Flexible Constraints and Qualitative Decision in A.I

The notion of constraint is basic in operations research and artificial intelligence. A constraint describes what are the potentially acceptable decisions (the solutions to a problem) and what are the absolutely unacceptable ones: it is an all-or-nothing matter. Moreover, no constraint can be violated, i.e., a constraint is classically considered as imperative. Especially the violation of a constraint cannot be compensated by the satisfaction of another one. If a solution violates a single constraint, it is regarded as unfeasible. The idea of flexible constraints is to keep the noncompensatory property of constraints, while introducing intermediary levels between feasibility and non-feasibility as well as levels in the imperativeness of constraints. A classical hard constraint C is represented by a classical set of solutions, i.e., only using degrees of membership μC(d) = 1 if choosing d is feasible and μC(d) = 0 if it is not. Introducing intermediary levels of feasibility on a linearly ordered valuation set S (with 0 as a bottom element and 1 as a top element), C is then called a fuzzy, or soft constraint: μC(d) = 1 means that a solution d totally satisfies C while μC(d) = 0 means that it totally violates C (d is unfeasible). If 0 < μC(d) < 1, d satisfies C only partially; μC(d) > μC(d') indicates that C is more satisfied by d than by d'. Hence, like an objective function, a fuzzy constraint rank-orders the feasible decisions. However, contrary to an objective function a fuzzy constraint also models a threshold (represented by the bottom level 0) beyond which a solution will be rejected. In fact, a fuzzy constraint can be viewed as the association of a constraint (defining the support of C) and a criterion which rank-orders the solutions satisfying the constraints. In this interpretive framework, a membership function, is similar to a qualitative utility function, or better, a value function. A soft constraint C' will be looser than another one C if and only if μC ≤ μC', that is, if any solution to C is at least as feasible for C'.