Axiomatizations of quasi-polynomial functions on bounded chains

Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We completely describe the function classes axiomatized by each of these properties, up to weak versions of monotonicity in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasi-polynomial function which appears as a natural extension of the well-established notion of polynomial function. We give further axiomatizations for this class both in terms of functional equations and natural relaxations of homogeneity and median decomposability. As noteworthy particular cases, we investigate those subclasses of quasi-term functions and quasi-weighted maximum and minimum functions, and provide characterizations accordingly

Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices

We give several characterizations of discrete Sugeno integrals over bounded distributive lattices, as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted minimum and maximum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions

Decision-making with Sugeno integrals: Bridging the gap between multicriteria evaluation and decision under uncertainty

International audienceThis paper clarifies the connection between multiple criteria decision-making and decision under uncertainty in a qualitative setting relying on a finite value scale. While their mathematical formulations are very similar, the underlying assumptions differ and the latter problem turns out to be a special case of the former. Sugeno integrals are very general aggregation operations that can represent preference relations between uncertain acts or between multifactorial alternatives where attributes share the same totally ordered domain. This paper proposes a generalized form of the Sugeno integral that can cope with attributes which have distinct domains via the use of qualitative utility functions. It is shown that in the case of decision under uncertainty, this model corresponds to state-dependent preferences on act consequences. Axiomatizations of the corresponding preference functionals are proposed in the cases where uncertainty is represented by possibility measures, by necessity measures, and by general order-preserving set-functions, respectively. This is achieved by weakening previously proposed axiom systems for Sugeno integrals

Representations and characterizations of polynomial functions on chains

We are interested in representations and characterizations of lattice polynomial functions f:L^n -> L, where L is a given bounded distributive lattice. In companion papers [arXiv 0901.4888, arXiv 0808.2619], we investigated certain representations and provided various characterizations of these functions both as solutions of certain functional equations and in terms of necessary and sufficient conditions. In the present paper, we investigate these representations and characterizations in the special case when L is a chain, i.e., a totally ordered lattice. More precisely, we discuss representations of lattice polynomial functions given in terms of standard simplices and we present new axiomatizations of these functions by relaxing some of the conditions given in [arXiv 0901.4888, arXiv 0808.2619] and by considering further conditions, namely comonotonic minitivity and maxitivity

Imprecise data fusion

Possibility theory offers a natural setting for representing imprecise data and poor information. This theory turns out to be quite useful for the purpose of pooling pieces of information stemming from several sources (for instance, several experts, sensors, or databases) . Indeed it looks more flexible than probability theory for the representation of aggregation modes that do not express averaging processes . This paper tentatively explains why possibility theory is appealing for the fusion of imprecise data, and it describes several aggregation modes it allows, along with their underlying assumptions . The existence of adaptive combination rules are pointed out, that take into account the level of conflict between the sources . This approach sounds natural in the pooling of expert opinions . It is suggested here that, under some assumptions, it might also be useful in sensor data fusion .La théorie des possibilités offre un cadre formel naturel pour la représentation de données imprécises, d'informations pauvres. Cette théorie prend tout son intérêt quand il s'agit d'agréger des informations issues de plusieurs sources (par exemple un groupe d'experts, un ensemble hétérogène de capteurs, plusieurs bases de données). En effet elle s'avère être beaucoup plus souple que la théorie des probabilités pour décrire des modes d'agrégation qui ne correspondent pas à des moyennes. Dans cet article on tente d'expliquer pourquoi la théorie des possibilités est intéressante dans le problème de fusion d'informations imprécises, et on décrit les modes d'agrégation qu'elle permet de représenter, avec les hypothèses qui les sous-tendent. On indique notamment l'existence d'opérations de combinaison adaptatives qui prennent en compte le niveau de conflit entre les sources. Cette approche semble justifiée pour l'agrégation d'opinions d'experts. On suggère ici qu'elle peut, dans certaines conditions, être utilisée pour la fusion multi-capteur

