A credit union wants to make investments in the following areas. The information

Question

A credit union wants to make investments in the following areas.

The information presented in the table below indicates the area of investment and the corresponding yearly return.

Type of investment and Annual Rate of Return(%)

Vehicle Loans 7%

Consumer Loans 9%

Other secured Loans 10%

Signature Loans 11%

Risk free Securities 8%

The firm will have a maximum of $2,500,000 available for investment during the coming year. In order to create a diverse portfolio the credit union has established the following investment restrictions:

Risk free securities may not exceed 30% of the total funds invested, but must comprise at least 5% of the total funds invested.

Signature loans may not exceed 12% of the funds invested in all loans (vehicle, consumer, other secured loans, and signature loans)

Consumer loans plus other secured loans may not exceed the vehicle loans

Other secured loans plus signature loans may not exceed the funds invested in risk free securities.

Formulate a linear programming model that will help the credit union decide on how best to allocate funds among each alternative to maximize annual return

How should the $2,500,000 be allocated to each alternative to maximize annual return?What is the annual return?

Explanation / Answer

All linear programs are

Maximize Objective Function , subject to Constraints or Minimize Objective Function subject to Constraints.

This is a maximization problem since firm is making loans for purpose of
maximizing investment income.


So Define variables
Let each unknown equal amount invested or loaned in given category

Type of investments:-

Vehicle Loans...........a with 7% rate of return
Consumer Loans...........b with 9% rate of return
Other secured Loans.......cwith 10% rate of return
Signature Loans.............d with 11%% rate of return
Risk free Securities ........e with 8% rate of return

.


The objective function is

Maximize

7a + 9b+10c+11d+ 8e


I multiplied the real objective function by 100 since
real one is 0.07a+0.09b+.. however multiplying by
100 makes the numbers easier to deal with.

Contrainsts are

a+b+c+d+e= 2,500,000 Firm has 2 million five hundred thousand to invest



Risk free securities may not exceed 30% of the total funds invested, but must comprise at least 5% of the total funds invested.

0.05(a+b+c+d+e)s0.3(a+b+c+d+e)-This has to be reformulated so that a constant appears on right side

Signature loans may not exceed 12% of the funds invested in all loans (vehicle,consumer, other secured loans, and signature loans)

s0.12(a+b+c+d+e) - This has to be reformulated so that a constant appears on right side

Consumer loans plus other secured loans may not exceed the vehicle loans

b+ca -This has to be reformulated so that a constant appears on right side means

-a+b+c0
or
a-b-c0


Other secured loans plus signature loans may not exceed the funds invested in risk free securities

a+be -This has to be reformulated so that a constant appears on right side

Vehicle Loans...........a with 7% rate of return
Consumer Loans...........b with 9% rate of return
Other secured Loans.......cwith 10% rate of return
Signature Loans.............d with 11%% rate of return
Risk free Securities ........e with 8% rate of return

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