A financial advisor is about to build an investment portfolio for a client who has $100000 to invest. The four investments available are A,B,C, and D. Investment A will earn 4 percent and has risk of two “points” per $ 1000 invested. B earns 6 percent with 3 risk points; C earns 9 percent with 7 risk points; and D earns 11 percent with a risk of 8. The client has put the following conditions on the investments: A is to be no more than one-half of the total invested. A cannot be less than 20 percent of the total investment. D cannot be less than C. Total risk points must be at or below 1,000. Determine the amount of each investment that should be purchased.
Note : Focus on the unit of measure that you use for the investments.

A financial advisor is about to build an investment portfolio for a client who has $100000 to invest. The four investments available are A,B,C, and D. Investment A will earn 4 percent and has risk of two “points” per $ 1000 invested. B earns 6 percent with 3 risk points; C earns 9 percent with 7 risk points; and D earns 11 percent with a risk of 8. The client has put the following conditions on the investments: A is to be no more than one-half of the total invested. A cannot be less than 20 percent of the total investment. D cannot be less than C. Total risk points must be at or below 1,000. Determine the amount of each investment that should be purchased.
Note : Focus on the unit of measure that you use for the investments.


This looks like a linear programming problem.

Let A, B, C, D be the respective amounts invested.

Objective function could be to maximize interest:

.04A+.06B+.09C+.11D=z

Constraints:

A+B+C+D=100,000

A\geq 20,000

A\leq 50,000

D\geq C

\frac{2A}{1000}+\frac{3B}{1000}+\frac{7C}{1000}+
\frac{8D}{1000}\leq 1000

Run this through the Excel solver. This gives A=20,000, B=0, C=0, D=80,000. $9,600 in interest and 680 points.

To do this by hand would be tedious. One could use the M-method by adding the appropriate slack(F,H), surplus(E,G), and artificial variables(R1, R2, R3) and penalizing the objective function.

Rewrite in the appropriate form to set up the tableau:

z=.04A+.06B+.09C+.11D-MR_{1}-MR_{2}-MR_{3}

A+B+C+D+R_{1}=100,000

A-E+R_{2}=50,000

A+F=20,000

-C+D-G+R_{3}=0

\frac{A}{500}+\frac{3B}{1000}+\frac{7C}{1000}+
\frac{D}{125}+H=1000

If this must be done by hand by building the tableau, start by multiplying the R rows by -M and adding to the z-row. This eliminates the M's in the z row.

Then, find the entering and leaving variables and do the numerous, tedious, iterations. That is where Excel Solver, LINGO, TORA and other optimization software comes in handy.

The tableau is:

\begin{array}\text{Basic}&A&B&C&D&E&F&G&H&R_{1}&R_{2}&R_{3}&RHS\\z&-.04&-.06&-.09&-.11&0&0&0&0&M&M&M&0\\
R_{1}&1&1&1&1&0&0&0&0&1&0&0&100,000\\R_{2}&1&0&0&0&-1&0&0&0&0&1&0&50,000\\R_{3}&0&0&0&1&0&0&-1&0&0&0&1&0\\F&1&0&0&0&0&1&0&0&0&0&0&20,000\\
H&\frac{1}{500}&\frac{3}{1000}&\frac{7}{1000}& \frac{1}{125}&0&0&0&1&0&0&0&1000\end{array}

This is Operations Research. A highly demanded field for math majors. If one gets a math degree, one can get a job in OR. Before computers, most with math degrees were relegated to teaching. Not anymore. Look up M-method and Simplex method.
My grad work required OR classes, so I know a little. Perhaps enough to be dangerous :):)

Here is the code to run this in GAMS. GAMS is optimization software that is available free online: http://www.gams.com/download/

If you download it, copy and paste this into a new program.

Variables
I total interest
A Amount invested in A
B amount invested in B
C amount invested in C
D amount invested in D

Positive variable A;
Positive variable B;
Positive variable C;
Positive variable D;

Equations
interest shows total interest
constraintA total invested equals 100000
constraintB amount invested in A less than or equal to 50000
constraintC amount invested in A must be greater than 20000
constraintD amount invested in D must be greater than C
points total points accumulated;

interest.. I =e= .04*A+.06*B+.09*C+.11*D;
constraintA.. A+B+C+D =e= 100000;
constraintB.. A =l= 50000;
constraintC.. A =g= 20000;
constraintD.. D =g= C;
points.. A/500+3*B/1000+7*C/1000+D/125 =l= 1000;

Model interest1 /all/;

solve interest1 using lp maximizing I;

display A.l, B.l, C.l, D.l, interest.l, points.l;

Thanks for the additional info! :)

You're welcome. I doubt if the OP comes back to even look it over. GAMS is easy to program. Download it. It's free. There is a red arrow at the top to run the program once it is entered. Excel solver is also nice once one gets the hang of it.