ILOG OPL Development Studio Language Reference Manual

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Logical constraints for CPLEXLogical constraints are one particular kind of discrete or numerical constraints. OPL andCPLEX can translate logical constraints automatically into their transformed equivalent thatthe discrete (MIP) or continuous (LP) optimizers of ILOG CPLEX can process efficiently.This section describes all the available logical constraints, as well as the logical expressionsthat can be used in logical constraints. Logical constraints are available in constraintprogramming models without linearization.For an example of how OPL uses logical constraints, see Tutorial: Using CPLEX logicalconstraints in the Language User’s Manual.In this section, you will learn:♦What are logical constraints?♦What can be extracted from a model with logical constraints?♦Which nonlinear expressions can be extracted?♦Logical constraints for counting♦How are logical constraints extracted?What are logical constraints?For ILOG CPLEX, a logical constraint combines linear constraints by means of logicaloperators, such as logical-and, logical-or, negation (not), conditional statements (if ...then ...) to express complex relations between linear constraints. ILOG CPLEX can alsohandle certain logical expressions appearing within a linear constraint. One such logicalexpression is the minimum of a set of variables. Another such logical expression is the absolutevalue of a variable. There’s more about logical expressions in Which nonlinear expressionscan be extracted?.What can be extracted from a model with logical constraints?Logical constraints extracted by CPLEX lists what logical constraints CPLEX can extract.Logical constraints extracted by CPLEXSymbol&&| |!MeaningLogical ANDLogical ORLogical NOT126I 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
Symbol=>!===MeaningImplyDifferent fromEquivalenceAll those constructs accept as their arguments other linear constraints or logical constraints,so you can combine linear constraints with logical constraints in complicated expressions inyour application.Which nonlinear expressions can be extracted?Some expressions are easily recognized as nonlinear, for example, a function such asx^2 + y^2 = 1However, other nonlinearities are less obvious, such as absolute value as a function. In a veryreal sense, MIP is a class of nonlinearly constrained problems because the integrality restrictiondestroys the property of convexity which any linear constraints otherwise might possess.Because of that characteristic, certain (although not all) nonlinearities are capable of beingconverted to a MIP formulation, and thus can be solved by ILOG CPLEX. The followingnonlinear expressions are accepted in an OPL model:♦min and minl : the minimum of several numeric expressions♦max and maxl : the maximum of several numeric expressions♦abs : the absolute value of a numeric expression♦piecewise : the piecewise linear combination of a numeric expression♦A linear constraint can appear as a term in a logical constraint.In fact, ranges containing logical expressions can, in turn, appear in logical constraints. It isimportant to note here that only linear constraints can appear as arguments of logical constraintsextracted by ILOG CPLEX. That is, quadratic constraints are not handled in logical constraints.Similarly, quadratic terms can not appear as arguments of logical expressions such as min,max, abs, and piecewise.Logical constraints for countingIn many cases it is even unnecessary to allocate binary variables explicitly in order to gainthe benefit of linear constraints within logical expressions. For example, optimizing how manyitems appear in a solution is often an issue in practical problems. Questions of counting (howmany?) can be represented formally as cardinality constraints. Suppose that your applicationincludes three variables, each representing a quantity of one of three products, and assumeI 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 L127
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CopyrightCOPYRIGHT NOTICECopyright
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C O N T E N T STable of contentsLan
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Language Reference ManualThis manua
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OPL, the modeling languagePresents
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ModelsDescribes the overall structu
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}ctMaxTotal:Gas + Chloride
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Data typesDescribes basic data type
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Basic data typesDescribes integers,
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FloatsShows how to declare floats i
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Data structuresDescribes how the ba
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The range float declarationA range
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{Edge} Edges = {, , };int a[Edges]
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TuplesData structures in OPL can al
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In Declaring a tuple using a set of
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SetsDefinitionSets are non-indexed
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Sorted and ordered setsBy default,
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The data shows that the team includ
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Data sourcesDescribes data and data
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Data initializationDefines internal
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{Ship} shipData = ...;and assume th
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* .dat file */a = #[“Monday”: 1
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}}the same can be expressed with th
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Initializing tuplesYou initialize t
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Initializing setsYou can initialize
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As generic setsOPL supports generic
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Initialization and memory allocatio
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Database initializationDescribes ho
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Supported databasesSupported databa
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Reading from a databaseIn OPL, data
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DBconnection db("oracle9", connecti
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Note: OPL supports the same element
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Spreadsheet Input/OutputDescribes h
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Connection to a spreadsheetThe spre
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♦Sets of tuples: The data has to
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Data consistencyDefines the purpose
