Question

ABC Savings and Loan provides five kinds of loans. These loans, with the yearly interest rate charged to customers, are shown in the table below:

Type of Loan Interest charge (%)

Commercial loans 15

First mortgage 10

Home improvements   13

Second mortgage 14

Short-term revolving loan 18

Currently ABC has $53 million available to provide loans to customers. ABC does not need to use all the money for customer loans. Any money left can be invested in an investment X which yields an annual return of 14% . Its objective is to maximize total annual interest earned from the loan plus the return from the investment X.
ABC has found that the demand for short-term revolving loans never exceeds $5 million. ABC wants to satisfy the following Policies and regulations:

• Home improvement loans cannot be higher than 20 percent of first mortgage loans.• Commercial loans must be smaller than or equal to the second mortgage loans.
• The bank must provide at least 25 percent of the loans in First Mortgage.
• Short-term loans cannot exceed $5 million.

Develop a linear programming model for ABC.

Answer 1

Let the amount invested in Commercial loans be Xc, First mortgage Loans be Xf, Home improvement Loans be Xh, Second mortgage be Xs, Short term revolving loans be Xr, Balance amount in X be Xx

Based on the Annual interest charge given, total interest = 15%*Xc + 10%*Xf + 13%*Xh + 14%*Xs + 18%*Xr

We have to maximize this return, hence, we get Objective function as

Maximize Total Interest I = 15%*Xc + 10%*Xf + 13%*Xh + 14%*Xs + 18%*Xr + 14%*Xx

Various constraints given can be written as follows:

Xc + Xf + Xh + Xs + Xr <= 53,000,000.................Constraint for amount available for providing loans

Xx = 53,000,000 - (Xc + Xf + Xh + Xs + Xr).........Balance amount in X

Xr <= 5,000,000...................................................Constraint for Short term revolving loans

Xh <= 20%*Xf......................................................Constraint for amount in home improvement loans vs. first mortgage

Xc <= Xs..............................................................Constraint for Commercial loans vs Second mortgage

Xs >= (Xc + Xf + Xh + Xs + Xr)............................Constraint for minimum amount in Second mortgage

Xc, Xf, Xh, Xs, Xr, Xx >= 0...................................Non-negative constraint

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