<s>
The	O
quadratic	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
(	O
QAP	O
)	O
is	O
one	O
of	O
the	O
fundamental	O
combinatorial	O
optimization	O
problems	O
in	O
the	O
branch	O
of	O
optimization	O
or	O
operations	O
research	O
in	O
mathematics	O
,	O
from	O
the	O
category	O
of	O
the	O
facilities	O
location	O
problems	O
first	O
introduced	O
by	O
Koopmans	O
and	O
Beckmann	O
.	O
</s>
<s>
The	O
problem	O
statement	O
resembles	O
that	O
of	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
,	O
except	O
that	O
the	O
cost	O
function	O
is	O
expressed	O
in	O
terms	O
of	O
quadratic	O
inequalities	O
,	O
hence	O
the	O
name	O
.	O
</s>
<s>
The	O
formal	O
definition	O
of	O
the	O
quadratic	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
is	O
as	O
follows	O
:	O
</s>
<s>
Given	O
two	O
sets	O
,	O
P	O
(	O
"	O
facilities	O
"	O
)	O
and	O
L	O
(	O
"	O
locations	O
"	O
)	O
,	O
of	O
equal	O
size	O
,	O
together	O
with	O
a	O
weight	B-Algorithm
function	I-Algorithm
w	O
:	O
P	O
P	O
→	O
R	O
and	O
a	O
distance	O
function	O
d	O
:	O
L	O
L	O
→	O
R	O
.	O
Find	O
the	O
bijection	B-Algorithm
f	O
:	O
P	O
→	O
L	O
(	O
"	O
assignment	O
"	O
)	O
such	O
that	O
the	O
cost	O
function	O
:	O
</s>
<s>
Usually	O
weight	O
and	O
distance	O
functions	O
are	O
viewed	O
as	O
square	O
real-valued	O
matrices	B-Architecture
,	O
so	O
that	O
the	O
cost	O
function	O
is	O
written	O
down	O
as	O
:	O
</s>
<s>
where	O
is	O
the	O
set	O
of	O
permutation	O
matrices	B-Architecture
,	O
is	O
the	O
weight	O
matrix	O
and	O
is	O
the	O
distance	O
matrix	O
.	O
</s>
<s>
The	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
(	O
TSP	O
)	O
may	O
be	O
seen	O
as	O
a	O
special	O
case	O
of	O
QAP	O
if	O
one	O
assumes	O
that	O
the	O
flows	O
connect	O
all	O
facilities	O
only	O
along	O
a	O
single	O
ring	O
,	O
all	O
flows	O
have	O
the	O
same	O
non-zero	O
(	O
constant	O
)	O
value	O
and	O
all	O
distances	O
are	O
equal	O
to	O
the	O
respective	O
distances	O
of	O
the	O
TSP	O
instance	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
original	O
plant	O
location	O
formulation	O
,	O
QAP	O
is	O
a	O
mathematical	O
model	O
for	O
the	O
problem	O
of	O
placement	O
of	O
interconnected	O
electronic	O
components	O
onto	O
a	O
printed	O
circuit	O
board	O
or	O
on	O
a	O
microchip	O
,	O
which	O
is	O
part	O
of	O
the	O
place	O
and	O
route	O
stage	O
of	O
computer	B-Application
aided	I-Application
design	I-Application
in	O
the	O
electronics	O
industry	O
.	O
</s>
