Eurofuse 2011

One of the main advantages of the Fuzzy Rule Based Classification Systems (FRBCSs) is the high interpretability of the model. However, the disadvantage of these systems may be their lack of accuracy when dealing some complex systems, due to the inflexibility of the concept of linguistic variable, which imposes hard restrictions to the fuzzy rule structure. For example, sometimes when the classes are overlapped, we have not exact knowledge about the membership degree of some elements to the fuzzy sets that characterize the attributes defining the class.

This situation suggests the possibility to represent the membership degrees of the objects to the fuzzy set by means of an interval. That is, to employ the Interval- Valued Fuzzy Sets (IVFSs) to characterize the linguistic labels that compound the attributes of the problems. IVFSs allow us to take into account the effect of the ignorance of the experts in the membership function definition.

The aim of this talk is to shown the performance of FRBCSs by extending the Knowledge Base with the application of the concept of IVFSs. The modeling of the linguistic labels by means of IVFSs implied an adaptation of the original fuzzy reasoning method to allow us to handle the uncertainty that is inherent to the definition process of the membership functions. We define new reasoning methods meaning use of the interval-valued restricted equivalence functions to increase the relevance of the rules in which the equivalence of the interval membership degrees of the patterns and the ideal membership degrees is greater, which is a desirable behavior. Furthermore, the parametrized construction of this fuzzy reasoning method allows the choice of the optimal function for each variable to be performed, which could involve a potential improvement of the system behavior. These parameters will be tuned using genetic algorithms in order to further improve the performance of the systems in a general framework.

We will show different experimental studies showing the usefulness of the IVFSs for enhancing the FRBCSs performance.

Poverty reduction is without doubt a goal of development policy in most countries. To evaluate the evolution of poverty over time in some particular region, the differences of poverty across different countries or the effect of different policies in the alleviation of poverty, one should be first able to measure poverty.

We recall several types of discrete fuzzy integrals and their recent generalizations based on level-dependent capacities. In multicriteria decision support these functionals can be seen as special utility functions. We include some examples of axiomatically defined utility functions and show their relationship to fuzzy integrals.

Sugeno integrals are well-known qualitative aggregation functions in multiple criteria decision making. They return a global evaluation between the minimum and the maximum of the input criteria values. They can model sophisticated aggregation schemes through a system of priorities that applies to any subset of criteria and can take into account some kind of synergy inside subsets of criteria. Although a given Sugeno integral specifies a particular way of implicitly describing a set of entities reaching some global satisfaction level, it is hard to figure out what is the underlying explicit meaning of such an integral in practice (even if the priority level associated to each subset of criteria has a precise meaning). The paper proposes an answer to this problem. Any capacity on a finite set can be represented by a special possibilistic logic base containing positive prioritised clauses, and conversely any possibilistic logic base can represent a set-function. Moreover, Sugeno integral can be represented by a possibilistic logic base expressing how it behaves (thanks to a mapping between the scale and a set of logical atoms reflecting the different values for each criterion). Viewing a Sugeno integral as a set of prioritized logically expressed goals has not only the advantage to make the contents of a Sugeno integral more readable, but it also prompts Sugeno integrals into the realm of logic, and makes it possible to define entailment between them.

In Shafer evidence theory some belief functions, called separable belief functions, can be decomposed in terms of simple support functions. Moreover this decomposition is unique. Recently, a qualitative counterpart to Shafer evidence theory has been proposed. The mass functions in Shafer (addition-based) evidence theory are replaced by basic possibilistic assignments. The sum of weights is no longer 1, but their maximum is equal to 1. In such a context, a maxitive counterpart to belief functions, called possibilistic belief functions can be defined, replacing the addition by the maximum. The possibilistic evidence framework provides a general setting for describing imprecise possibility and necessity measures. This paper investigates a qualitative counterpart of the result about the decomposition of belief functions. Considering the qualitative Möbius transform, conditions for the existence of a decomposition of possibilistic belief functions into simple support functions are presented. Moreover the paper studies the unicity of such a decomposition.

In this paper we give a definition of an OWA operator on any complete lattice that generalizes the notion of an OWA operator in the real case. In addition we introduce a class of functions defined on lattice intervals by weakening the generalized Atanassov’s

K

α

operators.We show that under certain conditions these functions provide a binary OWA operator.

Hedges play an important role in fuzzy theory, although there are relatively few articles on them. Our aim is to provide a theoretical basis not only for hedges, but also for every type of unary operator. One of them is the negation operator, which was presented in an article [14] concerning the DeMorgan class. In our study we will develop unary operators related to other binary operators by demanding that they satisfy certain properties.

