County Credit Union (CCU) has $1 million in new funds that must be allocated to
ID: 3773498 • Letter: C
Question
County Credit Union (CCU) 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 credit union’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 CCU should allocate to each type of loan 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 CCU 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? How much would the annual percentage return change?
Explanation / Answer
Linear Programming:
The method that is implemented to get to the optimal solution is known as Linear Programming.
This is done by study of analysis of the program and forms the result in terms of coefficient optimization.
From the provided information of the County Credit Union (CCU) annual rates of returns on loans for home(H) – 7%, personal(P) – 12% and automobiles(A) – 9% are defined.
The optimal solution to obtain maximum profit by using linear programming can be formulated as,
Let the above equation be (1).
Define constrains for the above linear programming which confirms the degree of accomplishment of function objective.
The constrains for the equation (1) is given as,
Amount given is $1 million. So, overall loans amount is $1million.
Thereby,
Constrain 1 (for new funds):
Constrain 2 (minimum amount of Home loan):
Constrain 3 (requirement of personal loan):
b)
Analysis made on the Computer solution:
Now, by using the data from the question, allocate the amount to each loan as done by the CCU.
So, the allocation is done as,
Now, replace the allocated loans values in equation (1).
So, the maximum profit obtained will be,
Therefore, the maximum profit is $88750 that is total annual return. The annual percentage of return is 8.8750%.
c)
Now, annual interest rate on home loans is increased by 9%. So, the objective coefficient range of Home loans “No lower limit to 0.101”.
Since, 9% is within the range.
Therefore, the solution obtained in part(b) will not change.
d)
It is provided that the effect of total amount of new funds increased by $10000.
The dual value of constrain (1) will be 0.089.
In constrain (1), the value at the right hand side range will be 0 to No upper limit.
So,
Therefore, the increase in the total annual return will be up to $890.
e)
Calculation of change of annual returns and annual percentage after the committee agrees to relax the requirement to atleast 40% of the new funds allocated by 1% to home loans is,
Consider the second constrain (2) is.
The new change of funds to obtain minimum home loans is given as,
Now, the new optimal solution is obtained as follow: