A financial advisor is about to build an investment portfolio for a client who h

Question

A financial advisor is about to build an investment portfolio for a client who has $400,000 to invest. The four investments available are A, B, C, and D. Investment A will earn four percent and has a risk of two "points" per $1,000 invested. Investment B earns six percent with five risk points; investment C earns nine percent with nine risk points; and investment D earns 11 percent with a risk of eighteen. The client has put the following conditions on the investments: 1) A is to be no more than 40 percent of the total invested; 2) A cannot be less than 20 percent of the total investment; 3) D cannot be less than C; and 4) Total risk points must be at or below 450.

Formulate this as a linear programming model, that is, i) define your decision variables, ii) state the objective function, iii) write down all of the constraints and put them in a standard LP format.

Explanation / Answer

Decision variables :

Let w, x,y&z are the amounts invested in A,B,C and D resp.

Contribution:

The total return on investment should be maximum

So

4% is return on A , 6% is the return on B , 9% on C and 11% on D.

So the total return in $= 4%×w +6%×x+9%×y+11%×z = Maximum

Constraints are :

1) All the variables cannot be negative.

So w>=0

X>=0

Y>=0

Z>=0

2) A is no more than 40% of total invested

I.e w <=40%× (w+x+y+z)

3) A cannot be less than 20 % of total investment  

I.e

w >= 20%× (w+x+y+z)

4) D cannot be less than C

I.e z>= y

5) total all the amounts invested can be less than equal to available amount of $400000

I.e

w+x+y+z<= 400000

6) total risk points must be at or below 450

Total risk points of A is 2 for $1000. So total risk points of A is w/ 2

Total risk points of B is 5 per $1000. So the total risk points of B is x/5

Total risk points of C is 9 per $1000. So the total risk points of C is y/9

Total risk points of D is 18 per $1000. So the total risk points of D is z/ 18.

Now condition is

w/2+x/5+y/9+z/18 <=450

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