A new family of implication operators, called probabilistic implications, are discussed. The suggested implications are based on conditional copulas and make a bridge between probability theory and fuzzy logic. It is shown that probabilistic fuzzy implications have some interesting properties, especially those connected with the dependence structure of the underlying environment. Therefore, it seems that probabilistic implications might be a useful tool in approximate reasoning, knowledge extraction and decision making.

The main contribution of this paper is concerned with the robustness of

N

-dual connectives in fuzzy reasoning. Starting with an evaluation of the sensitivity in n-order function on [0,1], we apply the results in the D-coimplication classes. The paper formally states that the robustness of pairs of mutual dual

n

-order functions can be compared, preserving properties and the ordered relation of their arguments.

A (crisp) binary relation is transitive if and only if its dual relation is negatively transitive. In preference modelling, if a weak preference relation is complete, the associated strict preference relation is its dual relation. It follows from here this well-known result: given a complete weak preference relation, it is transitive if and only if its strict preference relation is negatively transitive.

In the context of fuzzy relations, transitivity is traditionally defined by a t-norm and negative transitivity, by a t-conorm. In this setting, it is also well known that a (valued) binary relation is

T

-transitive if and only if its dual relation is negatively

S

-transitive where

S

stands for the dual t-conorm of the t-norm

T

. However, in this context there are several proposals to get the strict preference relation from the weak preference relation. Also, there are different definitions of completeness. In this contribution we depart from a reflexive fuzzy relation. We assume that this relation is transitive with respect to a conjunctor (a generalization of t-norms). We consider almost all the possible generators and therefore all the possible strict preference relations obtained from the reflexive relation and we provide a general expression for the negative transitivity that those relations satisfy.

n-dimensional fuzzy sets are an extension of fuzzy sets that includes interval-valued fuzzy sets and interval-valued Atanassov intuitionistic fuzzy sets. The membership values of n-dimensional fuzzy sets are n-tuples of real numbers in the unit interval [0,1], called n-dimensional intervals, ordered in increasing order. The main idea in n-dimensional fuzzy sets is to consider several uncertainty levels in the memberships degrees. Triangular norms have played an important role in fuzzy sets theory, in the narrow as in the broad sense. So it is reasonable to extend this fundamental notion for n-dimensional intervals. In interval-valued fuzzy theory, interval-valued t-norms are related with t-norms via the notion of t-representability. A characterization of t-representable interval-valued t-norms is given in term of inclusion monotonicity. In this paper we generalize the notion of t-representability for n-dimensional t-norms and provide a characterization theorem for that class of n-dimensional t-norms.

In this article we propose a method to construct aggregation functions on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers contained in the finite chain

L

 = {0,1, ⋯ ,

n

} from a couple of aggregation functions also defined on

L

. In addition, if the pair of discrete aggregation functions fulfills several properties such as associativity, commutativity or idempotence, we show that this new operator will satisfy these properties too. The particular case of uninorms is studied showing that some properties and part of the structure of the uninorms is preserved under the presented construction method. Finally, we provide an application of this last operator in a decision-making problem.

In this paper a method of defining aggregation functions from fuzzy negations is introduced.Any aggregation function obtained from a fuzzy negation by this method is proved to be a commutative semicopula and some properties are investigated. In particular, it is proved that by this method some well known examples of copulas and t-norms can be obtained. Moreover, any commutative semicopula constructed by this method can be always obtained from a negation

N

which is symmetric with respect to the diagonal. Then, those fuzzy negations

N

for which the corresponding semicopula is a copula are characterized. Also, several examples of negations N are given such that the corresponding semicopula is a t-norm.

The study of discrete aggregation functions (those defined on a finite chain) with some kind of smoothness has been extensively developed in last years. Smooth t-norms and t-conorms, nullnorms and some kinds of uninorms, copulas and quasi-copulas have been characterized in this context. In this paper discrete aggregation functions with the kernel property (which implies the smoothness property) are investigated. Some properties and characterizations, as well as some construction methods for this kind of discrete aggregation functions are studied. It is also investigated when the marginal functions of a discrete kernel aggregation function fully determine it.

We introduce a transformation that acts on binary aggregation functions and that generalizes the transformation that maps copulas, a well-studied class of binary aggregation functions with a profound probabilistic interpretation, to their associated survival copulas. The new transformation, called double flipping, is the composition of two elementary flipping transformations introduced earlier, each operating on one of the arguments of the aggregation function. We lay bare the relationships between these elementary flipping operations and double flipping. We characterize different subclasses of flippable aggregation functions, in particular aggregation functions that have an absorbing element or that have a neutral element. In this investigation, the key role played by quasi-copulas and their dual operations is highlighted. These findings support the introduction of the term survival aggregation function.

