Variable
Declaration and Attributes
Declaration and attributes
Variables have some additional attributes above those of parameters. These extra attributes are used to steer a solver, or to hold additional information about solution values provided by the solver. The possible attributes of variables are given in this table.
Attribute 
Valuetype 
See also page 


indexdomain 


range 


constantexpression 


unitexpression 


expression 


expression 


expression 



The Set Property attribute, The Parameter Property attribute 




constraintrelatedsensitivityproperty 


string 


comment string 


expression 
The Set Definition attribute, The Parameter Definition attribute 

expression 
Uncertainty Related Properties and Attributes, Stochastic Parameters and Variables 

expression 
Uncertainty Related Properties and Attributes, The Robust Dependency attribute 
Index domain for variables
By specifying the IndexDomain
attribute you can restrict the domain
of a variable in the same way as that of a parameter. For variables,
however, the domain restriction has an additional effect. During the
generation of individual constraints AIMMS will reduce the size of the
generated mathematical program by including only those variables that
satisfy all domain restrictions.
The Range
attribute
The values of the Range
attribute of variables determine the bounds
that are passed on to the solver. In addition, during an assignment, the
Range
attribute restricts the range of allowed values that can be
assigned to a particular interval (as for parameters). The possible
values for the Range
attribute are:
one of the predefined ranges
Real
,Nonnegative
,Nonpositive
,Integer
orBinary
,any one of the interval expressions
[
\(a,b\)]
,[
\(a,b\))
,(
\(a,b\)]
or(
\(a,b\))
, where \(a\) and \(b\) can be a constant number,inf
,inf
, or a parameter reference involving some or all of the indices on the index list of the declared variable,any enumerated integer set expression, e.g.
{
\(a\)..
\(b\)}
with \(a\) and \(b\) as above, oran integer set identifier.
If you specify Real
, Nonnegative
, Nonpositive
, or an
interval expression, AIMMS will interpret the variable as a continuous
variable. If you specify Integer
, Binary
or an integer set
expression, AIMMS will interpret the variable as a binary or integer
variable.
Example
The following example illustrates a simple variable declaration.
Variable Transport {
IndexDomain : (i,j) in Connections;
Range : [ MinTransport(i), Capacity(i,j) ];
}
The declaration of the variable Transport(i,j)
sets its lower bound
equal to MinTransport(i)
and its upper bound to Capacity(i,j)
.
When generating the mathematical program, the variable Transport
will only be generated for those tuples (i
,j
) that lie in the
set Connections
. Note that the specification of the lower bound only
uses a subdomain (i
) of the full index domain of the variable
(i,j
).
The .Lower and .Upper suffices
Besides using the Range
attribute to specify the lower and upper
bounds, you can also use the .Lower and .Upper suffices in
assignment statements to accomplish this task. The .Lower and
.Upper suffices are attached to the name of the variable, and, as a
result, the corresponding bounds are defined for the entire index
domain. This may lead to increased memory usage when variables share
their bounds for slices of the domain. For this reason, you are advised
to use the Range
attribute as much as possible when specifying the
lower and upper bounds.
When allowed
You can only make a bound assignment with either the .Lower or
.Upper suffix when you have not used a parameter reference (or a
nonconstant expression) at the corresponding position in the Range
attribute. Bound assignments via the .Lower and .Upper suffices
must always lie within the range specified in the Range
attribute.
Example
Consider the variable Transport
declared in the previous example.
The following assignment to Transport.Lower(i,j)
is not allowed,
because you have already specified a parameter reference at the
corresponding position in the Range
attribute.
Transport.Lower(i,j) := MinTransport(i) ;
On the other hand, given the following declaration,
Variable Shipment {
IndexDomain : (i,j) in Connections;
Range : Nonnegative;
}
the following assignment is allowed:
Shipment.Lower(i,j) := MinTransport(i);
AIMMS will produce a runtime error message if any value of
MinTransport(i)
is less than zero, because this violates the bound
in the Range
attribute of the variable Shipment
.
The Default
attribute
Variables that have not been initialized, evaluate to a default value
automatically. These default values are also passed as initial values to
the solver. You can specify the default value using the Default
attribute. The value of this attribute must be a constant expression.
If you do not provide a default value, AIMMS will use a default of 0.
The Unit attribute
Providing a unit for every variable and constraint in your model will help you in a number of ways.
AIMMS will help you to check the consistency of all your constraints and assignments in your model, and
AIMMS will use the units to scale the model that is sent to the solver.
Proper scaling of a model will generally result in a more accurate and robust solution process. You can find more information on the definition and use of units to scale mathematical programs in Units of Measurement.
The Definition
attribute
It is not unusual that symbolic constraints in a model are equalities
defining just one variable in terms of others. Under these conditions,
it is preferable to provide the definition of the variable through its
Definition
attribute. As a result, you no longer need to specify
extra constraints for just variable definitions. In the constraint
listing, the constraints associated with a defined variable will be
listed with a generated name consisting of the name of the variable with
an additional suffix _definition
.
Example
The following example defines the total cost of transport, based on unit transport cost and actual transport taking place.
Variable TransportCost {
Definition : sum( (i,j), UnitTransportCost(i,j)*Transport(i,j) );
}
The Priority
, Nonvar
and RelaxStatus
Attributes
The Priority
attribute
With the Priority
attribute you can assign priorities to integer
variables (or continuous variables when using the solver BARON). The
value of this attribute must be an expression using some or all of the
indices in the index domain of the variable, and must be nonnegative and
integer. All variables with priority zero will be considered last by the
branchandbound process of the solver. For variables with a positive
priority value, those with the highest priority value will be considered
first.
The .Priority suffix
Alternatively, you can specify priorities through assignments to the
.Priority suffix. This is only allowed if you have not specified the
Priority
attribute. In both cases, you can use the .Priority
suffix to refer to the priority of a variable in expressions.
Use of priorities
