The U-Save Loan Company is planning its operations for the next year. The company makes four types of loans, listed below, together with the annual return (in percent) to the company.

Legal requirements and company policies place the following limits on the various types of loans.

Signature loans may not exceed 10 percent of the total amount of loans. The amount of signature and furniture loans together may not exceed 20 percent of the total amount of loans. First mortgages must be at least 40 percent of the total mortgages and at least 20 percent of the total amount of loans. The company can lend a maximum of 1.5 million dollars next operating period.

Type Annual Return (percent)

Signature loan 15

Furniture loan 12

Second Home Mortgage 10

First Home Mortgage 7

U-Save wants to know the amounts it should invest in each type of loan to maximize its total annual return.

X1 = Dollars invested in Signature Loans

X2 = Dollars invested in Furniture Loans

X3 = Dollars invested in Second Home Mortgages

X4 = Dollars invested in First Home Mortgages

MAX dollar return: .15X1 + .12X2 + .10X3 + .07X4

Signature loan limit: X1 <= .10(X1 + X2 + X3 + X4)

Signature & Furniture loan limit: X1 + X2 <= .20(X1 + X2 + X3 + X4)

1st mortgage limit: X4 >= .40(X3 + X4)

1st –to-total limit: X4 >= .20(X1 + X2 + X3 + X4)

Available funds: X1 + X2 + X3 + X4 <= 1,500,000

Non-negativity constraint: X1 , X2 , X3 , X4 >= 0

Signature loan limit: -.90X1 + .10X2 + .10X3 + .10X4 >= 0

Signature & Furniture loan limit: -.80X1 -.80X2 + .20X3 + .20X4>= 0

1st mortgage limit: .40X3 -.60X4 <= 0

1st –to-total limit: .20X1 + .20X2 + .20X3 -.80X4 <= 0

Available funds: X1 + X2 + X3 + X4 <= 1,500,000

Non-negativity constraint: X1 , X2 , X3 , X4 >= 0