<s>
In	O
applied	O
mathematics	O
,	O
branch	B-Algorithm
and	I-Algorithm
price	I-Algorithm
is	O
a	O
method	O
of	O
combinatorial	O
optimization	O
for	O
solving	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
(	O
ILP	O
)	O
and	O
mixed	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
(	O
MILP	O
)	O
problems	O
with	O
many	O
variables	O
.	O
</s>
<s>
The	O
method	O
is	O
a	O
hybrid	O
of	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
and	O
column	B-Algorithm
generation	I-Algorithm
methods	O
.	O
</s>
<s>
Branch	B-Algorithm
and	I-Algorithm
price	I-Algorithm
is	O
a	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
method	O
in	O
which	O
at	O
each	O
node	O
of	O
the	O
search	O
tree	O
,	O
columns	O
may	O
be	O
added	O
to	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
(	O
LP	B-Algorithm
relaxation	I-Algorithm
)	O
.	O
</s>
<s>
At	O
the	O
start	O
of	O
the	O
algorithm	O
,	O
sets	O
of	O
columns	O
are	O
excluded	O
from	O
the	O
LP	B-Algorithm
relaxation	I-Algorithm
in	O
order	O
to	O
reduce	O
the	O
computational	O
and	O
memory	O
requirements	O
and	O
then	O
columns	O
are	O
added	O
back	O
to	O
the	O
LP	B-Algorithm
relaxation	I-Algorithm
as	O
needed	O
.	O
</s>
<s>
The	O
algorithm	O
typically	O
begins	O
by	O
using	O
a	O
reformulation	O
,	O
such	O
as	O
Dantzig	B-Algorithm
–	I-Algorithm
Wolfe	I-Algorithm
decomposition	I-Algorithm
,	O
to	O
form	O
what	O
is	O
known	O
as	O
the	O
Master	O
Problem	O
.	O
</s>
<s>
This	O
involves	O
finding	O
a	O
column	O
that	O
has	O
a	O
negative	O
reduced	B-Algorithm
cost	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
the	O
pricing	O
problem	O
itself	O
may	O
be	O
difficult	O
to	O
solve	O
but	O
since	O
it	O
is	O
not	O
necessary	O
to	O
find	O
the	O
column	O
with	O
the	O
most	O
negative	O
reduced	B-Algorithm
cost	I-Algorithm
,	O
heuristic	O
and	O
local	O
search	O
methods	O
can	O
be	O
used	O
.	O
</s>
<s>
Each	O
time	O
a	O
column	O
is	O
found	O
with	O
negative	O
reduced	B-Algorithm
cost	I-Algorithm
,	O
it	O
is	O
added	O
to	O
the	O
Restricted	O
Master	O
Problem	O
and	O
the	O
relaxation	O
is	O
reoptimized	O
.	O
</s>
<s>
Most	O
branch	B-Algorithm
and	I-Algorithm
price	I-Algorithm
algorithms	O
are	O
problem	O
specific	O
since	O
the	O
problem	O
must	O
be	O
formulated	O
in	O
such	O
a	O
way	O
so	O
that	O
effective	O
branching	O
rules	O
can	O
be	O
formulated	O
and	O
so	O
that	O
the	O
pricing	O
problem	O
is	O
relatively	O
easy	O
to	O
solve	O
.	O
</s>
<s>
If	O
cutting	O
planes	O
are	O
used	O
to	O
tighten	O
LP	B-Algorithm
relaxations	I-Algorithm
within	O
a	O
branch	B-Algorithm
and	I-Algorithm
price	I-Algorithm
algorithm	O
,	O
the	O
method	O
is	O
known	O
as	O
branch	O
price	O
and	O
cut	O
.	O
</s>
<s>
The	O
branch	B-Algorithm
and	I-Algorithm
price	I-Algorithm
method	O
can	O
be	O
used	O
to	O
solve	O
problems	O
in	O
a	O
variety	O
of	O
application	O
areas	O
,	O
including	O
:	O
</s>
<s>
Vehicle	B-Algorithm
routing	I-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
Generalized	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
.	O
</s>