The solution algorithm (i.e. solver) for integer and mixedinteger programs initially solves without the integer restriction, and then adds this restriction one variable at a time according to their priority. By default, all integer variables have equal priority. Some decisions, however, have a natural order in time or space. For example, the decision to build a factory at some site comes before the decision to purchase production capacity for that factory. Obeying this order naturally limits the number of subsequent choices, and could speed up the overall search by the solution algorithm.
The NonvarStatus
attribute
You can use the NonvarStatus
attribute to tell AIMMS which variables
should be considered as parameters during the execution of a SOLVE
statement. The value of the NonvarStatus
attribute must be an
expression in some or all of the indices in the index list of the
variable, allowing you to change the nonvariable status of individual
elements or groups of elements at once.
Positive versus negative values
The sign of the NonvarStatus
value determines whether and how the
variable is passed on to the solver. The following rules apply.
If the value is 0 (the default value), the corresponding individual variable is generated, along with its specified lower and upper bounds.
If the value is negative, the corresponding individual variable is still generated, but its lower and upper bounds are set equal to the current value of the variable.
If the value is positive, the corresponding individual variable is no longer generated but passed as a constant to the solver.
When you specify a negative value, you will still be able to inspect the corresponding reduced cost values. In addition, you can modify the nonvariable status to zero without causing AIMMS to regenerate the model. When you specify a positive value, the size of the mathematical program is kept to a minimum, but any subsequent changes to the nonvariable status will require regeneration of the model constraints.
The .NonVar
suffix
Alternatively, you can change the nonvariable status through assignments
to the .NonVar
suffix. This is only allowed if you have not
specified the NonvarStatus
attribute. In both cases, you can use the
.NonVar
suffix to refer to the variable status of a variable in
expressions.
When to change the nonvariable status
By altering the nonvariable status of variables you are essentially reconfiguring your mathematical program. You could, for instance, reverse the role of an input parameter (declared as a variable with negative nonvariable status) and an output variable in your model to observe what input level is required to obtain a desired output level. Another example of temporary reconfiguration is to solve a smaller version of a mathematical program by first discarding selected variables, and then changing their status back to solve the larger mathematical program using the previous solution as a starting point.
The RelaxStatus
attribute
With the RelaxStatus
attribute you can tell AIMMS to relax the
integer restriction for those tuples in the domain of an integer
variable for which the value of the relax status is nonzero. AIMMS will
generate continuous variables for such tuples instead, i.e. variables
which may assume any real value between their bounds.
The .Relax suffix
Alternatively, you can relax integer variables by making assignments to
the .Relax suffix. This is only allowed if you have not specified
the RelaxStatus
attribute. In both cases, you can use the .Relax
suffix to refer to the relax status of a variable in expressions.
When to relax variables
When solving large mixed integer programs, the solution times may become unacceptably high with an increase in the number of integer variables. You can try to resolve this by relaxing the integer condition of some of the integer variables. For instance, in a multiperiod planning model, an accurate integer solution for the first few periods and an approximating continuous solution for the remaining periods may very well be acceptable, and at the same time reduce solution times drastically.
Effect on mathematical program type
As you will see in Solving Mathematical Programs, there are several types of
mathematical programs. By changing the nonvariable and/or relax status
of variables you may alter the type of your mathematical program. For
instance, if your constraints contains a nonlinear term x*y
, then
changing the nonvariable status of either x
or y
will change it
into a linear term. Eventually, this may result in a nonlinear
mathematical program becoming a linear one. Similarly, changing the
nonvariable or relax status of integer variables may at some point
change a mixed integer program into a linear program.
Variable Properties
Properties of variables
Variables can have one or more of the following properties: NoSave
,
Inline
, SemiContinuous
, ReducedCost
, CoefficientRange
,
ValueRange
, Stochastic
, and Adjustable
. They are described
in the paragraphs below.
Use of PROPERTY
statement
You can also change the properties of a variable during the execution of
your model by calling the PROPERTY
statement. Identifier properties
are changed by adding the property name as a suffix to the identifier
name in a PROPERTY
statement. When the value is set to off
, the
property no longer holds.
The NoSave
property
With the property NoSave
you indicate that you do not want to store
data associated with this variable in a case. This property is
especially suited for those identifiers that are intermediate quantities
in the model, and that are not used anywhere in the graphical enduser
interface.
Inline variables
With the property Inline
you can indicate that AIMMS should
substitute all references to the variable at hand by its defining
expression when generating the constraints of a mathematical program.
Setting this property only makes sense for defined variables, and will
result in a mathematical program with less rows and columns but with a
(possibly) larger number of nonzeros. After the mathematical program has
been solved, AIMMS will compute the level values of all inline variables
by evaluating their definition. However, no sensitivity information will
be available.
If bounds have been specified for an inline variable then these bounds will be ignored when the mathematical program is solved. To enforce the bounds, they should be specified in a constraint (after which the variable can be changed into a free variable).
If the nonvariable status of an inline variable has been set to a nondefault
value, either using the NonvarStatus
attribute or the .NonVar
suffix,
then this status will be ignored when the mathematical program is solved.
Fixing an inline variable to the current value of the variable should be done
using a constraint.
Semicontinuous variables
To any continuous or integer variable you can assign the property
SemiContinuous
. This indicates to the solver that this variable is
either zero, or lies within its specified range. Not all solvers support
semicontinuous variables. In the latter case, AIMMS will automatically
add the necessary constraints to the model.