Question 835823: Mr. Subhash has taken Rs. 100,000 from his father to invest them in a combination of only two stock portfolios with the maximum investment allowed in either portfolio set at Rs. 75,000. The first portfolio has an average return of 10% whereas the second has 20%. In terms of risk factors associated with these portfolios, the first has a risk rating of 4 (on a scale from 0 to 10), and the second has 9. Since he wants to maximize his return, he will not accept an average rate return below 12% or a risk above 6. Hence, he then faces the important question. How much should he invest in each portfolio?
(a)Formulate the above as linear programming problem.
(b)Solve (a) by graphical method.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x equal the amount of money to invest at 10%.
let y equal the amount of money to invest at 20%.
your objective function will be to maximize your return.
this function is therefore:
return = .10 * x + .20 * y
your constraint equations will be as follows:
x >= 0
y >= 0
this is because the amount invested can't be less than 0.
x + y <= 100,000
this is because the total amount to be invested cannot exceed 100,000.
x <= 75
y <= 75
this is because the maximum investment in either portfolio cannot be greater than 75,000.
4x + 9y <= 6(x+y)
this is because the average risk factor has to be less than or equal to 6.
.10x + .20y >= .12(x+y)
this is because the average rate of return cannot be less than 12%.
the problem has been formulated.
what is left is to graph and find the solution.
the graph of all these equations is shown below:
your feasible region is the shaded area on or below the line of 4x + 9y <= 6(x+y) and on or below the line of x + y <= 100 and on or to the left of the line of x <= 75 and on or above the line of .10x + .20y >= .12 * (x+y).
this area satisfies all the constraints.
the maximum return will be at the intersection points of the lines bounding this feasible region.
those points, and the value of the objective function at those points, are:
(0,0) 0 * .1 + 0 * .2 = 0
(60,40) 60 * .1 + 40 * .2 = 14 ***** maximum return
75,25) 75 * .1 + 25 * .2 = 12.5
75,18.75) 75 * .1 + 18.75 * .2 = 11.25
the maximum return is when you invest 60,000 at 10% and 40,000 at 20%.
this meets all the constraints of the project.
maximum investment in each portfolio is less than 75,000
all of the original 100,000 has been invested (requirement was less than or equal to 100,000).
the average rate of return is greater than 12%.
the average risk factor is less than or equal to 6.
it is actually right on 6. 60 * 4 + 40 * 9 = 600 / 100 equals an average risk factor of 6.
the graph shows the original constraint equations and also the same equations after solving for y in each of the ones that i thought would be useful to show that way.
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