1) A citizen has Rs. 10 lakhs to invest in various investment policies with the bank under certain constraints. 2) The bank aims to maximize returns from loans provided to customers from funds received from investors and RBI totaling Rs. 160 crores under loan amount constraints. 3) The linear programming model minimizes deviations from goals of maximizing returns for investors and bank within investment and loan limits.

1. PROBLEM
STATEMENT
A Citizen has a total amount of Rs. 10 Lakhs for different investment
policies in the People’s Bank of India bank with the hope to receive
high returns from the bank. The investor has the following constraints
to be satisfied while maximising the returns from his investments.
1. Total invested amount in Saving Account is at least 20% of the total fund.
2. Total invested amount in Fixed Deposit is at least 25% of the total fund.
3.Total invested amount in Public Provident Fund (PPF) and Senior Citizen
Savings Scheme (SCSS) is at least 10% of the total fund.
4.Total invested amount in each of Public Provident Fund (PPF) and Senior
Citizen Savings Scheme (SCSS) is not more than 25% the total fund.

2. For bank to maximize its returns on the total funds of Rs. 150 crores
from RBI and from the investments which it receives from 100
similar citizens of 10 lakhs they provide loans such as:
Home Loan
Personal Loan
Business Loan
Car Loan
Two Wheelers Loans
24x7 Loan against Securities
Education Loan
Loan Against Property
Gold Loan
Loan Against FD
Commercial Vehicle
Construction Equipment Loan

3. CONSTRAINTS
The total amount allocated to Home Loan, Personal Loan and Education Loan should be at least
25% of the total fund.
The total amount allocated to Loan against property, Gold Loan, 24x7 loan against securities, Loan
against FD should not be more than 20% of the total fund.
The total amount allocated to Car Loan, Two wheelers Loans should not be more than 15% of the
total fund.
The total amount allocated to Business Loan, Commercial Vehicle Construction Equipment Loan
should be at least 20% of the total fund.
The total amount allocated to Business loan is not more than 10%.
The total amount allocated to education loan is at least 10%
The total amount allocated to Home loan is at least 5%

4. CONSTRAINTS
The total amount allocated to Personal loan is at least 5%
The total amount allocated to Loan against property, Gold loan, Loan Against FD is at least
3.5%
The total amount allocated to each of Car loan, Two wheelers Loans is at least 7.5%
The total amount allocated to each of 24x7 loan against securities, Construction vehicle
Construction Equipment loan, Gold Loan, Loan against property, Loan against FD is at least
2.5%.

6. GO A LS
Goal 1: The total amount invested by each investor is at most 10 Lakhs
Goal 2: The returns of the investors should be at least 20% of the amount all
the investors have invested
Goal 3: The total amount available for the bank to lend as various types of
loans should not exceed the total amount of 10 crores received from 100
investors and the fund of 150 crores from the RBI
Goal 4: The returns of the bank should be at least 25% of the amount they
have lent in the form of loans to the customers

7. DECISION VARIABLES
y1 =The amount to be invested by investor in Savings account
y2 =The amount to be invested by investor in Fixed Deposit
y3 =The amount to be invested by investor in PPF
y4 =The amount to be invested by investor in SCSS
x1 =The amount to be lent by the bank to customers as Home Loan
x2 = The amount to be lent by the bank to customers as Personal Loan
x3 = The amount to be lent by the bank to customers as Business Loan
x4 =The amount to be lent by the bank to customers as Car Loan
x5= The amount to be lent by the bank to customers as Two Wheelers Loan
x6 =The amount to be lent by the bank to customers as 24x7 Loan against
securities
x7 =The amount to be lent by the bank to customers as Commercial
Vehicle Construction Equipment Loan
x8 =The amount to be lent by the bank to customers as Education Loan
x9 =The amount to be lent by the bank to customers as Loan Against
Property
x10 =The amount to be lent by the bank to customers as Gold Loan
x11 =The amount to be lent by the bank to customers as Loan against FD

8. D EV IA TIO N V A RIA B LES
diN = Amount the right-hand side of goal i is deficient
diP = Amount the right-hand side of goal i is exceeded
Each bank investor invested at most 10 Lakhs
There are 100 similar investors in and around that region who are investing at
most 10 Lakhs similarly.
Therefore, total amount invested by 100 investors = 100 * 10 Lakhs = 10 Crores
Total amount available with Bank to lend as loans = Amount received from
investors + Amount received from RBI = 10 crores + 150 crores = 160 crores