Recently, a new construction method of a fuzzy implication from two given ones, called

e

-generation method, has been introduced. This method allows to control, up to a certain level, the increasingness on the second variable of the fuzzy implication through an adequate scaling on that variable of the two given implications. In this paper, the main goal is to reproduce the same idea but now on the first variable of the fuzzy implication. The new implications, called

e

-vertical generated implications, are studied in detail focusing on the preservation of the most common properties of fuzzy implications from the initial ones to the constructed implication.

Properties related with aggregation operators functions have been widely studied in literature. Nevertheless, few efforts have been dedicated to analyze those properties related with the family of operators in a global way. What should be the relationship among the members of a family of aggregation operators? Is it possible to build the aggregation of

n

data with aggregation operators of lower dimension? Should it exist some consistency in the family of aggregation operators? In this work, we analyze two properties of consistency in a family of aggregation operators:

Stability

and

Structural Relevance

. The stability property for a family of aggregation operators tries to force a family to have a stable/continuous definition in the sense that the aggregation of

n

items should be similar to the aggregation of

n

 + 1 items if the last item is the aggregation of the previous

n

items. Following this idea some definitions and results are given. The second concept presented in this work is related with the construction of the aggregation operator when the data that have to be aggregated has an inherent structure. The Structural Relevance property tries to give some ideas about the construction of the aggregation operator when the items are related by means of a graph.

Ontologies represent a method of sharing and reusing knowledge on the semantic web. Moreover, fuzzy ontologies, i.e., the combination of fuzzy logic and ontologies, may be an interesting tool for representing domain knowledge with the aim of solving problems where uncertainty is present. This paper presents three fuzzy-based ontology models for knowledge representation. These ontologies have been obtained after the automatic analysis of a collection of relevant documents that are related to a specific subject. Some experiments have been carried out to illustrate the feasibility of these approaches.

This paper analyzes the relationship between fuzziness and bipolarity, notions which were devised to address different kinds of uncertainty: linguistic imprecision, in the former, and knowledge relevance and character or polarity, in the latter. Although different types of fuzziness and bipolarity have been defined, these relations are not always clear. This paper proposes the use of four-valued extensions to provide a formal method to rigorously define and compare the semantics and logical structure of diverse combinations of fuzziness and bipolarity types. As a result, this paper claims that these notions and their different types are independent and not semantically equivalent despite its possible formal equivalence.

Decision makers are increasingly involved in complex real decisions that require multiple viewpoints. A specific case of this fact is the evaluation of sustainable policies related to environment and energy sectors. In this evaluation process, multiple experts are involved to assess a set of scenarios, according to multiple criteria that might have different nature. These evaluation processes aim to achieve an overall value for each scenario to obtain a ranking among them with the goal of identifying the best one. In this evaluation process a key issue is the treatment of experts’ assessments for each criterion. Due to the uncertainty and vagueness in the judgments of the experts and the nature of the criteria, these assessments can be expressed in different information formats, generating an heterogeneous framework. There are diverse approaches to deal with this type of framework, the use of one approach or another could be crucial in the evaluation process, according to the necessities and requirements of the evaluation models regarding the expected results. In this contribution, we propose an evaluation model applied to energy policy selection based on the decision analysis which may use different approaches to deal with heterogeneous information. We present a comparative study of the proposed model using two different approaches to deal with heterogeneous information. Finally, we show the strengths and weaknesses of the evaluation model depending on the approach used to manage heterogeneous information

A supply chain is a set of geographically dispersed facilities that store and transform products, and that are connected by a transportation network. The main task of supply chain management is to design the supply chain so that a given set of objectives is achieved, for example by deciding the location and capacity of new production plants, or the location of warehouses. Since suppliers also play a key role in performance maximization, it is natural to integrate them in the supply chain as well [8]. For this reason, selection of potential suppliers has become a fundamental component of supply chain management; this is even more true in the globalized market of today. The problem of supplier selection can be easily understood as a multiple-criteria decision making (MCDM) problem: businesses express their preferences with respect to suppliers, which can then be ranked and selected. Doing so, however, does not take into account the temporal evolution of supplier performances, neither can it be easily applied when considering more than one customer. To overcome these problems, we introduce a model for supplier selection that extends the classic MCDM model by introducing feedback, and consider its application in the context of multiple customers by means of linear programming.

