ILOG OPL Development Studio Language Reference Manual

Interval variablesInformally speaking, an interval variable represents an interval of time during which somethinghappens (a task, an activity is carried out) and whose position in time is an unknown of thescheduling problem. An interval is characterized by a start value, an end value and a size. Animportant feature of interval variables is the fact that they can be optional; that is, one candecide not to consider them in the solution schedule. This concept is crucial in applicationsthat present at least some of the following features:♦optional activities (operations, tasks) that can be left unperformed (with an impact on thecost); examples include externalized, maintenance or control tasks♦activities that can execute on a set of alternative resources (machines, manpower) withpossibly different characteristics (speed, calendar) and compatibility constraints♦operations that can be processed in different temporal modes (for instance in series or inparallel)♦alternative modes for executing a given activity, each mode specifying a particularcombination of resources♦alternative processes for executing a given production order, a process being specified asa sequence of operations requiring resources♦hierarchical description of a project as a work-breakdown structure with tasks decomposedinto sub-tasks, part of the project being optional (with an impact on the cost if unperformed),and so forth.Formally, an interval variable a is a variable whose domain dom(a) is a subset ofreduced to a singleton; that is, if. An interval variable is said to be fixed if its domain isdenotes a fixed interval variable:♦interval is absent: = ; or♦interval is present:= [s,e)Absent interval variables have special meaning. Informally speaking, an absent intervalvariable is not considered by any constraint or expression on interval variables it is involvedin. For example, if an absent interval variable is used in a noOverlap constraint, the constraintwill behave as if the interval was never specified to the constraint. If an absent interval variablea is used in a precedence constraint between interval variables a and b this constraint doesnot impact interval variable b. Each constraint specifies how it handles absent interval variables.The semantics of constraints defined over interval variables is described by the properties thatfixed intervals must have in order the constraint to be true. If a fixed interval is present andsuch that = [s, e), we will denote s( ) its integer start value s, e( ) its integer end valuee and l( ) its positive integer length defined as e( )−s( ). The presence status x( ) will148I L O G O P L D E V E L O P M E N T S T U D I O L A N G U A G ER E F E R E N C E M A N U A L