<s>
Column	B-Algorithm
generation	I-Algorithm
or	O
delayed	B-Algorithm
column	I-Algorithm
generation	I-Algorithm
is	O
an	O
efficient	O
algorithm	O
for	O
solving	O
large	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
The	O
overarching	O
idea	O
is	O
that	O
many	O
linear	B-Algorithm
programs	I-Algorithm
are	O
too	O
large	O
to	O
consider	O
all	O
the	O
variables	O
explicitly	O
.	O
</s>
<s>
The	O
hope	O
when	O
applying	O
a	O
column	B-Algorithm
generation	I-Algorithm
algorithm	O
is	O
that	O
only	O
a	O
very	O
small	O
fraction	O
of	O
the	O
variables	O
will	O
be	O
generated	O
.	O
</s>
<s>
In	O
many	O
cases	O
,	O
this	O
method	O
allows	O
to	O
solve	O
large	O
linear	B-Algorithm
programs	I-Algorithm
that	O
would	O
otherwise	O
be	O
intractable	O
.	O
</s>
<s>
One	O
particular	O
technique	O
in	O
linear	B-Algorithm
programming	I-Algorithm
which	O
uses	O
this	O
kind	O
of	O
approach	O
is	O
the	O
Dantzig	B-Algorithm
–	I-Algorithm
Wolfe	I-Algorithm
decomposition	I-Algorithm
algorithm	O
.	O
</s>
<s>
Additionally	O
,	O
column	B-Algorithm
generation	I-Algorithm
has	O
been	O
applied	O
to	O
many	O
problems	O
such	O
as	O
crew	O
scheduling	O
,	O
vehicle	B-Algorithm
routing	I-Algorithm
,	O
and	O
the	O
capacitated	O
p-median	O
problem	O
.	O
</s>
<s>
This	O
can	O
be	O
done	O
by	O
finding	O
the	O
variable	O
with	O
the	O
most	O
negative	O
reduced	B-Algorithm
cost	I-Algorithm
(	O
assuming	O
without	O
loss	O
of	O
generality	O
that	O
the	O
problem	O
is	O
a	O
minimization	O
problem	O
)	O
.	O
</s>
<s>
If	O
no	O
variable	O
has	O
a	O
negative	O
reduced	B-Algorithm
cost	I-Algorithm
,	O
then	O
the	O
current	O
solution	O
of	O
the	O
master	O
problem	O
is	O
optimal	O
.	O
</s>
<s>
When	O
the	O
number	O
of	O
variables	O
is	O
very	O
large	O
,	O
it	O
is	O
not	O
possible	O
to	O
find	O
an	O
improving	O
variable	O
by	O
calculating	O
all	O
the	O
reduced	B-Algorithm
cost	I-Algorithm
and	O
choosing	O
a	O
variable	O
with	O
a	O
negative	O
reduced	B-Algorithm
cost	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
the	O
idea	O
is	O
to	O
compute	O
only	O
the	O
variable	O
having	O
the	O
minimum	O
reduced	B-Algorithm
cost	I-Algorithm
.	O
</s>
<s>
The	O
objective	O
function	O
of	O
the	O
subproblem	O
is	O
the	O
reduced	B-Algorithm
cost	I-Algorithm
of	O
the	O
searched	O
variable	O
with	O
respect	O
to	O
the	O
current	O
dual	O
variables	O
,	O
and	O
the	O
constraints	O
require	O
that	O
the	O
variable	O
obeys	O
the	O
naturally	O
occurring	O
constraints	O
.	O
</s>
<s>
The	O
column	B-Algorithm
generation	I-Algorithm
method	O
is	O
particularly	O
efficient	O
when	O
this	O
structure	O
makes	O
it	O
possible	O
to	O
solve	O
the	O
sub-problem	O
with	O
an	O
efficient	O
algorithm	O
,	O
typically	O
a	O
dedicated	O
combinatorial	O
algorithm	O
.	O
</s>
<s>
We	O
now	O
detail	O
how	O
and	O
why	O
to	O
compute	O
the	O
reduced	B-Algorithm
cost	I-Algorithm
of	O
the	O
variables	O
.	O
</s>
<s>
Consider	O
the	O
following	O
linear	B-Algorithm
program	I-Algorithm
in	O
standard	O
form	O
:	O
</s>
<s>
which	O
we	O
will	O
call	O
the	O
primal	O
problem	O
as	O
well	O
as	O
its	O
dual	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
:	O
</s>
<s>
These	O
solutions	O
verify	O
the	O
constraints	O
of	O
their	O
linear	B-Algorithm
program	I-Algorithm
and	O
,	O
by	O
duality	B-Algorithm
,	O
have	O
the	O
same	O
value	O
of	O
objective	O
function	O
(	O
)	O
which	O
we	O
will	O
call	O
.	O
</s>
<s>
If	O
and	O
are	O
respectively	O
the	O
coefficients	O
associated	O
with	O
the	O
variable	O
in	O
the	O
objective	O
function	O
and	O
in	O
the	O
constraints	O
then	O
the	O
linear	B-Algorithm
program	I-Algorithm
is	O
modified	O
as	O
follows	O
:	O
</s>
<s>
The	O
derivative	O
is	O
generally	O
called	O
the	O
reduced	B-Algorithm
cost	I-Algorithm
of	O
the	O
variable	O
and	O
will	O
be	O
denoted	O
by	O
in	O
the	O
following	O
.	O
</s>