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Data membership consistencyThe keyw
Page 75 and 76: AssertionsOPL provides assertions t
Page 77 and 78: execute{writeln(s2);s1.add(3);write
Page 79 and 80: Decision typesVariables in an OPL a
Page 81 and 82: tuple Route {City orig;City dest;}{
Page 83 and 84: Dynamic collection of elements into
Page 85 and 86: The all syntaxThen, the all syntax
Page 87 and 88: Appending arraysYou can also concat
Page 89 and 90: ExpressionsDescribes data and decis
Page 91 and 92: Data and decision variable identifi
Page 93 and 94: (condition)?thenExpr : elseExprwher
Page 95 and 96: Piecewise-linear functionsPiecewise
Page 97 and 98: eakpoint[k-1] < ship[o][d] 0; 0}(1,
Page 99 and 100: Figure The discontinuous piecewise
Page 101 and 102: x == y;signx == piecewise{0->0; 2->
Page 103 and 104: Functions over setsFunction Descrip
Page 105 and 106: ConstraintsSpecifies the constraint
Page 107 and 108: Using constraintsExplains how to ap
Page 109 and 110: Conditional constraintsIf-then-else
Page 111 and 112: Constraint labelsExplains why label
Page 113 and 114: Labeling constraintsTo label a cons
Page 115 and 116: subject to {forall( p in Products ,
Page 117 and 118: Limitations to constraint labelingN
Page 119 and 120: Compatibility between constraint na
Page 121 and 122: Types of constraintsDescribes const
Page 123 and 124: Discrete constraintsDiscrete constr
Page 125: Implicit constraintsImplicit constr
Page 129 and 130: Constraints available in constraint
Page 131 and 132: ♦Interval grouping constraint: al
Page 133 and 134: Formal parametersDescribes basic fo
Page 135 and 136: {forall(i in 1..8)forall(j in 1..8:
Page 137 and 138: Tuples of parametersOPL allows tupl
Page 139 and 140: int array[i in set1] = ((sum(j in s
Page 141 and 142: SchedulingDescribes how to model sc
Page 143 and 144: IntroductionILOG OPL Development St
Page 145 and 146: Piecewise linear and stepwise funct
Page 147 and 148: I L O G O P L D E V E L O P M E N T
Page 149 and 150: e equal to 1. For a fixed interval
Page 151 and 152: ExamplesFor examples of using inter
Page 153 and 154: Precedence constraints between inte
Page 155 and 156: A logical constraint between interv
Page 157 and 158: Important:The piecewise linear func
Page 159 and 160: ♦prev(p, a, b) states that if bot
Page 161 and 162: Cumulative functionsIn scheduling p
Page 163 and 164: Constraints on cumul function expre
Page 165 and 166: Syntax and examplesFor the syntax a
Page 167 and 168: ♦That everywhere over a given fix
Page 169 and 170: in state s. If algn e is true, it m
Page 171 and 172: NotationsThe main notations used th
Page 173 and 174: ILOG Script for OPLDescribes the st
Page 175 and 176: Language structurePresents the stru
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SyntaxWhat composes a scripting sta
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Compound statementsA compound state
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IdentifiersIdentifiers are used to
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Expressions in ILOG ScriptExpressio
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LiteralsLiterals can represent the
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Syntax of different types of expres
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Property accessThere are two ways o
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Function callsSyntactic shorthandSy
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Special operatorsSpecial operator s
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Other operatorsOther operators are
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StatementsA statement can be a cond
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SyntaxEffectfor (var i=0; i < a.len
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Outside a function definitionAt the
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Default valuesDefault value syntaxS
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Built-in values and functionsPresen
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NumbersNumber representations and f
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IntroductionNumbers can be expresse
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Hexadecimal numbersA hexadecimal nu
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Special numbersThere are three spec
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Number methodsThere is only one num
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Numeric constantsThe following nume
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Syntax~ xx > yx >>> yEffectBitwise
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ILOG Script stringsString represent
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Examples of string literals using e
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String propertiesThere is a single,
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Syntaxstring.toUpperCase()EffectExa
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String operatorsString operatorsSyn
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ILOG Script BooleansBoolean represe
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Automatic conversion to BooleanWhen
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Logical operatorsThe following Bool
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ILOG Script arraysArray representat
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Array constructorThe array construc
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Array methodsArray methodsSyntaxarr
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ObjectsObject representation and fu
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Defining methodsSince a method is r
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Object constructorObject constructo
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Built-in methodsThere is only one o
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DatesDate representation and functi
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Date constructorThe date constructo
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Date methodsDate methodsSyntaxdate.
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Date functionsDate functionsSyntaxD
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The null valueThe null value is a s
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ILOG Script functionsIn ILOG Script
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I N D E XIndexABabs, OPL function 9
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limitations in tuples 29ranges 22se
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GHIgeneric sets 51, 137groundbreakp
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NOfor the null value 259for the und
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sparsityand multidimensional arrays
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ILOG OPL Development Studio Language Reference Manual

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