The concept of hierarchical structure is central when considering complex systems. On one hand, many complex systems exhibit a hierarchical structure, on the other hand, the idea of defining a hierarchical structure to cope with the complexity of the system is widely present in literature related to many different fields. Hierarchies are also present in the field of aggregation with the definition of hierarchical aggregation processes. Broadly speaking, a hierarchical system is a system formed by several components (subsystems) structured at different levels, implying a sort of ranking or ordering relation among them. Positioning in the levels could be related to many different aspects or properties of the components: priority, abstraction, granularity, specificity, precision, etc.Computable aggregations have recently been introduced as a natural approach to classical aggregation functions, in which the emphasis is placed on the program (the implementation) that makes possible the aggregation instead of simply considering the function or the algorithm that permits the aggregation process. Considered as a program that implements an aggregation process, a computable aggregation is also suitable for being interpreted in terms of a hierarchical process. In this paper, the idea of hierarchical computable aggregation is considered, exploring those situations where the aggregation process involves some intrinsic structure that can be interpreted in hierarchical terms (like priorities and veto), as well as those other situations where the hierarchical approach is mostly related to computational considerations (like recursion and parallelization). These and other types of hierarchical computable aggregation will be presented and analyzed.

The idea of computable aggregation operators was introduced as a generalization of aggregation operators, allowing the replacement of the mathematical function usually considered for aggregation, by a program that performs the aggregation process. There are different reasons to justify this extension. One of them is the interest in exploring some computational properties not directly related to the aggregation itself but to its implementation (complexity, recursivity, parallelisation, etc). Another reason, the one driving to the present paper, is the need to define a framework where the quite common process of first sampling (over a large data set) and then aggregating the sample, could be analysed as a formal aggregation process. This process does not match with the idea of an aggregation function, due to its non-deterministic nature, but could easily be adapted to that of a (non-deterministic) computable aggregation. The idea of non-deterministic aggregation requires the extension of the concept of monotonicity (a key aspect of aggregation operators) to this new framework. The present paper will explore this kind of non-deterministic aggregation processes, first from an empirical point of view and then in terms of populations, adapting the idea of monotonicity to both of them and finally defining a common framework for its analysis.

Computable aggregation operators can be seen as a generalization of aggregation operators where the mathematical function is replaced by a program that performs the aggregation process. This extension allows the introduction of new aggregation processes not feasible under the classical framework. Particularly interesting are some non-deterministic processes widely considered to merge information. However, especially in non-deterministic processes, the extension of some of the well-known concepts for aggregation operators such as monotony, is needed. In this work, a new concept of monotonicity is proposed, from a probabilistic perspective, for non-deterministic computable aggregation operators. To be consistent, the concept coincides with the classical definition in the deterministic case. In addition, some cases of interest are analysed.

The concept of pre-aggregation function defined in [0,1]nhas been recently extended to that of generalized pre-aggregation function in the framework of a totally ordered set T with maximum and minimum value. To do so, the concept of monotonicity is transformed in that of conditioned monotonicity based on the chains in Tn, generalizing the idea of directional monotonicity. In the present paper we explore the concept of conditioned monotonicity considering some specific conditioning structures (covers, partitions and projections). On this basis we consider some situations where conditioned monotonicity ensures monotonicity. Finally we use these definitions and properties to define some pre-aggregation and aggregation functions that are applied to image preprocessing problems.

In this paper the relation between aggregation functions, algorithms and computer programs is revisited, extending the concept of recursive aggregation operator by means of the recursive computable aggregation. In particular, two different recursive computable aggregation are distinguished: on the one hand, the hard recursive computable aggregation, which appears when there is a unique recursive function generating the aggregation, and are fully related to associativity; and on the other hand, the soft recursive computable aggregation, which appears when the number of elements to be aggregated is needed. Some illustrative examples are provided.

One of the most common techniques to find the adequate weights in ordered weighted averaging (OWA) operators is based on the orness concept, where the weights are determined by maximizing the entropy (variation) for a fixed orness value. But such an entropy represents a dispersion measure for nominal variables, while weights in an OWA operator are essentially ordinal rather than nominal. Hence, in this paper, we propose a novel way to determine OWA weights based upon ordinal dispersion measures instead of an standard entropy measure. From this approach, we find an explicit formula for the weights, and we illustrate differences by means some multicriteria decision-making examples.

