Horizons

Horizons

A horizon in AIMMS is basically a set of planning periods. The elements in a horizon are divided into three groups, also referred to as time blocks. The main group of elements comprise the planning interval. Periods prior to the planning interval form the past, while periods following the planning interval form the beyond. When variables and constraints are indexed over a horizon, AIMMS automatically restricts the generation of these constraints and variables to periods within the planning interval.

Effect on constraints and assignments

Whenever you use a horizon to construct a time-dependent model, AIMMS has the following features:

• constraints are excluded from the past and beyond periods,

• variables are assumed to be fixed for these periods, and

• assignments and definitions to variables and parameters are, by default, only executed for the periods in the planning interval.

Horizon attributes

Horizons, like calendars, have a number of associated attributes, some of which are inherited from sets. They are summarized in this table.

Table 69 Horizon attributes

Attribute	Value-type	See also page	Mandatory
SubsetOf	subset-domain	The SubsetOf attribute
Index	identifier-list	The Index attribute	yes
Parameter	identifier-list	The Parameter attribute
Text	string	The Text attribute
Comment	comment string	The Comment attribute
Definition	set-expression	The Definition attribute	yes
CurrentPeriod	element		yes
IntervalLength	integer-reference

Horizon attributes explained

The CurrentPeriod attribute denotes the first period of the planning interval. The periods prior to the current period belong to the past. The integer value associated with the attribute IntervalLength determines the number of periods in the planning interval (including the current period). Without an input value, all periods from the current period onwards are part of the planning interval.

Definition is mandatory

AIMMS requires you to specify the contents of a Horizon uniquely through its Definition attribute. The ordering of the periods in the horizon is fully determined by the set expression in its definition. You still have the freedom, however, to specify the Horizon as a subset of another set.

Example

Given a scalar parameter MaxPeriods, the following example illustrates the declaration of a horizon.

Horizon ModelPeriods {
    Index          : h;
    Parameter      : IntervalStart;
    CurrentPeriod  : IntervalStart;
    Definition     : ElementRange( 1, MaxPeriods, prefix: "p-" );
}

If, for instance, the scalar MaxPeriods equals 10, this will result in the creation of a period set containing the elements 'p-01',…,'p-10'. The start of the planning interval is fully determined by the value of the element parameter IntervalStart, which for instance could be equal to 'p-02'. This will result in a planning interval consisting of the periods 'p-02',…,'p-10'.

Example of use

Consider the parameter Demand(h) together with the variables Production(h) and Stock(h). Then the definition of the variable Stock can be declared as follows.

Variable Stock {
    IndexDomain    : h;
    Range          : NonNegative;
    Definition     : Stock(h-1) + Production(h) - Demand(h);
}

When the variable Stock is included in a mathematical program, AIMMS will only generate individual variables with their associated definition for those values of h that correspond to the current period and onwards. The reference Stock(h-1) refers to a fixed value of Stock from the past whenever the index h points to the current period. The values associated with periods from the past (and from the beyond if they were there) are assumed to be fixed.

Accessing past and beyond

To provide easy access to periods in the past and the beyond, AIMMS offers three horizon-specific suffices. They are:

• the Past suffix,

• the Planning suffix, and

• the Beyond suffix.

These suffices provide access to the subsets of the horizon representing the past, the planning interval and the beyond.

Horizon binding rules

When you use a horizon index in an index binding operation (see Index Binding), AIMMS will, by default, perform that operation only for the periods in the planning interval. You can override this default behavior by a local binding using the suffices discussed above.

Example

Consider the horizon ModelPeriods of the previous example. The following assignments illustrate the binding behavior of horizons.

Demand(h)                          := 10; ! only periods in planning interval (default)
Demand(h in ModelPeriods.Planning) := 10; ! only periods in planning interval

Demand(h in ModelPeriods.Past)     := 10; ! only periods in the past
Demand(h in ModelPeriods.Beyond)   := 10; ! only periods in the beyond

Demand(h in ModelPeriods)          := 10; ! all periods in the horizon

Use of lag and lead operators

When you use one of the lag and lead operators +, ++, - or -- (see also Lag and Lead Element Operators) in conjunction with a horizon index, AIMMS will interpret such references with respect to the entire horizon, and not just with respect to the planning period. If the horizon index is locally re-bound to one of the subsets of periods in the Past or Beyond, as illustrated above, the lag or lead operation will be interpreted with respect to the specified subset.

Example

Consider the horizon ModelPeriods of the previous example. The following assignments illustrate the use of lag and lead operators in conjuction with horizons.

Stock(h)                              := Stock(h-1) + Supply(h) - Demand(h);
Stock(h | h in ModelPeriods.Planning) := Stock(h-1) + Supply(h) - Demand(h);

Stock(h in ModelPeriods.Planning)     := Stock(h-1) + Supply(h) - Demand(h);
Stock(h in ModelPeriods.Planning)     := Stock(h--1) + Supply(h) - Demand(h);

The first two assignments are completely equivalent (in fact, the second assignment is precisely the way in which AIMMS interprets the default binding behavior of a horizon index). For the first element in the planning interval, the reference h-1 refers to the last element of the past interval. In the third assignment, h-1 refers to a non-existing element for the first element in the planning interval, completely in accordance with the default semantics of lag and lead operators. In the fourth assignment, h-1 refers to the last element of the planning interval.

Data transfer on entire domain

Operations which can be applied to identifiers without references to their indices (such as the READ, WRITE or DISPLAY statements), operate on the entire horizon domain. Thus, for example, during data transfer with a database, AIMMS will retrieve or store the data for all periods in the horizon, and not just for the periods in the planning interval.