Tight representation of logical constraints as cardinality rules

We address the problem of nding a tight representation of com plex logical constraints in a mixed integer programming model by de scribing a convex hull representation of cardinality rules A cardinality rule asserts that if at least k of the propositions A Am are true then at least l of the propositions B Bn are true As rule based systems and other types of logic modeling grow in popu larity logical rules and propositions can play an increasingly important role in mathematical programming models Such simple logical constraints as if A is produced then either B or C must be produced have long been a part of mathematical programming But much more complex logic models are now being formulated and they can also be embedded in mathematical programming models A logic model is a description of a problem expressed in some logical for malism usually propositional or predicate logic from which facts about its solution can be deduced Expert systems and other rule based systems are logic models as are programs written in the logical programming language PROLOG It is useful to incorporate a logic model into a mathematical pro gramming model when neither model alone adequately describes reality or when the logic model contains constraints or heuristic rules that ease the The rst author is partially supported by AFOSR grant and ONR grant N J Both authors are partially supported by NSF grant solution of the mathematical model by reducing the number of alternatives that must be examined When logical constraints become a signi cant component of mathemat ical programming models the quality of their representation becomes an important issue We wish to address this problem by showing how best to represent a particular type of logical formula namely a cardinality rule in a mixed integer programming MIP model We focus on cardinality rules because we have observed that logical constraints take this form in a large variety of applications In fact McKinnon and Williams use cardinality conditions as a basic form in which to express logical constraints A cardinality rule has the form If at least or at most k of the propositions A Am are true then at least or at most l of the propositions B Bn are true An elementary cardinality rule is one in which each Ai and each Bj is a literal an atomic proposition or its negation and no atomic proposition occurs more than once The rules one typically nds in expert systems if A Am are all true then B is true are special cases of elementary cardinality rules as are cardinality clauses of the form at least or at most k of A Am are true The usual method for representing logical conditions in an MIP model has two stages They are rst rewritten as a conjunction of logical clauses which is to say in conjunctive normal form CNF A clause is a disjunction of literals such as x x x where means not and means or Each clause is then written as an inequality in variables which for this example is