<s>
In	O
mathematical	O
optimization	O
,	O
linear-fractional	B-Algorithm
programming	I-Algorithm
(	O
LFP	O
)	O
is	O
a	O
generalization	O
of	O
linear	B-Algorithm
programming	I-Algorithm
(	O
LP	O
)	O
.	O
</s>
<s>
Whereas	O
the	O
objective	O
function	O
in	O
a	O
linear	B-Algorithm
program	I-Algorithm
is	O
a	O
linear	B-Algorithm
function	I-Algorithm
,	O
the	O
objective	O
function	O
in	O
a	O
linear-fractional	O
program	O
is	O
a	O
ratio	O
of	O
two	O
linear	O
functions	O
.	O
</s>
<s>
A	O
linear	B-Algorithm
program	I-Algorithm
can	O
be	O
regarded	O
as	O
a	O
special	O
case	O
of	O
a	O
linear-fractional	O
program	O
in	O
which	O
the	O
denominator	O
is	O
the	O
constant	O
function	O
1	O
.	O
</s>
<s>
Formally	O
,	O
a	O
linear-fractional	O
program	O
is	O
defined	O
as	O
the	O
problem	O
of	O
maximizing	O
(	O
or	O
minimizing	O
)	O
a	O
ratio	O
of	O
affine	B-Algorithm
functions	I-Algorithm
over	O
a	O
polyhedron	O
,	O
</s>
<s>
Both	O
linear	B-Algorithm
programming	I-Algorithm
and	O
linear-fractional	B-Algorithm
programming	I-Algorithm
represent	O
optimization	O
problems	O
using	O
linear	O
equations	O
and	O
linear	O
inequalities	O
,	O
which	O
for	O
each	O
problem-instance	O
define	O
a	O
feasible	O
set	O
.	O
</s>
<s>
Fractional	O
linear	B-Algorithm
programs	I-Algorithm
have	O
a	O
richer	O
set	O
of	O
objective	O
functions	O
.	O
</s>
<s>
Informally	O
,	O
linear	B-Algorithm
programming	I-Algorithm
computes	O
a	O
policy	O
delivering	O
the	O
best	O
outcome	O
,	O
such	O
as	O
maximum	O
profit	O
or	O
lowest	O
cost	O
.	O
</s>
<s>
In	O
contrast	O
,	O
a	O
linear-fractional	B-Algorithm
programming	I-Algorithm
is	O
used	O
to	O
achieve	O
the	O
highest	O
ratio	O
of	O
outcome	O
to	O
cost	O
,	O
the	O
ratio	O
representing	O
the	O
highest	O
efficiency	O
.	O
</s>
<s>
Any	O
linear-fractional	O
program	O
can	O
be	O
transformed	O
into	O
a	O
linear	B-Algorithm
program	I-Algorithm
,	O
assuming	O
that	O
the	O
feasible	O
region	O
is	O
non-empty	O
and	O
bounded	O
,	O
using	O
the	O
Charnes-Cooper	O
transformation	O
.	O
</s>
<s>
Formally	O
,	O
the	O
linear	B-Algorithm
program	I-Algorithm
obtained	O
via	O
the	O
Charnes-Cooper	O
transformation	O
uses	O
the	O
transformed	O
variables	O
and	O
:	O
</s>
<s>
Let	O
the	O
dual	B-Algorithm
variables	I-Algorithm
associated	O
with	O
the	O
constraints	O
and	O
be	O
denoted	O
by	O
and	O
,	O
respectively	O
.	O
</s>
<s>
which	O
is	O
an	O
LP	O
and	O
which	O
coincides	O
with	O
the	O
dual	O
of	O
the	O
equivalent	O
linear	B-Algorithm
program	I-Algorithm
resulting	O
from	O
the	O
Charnes-Cooper	O
transformation	O
.	O
</s>
<s>
Since	O
an	O
LFP	O
can	O
be	O
transformed	O
to	O
an	O
LP	O
,	O
it	O
can	O
be	O
solved	O
using	O
any	O
LP	O
solution	O
method	O
,	O
such	O
as	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
(	O
of	O
George	O
B	O
.	O
Dantzig	O
)	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
,	O
or	O
interior-point	B-Algorithm
methods	I-Algorithm
.	O
</s>
