You are asked to invest up to $200,000 with maximal dividend. There are three stocks available: utilities, with 10% dividend, bonds with 5% dividend, technology, with 6% dividend. You cannot invest more than 50% of the total amount into one stock type, and utilities are restricted to 10% due to risks. Set up, but don't compute the LP problem in standard form.
This is what I did so far:
Let x, y , and z denote the amounts invested in utilities, bonds, and technology.
Find values of x, y and z that maximize
q = .10x+ .05y+ .06z
Subject to the constraints:
x+ y+ z = 200,000
x<= 20,000,
y<= 100,000,
z<= 100,000,
Also, x, y, and z >= 0.
Am I doing the problem wrong? I think it is in standard form but it says "up to" 200,000 so I think I messed up.
Best Answer
To set up your constraints I would use $x+y+z \le 200,000$. Together with your other constraints, this defines the feasible region, which will be a convex polyhedron bounded by planes that represent the extreme values of your constraints - one of these planes will be $x+y+z=200,000$.
When you maximize the objective function, you will find the optimal solution lies at a vertex of the feasible region, and this vertex will be on the plane $x+y+z=200,000$ (because a solution that leaves money uninvested can always be improved by investing the uninvested amount in either bonds or technology).