Question

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 8

Furniture loans 10

Other secured loans 11

Signature loans 12

Risk-free securities 9

The credit union will have $1.8 million 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 25% of the total funds available for investment. • Signature loans may not exceed 12% 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 $1.8 million be allocated to each of the loan/investment alternatives to maximize total annual return?

Answer 1

Let the investments in loan types be x?, x?, x?, x? and x? for the types 1.-5 as listed above.

The linear programming problem:
To Maximize the total returs e = { Total returns = 0.08x? + 0.10x? + 0.11x? + 0.12x? + 0.09x? }
The following are the constraints:

x? ? 0, x? ? 0, x? ? 0, x? ? 0, x? ? 0.

x? + x? + x? + x? + x? ? 1,800 000

x? ? 450,000 viz., 25% of 1,800 000

x? ? 0.1(x? + x? + x? + x? + x?); /12%

x? + x? ? x?; /3rd restriction

x? + x? ? x?; /4th restriction

Therefore

I) x? = 450,000

II)   x? = 1,800,000 x 12%

       x? = 216,000

III) x? = x? + x? ? x?

= 216,000 + x? = 450,000

x? = 450,000 - 216,000

x?? = 234,000

  

IV)   x? = x? + x? ? x?

234,000 + x? = 450,000

x? = 450,000 - 234,000

   x? = 216,000

V) x? = 450,000

x? = 450,000

x? = 216,000

x? = 234,000

x? = 216,000

x? = 450,000

Therefore Total Maximum returns

= x? + x? + x? + x? + x?

= 450,000 + 216,000 + 234,000 + 216,000 + 450,000

= 1,566,000