In this contribution our aim is to present a multicriteria linguistic decision making model in which experts might provide their assessments by using linguistic expressions based on comparative terms close to the expressions used by human beings in real world problems or single linguistic terms. To aggregate such a type of linguistic information two symbolic aggregation operators are introduced. Finally, an exploitation phase is proposed to build a preference relation among alternatives and then, a non-dominance choice degree is applied to obtain the solution set of alternatives.

In this work we present a construction method for interval-valued fuzzy preference relations from a fuzzy preference relation and the representation of the lack of knowledge or ignorance that experts suffer when they define the membership values of the elements of that fuzzy preference relation.We also prove that, with this construction method, we obtain membership intervals for an element which length is equal to the ignorance associated with that element. We then propose a generalization of Orlovsky’s non dominance method to solve decision making problems using interval-valued fuzzy preference relations.

Driven by a large number of potential applications in areas like bioinformatics, information retrieval and social network analysis, the problem setting of inferring relations between pairs of data objects has recently been investigated quite intensively in the machine learning community. To this end, current approaches typically consider datasets containing crisp relations, so that standard classification methods can be adopted. However, relations between objects like similarities and preferences are in many real-world applications often expressed in a graded manner. A general kernel-based framework for learning relations from data is introduced here. It extends existing approaches because both crisp and valued relations are considered, and it unifies existing approaches because different types of valued relations can be modeled, including symmetric and reciprocal relations. This framework establishes in this way important links between recent developments in fuzzy set theory and machine learning. Its usefulness is demonstrated on a case study in document retrieval.

In this work we propose a new multicriteria decision making algorithm for interval-valued fuzzy preference relations based on the use on a appropriate definition of interval-valued Choquet integrals. This algorithm allows to recover some of the best known usual fuzzy algorithms when the considered intervals are reduced to a single point. Since a key point in every decision making problem is that of the ordering, we propose a method to build orders based on the use of aggregation functions that, on one hand, allows to define several different total orders and, on the other hand, recovers some of the most commonly used total orders between intervals.

A well known problem that many sources of data nowadays cope with, is the problem of duplicate data. In general, we can represent a data source as a collection of objects. Deduplication then consists of two main problems: (a) finding duplicate objects and (b) processing those duplicate objects. This paper contributes to the study of the latter problem by investigating functions that map a multiset of objects to a single object. Such functions are called merge functions.We investigate the specific case where an object itself is a multiset. An interesting application of this case is the problem of multiple document summarization. Next to the basic definition of such merge functions, we focus on an important property borrowed from the (more general) field of information fusion: the majority rule.

Penalty and Barrier methods are normally used to solve Nonlinear Optimization Constrained Problems. The problems appear in areas such as engineering and are often characterized by the fact that involved functions (objective and constraints) are non-smooth and/or their derivatives are not know. This means that optimization methods based on derivatives cannot be used. A Java based API was implemented, including only derivative-free optimization methods, to solve both constrained and unconstrained problems, which includes Penalty and Barriers methods. In this work a new penalty function, based on Fuzzy Logic, is presented. This function imposes a progressive penalization to solutions that violate the constraints. This means that the function imposes a low penalization when the violation of the constraints is low and a heavy penalization when the violation is high. The value of the penalization is not known in beforehand, it is the outcome of a fuzzy inference engine. Numerical results comparing the proposed function with two of the classic penalty/barrier functions are presented. Regarding the presented results one can conclude that the proposed penalty function besides being very robust also exhibits a very good performance.

Principal component analysis (PCA) is a usefully tool for data compression and information extraction. It is often utilized in point cloud processing as it provides an efficient method to approximate local point properties through the examination of the local neighborhoods. This process does sometimes suffer from the assumption that the neighborhood contains only a single surface, when it may contain curved surface or multiple discrete surface entities, as well as relating the properties from PCA to real world attributes. This paper will present a new method that joins the fuzzy clustering algorithm with a local sliding PCA analysis to identify the non-linear relations and to obtain morphological information of the data. The proposed PCA-Fuzzy algorithm is performed on the neighborhood of the cluster center and normal approximations in order to estimate a tangent surface and the radius of the curvature that characterizes the trend and curvature of the data points or contour regions.

In this work we present a simple magnification algorithm for color images. It uses Interval-Valued Fuzzy Sets in such a way that every pixel has an interval membership constructed from its original intensity and its neighbourhood’s one. Based on that interval membership, a block is created for each pixel, so this is a block expansion method.

A digital image is an approximation of some real situation, and carries some uncertainty. In this work we model the ambiguity related to the brightness by associating an interval with each pixel, instead of a scalar brightness value. Then we adapt the Sobel method for edge detection to the new conditions of the image, leading to a representation of the edges in the shape of an interval-valued fuzzy set. To conclude, we illustrate the performance of the method and perform a qualitative comparison with the classical Sobel method on grayscale images.

