1. ## Linear programming

The question:

A bank is attempting to determine where its assets should be invested during the current year. At present \$500 million is available for investment in bonds, home loans, car loans,and personal loans. The annual rate of return on each type of investment is known to be:bonds, 7%; home loans, 8%; car loans, 12%; personal loans, 11%. In order to ensure that
the bank’s portfolio is not too risky, the bank’s investment manager has placed the following three restrictions on the bank’s portfolio:
(a) The amount invested in personal loans cannot exceed the amount invested in bonds.
(b) The amount invested in home loans cannot exceed the amount invested in car loans.
(c) No more than 25% of the total amount invested may be in personal loans.
The bank’s objective is to maximize the annual return on its investment portfolio. Formulate an LP (in standard form) that will enable the bank to meet this goal. Assume interest is calculated annually.

Here are my equations:

let b=bonds, h=home loans, c=car loans, p=personal loans:

maximise z=0.07b+0.08h+0.12c+0.11p
subject to:
b+h+c+p<=500x10^6
b,h,c,p>=0
p<=b
h<=c
125x10^6>=p

Note; x>=y means x is greater than or equal to y..etc