Adjustable Variables

Adjustable variables

An adjustable variable reflects a decision made after uncertain data has been revealed. In robust optimization this is interpreted as the adjustable variable taking some (explicit or implicit) functional form in terms of the uncertain data on which it depends. In AIMMS, you indicate that a Variable should be treated as adjustable by setting its Adjustable property.

The Dependency attribute

For any adjustable variable, AIMMS will create a Dependency attribute which you can use to specify on which uncertain parameters the variable depends. The attribute value must be a comma-separated list of mappings from an uncertain parameter to a binary parameter, indicating for which combination of indices a dependency exists on that uncertain parameter.

Linear decision rule only

AIMMS currently only supports the linear decision rule, which means any adjustable variable will be expressed as an affine relation in terms of the uncertain parameters which it depends on. More explicitly, if an adjustable variable \(x(t)\) depends on uncertain parameters \(d_r\), then, under the linear decision rule, AIMMS assumes that \(x(t)\) takes the form

\[x(t) = X_0(t) + \sum_r X_r(t) d_r\]

where \(X_0(t)\) and \(X_r(t)\) are newly introduced intermediate variables, the value of which is determined by solving the robust counterpart. As such, the value of an adjustable variable is not fully determined by the solver. It can be computed afterwards for a given realization of the uncertain parameters. AIMMS will automatically generate the affine relation based on the dependencies you indicated in the Dependency attribute, without the need for you to introduce the appropriate intermediate variables.

Requirements for adjustable variables

In order for AIMMS to be able to generate the robust counterpart of a robust optimization model, the model must satisfy the fixed recourse condition, i.e., the coefficients of any adjustable variables in your model must not depend on uncertain parameters. In addition, for AIMMS to be able to generate the robust counterpart, adjustable variables may not occur in chance constraints. Also, adjustable variables cannot be integer.

The .Adjustable suffix for variables

The collection of intermediate variables introduced during this process, automatically becomes available through the .Adjustable attribute of the adjustable variable at hand, followed by the name of the uncertain parameter involved. That is, if an adjustable variable x(i) depends on an uncertain parameter a(j), then the corresponding intermediate variable is available as the expression x.Adjustable.a(i,j). In addition, a variable x.Adjustable.Constant(i) will be created to account for the constant part of the affine relation. If necessary, you can bound these variables through the .Lower and .Upper suffices, or you can formulate additional constraints on these variables.

Example

Consider the following declarations

Variable Stock {
    IndexDomain  :  t;
    Property     :  Adjustable;
    Dependency   :  Demand(t2) : StockDemandDependency(t,t2);
}
Parameter Demand {
    IndexDomain  :  t;
    Property     :  Uncertain;
}
Parameter StockDemandDependency {
    IndexDomain  :  (t,t2);
    Definition   :  1 $ (t2 < t);
}

These declarations yield that the adjustable variable Stock(t) depends on the uncertain parameter Demand(t2) for all elements t2 smaller than t. Given these declarations, AIMMS will generate the following definition for Stock(t)

Stock(t) = Stock.Adjustable.Constant(t) +
           sum(t2 | StockDemandDependency(t,t2), Stock.Adjustable.Demand(t,t2)*Demand(t2))

If the data for Demand(t) becomes available, you can use the computed values of Stock.Adjustable.Demand(t,t2) and Stock.Adjustable.Constant to compute the value of Stock(t).

Warning: using same indices

You should be aware that using the same indices in the Dependency attribute and the index domain of the adjustable variable will restrict the dependencies that are generated. For example, assume we have the following declarations

Variable Stock {
    IndexDomain  :  t;
    Property     :  Adjustable;
    Dependency   :  Demand(t);
}
Parameter Demand {
    IndexDomain  :  t;
    Property     :  Uncertain;
}

Given these declarations, AIMMS will generate the following definition for Stock(t)

Stock(t) = Stock.Adjustable.Constant(t) + Stock.Adjustable.Demand(t)*Demand(t)

If you want Stock(t) to depend on all possible Demand then you should use a different index in the Dependency attribute, e.g.,

Variable Stock {
    IndexDomain  :  t;
    Property     :  Adjustable;
    Dependency   :  Demand(t2);
}

Evaluating adjustable variables

To compute the values of an adjustable variable for a given realization of the uncertain parameters of the robust optimization model, you do not have to explicitly add the appropriate definitions to your model. AIMMS offers the function GMP::Robust::EvaluateAdjustableVariables, discussed in Supporting Functions for Robust Optimization Models, to automatically compute these values for you.

Adjustable Variables

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