The employee credit union at State University is planning the allocation of fund

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:

The credit union will have $2 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 11% 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 million be allocated to each of the loan/investment alternatives to maximize total annual return?

What is the projected total annual return?

Annual Return = $

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

Explanation / Answer

Let x, x, x, x and x are the investments in loans types 1.-5. as listed above. Then we'll have the following linear programming problem:
Maximize { Total_return = 0.08x + 0.10x + 0.11x + 0.12x + 0.09x }
subject to constraints: x 0, x 0, x 0, x 0, x 0,
x + x + x + x + x 2000000
x 500 000; /25% of 2000000/;
x 0.11(x + x + x + x + x); /11% of all/;
x + x x; /3rd restriction/;
x + x x; /4th restriction/

Solving the above problem with simplex-method we get the optimal solution:
x = 650900, x =349099, x = 301802, x =198198, x =500000,
Maximal total return = 188964

Tableau 1 0.08 0.1 0.11 0.12 0.09 0 0 0 0 0 0 0 0 0 0 Base Cb P0 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P6 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 P7 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 P8 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 P9 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 P10 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 P11 0 2000000 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 P12 0 500000 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 P13 0 0 -0.11 -0.11 -0.11 1 -0.11 0 0 0 0 0 0 0 1 0 0 P14 0 0 -1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 P15 0 0 0 0 1 1 -1 0 0 0 0 0 0 0 0 0 1 Z 0 -0.08 -0.1 -0.11 -0.12 -0.09 0 0 0 0 0 0 0 0 0 0

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