Integer Prog. 64
IE301 Operations Research I
–
Spring 2015
●
TREE
Subproblem 1
Z=165/4
X1=15/4
X2=9/4
Subproblem 2
Subproblem 3
t=1
x1≥4
x1≤ 3
arc
node
chronological order in which
the problems are solved

9.3 Branch and Bound Method for Solving Pure
Integer Programming Problems
EXAMPLE 9 Branch-and-Bound
Method
●
Every point is included in either
region
●
Feasible regions have no points
in common.
●
x1
x2
1
1
2
2
3
3
4
5
6
IP feasible point
4
5
6
7
8
9
Subproblem 2
Figure 12 Feasible region for subprblems 2 and 3
x1
x2
1
1
2
2
3
3
4
5
6
IP feasible point
Subproblem 3
4
5
6
7
8
9
Optimal LP solution
to subproblem 1
X1=3.75
X2=2.25
Figure 12 Feasible region for subprblems 2 and 3
A
B
C
F
E
D
G
Integer Prog. 65
IE301 Operations Research I
–
Spring 2015

9.3 Branch and Bound Method for Solving Pure
Integer Programming Problems
EXAMPLE 9
Branch-and-Bound
Method
●
We now choose any subproblem
that has not been solved as an LP
●
Choose subproblem 2
Z=41, x1=4,
x2=9/5 (Point C)
●
The subproblem solution to
subproblem 2
did not
yield an all
integer solution.
●
This means we may need to branch
further under subproblem 2.
●
Our accomplishments are
summarized in figure 13
●
Subproblem 1
Z=165/4
X1=15/4
X2=9/4
Subproblem 2
Z=41
X1=4
X2=9/5
Subproblem 3
t=1
t=2
x1≥4
x1≤ 3
Subproblem 2 did not yield an all integer
solution
Figure 13 Telfa subproblems 1 and 2 Solved
Integer Prog. 66
IE301 Operations Research I
–
Spring 2015

9.3 Branch and Bound Method for Solving Pure
Integer Programming Problems
EXAMPLE 9 Branch-and-Bound Method
●
From figure 12 we find that the optimal
solution to the subproblem 3 is
point F: z=39, x1=x2=3
●
Subproblem 3 yield
an all integer
solution
with z-value=39.
●
A solution obtained by solving a
subproblem in which all variables have
integer values is a
candidate solution
.
So, there is no need to go further in
Subproblem 3 branch.
●
Because the candidate solution may be
optimal, we must keep it, until a better
feasible solution to the IP is found.
●
The z-value (z=39) for the candidate
solution is a
lower bound
on the
original IP.
●
●
x1
x2
1
1
2
2
3
3
4
5
6
IP feasible point
Subproblem 3
4
5
6
7
8
9
Optimal LP solution
to subproblem 1
X1=3.75
X2=2.25
Subproblem 2
Figure 12 Feasible region for subprblems 2 and 3
A
B
C
F
E
D
G
Integer Prog. 67
IE301 Operations Research I
–
Spring 2015

9.3 Branch and Bound Method for Solving Pure
Integer Programming Problems
EXAMPLE 9 Branch-and-Bound Method
●
The subproblem solution to subproblem 2 did not yield an all integer
solution.
●
But subproblem 2 still has the potential to give an integer solution
better than the current lower bound=39 (found by Subproblem 3)
●
So we choose to use subproblem 2 to create two new subproblems.
●
We choose fractional valued variable in the optimal solution to
subproblem 2 and then branch on that variable.