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.