Inspired by the Zadeh approach to fuzzy connectives in fuzzy set theory and by some applications, we introduce and study set-based extended functions, and in particular, set-based extended aggregation functions. These functions reflect neither reordering nor repetition of input values, and, linking different arities, they introduce serious constraints for extended functions. A complete characterization of set-based extended (aggregation) functions is given, and some constructions of such functions are also proposed, including several examples.

It is common to find index construction methods based on crisp techniques and linear models. However, these procedures are not always adequate to capture the essential model of an index. Index development can be approached as a nested or hierarchical aggregation process, where part of the data can have a different level of importance, and the other part can be unstructured. In addition, the information can be studied under a fuzzy approach. This paper proposes an index construction method using the notion of consistency in aggregation operator families to ensure robustness, in addition to considering a hierarchical structure with different level of importance of input information. Two applications are presented, a measure of social and advanced uses of the Internet considering attitudinal, demographic, social and technological factors, and a measure of the child's family environment considering different aspects.

Hesitant Fuzzy Sets are useful to represent the information given by several experts. However, this possibility usually require to deal with sets of values with different cardinality and the necessity to compare them, it is, not all experts evaluate all the elements in the universe. In this paper it is proposed a new nomenclature for Hesitant Fuzzy Sets called Hesitant Fuzzy Sets using Lists. It is proposed a new partial order relation between lists of membership degrees. This partial order relation allows to handle lists with different lengths. In addition, Hesitant Fuzzy Relations on Lists operations are defined and the T-transitive closure of a Hesitant Fuzzy Relation is given. It is proved this T-transitive closure always exists and it is unique. Finally, an algorithm to compute the T-transitive closure of a Hesitant Fuzzy Relation is proposed and several examples are given.

In this paper we stress the relevance of those fuzzy models that impose a couple of simultaneous views in order to represent concepts. In particular, we point out that the basic model to start with should contain at least two somehow opposite valuations plus a number of neutral concepts that are generated from the semantic relationship between those two opposites. Such a basic model should be distinguished from some other similar approaches that can be found in the literature, and that may bring some difficulties in intuition, partially because of their denomination. The general term “paired fuzzy sets” is then proposed together with the notion of sub-antonym, to be considered as a particular case of opposition relationship.

In this paper, we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and enumerate some of the applications in which they have been used.

In this paper we want to stress the relevance of paired fuzzy sets, as already proposed in previous works of the authors, as a family of fuzzy sets that offers a unifying view for different models based upon the opposition of two fuzzy sets, simply allowing the existence of different types of neutrality associated to the different semantic relationships that may hold between opposite references. This scheme should be seen as a basic model for knowledge acquisition, which eventually will lead to a better understanding of the relationship of different knowledge representation models and to the acquisition of more complex valuation scales.

In a trilevel hierarchical decision problem, the objectives and variables of each decision entity at one level are controlled, in part, by the decision entities at other levels. The choice of values for the decision variables at each level may influence the decisions made at other levels, and may thereby improve/reduce the objective for each level. When multiple decision entities are involved at the middle and bottom levels of a trilevel decision problem, the top-level entity's decision will be affected, not only by these followers' individual reactions, but also by the relationships between them. We call this problem a multifollower trilevel (MFTL) decision. This paper firstly defines and analyzes various kinds of relationships between decision entities in an MFTL decision problem. We then propose an MFTL decision making framework, in which 64 standard MFTL decision situations and their possible combinations are identified. To model these MFTL decision situations, we developed an innovative decision entity-relationship diagram (DERD) approach. We also established a general model for MFTL decision making and a set of standard MFTL decision models using trilevel programming. A trilevel decision support system (TLDSS) software has also been developed to transfer a DERD into a programming model. Finally, a case study illustrates typical MFTL decision making models and their development, using both DERD and programming approaches.

In this paper we present an efficient hierarchical clustering algorithm for relational data, being those relations modeled by a graph. The hierarchical clustering approach proposed in this paper is based on divisive and link criteria, to break the graph and join the nodes at different stages. We then apply this approach to a community detection problems based on the well-known edge line betweenness measure as the divisive criterium and a fuzzy similarity relation as the link criterium. We present also some computational results in some well-known examples like the Karate Zachary club-network, the Dolphins network, Les Miserables network and the Authors centrality network, comparing these results to some standard methodologies for hierarchical clustering problem, both for binary and valued graphs.

