Stochastic Parameters and Variables
The set AllStochasticScenarios
To allow the storage of scenario-dependent parameter and variable data
for multiple scenarios in a stochastic model, all such scenarios should
be added to the predefined set AllStochasticScenarios. If your
application contains multiple stochastic models-each with different
scenario sets-the set AllStochasticScenarios should be the union of
all these scenario sets. For each stochastic model you can then define
an associated subset of AllStochasticScenarios to use with that
particular stochastic model.
Stochastic parameters
Stochastic events are modeled in AIMMS as numeric Parameters for
which the Stochastic property has been set (see also
Parameter Declaration and Attributes). For stochastic parameters AIMMS provides an
additional .Stochastic suffix, which you can use to store
scenario-dependent stochastic event outcomes. The data stored in the
suffix is used by AIMMS when generating the stochastic model. The index
domain of the .Stochastic suffix is, therefore, the set
AllStochasticScenarios plus the original domain of the parameter.
Example
Consider the following declarations
Set MyScenarios {
SubsetOf : AllStochasticScenarios;
Index : sc;
}
Parameter Demand {
IndexDomain : (c,t);
Property : Stochastic;
}
These declarations will cause AIMMS to create a .Stochastic suffix
for the parameter Demand(c,t). To use, or assign values to,
Demand.Stochastic, you must use an additional index into (a subset
of) AllStochasticScenarios. The following statement provides an
example of such a statement.
Demand.Stochastic(sc,c,t) := Uniform(10,20);
If a constraint contains a reference to the parameter Demand, AIMMS
will use the data in Demand.Stochastic to generate the appropriate
demand constraint for every scenario.
Stochastic variables
By setting the Stochastic property for a Variable in your model,
you indicate to AIMMS that this variable may have multiple,
scenario-dependent, solutions when used in a stochastic model.
Consequently, when generating a matrix for the stochastic model, a
column will be generated conceptually for every single scenario.
The Stage attribute
For stochastic variables you must also specify the mandatory Stage
attribute. Through the Stage attribute you specify the stage at the
end of which the decision corresponding to the stochastic variable is to
be taken. The value of the Stage attribute must be an explicit
positive integer value, or a parameter reference involving some or all
of the indices on the index list of the declared variable.
Non-anticipativity constraints…
As discussed in the previous section, for every scenario \(s_0\), a
stochastic variable \(x\) gets its value \(x_{s_0}\) at the end
of stage \(n\) as specified in the Stage attribute of the
variable. In addition, its value is based on the specific outcomes of
the stochastic events for that scenario taking place during stages
\(1,\dots,n\), but only on the distribution of the stochastic event
outcomes for any further stages. Therefore, the value \(x_s\) must
be equal to \(x_{s_0}\) for every other scenario \(s\) that
passes through the same node in the scenario tree at the end of stage
\(n\) as \(s_0\). The constraints enforcing this equality are
called non-anticipativity constraints-they do not allow the solution
to anticipate on stochastic outcomes that lie beyond the stage as
specified by the Stage suffix.
…enforced explicitly or implicitly
When generating a stochastic model, AIMMS will automatically enforce the non-anticipativity constraints, either by explicitly adding them to the generated matrix, or implicitly by substituting a single representative \(x_{s_0}\) for every other variable \(x_s\). While enforcing non-anticipativity in an implicit manner will drastically reduce the matrix size, an explicit representation may be helpful for solvers able to decompose the generated matrix.
Non-stochastic variables
If a variable in a stochastic model has not been declared stochastic, it is deterministic in the sense that it assumes the same value for every scenario, as is the case with first stage variables.
The .Stochastic suffix for variables
Variables can also have a .Stochastic suffix in AIMMS. It follows the same rules for its index domain as the .Stochastic suffix of parameters. AIMMS uses the .Stochastic suffix of variables to store the solution data of a stochastic model after solving it. However, contrary to stochastic parameters, AIMMS will not only create the .Stochastic suffix for stochastic variables, but for all variables that are involved in a stochastic model.
Contents of .Stochastic suffix
The values stored in the .Stochastic suffix after solving a stochastic model for each type of variable are as follows:
for stochastic variables, the .Stochastic suffix will contain the solution of the variable for each scenario,
for the objective variable, the .Stochastic suffix will contain the contribution to the objective of each scenario, as well as the weighted objective value of the stochastic model itself,
for any other non-stochastic variable, the .Stochastic suffix will contain the deterministic solution of that variable for the stochastic model.
As the solution of a stochastic model is entirely stored in the .Stochastic suffix, the solution of the underlying deterministic model remains completely intact after solving the stochastic model. This makes it easy to visually, and/or programmatically, compare the solutions of the deterministic and stochastic model.
Non-stochastic solution data
As the objective value and solution of the non-stochastic variables of
the stochastic model cannot be coupled directly with one specific
scenario in the scenario set, AIMMS creates an extra element in the set
AllStochasticScenarios for this purpose. You must specify the name
of this element when solving the stochastic model (see also
Solving Stochastic Models).