Sets and Indices

General

Sets and indices give your AIMMS model dimension and depth by providing a mechanism for grouping parameters, variables, and constraints. Sets and indices are also used as driving mechanism in arithmetic operations such as summation. The use of sets for indexing expressions helps to describe large models in a concise and understandable way.

Example

Consider a set of Cities and an identifier called Transport defined between several pairs of cities \((i,j)\), representing the amount of product transported from supply city \(i\) to destination city \(j\). Suppose that you are interested in the quantities arriving in each city. Rather than adding many individual terms, the following mathematical notation, using sets and indices, concisely describes the desired computation of these quantities.

\[(\forall j \in Cities) \qquad Arrival_j = \sum_{i \in Cities} Transport_{ij}.\]

This multidimensional index notation forms the foundation of the AIMMS modeling language, and can be used in all expressions. In this example, i and j are indices that refer to individual Cities.

Several types of sets

A set in AIMMS

• has either strings or integers as elements,

• is either a simple set, or a relation, and

• is either indexed or not indexed.

String versus integer

Sets can either have strings as elements (such as the set Cities discussed above), or have integers as elements. An example of an integer set could be a set of Trials represented by the numbers \(1,\dots,n\). The resulting integer set can then be used to refer to the results of each single experiment.

Simple versus relation

A simple set is a one-dimensional set, such as the set Cities mentioned above, while a relation or multidimensional set is the Cartesian product of a number of simple sets or a subset thereof. An example of a relation is the set of possible Routes between supply and destination cities, which can be represented as a subset of the Cartesian product Cities \(\times\) Cities.

Indexing as basic mechanism

Sets in AIMMS are the basis for creating multidimensional identifiers in your model. Through indices into sets you have access to individual values of these identifiers for each tuple of elements. In addition, the indexing notation in AIMMS is your basic mechanism for expressing iterative operations such as repeated addition, repeated multiplication, sequential search for a maximum or minimum, etc.

Indexed sets

Simple sets may be indexed. An indexed set is a family of sets defined for every element in the index domain of the indexed set. An example of an indexed set is the set of transport destination cities defined for each supply city. On the other hand, the set Cities discussed above is not an indexed set.

Sorting of sets

The contents of any simple can be sorted in AIMMS. Sorting can take place either automatically or manually. Automatic sorting is based on the value of some expression defined for all elements of the set. By using an index into a sorted subset, you can access any subselection of data in the specified order. Such a subselection may be of interest in your end-user interface or at a certain stage in your model.