Subject: Modeling
Problem*
A bank is attempting to determine where its assets should be invested during the current year. At present, $500,000 is available for investment in bonds, home loans, auto loans, and personal loans. The annual rate of return on each type of investment is known to be: bonds, 10%; home loans, 16%; auto loans, 13%; personal loans, 20%. 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:
The bank's objective is to maximize the annual return on its investment portfolio. Formulate an LP that will enable the bank to meet this goal.
- The amount invested in personal loans cannot exceed the amount invested in bonds.
- The amount invested in home loans cannot exceed the amount invested in auto loans.
- No more than 25% of the total amount invested may be in personal loans.
* Source: Operations Research, W. L. Winston, 4th edition, Duxbury 2004. Page 92, Problem #3.
SOLUTION
Let
: Dollars invested in bonds : Dollars invested in home loans : Dollars invested in auto loans : Dollars invested in personal loans The LP model simply follows the given restriction and desired performance measure .
, , , The LINDO output for the problem is given below. Based on this solution, The bank should invest equal amounts of funds ($125,000) in each of the four options with the expected return of $73,750.
MAX 0.1 X1 + 0.16 X2 + 0.13 X3 + 0.2 X4 SUBJECT TO 2) X1 + X2 + X3 + X4 <= 500000 3) - X1 + X4 <= 0 4) X2 - X3 <= 0 5) - 0.25 X1 - 0.25 X2 - 0.25 X3 + 0.75 X4 <= 0 END LP OPTIMUM FOUND AT STEP 4 OBJECTIVE FUNCTION VALUE 1) 73750.00 VARIABLE VALUE REDUCED COST X1 125000.000000 0.000000 X2 125000.000000 0.000000 X3 125000.000000 0.000000 X4 125000.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 0.147500 3) 0.000000 0.045000 4) 0.000000 0.015000 5) 0.000000 0.010000 NO. ITERATIONS= 4
