operational research

1. 1
An Integer Programming Problem is an LP in which
some or all of the variables are required to be non-
negative integer
When all variables are required to be integers, it is Pure
Integer Programming Problem
When some of the variables are required to be integers, it is
Mixed Integer Programming Problem
An Integer Programming Problem in which all the variables
must equal 0 or 1 is called 0-1 IP
Without Integer Constraint on variables, it is LP
relaxation of the IP
The feasible region for any IP must be contained in the
feasible region for the corresponding LP relaxation
INTEGER PROGRAMMING (IP)
EM 505 Operations Research
Examples
Pure Integer Programming Problem
Max Z = 3A + 2B
s.t. A + B  6
A, B  0 A,B integer
Mixed Integer Programming Problem
Max Z = 3A + 2B
s.t. A + B  6
A, B  0 A integer
0-1 IP
Max Z = A - B
s.t. A + 2B  2
2A - B  1
A, B = 0 or 1
EM 505 Operations Research

2. 2
Integer Programming
Case-I
Max Z = 21A + 11B
s.t. 7A + 4B  13
A, B  0 A,B integer
Case-II
Max Z = 4A + B
s.t. 2A + B  5
2A +3B = 5
A, B  0 A,B integer
EM 505 Operations Research
Graph of an Integer Programming Problem

3. 3
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FIRE STATION PROBLEM
CDGK intends to build Fire Stations for six towns. CDGK must determine where
to build fire stations.
CDGK wants to build the minimum number of fire stations needed to ensure
that at least one fire station is within 15 minutes of each town.
The time in minutes required to drive between the towns in CDGK are given
here under:
Town1 Town2 Town3 Town4 Town5 Town6
Town1 00 10 20 30 30 20
Town2 10 00 25 35 20 10
Town3 20 25 00 15 30 20
Town4 30 35 15 00 15 25
Town5 30 20 30 15 00 14
Town6 20 10 20 25 14 00
Formulate an IP that will tell how many Fire Stations should be built
and where they should be located.
First we have to ensure that there is a Fire Station within
15 minutes of each town
Town1 1 2
Town2 1 2 6
Town3 3 4
Town4 3 4 5
Town5 4 5 6
Town6 2 5 6
EM 505 Operations Research

4. 4
Let Xi is a Fire station built in Town i. It is 0-1 variable
Min z= X1 + X2 + X3 + X4 + X5 + X6
s.t X1 + X2  1 Town 1 constraint
X1 + X2 + X6  1 Town 2 constraint
X3 + X4  1 Town 3 constraint
X3 + X4 + X5  1 Town 4 constraint
X4 +X5 + X6  1 Town 5 constraint
X2 + X5 + X6  1 Town 6 constraint
Xi= 0 or 1
EM 505 Operations Research
INTEGER PROGRAMMING
FIRE STATION PROBLEM
X1 X2 X3 X4 X5 X6 MINIMIZE Number of Fire Station
0 1 0 1 0 0 2
1 1 1 1 Town 1 constraint
1 1 1 1 1 Town 2 constraint
1 1 1 1 Town 3 constraint
1 1 1 1 1 Town 4 constraint
1 1 1 1 1 Town 5 constraint
1 1 1 1 1 Town 6 constraint
EM 505 Operations Research

5. 5
Branch and Bound Algorithm for Solving
Integer Linear Programming
FURNITURE PROBLEM
A company manufactures tables and chairs.
A table requires 1 hour of labor and 9 square board feet of wood,
and a chair requires 1 hour labor and 5 square feet of wood
Currently, 6 hours of labor and 45 square board feet of wood are
available.
Each table contribute $8 to profit and each chair contribute $5 to
profit.
Formulate and solve an IP to maximize the company’s profit.
EM 505 Operations Research

6. 6
FURNITURE PROBLEM
A Number of tables manufactured
B Number of chairs manufactured
Max Z = 8A + 5B
s.t. A + B  6
9A + 5B  45
A, B  0 A,B integer
LP Relaxation Z=165/4 A=15/4 B=9/4
EM 505 Operations Research
INTEGER PROGRAMMING
FURNITURE PROBLEM
A Number of tables manufactured
B Number of chairs manufactured
A B $
Profit 8 5 40
5 0
Labor 1 1 5 6
Wood 9 5 45 45
EM 505 Operations Research

7. 7
Branch_Bound.pdf
Problem
Max Z = 21x1+11x2
s.t. 7x1+4x2 ≤13
x1 ≥0, x2 ≥0
x1 ,x2 are integers

8. 8