Jérôme Fortin

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.

A generalized vertex method for computing with fuzzy intervals

We introduce a new method for computing functions of fuzzy intervals under various monotonicity assumptions on the concerned functions. Our method makes exact computation for all possibility degrees, without resorting to /spl alpha/-cuts. We formally present the notion of left and right profiles of fuzzy intervals as a tool for fuzzy interval computation. Several results show that interval analysis methods can be directly adapted to fuzzy interval computation where end point of intervals are changed into left and right profiles. Our approach is illustrated by numerous simple examples all along the paper, and a special section is devoted to the application of these concepts to different known problems.

The Empirical Variance of a Set of Fuzzy Intervals

The profile method gives a tool to perform fuzzy interval computation under a condition of local monotony of considered functions. This is a plain extension of interval analysis to fuzzy intervals, viewed as pairs of fuzzy bounds. This method yields exact results without applying interval analysis to alpha-cuts. After a refresher on the notion of profile and its use in fuzzy interval analysis, we adapt the profile method to the computation of the empirical variance of a tuple of fuzzy intervals. To this end, we first reconsider results obtained by Ferson et al. on computation of the empirical variance of a set of intervals. Finally we apply our results to the definition of the variance of a single fuzzy interval, viewed as a family of its alpha-cuts, and compare this definition to previous ones

Interval analysis in scheduling

This paper reconsiders the most basic scheduling problem, that of minimizing the makespan of a partially ordered set of activities, in the context of incomplete knowledge. While this problem is very easy in the deterministic case, its counterpart when durations are interval-valued is much trickier, as standard results and algorithms no longer apply. After positioning this paper in the scope of temporal networks under uncertainty, we provide a complete solution to the problem of finding the latest starting times and floats of activities, and of locating surely critical ones, as they are often isolated. The minimal float problem is NP-hard while the maximal float problem is polynomial. New complexity results and efficient algorithms are provided for the interval-valued makespan minimization problem.

Computational methods for determining the latest starting times and floats of tasks in interval-valued activity networks

In project management, three quantities are often used by project managers: the earliest starting date, the latest starting date and the float of tasks. These quantities are computed by the Program Evaluation and Review Techniques/Critical Path Method (PERT/CPM) algorithm. When task durations are ill known, as is often the case at the beginning of a project, they can be modeled by means of intervals, representing the possible values of these task durations. With such a representation, the earliest starting dates, the latest starting dates and the floats are also intervals. The purpose of this paper is to give efficient algorithms for their computation. After recalling the classical PERT/CPM problem, we present several properties of the concerned quantities in the interval-valued case, showing that the standard criticality analysis collapses. We propose an efficient algorithm based on path enumeration to compute optimal intervals for latest starting times and floats in the general case, and a simpler polynomial algorithm in the case of series-parallel activity networks.

Criticality analysis of activity networks under interval uncertainty

This paper reconsiders the Project Evaluation and Review Technique (PERT) scheduling problem when information about task duration is incomplete. We model uncertainty on task durations by intervals. With this problem formulation, our goal is to assert possible and necessary criticality of the different tasks and to compute their possible earliest starting dates, latest starting dates, and floats. This paper combines various results and provides a complete solution to the problem. We present the complexity results of all considered subproblems and efficient algorithms to solve them.

Interval PERT and Its Fuzzy Extension

In project or production management, an activity network is classically defined by a set of tasks (activities) and a set of precedence constraints expressing which tasks cannot start before others are completed. When there are no resource constraints, we can display the network as a directed acyclic graph. With such a network the goal is to find critical activities, and to determine optimal starting times of activities, so as to minimize the makespan. The first step is to determine the earliest ending time of the project. This problem was posed in the fifties, in the framework of project management, by Malcolm et al. [32] and the basic underlying graph-theoretic approach, called Project Evaluation and Review Technique, is now popularized under the acronym PERT. The determination of critical activities is carried out via the so-called critical path method (Kelley [29]). The usual assumption in scheduling is that the duration of each task is precisely known, so that solving the PERT problem is rather simple. However, in project management, the durations of tasks are seldom precisely known in advance, at the time when the plan of the project is designed. Detailed specifications of the methods and resources involved for the realization of activities are often not available when the tentative plan is made up. This difficulty has been noticed very early by the authors that introduced the PERT approach. They proposed to model the duration of tasks by probability distributions, and tried to estimate the mean value and standard deviation of earliest starting times of activities. Since then, there has been an extensive literature on probabilistic PERT (see Adlakha and Kulkarni [1] and Elmaghraby [18] for a bibliography and recent views). Even if the task duration times are independent random variables, it is admitted that the problem of finding the distribution of the ending time of a project is intractable, due to the dependencies induced by the topology of the network [25]. Another difficulty, not always pointed out, is the possible lack of statistical data validating the choice of activity duration distributions. Even if statistical data are available, they may be partially inadequate because each project takes place in a specific environment, and is not the exact replica of past projects.

Default conceptual graph rules, atomic negation and Tic-Tac-Toe

In this paper, we explore the expressivity of default CG rules (a CG-oriented subset of Reiters default logics) through two applications. In the first one, we show that default CG rules provide a unifying framework for CG rules as well as polarized CGs (CGs with atomic negation). This framework allows us to study decidable subclasses of a new language mixing CG rules with atomic negation. In the second application, we use default CG rules as a formalism to model a game, an application seldom explored by the CG community. This model puts into light the conciseness provided by defaults, as well as the possibilities they offer to achieve efficient reasonings.

Some methods for evaluating the optimality of elements in matroids with ill-known weights

In this paper a class of matroidal combinatorial optimization problems with imprecise weights of elements is considered. The imprecise weights are modeled by intervals and fuzzy intervals. The concepts of possible and necessary optimality under imprecision are recalled. Some efficient methods for evaluating the possible and necessary optimality of elements in the interval-valued problems are proposed. Some efficient algorithms for computing the exact degrees of possible and necessary optimality of elements in the fuzzy-valued problems are designed.

Efficient methods for computing optimality degrees of elements in fuzzy weighted matroids

In this paper some effective methods for calculating the exact degrees of possible and necessary optimality of an element in matroids with ill-known weights modeled by fuzzy intervals are presented.