<s>
Job-shop	B-Algorithm
scheduling	I-Algorithm
,	O
the	O
job-shop	B-Algorithm
problem	I-Algorithm
(	O
JSP	O
)	O
or	O
job-shop	B-Algorithm
scheduling	I-Algorithm
problem	O
(	O
JSSP	O
)	O
is	O
an	O
optimization	O
problem	O
in	O
computer	B-General_Concept
science	I-General_Concept
and	O
operations	O
research	O
.	O
</s>
<s>
It	O
is	O
a	O
variant	O
of	O
optimal	B-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
.	O
</s>
<s>
In	O
a	O
general	O
job	O
scheduling	B-Application
problem	O
,	O
we	O
are	O
given	O
n	O
jobs	O
J1	O
,	O
J2	O
,...,	O
Jn	O
of	O
varying	O
processing	O
times	O
,	O
which	O
need	O
to	O
be	O
scheduled	O
on	O
m	O
machines	O
with	O
varying	O
processing	O
power	O
,	O
while	O
trying	O
to	O
minimize	O
the	O
makespan	B-Algorithm
–	O
the	O
total	O
length	O
of	O
the	O
schedule	O
(	O
that	O
is	O
,	O
when	O
all	O
the	O
jobs	O
have	O
finished	O
processing	O
)	O
.	O
</s>
<s>
In	O
the	O
specific	O
variant	O
known	O
as	O
job-shop	B-Algorithm
scheduling	I-Algorithm
,	O
each	O
job	O
consists	O
of	O
a	O
set	O
of	O
operations	O
O1	O
,	O
O2	O
,...,	O
On	O
which	O
need	O
to	O
be	O
processed	O
in	O
a	O
specific	O
order	O
(	O
known	O
as	O
precedence	O
constraints	O
)	O
.	O
</s>
<s>
The	O
name	O
originally	O
came	O
from	O
the	O
scheduling	B-Application
of	O
jobs	O
in	O
a	O
job	O
shop	O
,	O
but	O
the	O
theme	O
has	O
wide	O
applications	O
beyond	O
that	O
type	O
of	O
instance	O
.	O
</s>
<s>
This	O
problem	O
is	O
one	O
of	O
the	O
best	O
known	O
combinatorial	O
optimization	O
problems	O
,	O
and	O
was	O
the	O
first	O
problem	O
for	O
which	O
competitive	B-Algorithm
analysis	I-Algorithm
was	O
presented	O
,	O
by	O
Graham	O
in	O
1966	O
.	O
</s>
<s>
Best	O
problem	O
instances	O
for	O
basic	O
model	O
with	O
makespan	B-Algorithm
objective	O
are	O
due	O
to	O
Taillard	O
.	O
</s>
<s>
In	O
the	O
standard	O
three-field	B-Algorithm
notation	I-Algorithm
for	I-Algorithm
optimal	I-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
problems	I-Algorithm
,	O
the	O
job-shop	O
variant	O
is	O
denoted	O
by	O
J	O
in	O
the	O
first	O
field	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
problem	O
denoted	O
by	O
"	O
J3||	O
"	O
is	O
a	O
3-machines	O
job-shop	B-Algorithm
problem	I-Algorithm
with	O
unit	O
processing	O
times	O
,	O
where	O
the	O
goal	O
is	O
to	O
minimize	O
the	O
maximum	O
completion	O
time	O
.	O
</s>
<s>
Objective	O
function	O
can	O
be	O
to	O
minimize	O
the	O
makespan	B-Algorithm
,	O
the	O
Lp	O
norm	O
,	O
tardiness	O
,	O
maximum	O
lateness	O
etc	O
.	O
</s>
<s>
Jobs	O
may	O
have	O
constraints	O
,	O
for	O
example	O
a	O
job	O
i	O
needs	O
to	O
finish	O
before	O
job	O
j	O
can	O