In this paper we propose an efficient and polynomial hierarchical clustering technique for unsupervised classification of items being connected by a graph. The output of this algorithm shows the cluster evolution in a divisive way, in such a way that as soon as two items are included in the same cluster they will join a common cluster until the last iteration, in which all the items belong to a singleton cluster. This output can be viewed as a fuzzy clustering in which for each alpha cut we have a standard cluster of the network. The clustering tool we present in this paper allows a hierarchical clustering of related items avoiding some unrealistic constraints that are quite often assumed in clustering problems. The proposed procedure is applied to a hierarchical segmentation problem in astronomical images.

Segmentation algorithms are well known in the field of image processing. In this work we propose an efficient and polynomial algorithm for image segmentation based on fuzzy set theory. The main difference with the classical segmentation algorithms is in the output given by the segmentation process. Since the classical output for segmentation algorithms give us the homogeneous regions in the image, our proposal is to produce an hierarchical information (in a similar way as a dendrogam does in classical clustering methods) of how the groups are formed in the image, from the initial situation in which all pixels are in the same group to the final situation in which the whole image is divided in the minimal information units.

This paper studies the meaning of fuzzy preferences under the light of intuitionistic fuzzy sets. Intuitionism is revisited and intuitionistic fuzzy preference relations are defined. A linguistic interpretation of strict and weak fuzzy preferences is then developed where indecision can be explicitly analyzed with the help of a bilattice. Uncertainty is finally examined as a measure of ignorance associated to the intensity value of the preference predicate, where some information measures are applied over its complete structure.

The currently-in-use definition of aggregation function is analyzed in this paper, noting that the introduced variability in the dimension of information does not avoid some obvious dysfunctions. In particular, a potential abuse of the mathematical formalism underlies such a definition, which could lead to solve a complex concept by means of a formal mathematical expression. In this paper we propose an alternative definition making emphasis on the practical implementation of aggregation functions, taking into account the objectives and limitations observed in the application of aggregation functions within the fuzzy context.

In this paper, we introduce a novel system for recognition of partially occluded and rotated images. The system is based on a hierarchical network of integrate-and-fire spiking neurons with random synaptic connections and a novel organization process. The network generates integrated output sequences that are used for image classification. The proposed network is shown to provide satisfactory predictive performance given that the number of the recognition neurons and synaptic connections are adjusted to the size of the input image. Comparison of synaptic plasticity activity rule (SAPR) and spike timing dependant plasticity rules, which are used to learn connections between the spiking neurons, indicates that the former gives better results and thus the SAPR rule is used. Test results show that the proposed network performs better than a recognition system based on support vector machines.

In this work we propose an efficient and polynomial algorithm for the graph segmentation problem based on the coloring problem for graphs. The work here presented extend the algorithm published in making possible the segmentation to any class of graph (not only fuzzy-valued planar graphs) and also improving the computational complexity of the previous work.

In this paper we study in depth certain properties of interval type 2 fuzzy sets. In particular we recall a method to construct different interval type 2 fuzzy connectives starting from an operator. We further study the law of contradiction and the law of excluded middle for these sets. Furthermore we analyze the properties: idempotency, absorption, and distributiveness.

Since many complex decision making problems can be solved solely by means of an appropriate algorithm, checking the quality of such algorithm is a key issue, even more relevant in the presence of fuzzy uncertainty. In this paper we postulate that the design and formal specification of algorithms can be translated into a fuzzy framework introducing fuzzy first order logic and assert transformations. Following the classical crisp scheme we first formalize the concepts of a fuzzy algorithm specification and a fuzzy computing state, and then a new fuzzy computational logic is presented, so we can derive a computational reasoning for correctness of algorithms. A proposal for the evaluation and setting of suitable degrees of truth to computing states is also introduced.

