution
1. Make a graph of your decision variables. I put A on the horizontal axis and B on a vertical axis.
2. graph each of your contraint lines.
First set A to zero and note point B (0,50,000), then put B to zero and note point A (50,000, 0). Connect the points with a line.
Note the area of the graph that satisfies the constraint. This area will be below the line for the first constraint for example. That is the A + B <= 50,000 constraint.
Next the same for the >= 20,000 constraint. Note the solution set will be below this line.
The A, B positive are also constraint lines.
3. Shade the area of the graph that satisfies all of these constraints (the feasible sides of the constrain lines). It should look like a triangle in this case.

4. Look at the corners of your shaded area. Note the values of A and B at each point. Plug these values into the objective function and see which point results in the highest objective function. That is your solution (beware of multiple solutions).

Done!


Another way to find the optimal point (but more difficult to explain in text):


4. convert your objective function into a line with a slope and intercept (you can keep intercept as Z since it is not important)

Z = 0.1*A + 0.2*B

B = -0.5*A + 5*Z

The slope of the objective function line is -0.5. Try this: place your pencil on the paper and give it a slope of -0.5 to visualize the objective function line. As you move this objective function line through the feasible area of your graph you eventualy get to one edge of the feasible space. That is your optimal values of A and B.

Note that all parts of this problem are important but it can be solved intuitively as well in this case. If the problem had said that A produces 10% return and B 8% then it would change your answer. It's obvious in this case but perhaps another set of constraints it would not be clear what the solution would be.

There are a lot of different types of problems that can be set up this way.
 

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haole from the mainland
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Opus51569 said:
Focus on this part. The rest is a smoke-screen.

A has to be 20,000 more than B
He wants to invest as much as possible in B to maximize his return

A=35,000
B=15,000
Opus is right. While it appears some complicated function is needed here, what's really needed is just a bit of logic.

Insight triumphs over brute force. That, my dear fireplug, is why this type of course is required for a criminal justice degree.
 

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jorgy said:
Opus is right. While it appears some complicated function is needed here, what's really needed is just a bit of logic.

Insight triumphs over brute force. That, my dear fireplug, is why this type of course is required for a criminal justice degree.
Aye and also the ability to discern what information is useful and what is just noise.
 

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Cowboy up
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All parts of this case are important to the solution. There is no smoke or noise in this example. The yields of A and B are important to determine the return from investing in each one. That is why you can conclude to invest in as much of company B as possible. It's the objective function.

I agree though that you are unlikely to be asked to solve a problem that you cannot reason through along with trying a few values. Hopefully what I wrote about the objective function and constraints will be helpful with thinking it through for these problems in general.
 

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Art853 said:
All parts of this case are important to the solution. There is no smoke or noise in this example. The yields of A and B are important to determine the return from investing in each one. That is why you can conclude to invest in as much of company B as possible. It's the objective function.

I agree though that you are unlikely to be asked to solve a problem that you cannot reason through along with trying a few values. Hopefully what I wrote about the objective function and constraints will be helpful with thinking it through for these problems in general.
True, no noise in this example.
 

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What the what???
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Art853 said:
All parts of this case are important to the solution. There is no smoke or noise in this example. The yields of A and B are important to determine the return from investing in each one. That is why you can conclude to invest in as much of company B as possible. It's the objective function.
So, we're really going to debate the eccentricities of the problem itself??? Okay.

Of course there is smoke and noise in this example. The problem could have simply stated that B offers a greater return on Natsano's investment. The fact that 10% and 20% are given as specific values is extraneous information designed to test the student's ability to discern what is necessary to solve the problem and what is not. If the student realized that all that was needed to solve the problem is:

A+B=50000
A is 20000 more than B
B must be as large a number as possible

great. If the student spent even a moment calculating the specific rates of return because they saw specific values and thought that might be important...they were distracted by the smoke and noise. For that matter, if they thought that "Natsano" is an interesting name...they were distracted by the smoke and noise... :)
 

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Seat's not level
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Legal math...

$50,000 bail = $5,000 cash
10 Year sentence = 7 years if you act like a chitt, or 3 years if you act good

For advanced math you need to figure out how much to bribe the cop, the DA or the judge to get off compeltely...

So the answer is, just pick a friggin' number
 

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What the what???
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See. Chain gets it. :)

It doesn't matter that the criminals name is Big Larry. It doesn't matter that the going rate for bribing a judge is generally 15% more than bribing a DA. It doesn't matter that "act good" in this context means assuming the position and going to your "happy place"...
 

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danl1 said:
EXCEPT:

If you need a math guru, you need Wolfram Alpha:

http://www.wolframalpha.com/

Type this into their search box:

maximize .1A+.2B where A=20000+B and A+B = 50000

goofyspookycoolsmartcomputerstuff. Them wolfram folks are scary.
Good God I used Wolfram the other day when I was doing a math tutorial just because I forgot some calculus stuff. Brings back everything....
 
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