Hélène Fargier

Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty

In classical Constraint Satisfaction Problems (CSPs) knowledge is embedded in a set of hard constraints, each one restricting the possible values of a set of variables. However constraints in real world problems are seldom hard, and CSPs are often idealizations that do not account for the preference among feasible solutions. Moreover some constraints may have priority over others. Lastly, constraints may involve uncertain parameters. This paper advocates the use of fuzzy sets and possibility theory as a realistic approach for the representation of these three aspects. Fuzzy constraints encompass both preference relations among possible instantiations and priorities among constraints. In a Fuzzy Constraint Satisfaction Problem (FCSP), a constraint is satisfied to a degree (rather than satisfied or not satisfied) and the acceptability of a potential solution becomes a gradual notion. Even if the FCSP is partially inconsistent, best instantiations are provided owing to the relaxation of some constraints. Fuzzy constraints are thus flexible. CSP notions of consistency and k-consistency can be extended to this framework and the classical algorithms used in CSP resolution (e.g., tree search and filtering) can be adapted without losing much of their efficiency. Most classical theoretical results remain applicable to FCSPs. In the paper, various types of constraints are modelled in the same framework. The handling of uncertain parameters is carried out in the same setting because possibility theory can account for both preference and uncertainty. The presence of uncertain parameters leads to ill-defined CSPs, where the set of constraints which defines the problem is not precisely known.

Semiring-Based CSPs and Valued CSPs: Frameworks, Properties,and Comparison

In this paper we describe and compare two frameworks for constraint solving where classical CSPs, fuzzy CSPs, weighted CSPs, partial constraint satisfaction, and others can be easily cast. One is based on a semiring, and the other one on a totally ordered commutative monoid. While comparing the two approaches, we show how to pass from one to the other one, and we discuss when this is possible. The two frameworks have been independently introduced in ijcai95,jacm and schiex-ijcai95.

Fuzzy scheduling: Modelling flexible constraints vs. coping with incomplete knowledge

An overview of some fuzzy set-based approaches to scheduling is proposed,emphasizing two distinct uses of fuzzy sets: representing preference profiles and modelling uncertainty distributions. The first setting leads to a valued,noncompensatory generalization of constraint-directed scheduling. The other setting yields a possibility-theoretic counterpart of PERT,where probability distributions of activity durations are changed into possibility distributions,for the purpose of modelling incomplete information. It is pointed out that a special case of the latter,interval-valued PERT,is a difficult,ill-known problem,regarding the determination of critical activities,latest starting times and floats. Lastly when flexible constraints and uncertain processing times are to be jointly considered,the use of possibilistic decision theory leads to the computation of robust schedules. � 2002 Elsevier Science B.V. All rights reserved.

Consistency restoriation and explanations in dynamic CSPs----application to configuration

Most of the algorithms developed within the Constraint Satisfaction Problem (CSP) framework cannot be used as such to solve interactive decision support problems, like product configuration. Indeed, in such problems, the user is in charge of assigning values to variables. Global consistency maintaining is only one among several functionalities that should be offered by a CSP-based platform in order to help the user in her task; other important functionalities include providing explanations for some users choices and ways to restore consistency.

Fuzzy constraints in job-shop scheduling

This paper proposes an extension of the constraint-based approach to job-shop scheduling, that accounts for the flexibility of temporal constraints and the uncertainty of operation durations. The set of solutions to a problem is viewed as a fuzzy set whose membership function reflects preference. This membership function is obtained by an egalitarist aggregation of local constraint-satisfaction levels. Uncertainty is qualitatively described in terms of possibility distributions. The paper formulates a simple mathematical model of job-shop scheduling under preference and uncertainty, relating it to the formal framework of constraint-satisfaction problems in artificial intelligence. A combinatorial search method that solves the problem is outlined, including fuzzy extensions of well-known look-ahead schemes.

Uncertainty in Constraint Satisfaction Problems: a Probalistic Approach

We propose a framework for dealing with probabilistic uncertainty in constraint satisfaction problems, associating with each constraint the probability that it is a part of the real problem (the latter being only partially known). The probability degrees on the relevance of the constraints enable us to define, for each instanciation, the probability that it is a solution of the real problem. We briefly give a methodology for the search of the best solution (maximizing this probability).

Fusion: General concepts and characteristics

The problem of combining pieces of information issued from several sources can be encountered in various fields of application. This paper aims at presenting the different aspects of information fusion in different domains, such as databases, regulations, preferences, sensor fusion, etc., at a quite general level. We first present different types of information encountered in fusion problems, and different aims of the fusion process. Then we focus on representation issues which are relevant when discussing fusion problems. An important issue is then addressed, the handling of conflicting information. We briefly review different domains where fusion is involved, and describe how the fusion problems are stated in each domain. Since the term fusion can have different, more or less broad, meanings, we specify later some terminology with respect to related problems, that might be included in a broad meaning of fusion. Finally we briefly discuss the difficult aspects of validation and evaluation. © 2001 John Wiley & Sons, Inc.

The calculus of fuzzy restrictions as a basis for flexible constraint satisfaction

The authors propose a unified treatment of prioritized and flexible constraints, both being represented by possibility distributions. An approach based on possibility theory is described for representing and solving such fuzzy constraint satisfaction problems (FCSP) involving both types of constraints. Arc- and path-consistency-based methods for constraint satisfaction problems are extended to this possibility theory framework. An illustrative example is given.<<ETX>>

Gradual Numbers and Their Application to Fuzzy Interval Analysis

In this paper, we introduce a new way of looking at fuzzy intervals. Instead of considering them as fuzzy sets, we see them as crisp sets of entities we call gradual (real) numbers. They are a gradual extension of real numbers, not of intervals. Such a concept is apparently missing in fuzzy set theory. Gradual numbers basically have the same algebraic properties as real numbers, but they are functions. A fuzzy interval is then viewed as a pair of fuzzy thresholds, which are monotonic gradual real numbers. This view enables interval analysis to be directly extended to fuzzy intervals, without resorting to alpha-cuts, in agreement with Zadehs extension principle. Several results show that interval analysis methods can be directly adapted to fuzzy interval computation where end- points of intervals are changed into left and right fuzzy bounds. Our approach is illustrated on two known problems: computing fuzzy weighted averages and determining fuzzy floats and latest starting times in activity network scheduling.

Refinements of the maximin approach to decision-making in a fuzzy environment

Abstract The most popular approach to decision-making in the setting of fuzzy sets is the maximin ranking of solutions. This method is natural when interpreting the fuzzy sets as flexible constraints that cannot compensate with one another. However the obtained ranking of solutions is very coarse. Two kinds of refinements to this ordering are introduced: a partial ordering according to the least satisfied discriminating constraint, and a lexicographical ranking. The latter refines the former and combines utilitarist and egalitarist points of view on the aggregation of feasibility degrees. These orderings are characterized in several ways and their representation by means of two-place numerical functions is studied. Dual refinements of the maximax ranking are provided.