The U-Save Loan Company is planning its operations for the next year. The compan
ID: 3253883 • Letter: T
Question
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
Explanation / Answer
ans:
the company can lend a maximum of 1.5 million dollar next operation period
total amount of loan=1.5 million dollars=15,00,000.
signature loan may not exeed 10% of the total amount of loan
signature loan limit X1<=0.10(X1+X2+X3+X4)
<=0.10(15,00,000)
<=150000
so X1=150000
signature and furniture loan limit should not exceed the 20% of the total amount of loan
X1+ X2<=0.20(1500000)
150000+X2<=300000
X2<=300000-150000
X2<=150000
so X2=150000
first mortage atleast 20% of the total amount of loan
X4>=0.20(1500000)
X4>=300000
assume X4=600000
available fundsX1+X2+X3+X4<=1500000
150000+150000+X3+600000<=1500000
x3<=600000
X3=600000
X1=150000 X2=150000 X3=600000 X4=600000
non negative constantsX1,X2,X3,X4>=0
150000,150000,600000,600000>=0 proved
signature loan limit -0.90X1+0.10X2+0.10X3+0.10X4>=0
-135000+15000+60000+60000>=0
-135000-135000>=0
0>=0 PROVED
signature and furniture loan limit-0.80X1-0.80X2+0.20X3+0.20X4>=0
-120000-120000+120000+120000>=0
0>=0 proved
1 st mortage limit 0.40X3-0.60X4<=0
2,40,000-3,60,000<=0
-12000<=0 proved
1 st total limit 0.2X1+0.20X2+0.20X3-0.80X4<=0
30000+30000+120000-480000<=0
180000-480000<=0
-300000<=0 Proved
finally MAX dollar return=0.15X1+0.12X2+0.10X3+0.7X4
=0.15(150000)+0.12(150000)+0.10(600000)+0.7(600000)
=22500+18000+60000+420000
=520500
X1<=150000 dollars invested in signature loan
X2<= 150000 dollars invested in furniture loan
X3>=600000 dollars invested in second mortgages//X3 value must be more tham 3,00,000 . i consider 600000
x4>=600000 dollars invested in first mortgages
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