2. Adirondack Savings Bank (ASB) has $1 million in new funds that must be alloca
ID: 3202894 • Letter: 2
Question
2. 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 9% 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 lean in order to maximize the total annual return for the new funds.
b)How much should be allocated to each type of loan? What is the total annual return? What is the annual percentage return?
c)If the interest rate on home loans increased to 9%, would the amount allocated to each type of loan change? Explain.
d)Suppose the total amount of new funds available was increased by $10,000. What effect would this have on the total annual return? Explain.
e)Assume that ASB has the original $1 million in new funds available and that the planning committee has agreed to relax the requirements that at least 40% of the new funds must be allocated to home loans by 1%. How much would the annual return change? How much would the annual percentage return change?
3. Benson Electronics manufactures three components used to produce cell telephones and other communication devices. In a given production period, demand for the three components may exceed Benson’s manufacturing capacity. In this case, the company meets demand by purchasing the components from another manufacturer at an increased cost per unit. Benson’s manufacturing cost per unit and purchasing cost per unit for the three components are as follows:
Source
Component 1
Component 2
Component 3
Manufacture
$4.50
$5.00
$2.75
Purchase
$6.50
$8.80
$7.00
Manufacturing times in minutes per unit for Benson’s three departments are as follows:
Department
Component 1
Component 2
Component 3
Production
2
3
4
Assembly
1
1.5
3
Testing & Packaging
1.5
2
5
For instance, each unit of component 1 that Benson manufactures requires 2 minutes of production time, 1 minute of assembly time, and 1.5 minutes of testing and packaging time. For the next production period, Benson has capacities of 360 hours in the production department, 250 hours in the assembly department, and 300 hours in the testing and packaging department.
a)Formulate a linear programming model that can be used to determine how many units of each component to manufacture and how many units of each component to purchase. Assume that component demands that must be satisfied are 6000 units for component 1, 4000 units for component 2, and 3500 units for component 3. The objective is to minimize the total manufacturing and purchasing costs.
b)What is the optimal solution? How many units of each component should be manufactured and how many units should be purchased?
c)Which departments are limiting Benson’s manufacturing quantities? Use the dual price to determine the value of an extra hour in each of these departments.
d)Suppose that Benson had to obtain one additional unit of component 2. Discuss what the dual price for the component 2 constraint tells us about the cost to obtain the additional unit.
Source
Component 1
Component 2
Component 3
Manufacture
$4.50
$5.00
$2.75
Purchase
$6.50
$8.80
$7.00
Explanation / Answer
Linear programming is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.
So in the above question we are required to maximise the total annual return for the new funds alvailable with Adirondack Savings Bank (ASB).
Data that is available to us is
a) Now lets formulate a linear programming model to maximize the profit
Over Objective function is
MAX = 0.07*H + 0.12*P + 0.09*A
As this will maximize the total annual return
Where
H: Home Loans Amount
P: Personal Loans Amount
A: Automobile Loans Amount
Constrains are:
b) Now when we solve it we get
H= $400,000
P= $225,000
A= $375,000
Total Annual Return= $88,750
Annual Percentage Return = 8.875%
c) The amount allocation will not change even when the interest rate on home loans increased to 9%, as to maximize return we need to allocate maximum possible to personal loans (12% return) and the amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans. So we need to maximize Automobile loan to maximize personal loan. So even now that Home loan and Personal loan loan have same return, amount allocated would not change.
d) If the amount is increased by $10,000, the total annual return will change
As Constrains H + P + A = 1000000 will change to H + P + A = 1010000
and when we solve again
amount allocation will come out to be
H= $400,000
P= $228,750
A= $381,250
Total Annual Return= $87,762.5
So it can be seen that Annual return has increased.
e) Relaxing the requirements that at least 40% of the new funds must be allocated to home loans by 1% will change the constrain H >= 400000 to H >= 390000.
Now solving it will give values as
H= $390,000
P= $232,500
A= $387,500
Total Annual Return= $90,075
Annual Percentage Return = 9.0075%
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