be	O
started	O
(	O
see	O
workflow	B-Operating_System
)	O
.	O
</s>
<s>
Since	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
is	O
NP-hard	O
,	O
the	O
job-shop	B-Algorithm
problem	I-Algorithm
with	O
sequence-dependent	O
setup	O
is	O
clearly	O
also	O
NP-hard	O
since	O
the	O
TSP	O
is	O
a	O
special	O
case	O
of	O
the	O
JSP	O
with	O
a	O
single	O
job	O
(	O
the	O
cities	O
are	O
the	O
machines	O
and	O
the	O
salesman	O
is	O
the	O
job	O
)	O
.	O
</s>
<s>
The	O
disjunctive	O
graph	O
is	O
one	O
of	O
the	O
popular	O
models	O
used	O
for	O
describing	O
the	O
job-shop	B-Algorithm
scheduling	I-Algorithm
problem	O
instances	O
.	O
</s>
<s>
The	O
job-shop	B-Algorithm
problem	I-Algorithm
is	O
to	O
find	O
an	O
assignment	O
of	O
jobs	O
such	O
that	O
is	O
a	O
minimum	O
,	O
that	O
is	O
,	O
there	O
is	O
no	O
such	O
that	O
.	O
</s>
<s>
Scheduling	B-Application
efficiency	O
can	O
be	O
defined	O
for	O
a	O
schedule	O
through	O
the	O
ratio	O
of	O
total	O
machine	O
idle	O
time	O
to	O
the	O
total	O
processing	O
time	O
as	O
below	O
:	O
</s>
<s>
Here	O
is	O
the	O
idle	O
time	O
of	O
machine	O
,	O
is	O
the	O
makespan	B-Algorithm
and	O
is	O
the	O
number	O
of	O
machines	O
.	O
</s>
<s>
Notice	O
that	O
with	O
the	O
above	O
definition	O
,	O
scheduling	B-Application
efficiency	O
is	O
simply	O
the	O
makespan	B-Algorithm
normalized	O
to	O
the	O
number	O
of	O
machines	O
and	O
the	O
total	O
processing	O
time	O
.	O
</s>
<s>
In	O
fact	O
,	O
it	O
is	O
quite	O
simple	O
to	O
concoct	O
examples	O
of	O
such	O
by	O
ensuring	O
that	O
two	O
machines	O
will	O
deadlock	B-Operating_System
,	O
so	O
that	O
each	O
waits	O
for	O
the	O
output	O
of	O
the	O
other	O
's	O
next	O
step	O
.	O
</s>
<s>
Graham	O
had	O
already	O
provided	O
the	O
List	O
scheduling	B-Application
algorithm	O
in	O
1966	O
,	O
which	O
is	O
-competitive	O
,	O
where	O
m	O
is	O
the	O
number	O
of	O
machines	O
.	O
</s>
<s>
Also	O
,	O
it	O
was	O
proved	O
that	O
List	O
scheduling	B-Application
is	O
optimum	O
online	O
algorithm	O
for	O
2	O
and	O
3	O
machines	O
.	O
</s>
<s>
The	O
Coffman	B-Operating_System
–	I-Operating_System
Graham	I-Operating_System
algorithm	I-Operating_System
(	O
1972	O
)	O
for	O
uniform-length	O
jobs	O
is	O
also	O
optimum	O
for	O
two	O
machines	O
,	O
and	O
is	O
-competitive	O
.	O
</s>
<s>
Currently	O
,	O
the	O
best	O
known	O
result	O
is	O
an	O
algorithm	O
given	O
by	O
Fleischer	O
and	O
Wahl	O
,	O
which	O
achieves	O
a	O
competitive	B-Algorithm
ratio	I-Algorithm
of	O
1.9201	O
.	O
</s>
<s>
Taillard	O
instances	O
has	O
an	O
important	O
role	O
in	O
developing	O
job-shop	B-Algorithm
scheduling	I-Algorithm
with	O
makespan	B-Algorithm
