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 7% for home loans, 12% for personal loans, and 10% 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

Sol:

Let the amount allotted to home loans be “x”, to personal loans be “y” and to automobile loans be “z”.

Objective function = 0.07x+0.12y+0.09z. This has to be maximized.

Constraints are:

1. x>=400,000 (at least 40% of the new funds must be allocated to home loans)

2. y<=0.6z (amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans)

3. x+y+z<=1,000,000 (Total amount of funds is $1 million)

Solving this in excel using the solver function we get:

	Amount allocated to:	(in $)
	Home loans	400,000.00
	Personal loans	225,000.00
	Automobile loans	375,000.00
					Formula
	Total return	88,750.00			0.07x+0.12y+0.09z
	Constraints
1	400,000.00	>=	400,000.00		x>=400,000
2	225,000.00	<=	225,000.00		y<=0.6z
3	1,000,000.00	<=	1,000,000.00		x+y+z<=1,000,000

Thus $400,000 is allotted to home loans, $225,000 is allotted to personal loans, and $375,000 is allotted to automobile loans.

Total annual return = $88,750

Annual % return = 88,750/1,000,000 = 8.875%

If home loans rate becomes 9% then the objective function will become: 0.09x+0.12y+0.09z. Revised solution:

	Amount allocated to:	(in $)
	Home loans	400,000.00
	Personal loans	225,000.00
	Automobile loans	375,000.00
					Formula
	Total return	96,750.00			0.09x+0.12y+0.09z
	Constraints
1	400,000.00	>=	400,000.00		x>=400,000
2	225,000.00	<=	225,000.00		y<=0.6z
3	1,000,000.00	<=	1,000,000.00		x+y+z<=1,000,000

So, even when the interest rate on home loans is increased to 9% the amount allocated to each type of loan does not change. They remain the same.

If total amount of funds available = 1,000,000+10,000 = $1,010,000 then the new solution (with 7% on home loan i.e. the original rates) then the solution will be:

	Amount allocated to:	(in $)
	Home loans	404,000.00
	Personal loans	227,250.00
	Automobile loans	378,750.00
					Formula
	Total return	89,637.50			0.07x+0.12y+0.09z
	Constraints
1	404,000.00	>=	404,000.00		x>=404,000
2	227,250.00	<=	227,250.00		y<=0.6z
3	1,010,000.00	<=	1,010,000.00		x+y+z<=1,010,000

Thus total annual return amount increases to $89,637.50 and total annual % = 89637.5/1010000 = 8.750% remains the same.

If only 39% (i.e. 40%-1%) of funds are now allotted to home loans then:

Amount allocated to:	(in $)
Home loans	390,000.00
Personal loans	228,750.00
Automobile loans	381,250.00
				Formula
Total return	89,062.50			0.07x+0.12y+0.09z
Constraints
390,000.00	>=	390,000.00		x>=390,000
228,750.00	<=	228,750.00		y<=0.6z
1,000,000.00	<=	1,000,000.00		x+y+z<=1,000,000

Thus annual return will become $89,062.50 and annual % return will be = 89062.5/1 million = 8.90625%

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