A review of applications of fuzzy sets to safety and reliability engineering

Safety and reliability are rigorously assessed during the design of dependable systems. Probabilistic risk assessment (PRA) processes are comprehensive, structured and logical methods widely used for this purpose. PRA approaches include, but not limited to Fault Tree Analysis (FTA), Failure Mode and Effects Analysis (FMEA), and Event Tree Analysis (ETA). In conventional PRA, failure data about components is required for the purposes of quantitative analysis. In practice, it is not always possible to fully obtain this data due to unavailability of primary observations and consequent scarcity of statistical data about the failure of components. To handle such situations, fuzzy set theory has been successfully used in novel PRA approaches for safety and reliability evaluation under conditions of uncertainty. This paper presents a review of fuzzy set theory based methodologies applied to safety and reliability engineering, which include fuzzy FTA, fuzzy FMEA, fuzzy ETA, fuzzy Bayesian networks, fuzzy Markov chains, and fuzzy Petri nets. Firstly, we describe relevant fundamentals of fuzzy set theory and then we review applications of fuzzy set theory to system safety and reliability analysis. The review shows the context in which each technique may be more appropriate and highlights the overall potential usefulness of fuzzy set theory in addressing uncertainty in safety and reliability engineering

Weighted minimum and maximum operations in fuzzy set theory

International audienceThis short note comments on the interpretation of weighted minimum and maximum operations introduced in a previous paper as medians, and corrects a misleading statement about it

Evolution de profils multi-attributs, par apprentissage automatique et adaptatif dans un système de recommandation pour l'aide à la décision

