SOLUTION: This is Linear Programming Pls help us! :( MILESTONE: Linear Program After making various presentations to potential investors, MC has finally hit the jackpot. An investor w

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Question 1116232: This is Linear Programming Pls help us! :(
MILESTONE: Linear Program
After making various presentations to potential investors, MC has finally hit the jackpot. An investor wants to invest 100,000,000 pesos in MC.
After doing all your staff work in the previous weeks, the two of you have been tasked to find the most efficient way to invest the money. This involves making the highest profit possible from the money you have invested. You have decided to consult your friend Olivia, who is also a financial investor.
Olivia discusses that you need to use a mathematical method called Linear Programming to solve this problem. She explains that Linear Programming is a method that has to be set up very carefully. It involves creating a function to optimize and modelling various constraints using linear equations. She has decided to present you the profitability percentages of MC, given that she has already done prior work with MC.
The startup arm is projected to return at least 12%. To minimize risk, you must invest no more than 30,000,000 pesos. The financial stocks return 3%, while the retail arm returns 5%.
For tax reasons, you must invest at least 3 times more in the financial stocks than the retail arm.
Your task is to set up the linear program. You have to define the following:
1. The variables that will be used
2. The optimization function that will be used
a. What kind of optimization will be done: minimization or maximization?
3. The constraints that the linear program will be subjected to
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
assume the following:

x = amount invested in the startup arm.
y = amount invested in stocks
z = amount invested in retail arm.

your objective function, which you will want to maximize, will be:

return = .12 * x + .03 * y + .05 * z

your constraints will be:

x <= 30 million

this means the amount invested in the startup arm will be less than or equal to 30 million pesos.

y + z <= 70 million

this means the amount invested in stocks and the retail arm will be less than or equal to 70 million.

x + y + z = 100

this means that your total investment must be equal to 100 million.
this was probably unnecessary since you wanted to maximize return and the only way to maximize return was to invest all of the money you had available to invest, but it didn't hurt to put it in there.
i ran it with and without this requirement in excel and got the same answer.

y >= 3z converted to y - 3z >= 0

this means that the investment in stocks must be greater than or equal to the investment in the retail arm.
the optimization formulas require the variables be on the left side of the inequality and the constants be on the right side of the inequality, therefore the equivalent inequality of y - 3z >= 0 was used instead of the original inequality of y >= 3z.
this was accomplished by subtracting 3z from both sides of the original inequality.

i put the objective function and the constraints in excel and used the excel solver to come up with a solution.

here's what excel came up with.

the solver was set up as follows:

$$$

with options set up as follows:

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with results shown as follows:

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the results indicate that:

30 million is invested in the startup arm.
this is the max that can be invested there.
this is reasonable since the startup arm will provide the greatest return.

52.5 million is invested in stocks and 17.5 is invested in the retail arm.
this adds up to 70 million which is the balance of the 100 million that needed to be invested.
52.5 million divided by 17.5 million = 3.
this was required since the amount inveswted in stocks had to be 3 times the amount invested in the retail arm.

the results look reasonable.

i also found linear programming optimization routine using simplex method online and came up with the same solution.

that calculator was found at https://www.zweigmedia.com/RealWorld/simplex.html

my inputs and outputs to that software tool are shown below:

$$$
$$$
$$$
$$$
$$$

your solution to the problem appears to be:

1. The variables that will be used.

you will use x, y, and z.

x = amount invested in startup arm.
y = amount invested in stocks.
z = amount invested in retail arm.

2. The optimization function that will be used

return = .12x + .03y + .05z


a. What kind of optimization will be done: minimization or maximization?

maximization

3. The constraints that the linear program will be subjected to

x <= 30 million.
y + z <= 70 million
x + y + z = 100 million
y >= 3z converted in form to y - 3z >= 0