Question

1) X Bank has $1 million in new funds that must be allocated to mortgage (home) loans, personal loans, and automobile loans. Assume that X Bank will allocate all the funnd. The annual rates of return for the three types of loans are 8% for home loans, 9% for personal loans, and 11% for automobile loans respectively. The bank’s planning committee has decided that at leat 30% of the new funds must be allocated to the mortgage loans. In addition, the planning committee has specified that the amount allocated to personal loans cannot exceed 50% of the amount allocated to automobile loans.

a) Formulate a linear programming model that can be used to determine the amount of funds X Bank should allocate to each type of loan in order to maximize the total annual return for the new funds.

b) Solve the problem with Excel and find the optimal solution and Interpret your result.

c) What are the objective coefficient ranges for the three components? Interpret these ranges.

d) What are the right-hand-side ranges? Interpret these ranges. e) If more amount could be allocated to loans, how much would it be worth?

Note :Be fast pls i love chegg i need interpretion part because so important :)

Answer 1

1)

a)

Let, h = amount allocated to home loan, p = amount allocated to personal loan and a = amount allocated to automobile loan

Objective is to maximize return => Z = Max 0.08h+0.09p+0.11a

subject to,

h+p+a = 1000000 (Total fund)
h >= 300000 (Min home loan)
p <= 0.5a (personal and auto loan ratio)

or, p-0.5a <= 0

h,p,a >= 0

b)

Solving in solver we get,

h = 300000, p = 0, a = 700000

maximized return = 101000

Solver screenshot

Solver formula

Solver window

Sensitivity report

c)Objective coefficient ranges:

lower range = objective coefficient-allowable decrease and upper range = objective coefficient+allowable increase

For home loan, lower range = 0.08-infinity = -infinity, Upper range = 0.08+0.03 = 0.11
For personal loan, lower range = 0.09-infinity = -infinity, Upper range = 0.09+0.02 = 0.11
For automobile loan, lower range = 0.11-0.02 = 0.09, Upper range = 0.11+infinity = + infinity

[ if a range is -infinity it means decrease by any amount and a range of + infinity means increase by any amount would keep the range valid]

d)

Right hand side ranges:

lower range = constraint R.H. Side-allowable decrease and upper range = constraint R.H. Side +allowable increase

For Total fund constraint, lower range = 1000000-700000= 300000 Upper range = 1000000+infinity = infinity
For Min home loan constraint, lower range = 300000-300000 = 0, Upper range = 300000+700000 = 1000000
For personal and auto loan ratio constraint, lower range =0-350000 = -350000 Upper range = 0+infinity = + infinity

[ if a range is -infinity it means decrease by any amount and a range of + infinity means increase by any amount would keep the range valid]

e) every additional amount of loan is worth of $0.11 which happens to be shadow price of Total fund constraint