optimal solution would have been to produce
570 compacts and 1,715 midsize cars.
1
2
3
1
2
3
6,000
0,
2,000,
0
0,
1,
0
Z
x
x
x
y
y
y

69
Suppose we want to ensure that
> 0
implies
. Then we include the
following constraint in the formulation:
Here,
M
is a large positive number, chosen large enough
so that
f
<
M
and –
g
<
M
hold for all values of
that satisfy the other constraints in the problem.
If f > 0, y must be zero, g >= 0.
This is called an if-then constraint.
x
x
x
n
f
,...,
,
2
1
0
,...,
,
2
1
x
x
x
n
g
My
g
x
x
x
n
,...,
,
2
1
y
M
f
x
x
x
n
1
,...,
,
2
1
1
0
or
y
If-then constraint

70
1.3 The Branch-and-Bound Method for
Solving Pure Integer Programming Problems
In practice, most IPs are solved by some
versions of the branch-and-bound
procedure.
Branch-and-bound methods implicitly
(cleverly) enumerate all possible solutions to
an IP.
By solving a single subproblem, many possible
solutions may be eliminated from
consideration.
Subproblems are generated by branching on an
appropriately chosen fractional-valued variable.

71
An important observation
If you solve the LP relaxation of a pure IP and
obtain a solution in which
all variables are
integers
, then the optimal solution to the LP
relaxation is also the optimal solution to the IP.
For example, the following IP:
1
2
1
2
1
2
max
3
2
. .2
6
,
0,int
Z
x
x
s t
x
x
x
x
eger

72
Cont’d
Then the LP relaxation is
1
2
1
2
1
2
max
3
2
. .2
6
,
0
Z
x
x
s t
x
x
x
x
The optimal solution to the LP relaxation is x
1
=0, x
2
=6, z=12.
Because this solution gives integer values to all variables, the
preceding observation implies that this solution is also the op-
timal solution to the IP.

73
Cont’d
Observe that the feasible region for the IP is a subset of
the points in the LP relaxation’s feasible region. Thus,
the optimal z-value for the IP cannot be larger than the
optimal z-value for the LP relaxation.
This means that the optimal z-value for the IP must be
≤12.
But the point x
1
=0,x
2
=6 is also feasible for the IP.
This implies that (x
1
=0,x
2
=6) must be optimal for the
IP.

74
An example
The Telfa Corporation manufactures tables and
chairs.
A table requires 1 hour of labor and 9 square
board feet of wood, and a chair requires 1 hour of
labor and 5 square board feet of wood.
Currently, 6 hours of labor and 45 square board
feet of wood are available.
Each table contributes $8 to profit, and each chair
contributes $5 to profit.
Formulate and solve an IP to maximize Telfa’s
profit.

75
Cont’d
Let x
1
:=number of tables manufactured
x
2
:=number of chairs manufactured.
The IP model is
1
2
1
2
1
2
1
2
max
8
5
. .
6
9
5
45
,
0;int e
Z
x
x
s t x
x
x
x
x
x
ger

76
Cont’d
The branch-and-bound method begins by solving the LP
relaxation of the IP. If all the decision variables assume
integer values in the optimal solution to the LP
relaxation, then the optimal solution to the LP
relaxation will also be the optimal solution to the IP.