In a previous paper, the authors proposed an alternative approach to classical dimension theory, based upon a general representation of strict preferences not being restricted to partial order sets. Without any relevant restriction, the proposed approach was conceived as a potential powerful tool for decision making problems where basic information has been modeled by means of valued binary preference relations. In fact, assuming that each decision maker is able to consistently manage intensity values for preferences is a strong assumption even when there are few alternatives being involved (if the number of alternatives is large, the same criticism applies to crisp preferences). Any representation tool, as the one proposed by the authors, will in principle play a key role in order to help decision makers to understand their preference structure. In this paper we introduce an alternative approach in order to avoid certain complexity issues of the initial proposal, allowing a close representation easier to be obtained in practice.

In this paper we address the problem of inconsistency in preference relations, pointing out the relevance of a meaningful representation in order to help decision maker to capture such inconsistencies. Dimension theory framework, despite its computational complexity, is considered here, pursuing in principle a decomposition of arbitrary preference relations in terms of linear orderings of alternatives. But we shall then stress that consistency should not be necessarily associated to a linear ordering. In this way, alternative decompositions of a preference relation can be proposed to decision maker, allowing an effective search for a useful representations of alternatives in terms of possible criteria. Such decompositions of our preference relations will then become the basis of a future decision aid model, always with the restricted aim of allowing the decision maker a better understanding of the problem. Inconsistencies may be not simply suppressed but understood, since they may contain relevant information.

Land cover analysis by means of remotely sensing images quite often suggests the existence of fuzzy classes, where no clear borders or particular shapes appear. In this paper we present an image classification aid algorithm which shows as its main output a processed image where each pixel is being colored according to the degree of similitude to their respective surrounding pixels. Such a processed image is therefore suggesting possible classes, to be implemented in a more sophisticated image classification process. A key underlying argument for this approach is the relevance of painting techniques in order to help decision makers to understand complex information relative to fuzzy image classification.

Geographical information (including remotely sensed data) is usually imprecise, meaning that the boundaries between different phenomena are fuzzy. In fact, many classes in nature show internal gradual differences in species, health, age, moisture, as well other factors. If our classification model does not acknowledge that those classes are heterogeneous, and crisp classes are artificially imposed, a final careful analysis should always search for the consequences of such an unrealistic assumption. We consider the unsupervised algorithm presented by A. del Amo et al. (2000), and its application to a real image in Sevilla province (south Spain). Results are compared with those obtained from the ERDAS ISO-DATA classification program on the same image, showing the accuracy of our fuzzy approach. As a conclusion, it is pointed out that whenever real classes are natural fuzzy classes, with gradual transition between classes, then its fuzzy representation will be more easily understood, and therefore accepted by users.

An associative binary connective allows the evaluation of arbitrary finite sequences of items by means of a one-by-one sequential process. In this paper we develop an alternative approach for those nonassociative connectives, allowing a sequential definition by means of binary fuzzy connectives. It will be then stressed that a connective rule should be understood as a consistent sequence of binary connective operators. ©1999 John Wiley & Sons, Inc.

We will continue here the research work, the goal of which was to introduce the notion of nondeterministic aggregation operators and study their properties, even in relation to classification systems and the associated learning problem. Here we will concentrate mostly on the notion of the nondeterministic aggregation system and its relation with deterministic ones. We will also see how such a model extends a discretized version of a model of participatory learning with an arousal background mechanism. © 1998 John Wiley & Sons, Inc.13: 181–192, 1998

In this paper we will analyze some computational problems related with ordered hierarchical aggregations of OWA operators as defined by the authors in a previous paper. In particular, we will provide polynomial algorithms to maximize or minimize the degrees of dispersion and orness of the hierarchical OWA aggregation.

This article deals with hierarchies of operators. In particular, we will analyze the problem of amalgamating individual opinions into a single group opinion, based upon hierarchical intensity aggregation rules. Our main goal is to decide whether hierarchical amalgamations are supported from an ethical and rational point of view. We will consider two different hierarchical procedures: cover-based hierarchical aggregations and ordered hierarchical aggregations, focusing here our attention to ordered hierarchical aggregations of OWA operators. In particular, we will obtain that basic key properties propagate under both hierarchical amalgamation rules, in the sense that the procedure itself will verify such properties whenever they are assumed for every partial amalgamation defining such a hierarchical procedure. © 1994 John Wiley & Sons, Inc.