Question

Adirondack Savings Bank (ASB) has $1 million in new funds that must be allocated to home loans, personal loans, and automobile loans. The annual rates of return for the three types of loans are 5% for home loans, 12% for personal loans, and 11% for automobile loans. The bank’s planning committee has decided that at least 40% of the new funds must be allocated to home loans. In addition, the planning committee has specified that the amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans.

(a)	Formulate a linear programming model that can be used to determine the amount of funds ASB should allocate to each type of loan to maximize the total annual return for the new funds. If the constant is "1" it must be entered in the box. If your answer is zero enter “0”.
	Let H = amount allocated to home loans P = amount allocated to personal loans A = amount allocated to automobile loans
	Max H + P + A s.t. H + P + A ≥ Minimum Home Loans H + P + A ≤ Personal Loan Requirement H + P + A = Amount of New Funds
(b)	How much should be allocated to each type of loan?
	Loan type Allocation Home $ Personal $ Automobile $
	What is the total annual return?
	If required, round your answer to nearest whole dollar amount.
	$
	What is the annual percentage return?
	If required, round your answer to two decimal places.
	%
(c)	If the interest rate on home loans increases to 9%, would the amount allocated to each type of loan change?
	- Select your answer -YesNoItem 21
	Explain.
	The input in the box below will not be graded, but may be reviewed and considered by your instructor.
(d)	Suppose the total amount of new funds available is increased by $10,000. What effect would this have on the total annual return? Explain.
	If required, round your answer to nearest whole dollar amount.
	An increase of $10,000 to the total amount of funds available would increase the total annual return by $ .
(e)	Assume that ASB has the original $1 million in new funds available and that the planning committee has agreed to relax the requirement that at least 40% of the new funds must be allocated to home loans by 1%. How much would the annual return change?
	If required, round your answer to nearest whole dollar amount.
	$
	How much would the annual percentage return change?
	If required, round your answer to two decimal places.
	%

Answer 1

Answer:

Given data

ASB has $1 million in new funds

5% for home loans

12% for personal loans

11%for automobile loans

The bank’s planning committee has decided that at least 40% of the new funds must be allocated to home loans. In addition, the planning committee has specified that the amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans.

a)

A linear program model is a mathematical model with a linear objective function , a set of linear constraints , and non negative variables .Let H be the amount allocated to home loans ,

P be the amount allocated to personal loans , and

A be the amount allocated to automobile loans.

Max	0.05H+0.12P+0.11A
s.t.
	H+P+A=1,000,000	Amount of new funds
	0.6H-0.4P-0.4A 0	Minimum Home loans
	P - 0.6A 0	Personal loan Requirement
	H,P,A 0

b)Use EXCEL to solve for the optimal solution and sensitivity report using the following steps:

1 ) Select the Data tab from the Ribbon

2) Select Solver from the Analysis Group

3) When the Solver parameters dialog appears:

Enter the objective function

Select the To: Max option

Enter the decision variables into the By changing Variables Cells box

Select Add

4) When the Add Constraint dialog box appears:

Enter the constraints in the Cell Refrence box

Select <=

Enter The right -hand side in the Constraint box

Click Ok

5) When the solver parameters dialog box reappears:

Click the checkbox for make Unconstrained Variables Non -negative

6) Select the select a solving Method drop - down button .

Select Simplex LP*

7) Click Sove

8) When the Solver Results dialog box appears:

Select keep Solver Solution

Select Sensitivity in the Reports box

Click OK

The optimal soliution is H = 4,00,000

P = 2,25,000

and A = 3,75,000.

Substitute these values into the objective function to find its value.

0.05 H + 0.12 P + 0.11 A = 0.05(4,00,000) +0.12(2,25,000)+0.11(3,75,000)

= 20000+27000+41250

= 88,250

Annual percentage return is the ratio of the annual return divided by the amount of funds expressed as a percentage.

= 8.825%

c) From the sensitivity report created by EXCEL , the objective coefficient range for H is no lower limit to 0.101.

Since 0.09 is within the range , the optimal solution remains H = 4,00,000

P = 2,25,000

and A = 3,75,000

d) From the sensitivity report created by EXCEL , the shadow price for new funds is 0.089.

The right -hand -side range ffor new funds is 0 to no upper limit. Then if new funds available increased by $10,000 the shadow price is applicable.

$10,000 * 0.089 = $ 890

THe total amount return will be incrase by $ 890

e) The new linear model is as follows:

Max	0.05H+0.12P+0.11A
s.t.
	H+P+A=1,000,000	Amount of new funds
	0.61H-0.39P-0.39A 0	Minimum Home loans
	P - 0.6A 0	Personal loan Requirement
	H,P,A 0

Use EXCEL to solve for the optimal solution and sensitivity report using the following steps:

1 ) Select the Data tab from the Ribbon

2) Select Solver from the Analysis Group

3) When the Solver parameters dialog appears:

Enter the objective function

Select the To: Max option

Enter the decision variables into the By changing Variables Cells box

Select Add

4) When the Add Constraint dialog box appears:

Enter the constraints in the Cell Refrence box

Select <=

Enter The right -hand side in the Constraint box

Click Ok

5) When the solver parameters dialog box reappears:

Click the checkbox for make Unconstrained Variables Non -negative

6) Select the select a solving Method drop - down button .

Select Simplex LP*

7) Click Sove

8) When the Solver Results dialog box appears:

Select keep Solver Solution

Select Sensitivity in the Reports box

Click OK

The new optimal solution is H = 3,90,000, P = 2,28,750,, and A = 3,81,250.

Subsitute these values into the objective function to find its value.

0.05H+0.12P+0.11A = 0.05(390000)+0.12(228750)+0.11(381250)

= 19,500+27,450+41,937.5

= 88,887.5

The change of annual return is the new value minus the old value.

= 88,887.5 - 88,250

= 637.5

Annual return would increase by $637.50

Annual percentage return is the ratio of the total annual return divided by the amount of funds expressed as a percentage .

= 8.88875%

The change of annual percentage return is the new value minus the old value.

= 8.88875 % - 8.825 %

= 0.06375 %

Annual percentage return would increase by 0.06375%.