objective	O
.	O
</s>
<s>
provided	O
optimal	O
algorithms	O
for	O
online	O
scheduling	B-Application
on	O
two	O
related	O
machines	O
improving	O
previous	O
results	O
.	O
</s>
<s>
The	O
simplest	O
form	O
of	O
the	O
offline	O
makespan	B-Algorithm
minimisation	O
problem	O
deals	O
with	O
atomic	O
jobs	O
,	O
that	O
is	O
,	O
jobs	O
that	O
are	O
not	O
subdivided	O
into	O
multiple	O
operations	O
.	O
</s>
<s>
Dorit	O
S	O
.	O
Hochbaum	O
and	O
David	O
Shmoys	O
presented	O
a	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
in	O
1987	O
that	O
finds	O
an	O
approximate	O
solution	O
to	O
the	O
offline	O
makespan	B-Algorithm
minimisation	O
problem	O
with	O
atomic	O
jobs	O
to	O
any	O
desired	O
degree	O
of	O
accuracy	O
.	O
</s>
<s>
The	O
basic	O
form	O
of	O
the	O
problem	O
of	O
scheduling	B-Application
jobs	O
with	O
multiple	O
(	O
M	O
)	O
operations	O
,	O
over	O
M	O
machines	O
,	O
such	O
that	O
all	O
of	O
the	O
first	O
operations	O
must	O
be	O
done	O
on	O
the	O
first	O
machine	O
,	O
all	O
of	O
the	O
second	O
operations	O
on	O
the	O
second	O
,	O
etc.	O
,	O
and	O
a	O
single	O
job	O
cannot	O
be	O
performed	O
in	O
parallel	O
,	O
is	O
known	O
as	O
the	O
flow-shop	B-Algorithm
scheduling	I-Algorithm
problem	O
.	O
</s>
<s>
Various	O
algorithms	O
exist	O
,	O
including	O
genetic	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
By	O
doing	O
so	O
,	O
we	O
have	O
reduced	O
the	O
m-Machine	O
problem	O
into	O
a	O
Two	O
Machining	O
center	O
scheduling	B-Application
problem	O
.	O
</s>
<s>
Machine	O
learning	O
has	O
been	O
recently	O
used	O
to	O
predict	O
the	O
optimal	O
makespan	B-Algorithm
of	O
a	O
JSP	O
instance	O
without	O
actually	O
producing	O
the	O
optimal	O
schedule	O
.	O
</s>
<s>
Preliminary	O
results	O
show	O
an	O
accuracy	O
of	O
around	O
80%	O
when	O
supervised	O
machine	O
learning	O
methods	O
were	O
applied	O
to	O
classify	O
small	O
randomly	O
generated	O
JSP	O
instances	O
based	O
on	O
their	O
optimal	O
scheduling	B-Application
efficiency	O
compared	O
to	O
the	O
average	O
.	O
</s>
<s>
Here	O
is	O
an	O
example	O
of	O
a	O
job-shop	B-Algorithm
scheduling	I-Algorithm
problem	O
formulated	O
in	O
AMPL	B-Language
as	O
a	O
mixed-integer	O
programming	O
problem	O
with	O
indicator	O
constraints	O
:	O
</s>
<s>
Flow-shop	B-Algorithm
scheduling	I-Algorithm
is	O
a	O
similar	O
problem	O
but	O
without	O
the	O
constraint	O
that	O
each	O
operation	O
must	O
be	O
done	O
on	O
a	O
specific	O
machine	O
(	O
only	O
the	O
order	O
constraint	O
is	O
kept	O
)	O
.	O
</s>
<s>
Open-shop	B-Algorithm
scheduling	I-Algorithm
is	O
a	O
similar	O
problem	O
but	O
also	O
without	O
the	O
order	O
constraint	O
.	O
</s>