9. FUNC TIO NA L C O NSTRA INTS
Amount invested in savings account: y1>= 2
Amount invested in Fixed Deposit: y2>= 2.5
Amount invested in PPF and SCSS (Minimum Limit): y3+y4>= 1
Amount invested in PPF (Maximum Limit): y3<= 2.5
Amount invested in SCSS (Maximum Limit): y4<= 2.5
Amount allocated to Home Loan, Personal Loan and Education Loan: x1+x2+x8>=40
Amount allocated to Loan against property, Gold Loan, 24x7 loan against securities, Loan against
FD: x9+x10+x6<=32
Amount allocated to Car Loan, two-wheelers Loans: x4+x5<=24
Amount allocated to Business Loan, Commercial Vehicle Construction Equipment Loan: x3+x7>=32
Amount allocated to Business loan: x3<=16
Amount allocated to Education loan: x8>=16
Amount allocated to Home Loan: x1>= 8
Amount allocated to Personal Loan: x2>= 8
Amount allocated to Loan against property, Gold loan, Loan Against FD: x9+x10+x11>= 5.6
Amount allocated to each Car loan, Two-wheelers Loans: x4>=12, x5>=12
Amount allocated to each of 24x7 loan against securities, Construction vehicle Construction
Equipment Loan, Gold Loan, Loan against property, Loan against FD: x6>=4, x7>=4, x9>=4,
x10>=4, x11>=4
Non-negativity constraints: yi, xi, di >=0, for all i

10. GO A LS
Goal 1: The total amount
invested by each investor is at
most 10 Lakhs:
y1+y2+y3+y4<=10
y1+y2+y3+y4+d1N-d1P =
10
Goal 2: The returns of the investors should be at
least 20% of the amount all the investors have invested:
0.035y1 +0.0715y2 +0.08y3 +0.086y4>=2
0.035y1 + 0.0715y2 + 0.08y3 + 0.086y4 +d2N -d2P = 2
Goal 3:The total amount available for the bank to lend as various types of loans should not
exceed the total amount of 10 crores received from 100 investors and the fund of 150 crores
from the RBI:
x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11<=160
X1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11 + d3N- d3P=160
Goal 4: The returns of the bank should be at least 25% of the amount they have lent in the form
of loans to the customers:
0.1025x1+0.1260x2+0.15x3+0.0865x4+0.07x5+0.065x6+0.0915x7+0.08x8+0.1x9+0.1150x10
+0.078x11>= 40
0.1025x1+0.1260x2+0.15x3+0.0865x4+0.07x5+0.065x6+0.0915x7+0.08x8+0.1x9+0.11
50x10+0.078x11 + d4N- d4P

11. OBJECTIVE FUNCTION
Minimize d1P + d2N + d3P +d4N

12. LINGO PROBLEM FORMULATION

13. LINGO SOLUTION

14. LINGO SOLUTION

15. LINGO SOLUTION

16. INTERPRETATION

17. VARIABLE VALUE INTERPRETATION
x1 8 8 cr should be lent by bank to customers as a Home Loan
x2 64 64 cr should be lent by bank to customers as a Personal Loan
x3 16 16 cr should be lent by bank to customers as a Business Loan
x4 12 12cr should be lent by bank to customers as a Car Loan
x5 12 12cr should be lent by bank to customers as a Two Wheeler Loan
x6 4 4 cr should be lent by bank to customers as a Loan Against Security
x7 16
16 cr should be lent by bank to customers as a Commercial Vehicle
Construction Equipment
x8 16 16 cr should be lent by bank to customers as a Educational Loan
x9 4 4 cr should be lent by bank to customers as a Loan Against Property
x10 4 4 cr should be lent by bank to customers as a Gold Loan
INTERPRETATION

18. INTERPRETATION
y1+y2+y3+y4<=10
y1+y2+y3+y4+d1N-d1P = 10
D1P = 0
D1N = 0
Goal 1
Hence, we can conclude that Goal 1 is
completely met and total amount
invested by each investor is 10 Lakhs
Goal 2
0.035y1 + 0.0715y2 + 0.08y3 + 0.086y4>=2
0.035y1 +0.0715y2 + 0.08y3
+ 0.086y4 +d2N -d2P = 2
Values in Lingo solution D2P =0
and D2N =1.30050
R.H.S. = 2 - D2N + D2P = 2 - 1.30050 = 0.6995
Hence, we can conclude that
Goal 2 is underachieved by
1.30050 crores i.e; each
investor will gain a profit of (2-
1.30050) crores/100 = 0.6995
crores/ 100 = Rs. 69,950

19. INTERPRETATION
Goal 3
x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11<=160
X1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11 +d3N- d3P=160
Values in Lingo solution
D3P =0
D3N = 0
Hence, we can conclude that the total amount available for the bank to lend
as various types of loans is exactly equal to 160 crores

20. INTERPRETATION
Goal 4
0.1025x1+0.1260x2+0.15x3+0.0865x4+0.07x5+0.065x6+0.0915x7+0.08x8+0
.1x9+0.1150 x10+0.078 x11>= 40
0.1025x1+0.1260x2+0.15x3+0.0865x4+0.07x5+0.065x6+0.0915x7+0.08x8+0
.1x9+0.1150x10+0.078x11 +d4N- d4P = 40
Values in Lingo solution
D4P =0
D4N = 22.6620
Hence, we can conclude that the Goal 4 is underachieved by 22.6620 crores
i.e; the bank will gain a profit of 40- 22.6620 = 17.338 crores.