<s>
In	O
the	O
field	O
of	O
mathematical	O
optimization	O
,	O
stochastic	B-Algorithm
programming	I-Algorithm
is	O
a	O
framework	O
for	O
modeling	O
optimization	O
problems	O
that	O
involve	O
uncertainty	O
.	O
</s>
<s>
The	O
goal	O
of	O
stochastic	B-Algorithm
programming	I-Algorithm
is	O
to	O
find	O
a	O
decision	O
which	O
both	O
optimizes	O
some	O
criteria	O
chosen	O
by	O
the	O
decision	O
maker	O
,	O
and	O
appropriately	O
accounts	O
for	O
the	O
uncertainty	O
of	O
the	O
problem	O
parameters	O
.	O
</s>
<s>
Because	O
many	O
real-world	O
decisions	O
involve	O
uncertainty	O
,	O
stochastic	B-Algorithm
programming	I-Algorithm
has	O
found	O
applications	O
in	O
a	O
broad	O
range	O
of	O
areas	O
ranging	O
from	O
finance	O
to	O
transportation	O
to	O
energy	O
optimization	O
.	O
</s>
<s>
The	O
basic	O
idea	O
of	O
two-stage	O
stochastic	B-Algorithm
programming	I-Algorithm
is	O
that	O
(	O
optimal	O
)	O
decisions	O
should	O
be	O
based	O
on	O
data	O
available	O
at	O
the	O
time	O
the	O
decisions	O
are	O
made	O
and	O
cannot	O
depend	O
on	O
future	O
observations	O
.	O
</s>
<s>
The	O
two-stage	O
formulation	O
is	O
widely	O
used	O
in	O
stochastic	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
The	O
general	O
formulation	O
of	O
a	O
two-stage	O
stochastic	B-Algorithm
programming	I-Algorithm
problem	O
is	O
given	O
by	O
:	O
</s>
<s>
and	O
,	O
moreover	O
,	O
the	O
two-stage	O
problem	O
can	O
be	O
formulated	O
as	O
one	O
large	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
(	O
this	O
is	O
called	O
the	O
deterministic	O
equivalent	O
of	O
the	O
original	O
problem	O
,	O
see	O
section	O
)	O
.	O
</s>
<s>
Optimizers	O
such	O
as	O
CPLEX	B-Application
,	O
and	O
GLPK	B-Application
can	O
solve	O
large	O
linear/nonlinear	O
problems	O
.	O
</s>
<s>
The	O
structure	O
of	O
a	O
deterministic	O
equivalent	O
is	O
particularly	O
amenable	O
to	O
apply	O
decomposition	O
methods	O
,	O
such	O
as	O
Benders	B-Algorithm
 '	I-Algorithm
decomposition	I-Algorithm
or	O
scenario	O
decomposition	O
;	O
</s>
<s>
A	O
stochastic	O
linear	B-Algorithm
program	I-Algorithm
is	O
a	O
specific	O
instance	O
of	O
the	O
classical	O
two-stage	O
stochastic	O
program	O
.	O
</s>
<s>
A	O
stochastic	O
LP	O
is	O
built	O
from	O
a	O
collection	O
of	O
multi-period	O
linear	B-Algorithm
programs	I-Algorithm
(	O
LPs	O
)	O
,	O
each	O
having	O
the	O
same	O
structure	O
but	O
somewhat	O
different	O
data	O
.	O
</s>
<s>
With	O
a	O
finite	O
number	O
of	O
scenarios	O
,	O
two-stage	O
stochastic	O
linear	B-Algorithm
programs	I-Algorithm
can	O
be	O
modelled	O
as	O
large	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
This	O
formulation	O
is	O
often	O
called	O
the	O
deterministic	O
equivalent	O
linear	B-Algorithm
program	I-Algorithm
,	O
or	O
abbreviated	O
to	O
deterministic	O
equivalent	O
.	O
</s>
<s>
For	O
example	O
,	O
to	O
form	O
the	O
deterministic	O
equivalent	O
to	O
the	O
above	O
stochastic	O
linear	B-Algorithm
program	I-Algorithm
,	O
we	O
assign	O
a	O
probability	O
to	O
each	O
scenario	O
.	O
</s>
<s>
For	O
a	O
given	O
sample	O
the	O
SAA	O
problem	O
is	O
of	O
the	O
same	O
form	O
as	O
a	O
two-stage	O
stochastic	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
with	O
the	O
scenarios	O
.	O
,	O
,	O
each	O
taken	O
with	O
the	O
same	O
probability	O
.	O
</s>
<s>
In	O
the	O
framework	O
of	O
two-stage	O
stochastic	B-Algorithm
programming	I-Algorithm
,	O
is	O
given	O
by	O
the	O
optimal	O
value	O
of	O
the	O
corresponding	O
second-stage	O
problem	O
.	O
</s>
<s>
Stochastic	B-Algorithm
dynamic	I-Algorithm
programming	I-Algorithm
is	O
frequently	O
used	O
to	O
model	O
animal	O
behaviour	O
in	O
such	O
fields	O
as	O
behavioural	O
ecology	O
.	O
</s>
<s>
Stochastic	B-Algorithm
dynamic	I-Algorithm
programming	I-Algorithm
is	O
a	O
useful	O
tool	O
in	O
understanding	O
decision	O
making	O
under	O
uncertainty	O
.	O
</s>
<s>
The	O
following	O
is	O
an	O
example	O
from	O
finance	O
of	O
multi-stage	O
stochastic	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
This	O
is	O
a	O
multistage	O
stochastic	B-Algorithm
programming	I-Algorithm
problem	O
,	O
where	O
stages	O
are	O
numbered	O
from	O
to	O
.	O
</s>
<s>
In	O
order	O
to	O
write	O
dynamic	B-Algorithm
programming	I-Algorithm
equations	O
,	O
consider	O
the	O
above	O
multistage	O
problem	O
backward	O
in	O
time	O
.	O
</s>
<s>
For	O
a	O
general	O
distribution	O
of	O
the	O
process	O
,	O
it	O
may	O
be	O
hard	O
to	O
solve	O
these	O
dynamic	B-Algorithm
programming	I-Algorithm
equations	O
.	O
</s>
<s>
All	O
discrete	O
stochastic	B-Algorithm
programming	I-Algorithm
problems	O
can	O
be	O
represented	O
with	O
any	O
algebraic	B-Application
modeling	I-Application
language	I-Application
,	O
manually	O
implementing	O
explicit	O
or	O
implicit	O
non-anticipativity	O
to	O
make	O
sure	O
the	O
resulting	O
model	O
respects	O
the	O
structure	O
of	O
the	O
information	O
made	O
available	O
at	O
each	O
stage	O
.	O
</s>
<s>
EMP	O
SP	O
(	O
Extended	O
Mathematical	O
Programming	O
for	O
Stochastic	B-Algorithm
Programming	I-Algorithm
)	O
-	O
a	O
module	O
of	O
GAMS	B-Application
created	O
to	O
facilitate	O
stochastic	B-Algorithm
programming	I-Algorithm
(	O
includes	O
keywords	O
for	O
parametric	O
distributions	O
,	O
chance	O
constraints	O
and	O
risk	O
measures	O
such	O
as	O
Value	O
at	O
risk	O
and	O
Expected	B-Algorithm
shortfall	I-Algorithm
)	O
.	O
</s>
