Question 1116232
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:


<img src ="http://theo.x10hosting.com/2018/050101.jpg" alt="$$$" >


with options set up as follows:


<img src ="http://theo.x10hosting.com/2018/050102.jpg" alt="$$$" >


with results shown as follows:


<img src ="http://theo.x10hosting.com/2018/050103.jpg" alt="$$$" >


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 <a href = "https://www.zweigmedia.com/RealWorld/simplex.html" target = "_blank">https://www.zweigmedia.com/RealWorld/simplex.html</a>


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


<img src ="http://theo.x10hosting.com/2018/050104.jpg" alt="$$$" >
<img src ="http://theo.x10hosting.com/2018/050105.jpg" alt="$$$" >
<img src ="http://theo.x10hosting.com/2018/050106.jpg" alt="$$$" >
<img src ="http://theo.x10hosting.com/2018/050107.jpg" alt="$$$" >
<img src ="http://theo.x10hosting.com/2018/050108.jpg" alt="$$$" >


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