Question

A linear programming computer package is needed. The employee credit union at State University is planning the allocation of funds for the coming year. The credit union makes four types of loans to its members. In addition, the credit union invests in risk-free securities to stabilize income. The various revenue-producing investments together with annual rates of return are as follows. Type of Loan/Investment Annual Rate of Return (%) Automobile loans Furniture loans Other secured loans Signature loans Risk-free securities The credit union will have $2,200,000 available for investment during the coming year. State laws and credit union policies impose the following restrictions on the composition of the loans and investments. • Risk-free securities may not exceed 30% of the total funds available for investment. • Signature loans may not exceed 10% of the funds invested in all loans (automobile, furniture, other secured, and signature loans). • Furniture loans plus other secured loans may not exceed the automobile loans. • Other secured loans plus signature loans may not exceed the funds invested in risk-free securities. How should the $2,200,000 be allocated to each of the loan/investment alternatives to maximize total annual return? Automobile loans Furniture loans S Other secured loans S Signature loans Risk-free securities $ What is the projected total annual return?

Answer 1

Let us first establish the variable

Let x1 = Automobile loans
x2 = Furniture loans
x3 = Other secured loans
x4 = Signature loans
x5 = Risk free securities

Maximize = 0.09x1 + 0.11x2 + 0.12x3 + 0.13x4 + 0.10x5

Constraints would be

x5 <= 660,000
x4 <= 0.10 (x1 + x2 + x3 + x4) or -0.10x1 - 0.10x2 - 0.10x3 + 0.90x4 <= 0
x1 + x2 + x3 + x4 + x5 = 2,200,000
x3 + x4 - x5 <= 0
-x1 + x2 + x3 <= 0

Here are the answers

Variables Variable Type Value Real 693000 Real 187000 Real 506000 Real 154000 660000 Real

Variable	Amount	Return	Abs Return
x1	$    693,000	9%	$    62,370
x2	$    187,000	11%	$    20,570
x3	$    506,000	12%	$    60,720
x4	$    154,000	13%	$    20,020
x5	$    660,000	10%	$    66,000
	$ 2,200,000	10.44%	$ 229,680