La prise en compte des profils utilisateurs ainsi que leurs évolutions, dans le domaine de l'aide à la décision, constitue actuellement dans la communauté des SIAD (Systèmes Interactifs d'Aide à la Décision) un enjeu important. En effet, la prise en compte du contexte lors de la décision est actuellement émergente pour les SIAD. Ces systèmes d'assistance offrent ainsi des conseils aux utilisateurs en se basant sur leur profil, qui représente leurs préférences à travers une liste de critères valués. Les principales contraintes viennent du fait qu'il est nécessaire que le système puisse amener de l'information pertinente de manière continue. Cela oblige donc à faire évoluer les profils des utilisateurs en fonction de leurs actions. Pour cela, le système ne doit pas seulement " comprendre " ce que l'utilisateur aime, mais également pourquoi. De plus, l'aide apportée aux utilisateurs évoluera donc dans le temps et également par rapport à l'utilisateur. Ainsi l'utilisateur aura à sa disposition une sorte d'assistant personnalisé. L'objectif du travail consiste à apporter une aide à l'activité de l'utilisateur en fonction de son profil. Pour cela, nous proposons de mettre en œuvre et de développer des algorithmes, basés sur des techniques issues du domaine de l'apprentissage, afin de faire évoluer le profil d'un utilisateur en fonction de ses actions. L'aide apportée à l'utilisateur par le système évoluera aussi en fonction de l'évolution de son profil. Le problème à traiter pour l'utilisateur est un problème de prise de décision. Pour ce problème, une assistance est apportée à l'utilisateur, et celle-ci se fait par un affinage des solutions potentielles. Cet affinage est effectué grâce à la mise en place d'un tri (ranking) évolutif des solutions qui sont présentées à l'utilisateur en fonction de son/ses profils. La réalisation d'un tel système nécessite l'articulation des trois principaux domaines de recherche ; qui sont l'Aide à la Décision multicritère, la Décomposition et Agrégation de préférence, et l'Apprentissage automatique. Les domaines de l'Aide à la Décision multicritère et de la Décomposition et Agrégation de préférence peuvent être aussi rassemblés en tant que Procédure d'Agrégation Multicritère (PAMC). Certaines méthodes d'Aide à la Décision multicritère sont mises en place ici et utilisent les données du profil afin d'apporter la meilleure aide possible à l'utilisateur. La décomposition est utilisée pour caractériser un objet afin de fournir à l'apprentissage les données nécessaires à son fonctionnement. L'agrégation quant à elle sert à obtenir une note sur un objet, et cela selon le profil de l'utilisateur, afin de pouvoir effectuer un classement (ranking). L'apprentissage sert à faire évoluer les profils des utilisateurs afin d'avoir toujours un profil représentant le plus fidèlement possible les préférences des utilisateurs. En effet les préférences des utilisateurs évoluant dans le temps, il est nécessaire de traiter ces changements afin d'adapter les réponses à apporter à l'utilisateur. Les contributions de cette thèse portent tout d'abord sur la définition, la construction et l'évolution d'un profil utilisateur (profiling évolutif) en fonction des actions explicites et implicites de l'utilisateur. Ce profiling évolutif est mis en œuvre au sein d'un système de recommandation utilisable sans base d'apprentissage, de manière synchrone et totalement incrémentale, et qui permet aux utilisateurs de changer rapidement de préférences et même d'être incohérents (rationalité limitée). Ce système, qui vient en complément d'un système de Recherche Information, a pour objectif d'établir un ordre total sur une liste d'éléments proposés à l'utilisateur (ranking), et ce en concordance avec les préférences de l'utilisateur. Ces contributions consistent également à la définition de techniques qui permettent d'apporter des parties de solutions à des verrous technologiques comme la désagrégation de critères et la prise en compte d'un nombre variable de critères dans le processus d'aide à la décision interactif, et ce sans définir au préalable de famille cohérente de critères sur laquelle est basée la décision. Plusieurs cadres applicatifs ont été définis afin d'évaluer le système par rapport à d'autres systèmes, mais également afin de tester ses performances de manière hors ligne avec des vraies données utilisateurs, ainsi qu'en ligne, en utilisant directement le système.Considering user profiles and their evolutions, for decision support is currently in the community of DSS (Decision Support Systems) an important issue. Indeed, the inclusion of context in the decision is currently emerging for DSS. Indeed the system offers advice to users based on their profile, which represents their preferences through a list of valued criteria. The main constraints come from the fact that the system need to continuously bring relevant information. It therefore requires changing user profiles thanks to their actions. So, the system must not only "understand" what the user likes, but also why. The users' assistance will evolve over time and therefore with the user. Thus the user has at his disposal a kind of personal assistant. The objective of this work is to provide assistance to the user's activity according to his profile. The objective is to develop an algorithm based on automatic techniques, in order to change the profile of a user based on his actions. The assistance provided to the user by the system will evolves according to the evolution of its profile. The problem addressed to the user is a problem of decision making. For this problem, assistance is provided to the user, and it is a refinement of potential solutions. This refining is done through the establishment of scalable scheduling solutions that are presented to the user depending on his / her profile. The realization of such a system requires the articulation of the three main areas of research which are the Multi-Criteria Decision Support, the Disaggregation and Aggregation of preferences, and Machine Learning. The fields of Decision Support and Multi Disaggregation and Aggregation preference can also be assembled as Multi-Criteria Aggregation Process (PAMC). Some methods of Multicriteria Decision Support are set up here and use profile data to provide the best possible support to the user. The decomposition is used to characterize an object to provide data to the learning algorithm required for its operation. Aggregation serves to score an object according to the user profile in order to rank the selected items. Machine Learning is used to change user profiles in order to always have a profile representing as closely as possible the preferences of users. Indeed user preferences change over the time, it is necessary to address these changes in order to adapt the answers to the user. The contributions of this thesis are firstly, the definition, construction and evolution of a user profile (evolutionary profiling) based on explicit and implicit user's actions. This evolutionary profiling is implemented within a recommender system usable without learning base, synchronously and completely incremental, and that allows users to quickly change their preferences and even to be inconsistent (bounded rationality). This system, which complements an Information System Research, aims to establish a total order on a list of items proposed to the user (ranking) and in accordance with his preferences. These also include the definition of techniques used to make parts of solutions to technological challenges as the disintegration of criteria and the inclusion of a variable number of criteria in the process of interactive decision support, and this without firstly defining coherent family of criteria on which the decision is based. Several application frameworks have been developed to evaluate the system and compare it to other systems, but also to test its performance with real user data in an offline mode, and in an online mode using directly the system

Fuzzy Techniques for Decision Making 2018

Zadeh's fuzzy set theory incorporates the impreciseness of data and evaluations, by imputting the degrees by which each object belongs to a set. Its success fostered theories that codify the subjectivity, uncertainty, imprecision, or roughness of the evaluations. Their rationale is to produce new flexible methodologies in order to model a variety of concrete decision problems more realistically. This Special Issue garners contributions addressing novel tools, techniques and methodologies for decision making (inclusive of both individual and group, single- or multi-criteria decision making) in the context of these theories. It contains 38 research articles that contribute to a variety of setups that combine fuzziness, hesitancy, roughness, covering sets, and linguistic approaches. Their ranges vary from fundamental or technical to applied approaches