In this paper we introduce two kinds of fuzzy histograms on the basis of fuzzy colors in a fuzzy color space and the notion of gradual number by Dubois and Prade. Fuzzy color spaces are a collection of fuzzy sets providing a suitable, conceptual quantization with soft boundaries of crisp color spaces. Gradual numbers assign numbers to values of a relevance scale, typically [0,1]. Contrary to convex fuzzy subsets of numbers (called fuzzy numbers, but corresponding to fuzzy intervals as an assignment of intervals to values of [0,1]), they provide a more precise representation of the cardinality of a fuzzy set. Histograms based on gradual numbers are particularly well-suited for serving as input to another process. On the contrary, they are not the best choice when showing the information to a human user. For this second case, linguistic labels represented by fuzzy numbers are a better alternative, so we define linguistic histograms as an assignment of linguistic labels to each fuzzy color. We provide a way to calculate linguistic histograms based on the compatibility between gradual numbers and linguistic labels. We illustrate our proposals with some examples.

In this work we propose an image reduction algorith based on local reduction operators. We analyze the construction of weak local reduction operators by means of aggregation functions and we analyze the effect of several aggregation functions in image reduction with original and noisy images.

Indoor location systems cannot rely on technologies such as GPS (Global Positioning System) to determine the position of a mobile terminal, because its signals are blocked by obstacles such as walls, ceilings, roofs, etc. In such environments the use of alternative techniques, such as the use of wireless networks, should be considered. The location estimation is made by measuring and analysing one of the parameters of the wireless signal, usually the received power. One of the techniques used to estimate the locations using wireless networks is fingerprinting. This technique comprises two phases: in the first phase data is collected from the scenario and stored in a database; the second phase consists in determining the location of the mobile node by comparing the data collected from the wireless transceiver with the data previously stored in the database. In this paper an approach for localisation using fingerprinting based on Fuzzy Logic and pattern searching is presented. The performance of the proposed approach is compared with the performance of classic methods, and it presents an improvement between 10.24% and 49.43%, depending on the mobile node and the Fuzzy Logic parameters.

Species-environment relationships are used for evaluating the current status of target species and the potential impact of natural or anthropogenic changes of their habitat. Recent researches reported that the results are strongly affected by the quality of a data set used. The present study attempted to apply pairwise comparisons to modelling fish habitat preference with Takagi-Sugeno-type fuzzy habitat preference models (FHPMs) optimized by a genetic algorithm (GA). The model was compared with the result obtained from the FHPM optimized based on mean squared error (MSE). Three independent data sets were used for training and testing of these models. The FHPMs based on pairwise comparison produced variable habitat preference curves from 20 different initial conditions in the GA. This could be partially ascribed to the optimization process and the regulations assigned. This case study demonstrates applicability and limitations of pairwise comparison-based optimization in an FHPM. Future research should focus on a more flexible learning process to make a good use of the advantages of pairwise comparisons.

In this paper a new tracking approach based in fuzzy concepts is introduced. The aim of this methodology is to incorporate in the proposed model the uncertainty underlying any problem of feature tracking, through the use of fuzzy sets. Several dynamic fuzzy sets are constructed according both cinematic (movement model) and non cinematic properties (image gray levels) that distinguish the feature. Meanwhile cinematic related fuzzy sets model the feature movement characteristics, the non cinematic fuzzy sets model the feature visible image related properties. The tracking task is performed through the fusion of these fuzzy models by means of a fuzzy inference engine. This way feature detection and matching steps are performed exclusively using inference rules on fuzzy sets.

In this paper a comparative analysis of several edge detectors based on diverse fuzzy morphologies is performed. In addition, two different processes in order to transform a fuzzy edge image to a thin binary edge image are studied, a recently introduced unsupervised hysteresis based on the determination of a “instability zone” on the histogram and a fuzzy Atanassov’s based threshold. The comparison is made according to some performance measures, such as Pratt’s figure of merit and the

ρ

-coefficient. The goodness of the employed binarization methods is studied depending on their capability to obtain the best threshold values according to these measures.

In many areas, fuzzy linguistic approaches have already shown their interest and successful results to express the preferences and the choices of a human. This paper focuses on the fuzzy linguistic 2-tuple representation model that is interesting and relevant when we need to express and to refer to linguistic assessments during the whole reasoning process. However, when data have a particular distribution on their axis, this model doesn’t fit well the needs anymore. We propose therefore a variant version of this representation model that allow for a more realistic distribution. We also show that an operation such as an arithmetic mean is easy to implement with it and gives consistent results.